2306.09896v1

# Demystifying GPT Self-Repair for Code Generation **Authors**: - Theo X. Olausson (MIT EECS & CSAIL &Jeevana Priya Inala) > Correspondence to . Work partially done while T.X.O. was at Microsoft Research. positioning ## Abstract Large Language Models (LLMs) have shown remarkable aptitude in code generation but still struggle on challenging programming tasks. Self-repair—in which the model debugs and fixes mistakes in its own code—has recently become a popular way to boost performance in these settings. However, only very limited studies on how and when self-repair works effectively exist in the literature, and one might wonder to what extent a model is really capable of providing accurate feedback on why the code is wrong when that code was generated by the same model. In this paper, we analyze GPT-3.5 and GPT-4’s ability to perform self-repair on APPS, a challenging dataset consisting of diverse coding challenges. To do so, we first establish a new evaluation strategy dubbed pass@t that measures the pass rate of the tasks against the total number of tokens sampled from the model, enabling a fair comparison to purely sampling-based approaches. With this evaluation strategy, we find that the effectiveness of self-repair is only seen in GPT-4. We also observe that self-repair is bottlenecked by the feedback stage; using GPT-4 to give feedback on the programs generated by GPT-3.5 and using expert human programmers to give feedback on the programs generated by GPT-4, we unlock significant performance gains. ## 1 Introduction Large language models (LLMs) have proven capable of generating code snippets from natural language specifications, but still struggle on complex coding challenges such as those found in competitions and professional software engineering interviews. Recent work has sought to improve performance by leveraging self-repair [Gupta et al., 2020, Le et al., 2022, Chen et al., 2023b, Zhang et al., 2023], in which the model introspects and corrects mistakes in its own code. Figure 1 shows a typical workflow of a self-repair based approach. First, given a specification, a program is sampled from a code generation model; this program is then executed on a suite of unit tests provided as part of the specification; if the program fails on any unit test, then the error message and the faulty program are given to a feedback generation model, which outputs a short explanation of why the code failed; finally, the feedback is passed to a repair model, which generates a fixed version of the program. On the surface, this is a very attractive idea. It allows the system to overcome mistakes caused by unfortunate samples during decoding; easily incorporates feedback during the repair phase from symbolic systems such as compilers, static analysis tools, and execution engines; and mimics the trial-and-error way in which human software engineers write code. <details> <summary>x1.png Details</summary>  ### Visual Description ## Diagram: Iterative Code Model Refinement with Feedback ### Overview This diagram illustrates a five-step iterative process for refining a "Code Model" based on execution feedback. It begins with a "User" interacting with an initial "Code Model," proceeds through an "Execution" phase where assertions are tested, uses a "Feedback Model" to learn from failures, updates the "Code Model," and concludes with a successful re-execution of the assertions. ### Components/Axes The diagram is structured horizontally, showing a left-to-right flow of components and processes. * **User**: Located on the far left, represented by a dark grey silhouette icon of a person's head and shoulders. Labeled "User" directly below the icon. * **Code Model (Initial)**: Positioned to the right of the "User." It is represented by a light grey circular container enclosing a network of seven interconnected blue circular nodes. The nodes are arranged with three on the left, one in the center, and three on the right, with lines connecting the left nodes to the center, and the center to the right nodes, forming a small, layered network structure. Labeled "Code Model" directly below the circle. * **Execution (Initial)**: Located to the right of the initial "Code Model." This component consists of three stacked, light grey rectangular blocks, each representing an assertion. * Top block: "assert f(x1) == y1" with a green checkmark icon on its right edge. * Middle block: "assert f(x2) == y2" with a green checkmark icon on its right edge. * Bottom block: "assert f(x3) == y3" with a red 'x' icon on its right edge. Labeled "Execution" directly below the stacked blocks. * **Feedback Model**: Positioned to the right of the initial "Execution" phase. It is represented by a light grey circular container enclosing a network of seven interconnected orange circular nodes. The internal structure of nodes and connections is identical to the "Code Model" but with orange nodes. Labeled "Feedback Model" directly below the circle. * **Code Model (Refined)**: Positioned to the right of the "Feedback Model." This component is visually identical to the initial "Code Model," featuring a light grey circular container with seven interconnected blue circular nodes. Labeled "Code Model" directly below the circle. * **Execution (Final)**: Located on the far right, to the right of the refined "Code Model." This component also consists of three stacked, light grey rectangular blocks, identical in text to the initial "Execution" blocks. * Top block: "assert f(x1) == y1" with a green checkmark icon on its right edge. * Middle block: "assert f(x2) == y2" with a green checkmark icon on its right edge. * Bottom block: "assert f(x3) == y3" with a green checkmark icon on its right edge. ### Detailed Analysis The diagram illustrates a sequential process through five numbered steps, indicated by black arrows: 1. **Step (1)**: An arrow points from the "User" to the initial "Code Model." This signifies the user providing input or initiating the "Code Model." 2. **Step (2)**: An arrow points from the initial "Code Model" to the "Execution" blocks. This indicates the "Code Model" is being executed or tested. * During this initial "Execution," two assertions pass: "assert f(x1) == y1" and "assert f(x2) == y2", indicated by green checkmarks. * One assertion fails: "assert f(x3) == y3", indicated by a red 'x'. 3. **Step (3)**: An arrow points from the initial "Execution" blocks (specifically, implying the results of the execution) to the "Feedback Model." This suggests that the execution results, particularly the failure, are fed into the "Feedback Model." The "Feedback Model" is visually distinct with orange nodes, implying a different state or function compared to the "Code Model." 4. **Step (4)**: An arrow points from the "Feedback Model" to the refined "Code Model." This indicates that the "Feedback Model" processes the feedback and uses it to update or refine the "Code Model." The "Code Model" reverts to its blue-node representation, signifying it's ready for re-execution. 5. **Step (5)**: An arrow points from the refined "Code Model" to the final "Execution" blocks. This signifies the refined "Code Model" is being executed again. * In this final "Execution," all three assertions pass: "assert f(x1) == y1", "assert f(x2) == y2", and "assert f(x3) == y3", all indicated by green checkmarks. ### Key Observations * The process is cyclical and iterative, demonstrating a feedback loop for improvement. * The "Code Model" is initially imperfect, as evidenced by one failed assertion in the first "Execution." * The "Feedback Model" (orange nodes) acts as an intermediary step, processing the results of the initial execution to inform the refinement of the "Code Model." * The "Code Model" is successfully refined, as all assertions pass in the subsequent "Execution." * The visual representation of the "Code Model" (blue nodes) remains consistent before and after refinement, suggesting its core structure is maintained, but its internal logic or parameters have been adjusted. The "Feedback Model" uses a distinct color (orange nodes) to differentiate its role. ### Interpretation This diagram illustrates a fundamental concept in software development, machine learning, or automated system design: an iterative process of testing, identifying failures, learning from those failures, and applying corrections to improve a system. The "User" initiates the process, likely by providing requirements or initial code. The "Code Model" represents the system under development, which could be a piece of software, an algorithm, or a machine learning model. The "Execution" phase simulates running the code or model with specific inputs (x1, x2, x3) and checking if the outputs (f(x1), f(x2), f(x3)) match expected values (y1, y2, y3) through assertions. The initial failure of "assert f(x3) == y3" highlights a defect or an area for improvement in the "Code Model." This failure is then fed into the "Feedback Model." The "Feedback Model" can be interpreted as an automated debugging system, a learning algorithm, or a human review process that analyzes the failure and determines how to modify the "Code Model." The change in node color from blue to orange for the "Feedback Model" visually distinguishes it as a separate, analytical, or learning component. Finally, the refined "Code Model" (back to blue nodes) is re-executed, and all assertions pass. This demonstrates that the feedback loop successfully identified and corrected the issue, leading to a more robust and correct system. This pattern is crucial for developing reliable systems, especially in domains like AI where models are continuously trained and fine-tuned based on performance metrics and errors. It suggests a self-correcting or self-improving system where failures are not terminal but rather opportunities for learning and refinement. </details> [ box/.style=rectangle, draw=black, align=justify, inner sep=3pt, tallbox/.style=box, fill=backcolor!50, shortbox/.style=box, fill=backcolor!50 ] [tallbox, text width=3.6cm, minimum height=3.2cm] (A) Given is a string $s$ representing the day of the week today. $s$ is one of SUN, MON, TUE, WED, THU, FRI, or SAT. After how many days is the next Sunday (tomorrow or later)? ⬇ # UNIT TESTS # (EXECUTABLE) assert f (’ MON ’) == 6 assert f (’ WED ’) == 4 assert f (’ SUN ’) == 7 ; [draw=none, left=0.2cm of A.west] (1); [shortbox, right=0.2cm of A.north east, text width=8cm, anchor=north west] (B) ⬇ def f (s): return (7 - [’ SUN ’, ... , ’ FRI ’, ’ SAT ’]. index (s)) %7 ; [draw=none, right=0.2cm of B.east] (2); [shortbox, below=0.2cm of B, text width=8cm] (C) Given input ’SUN’, the program returned 0, but the expected output was 7. ; [draw=none, right=0.2cm of C.east] (3); [shortbox, below=0.2cm of C, text width=8cm] (D) The code does not account for the case where the input is ’SUN’ and the output should be 7. This can be fixed by removing the modulo operation. ; [draw=none, right=0.2cm of D.east] (4); [shortbox, below=0.2cm of D, text width=8cm] (E) ⬇ def f (s): return (7 - [’ SUN ’, ... , ’ FRI ’, ’ SAT ’]. index (s)) # %7 ; [draw=none, right=0.2cm of E.east] (5); Figure 1: Self-repair with separate code and feedback models. First, a user gives a specification in the form of text and a suite of unit tests (1). Then, a code model (blue) generates a program (2). The program is checked against the unit tests using a symbolic execution engine, and an error message is returned (3). In order to provide more signal to the code model, textual feedback as to why this happened is provided by a feedback model (yellow; 4). Finally, this feedback is used by the code model to repair the program (5). However, it is important to remember that self-repair requires more invocations of the model, thus increasing the computational cost. In particular, whether self-repair is a winning strategy or not ultimately boils down to whether you would—at an equivalent compute budget—have had a greater chance of success if you had simply drawn more code samples i.i.d. from the model and checked them against the suite of unit tests provided as part of the task. Crucially, the effectiveness of self-repair depends not only on the model’s ability to generate code, which has been studied extensively in the literature, but also on its ability to identify how the code (generated by the model itself) is wrong with respect to the task specification. As far as we are aware, no previous or contemporary work has attempted to study the effect of this stage in detail. In this paper, we study the effectiveness of self-repair with GPT-3.5 [Ouyang et al., 2022, OpenAI, 2022] and GPT-4 [OpenAI, 2023] when solving competition-level code generation tasks. We begin by proposing a new evaluation strategy dubbed pass@t, in which the likelihood of obtaining a correct program (with respect to the given unit tests) is weighed against the total number of tokens sampled from the model. Using this instead of the traditional pass@k [Chen et al., 2021, Kulal et al., 2019] metric (which weighs pass rate against the number of trials), we are able to accurately compare performance gained through self-repair against any additional work done by the model when generating the feedback and carrying out the repair. Using this new evaluation strategy, we then carefully study the dynamics of the self-repair process under a range of hyper-parameters. Finally, given our primary objective of gaining insight into the state-of-the-art code generation models’ ability to reflect upon and debug their own code, we carry out a set of experiments in which we investigate the impact of improving the feedback stage alone. We do so by analyzing the impact of using a stronger feedback generation model than the code generation model (using GPT-4 to generate feedback for GPT-3.5 code model), as well as by carrying out a study in which human participants provide feedback on incorrect programs, in order to compare model-generated self-feedback to that provided by human programmers. From our experiments, we find that: 1. When taking the cost of doing inspection and repair into account, performance gains from self-repair can only be seen with GPT-4; for GPT-3.5, the pass rate with repair is lower than or equal to that of the baseline, no-repair approach at all budgets. 1. Even for the GPT-4 model, performance gains are modest at best ( $66\$ pass rate with a budget of 7000 tokens, $\approx$ the cost of 45 i.i.d. GPT-4 samples) and depend on having sufficient diversity in the initial programs. 1. Replacing GPT-3.5’s explanations of what is wrong with feedback produced by GPT-4 leads to better self-repair performance, even beating the baseline, no-repair GPT-3.5 approach ( $50\$ at 7000 tokens). 1. Replacing GPT-4’s own explanations with those of a human programmer improves repair significantly, leading to a 57% increase in the number of repaired programs which pass the tests. ## 2 Related work Program synthesis with large language models. The use of large language models for program synthesis has been studied extensively in the literature [Li et al., 2022, Austin et al., 2021, Chen et al., 2021, Le et al., 2022, Fried et al., 2023, Nijkamp et al., 2023, Chowdhery et al., 2022, Touvron et al., 2023, Li et al., 2023]. This literature has predominantly focused on evaluating models in terms of either raw accuracy or the pass@k metric [Kulal et al., 2019, Chen et al., 2021], often leveraging filtering techniques based on execution [Li et al., 2022, Shi et al., 2022] or ranking [Chen et al., 2021, Inala et al., 2022, Zhang et al., 2022] to reduce the number of samples which are considered for the final answer. In contrast, our work focuses on evaluating the models from the point of view of minimizing the number of samples that need to be drawn from the model in the first place. Our work is also different in that we assume access to the full collection of input-output examples, as is typically done in inductive synthesis [Kitzelmann, 2010, Polozov and Gulwani, 2015, Gulwani et al., 2017, Chen et al., 2019a, Ellis et al., 2021]. In particular, unlike some prior work [Li et al., 2022, Shi et al., 2022], we do not make a distinction between public tests used for filtering and private tests used to determine correctness, since our method does not involve filtering the outputs. Code repair. Statistical and learning-based techniques for code repair have a rich history in both the programming languages and machine learning communities, although they have traditionally been used predominantly to repair human-written code [Long and Rinard, 2016, Bader et al., 2019, Le Goues et al., 2021, Yasunaga and Liang, 2021, Chen et al., 2019b, Mesbah et al., 2019, Wang et al., 2018]. More recently, using repair as a post-processing step to improve code which was itself automatically synthesised has been used in the synthesis of both domain-specific languages [Gupta et al., 2020] and general-purpose code [Le et al., 2022, Yasunaga and Liang, 2021, 2020]. Our contribution differs from most prior work in this literature in the use of textual feedback for repair, which is possible thanks to the above mentioned rise in the use of LLMs for program synthesis. Contemporary work on LLM self-repair. Recognizing that there is much contemporary work seeking to self-repair with LLMs, we now briefly highlight a few such papers which are particularly close to our work. Zhang et al. [2023] explore self-repair without natural language feedback on APPS [Hendrycks et al., 2021] using a diverse range of fine-tuned models. They also experiment with prompt-based repair using Codex [Chen et al., 2021], InCoder [Fried et al., 2023], and CodeGen [Nijkamp et al., 2023]. Notably, their framework does not consider the cost associated with feedback and repair, which presents a significantly different perspective on self-repair. Similarly, Chen et al. [2023b] assess Codex’s ability to self-repair across a variety of tasks, in a framework that closely resembles that which we study in this work. However, their study differs from ours in terms of the models considered, the evaluation strategy, and, most importantly, the research goal, as we specifically aim to investigate the significance of the textual feedback stage. Self-repair, or frameworks with other names that are conceptually very similar to it, has also been used in contexts outside of code generation. Peng et al. [2023] use self-repair to mitigate hallucinations and improve factual grounding in a ChatGPT-based web search assistant, in which the model revises its initial response based on self-generated feedback. Similarly, Madaan et al. [2023] present a framework in which a model iteratively provides feedback on and revises its output until a stopping criterion is reached; they apply this framework to a range of tasks, including dialogue and code optimization. Ultimately, we see our work, in which we use the novel evaluation metric pass@t to investigate the significance of the textual feedback stage in competition-level self-repair, as being complementary to contemporary research which uses traditional metrics to evaluate self-repair in a broader context. We are eager to see what the implications of our results will be in these other domains. ## 3 Methodology ### 3.1 Self-Repair Overview As shown in Figure 1, our self-repair approach involves 4 stages: code generation, code execution, feedback generation, and code repair. We now formally define these four stages. Code generation. Given a specification $\psi$ , a programming model $M_{P}$ first generates $n_{p}$ samples i.i.d., which we denote $$ \{p_{i}\}_{i=1}^{n_{p}}\stackrel{{\scriptstyle i.i.d.}}{{\sim}}M_{P}(\psi) $$ Code execution. These $n_{p}$ code samples are then executed against a test bed. Recall from Section 2 that we assume that we have access to the full set of tests in executable form (see Section 5 for a brief discussion on the validity of this assumption in software engineering domains). Thus, we stop if any sample passes all of the tests, since a satisfying program has then been found. Otherwise, we collect the error messages $\{e_{i}\}_{i}$ returned by the execution environment. These error messages either contain the compile/runtime error information or an example input on which the program’s output differs from the expected one. An example is shown in Figure 1 (component 3). Feedback generation. Since the error messages from the execution environment are usually very high-level, they provide little signal for repair. Therefore, as an intermediate step, we use a feedback model to produce a more detailed explanation of what went wrong; Figure 1 (component 4) shows an example. Formally, in this stage, we generate $n_{f}$ feedback strings, $\{f_{ij}\}_{j}$ , for each wrong program, $p_{i}$ , as follows: $$ \{f_{ij}\}_{j=1}^{n_{f}}\stackrel{{\scriptstyle i.i.d.}}{{\sim}}M_{F}(\psi;p_{ i};e_{i}) $$ Having an explicit feedback generation step allows us to ablate this component so that we can study its significance in isolation. Code repair. In the final step, for each initial program $p_{i}$ and feedback $f_{ij}$ , $n_{r}$ candidate repaired programs are sampled from $M_{P}$ We use the same model for both the initial code generation and the code repair, since these are fundamentally similar tasks.: $$ \{r_{ijk}\}_{k=1}^{n_{r}}\stackrel{{\scriptstyle i.i.d.}}{{\sim}}M_{P}(\psi;p_ {i};e_{i};f_{ij}) $$ Repair tree. We call the tree of interleaved text and programs produced by this procedure—rooted in the specification $\psi$ , then branching into initial programs $p_{i}$ , each of which branches into feedback $f_{ij}$ and then repairs $r_{ijk}$ —a repair tree, $T$ (Figure 2). Caveat: jointly sampling feedback and repair. The general framework presented above does not require the programming model and feedback model to be the same, thus allowing for the use of specialized models in the system. However, when $M_{P}=M_{F}$ we jointly generate both the feedback and the repaired program in a single API call, since both GPT-3.5 and GPT-4 have a natural tendency to interleave text and code in their responses. See Appendix E for a detailed look at how the prompt differs between this and the previous setting. Formally, we denote this as $$ \{(f_{ij},r_{ij})\}_{j=1}^{n_{fr}}\stackrel{{\scriptstyle i.i.d.}}{{\sim}}M_{P }(\psi;p_{i};e_{i}) $$ <details> <summary>x2.png Details</summary>  ### Visual Description ## Diagram: Hierarchical Decomposition with Iterative Process Flow ### Overview This image presents a hierarchical tree-like diagram illustrating a multi-level decomposition process, accompanied by a vertical process flow description on the right side. The diagram shows a top-down branching structure, where a single root element expands into multiple sub-elements across three subsequent levels, with ellipses indicating omitted intermediate nodes at each branching stage. The right-side labels describe a sequence of operations: "Code Gen", "Feedback", and "Repair", each associated with a specific phase of the hierarchical decomposition. ### Components/Axes The diagram consists of circular nodes (representing entities or stages) and directed arrows (representing relationships or flow). There are no traditional axes or legends, but the right-side labels serve as a process legend. **Hierarchical Structure (Nodes and Edges):** * **Nodes:** All nodes are light grey circles containing black text labels. * **Level 1 (Root):** A single node at the top-center, labeled "ψ" (psi). * **Level 2:** Two visible nodes, "p₁" on the left and "p_n_p" on the right. An ellipsis "..." is placed horizontally between them, indicating a series of `n_p` intermediate nodes (p₂, ..., p_n_p-1) are omitted. * **Level 3:** * Under "p₁": Two visible nodes, "f₁₁" on the left and "f_1_n_f" on the right. An ellipsis "..." is placed horizontally between them, indicating `n_f` intermediate nodes (f₁₂, ..., f_1_n_f-1) are omitted. * Under "p_n_p": Two visible nodes, "f_n_p_1" on the left and "f_n_p_n_f" on the right. An ellipsis "..." is placed horizontally between them, indicating `n_f` intermediate nodes (f_n_p_2, ..., f_n_p_n_f-1) are omitted. * **Level 4 (Leaf Nodes):** * Under "f₁₁": Two visible nodes, "r₁₁₁" on the left and "r_1_1_n_r" on the right. An ellipsis "..." is placed horizontally between them, indicating `n_r` intermediate nodes (r₁₁₂, ..., r_1_1_n_r-1) are omitted. * Under "f_1_n_f": Two visible nodes, "r_1_n_f_1" on the left and "r_1_n_f_n_r" on the right. An ellipsis "..." is placed horizontally between them, indicating `n_r` intermediate nodes (r_1_n_f_2, ..., r_1_n_f_n_r-1) are omitted. * Under "f_n_p_1": Two visible nodes, "r_n_p_1_1" on the left and "r_n_p_1_n_r" on the right. An ellipsis "..." is placed horizontally between them, indicating `n_r` intermediate nodes (r_n_p_1_2, ..., r_n_p_1_n_r-1) are omitted. * Under "f_n_p_n_f": Two visible nodes, "r_n_p_n_f_1" on the left and "r_n_p_n_f_n_r" on the right. An ellipsis "..." is placed horizontally between them, indicating `n_r` intermediate nodes (r_n_p_n_f_2, ..., r_n_p_n_f_n_r-1) are omitted. * **Edges:** All connections are dark grey directed arrows pointing downwards, indicating a flow or derivation from a parent node to its child nodes. **Process Flow Labels (Right Side):** Positioned vertically along the right edge of the diagram, these labels describe a process sequence. * **Top:** A vertical blue arrow pointing downwards, labeled "Code Gen" in blue text. This arrow spans vertically, aligning with the first two levels of the hierarchical diagram (from "ψ" down to the "f" nodes). * **Middle:** A vertical orange arrow pointing downwards, labeled "Feedback" in orange text. This arrow is positioned below "Code Gen" and aligns vertically with the transition from the "f" nodes to the "r" nodes in the hierarchical diagram. * **Bottom:** A vertical blue arrow pointing downwards, labeled "Repair" in blue text. This arrow is positioned below "Feedback" and aligns vertically with the bottom-most "r" nodes in the hierarchical diagram. ### Detailed Analysis The diagram illustrates a four-level hierarchical decomposition. 1. **Level 1 to Level 2:** The root node "ψ" branches into a set of `n_p` nodes, represented by "p₁" through "p_n_p". This indicates a primary decomposition or generation step. 2. **Level 2 to Level 3:** Each "p_i" node further branches into a set of `n_f` nodes, represented by "f_i1" through "f_i_n_f". This signifies a secondary decomposition or feature extraction step for each 'p' component. 3. **Level 3 to Level 4:** Each "f_ij" node then branches into a set of `n_r` nodes, represented by "r_ij1" through "r_ij_n_r". This represents a tertiary decomposition, possibly leading to specific results or repair actions. The process flow labels on the right provide a contextual interpretation of these hierarchical levels: * The "Code Gen" phase (blue arrow) corresponds to the initial top-down decomposition from "ψ" to the "p" nodes and then to the "f" nodes. This suggests that the generation of code or a primary structure occurs in these initial stages. * The "Feedback" phase (orange arrow) aligns with the transition from the "f" nodes to the "r" nodes. This implies that after the initial generation (f-nodes), an evaluation or feedback mechanism is applied, leading to the "r" nodes. * The "Repair" phase (blue arrow) aligns with the "r" nodes themselves, or the final output derived from them. This suggests that the "r" nodes represent repair actions or refined outputs based on the feedback received. The consistent downward direction of all arrows in both the hierarchy and the process flow indicates a sequential, top-down progression. The use of subscripts `n_p`, `n_f`, and `n_r` implies that the number of branches at each level can vary, making the decomposition flexible. ### Key Observations * The diagram represents a recursive or iterative decomposition process, where a high-level entity is broken down into increasingly granular components. * The ellipsis notation "..." is used extensively to denote omitted intermediate nodes, implying a generalizable structure rather than a fixed number of branches. * The right-side labels provide a functional context to the hierarchical levels, mapping stages of decomposition to phases of a process (Code Generation, Feedback, Repair). * The color coding of the process flow labels (blue for "Code Gen" and "Repair", orange for "Feedback") might distinguish between primary forward actions (blue) and an intermediate evaluative or corrective step (orange). ### Interpretation This diagram likely illustrates a conceptual model for an automated system or a problem-solving methodology, possibly in the domain of software engineering, design automation, or AI-driven development. * **ψ (Psi):** Could represent the initial high-level problem statement, a system specification, or a desired outcome. * **p (Parameters/Parts/Processes):** The first level of decomposition (p₁ to p_n_p) suggests breaking down the initial problem into major parameters, components, or parallel processes. * **f (Features/Functions/Factors):** The second level of decomposition (f_i1 to f_i_n_f) indicates a further breakdown of each parameter/part into specific features, functions, or influencing factors. * **r (Results/Repairs/Refinements):** The final level (r_ij1 to r_ij_n_r) represents the most granular elements, which, in the context of the right-side labels, are likely specific results, repair actions, or refinements derived from the features. The process flow on the right suggests an iterative or cyclical approach: 1. **Code Gen:** The initial phase involves generating a solution or a system structure, represented by the decomposition from "ψ" down to the "f" nodes. This could be an automated code generation step, a design synthesis, or an initial hypothesis formulation. 2. **Feedback:** Once the initial generation is complete (at the "f" node level), a feedback mechanism is engaged. This implies evaluation, testing, or analysis of the generated components/features. This feedback then informs the next stage. 3. **Repair:** Based on the feedback, specific "repair" actions or refinements are performed, represented by the "r" nodes. This suggests an iterative refinement loop where the generated solution is adjusted or corrected to meet requirements or improve performance. The overall interpretation is a system that generates a solution through hierarchical decomposition, evaluates it, and then iteratively repairs or refines it based on feedback. The blue arrows for "Code Gen" and "Repair" might signify the primary forward progression of the system, while the orange "Feedback" arrow highlights a critical intermediate step for evaluation and guidance. This model emphasizes a structured, top-down approach to problem-solving with built-in mechanisms for quality assurance and improvement. </details> Figure 2: A repair tree begins with a specification $\psi$ (root node), then grows into initial programs, feedback, and repairs. ### 3.2 pass@t : pass rate vs. token count Since self-repair requires several dependent model invocations of non-uniform cost, this is a setting in which pass@ $k$ —the likelihood of obtaining a correct program in $k$ i.i.d. samples—is not a suitable metric for comparing and evaluating various hyper-parameter choices of self-repair. Instead, we measure the pass rate as a function of the total number of tokens sampled from the model, a metric which we call pass@t. Formally, suppose that you are given a dataset $D=\{\psi_{d}\}_{d}$ and a chosen set of values for the hyper-parameters $(M_{P},M_{F},n_{p},n_{f},n_{r})$ . Let $T_{d}^{i}\sim M(\psi_{d})$ denote a repair tree that is sampled as described in Section 3.1 for the task $\psi_{d}$ ; let $\text{size}(T_{d}^{i})$ denote the total number of program and feedback tokens in the repair tree; and say that $T_{d}^{i}\models\psi_{d}$ is true if, and only if, $T_{d}^{i}$ has at least one leaf program that satisfies the unit tests in the specification $\psi_{d}$ . Then the pass@t metric of this choice of hyper-parameters is defined as the expected pass rate at the number of tokens which you would expect to generate with this choice of hyper-parameters: | | $\displaystyle\texttt{pass@t}\triangleq\mathop{\mathbb{E}}_{\stackrel{{ \scriptstyle\psi_{d}\sim D}}{{T_{d}^{i}\sim M(\psi_{d})}}}\left[T_{d}^{i} \models\psi_{d}\right]\quad\textbf{at}\quad t=\mathop{\mathbb{E}}_{\stackrel{{ \scriptstyle\psi_{d}\sim D}}{{T_{d}^{i}\sim M(\psi_{d})}}}\left[\text{size}(T_ {d}^{i})\right]$ | | | --- | --- | --- | In our experiments, we plot bootstrapped estimates of these two quantities. To obtain these, we first generate a very large repair tree for each task specification, with: $N_{p}\geq n_{p}$ initial program samples; $N_{f}\geq n_{f}$ feedback strings per wrong program; and $N_{r}\geq n_{r}$ repair candidates per feedback string. Given a setting of $(n_{p},n_{f},n_{r})$ , we then sub-sample (with replacement) $N_{t}$ different repair trees from this frozen dataset. Finally, we compute the sample mean and standard deviation of the pass rate and the tree size over these $N_{t}$ trees. Estimating the pass@t in this way greatly reduces the computational cost of our experiments, since we can reuse the same initial dataset to compute the estimates for all of the various choices of $n_{p},n_{f}$ , and $n_{r}$ . We use $N_{p}=50$ for all experiments, and consider $n_{p}\leq 25$ for the self-repair approaches and $n_{p}\leq 50$ for the baseline, no-repair approach. Similarly, for the feedback strings, we use $N_{f}=25$ and $n_{f}\leq 10$ (except for Section 4.2, in which we only consider $n_{f}=1$ and therefore settle for $N_{f}=10$ instead). For the repair candidates, since we do joint sampling of feedback and repair in most of our experiments, we set $N_{r}=n_{r}=1$ . Finally, we use $N_{t}=1000$ for all settings. ## 4 Experiments In this section, we carry out experiments to answer the following research questions: (a) In the context of challenging programming puzzles, is self-repair better than i.i.d. sampling without repair for the models we consider? If so, under what hyper-parameters is self-repair most effective? (b) Would a stronger feedback model boost the model’s repair performance? (c) Would keeping a human in the loop to provide feedback unlock better repair performance even for the strongest model? We evaluate these hypotheses on Python programming challenges from the APPS dataset [Hendrycks et al., 2021]. The APPS dataset contains a diverse range of programming challenges paired with a suite of tests, making it a perfect (and challenging) setting to study self-repair in. To keep our experiments tractable, we evaluate on a subset of the APPS test set, consisting of 300 tasks. These tasks are proportionally sampled in accordance with the frequency of the different difficulty levels in the test set: 180 interview-level questions, 60 competition-level questions, and 60 introductory-level questions (listed in Appendix F). We use GPT-3.5 [Ouyang et al., 2022, OpenAI, 2022] and GPT-4 [OpenAI, 2023] as our models of choice, and implement self-repair using templated string concatenation with one-shot prompting; our prompts are given in Appendix E. When appropriate, we compare against a baseline without repair. This baseline, shown with a black line in the plots, is simply i.i.d. sampling from the corresponding model (e.g., GPT-4 when we explore whether GPT-4 is capable of self-repair). Based on preliminary experiments, we set the decoding temperature to $0.8$ for all the models to encourage diverse samples. <details> <summary>x3.png Details</summary>  ### Visual Description ## Chart Type: Scatter Plot with Fitted Curve ### Overview This image displays a scatter plot illustrating the relationship between "Mean number of tokens generated" and "Mean pass rate". The data points are categorized by two parameters, `n_p` (represented by color) and `n_fr` (represented by marker shape), and a fitted curve with a confidence interval is overlaid to show the general trend. ### Components/Axes **X-axis:** * **Title:** "Mean number of tokens generated" * **Range:** Approximately 0 to 10000 * **Major Tick Marks:** 0, 2000, 4000, 6000, 8000, 10000. The labels are rotated counter-clockwise by approximately 45 degrees. * **Minor Grid Lines:** Present at intervals of approximately 1000. **Y-axis:** * **Title:** "Mean pass rate" * **Range:** Approximately 0.0 to 1.0 * **Major Tick Marks:** 0.0, 0.2, 0.4, 0.6, 0.8, 1.0 * **Minor Grid Lines:** Present at intervals of approximately 0.1. **Legend (Top-Center, within the plot area):** The legend is divided into two columns, indicating two independent variables: `n_p` (number of passes) and `n_fr` (number of failures/retries, inferred from context). * **Left Column (Line colors for `n_p`):** * Brown line: `n_p = 1` * Goldenrod/Yellow line: `n_p = 2` * Teal/Light Blue line: `n_p = 5` * Blue line: `n_p = 10` * Dark Blue/Navy line: `n_p = 25` * **Right Column (Marker shapes for `n_fr`):** * Dark Grey circle: `n_fr = 1` * Dark Grey inverted triangle: `n_fr = 3` * Dark Grey square: `n_fr = 5` * Dark Grey triangle: `n_fr = 10` **Fitted Curve:** A solid dark grey line represents the overall trend, starting low and increasing, then flattening out. It is accompanied by a lighter grey shaded area, which likely indicates a confidence interval or standard error around the fitted curve. ### Detailed Analysis **Overall Trend of Fitted Curve:** The dark grey fitted curve shows a clear positive, non-linear relationship. It starts at a "Mean pass rate" of approximately 0.2 at low "Mean number of tokens generated" (around 0-200), rises steeply, and then gradually flattens out, approaching a "Mean pass rate" of about 0.5 to 0.53 as the "Mean number of tokens generated" increases towards 10000. The light grey shaded area around the curve suggests a relatively narrow confidence interval, indicating good fit or low uncertainty for the trend. **Data Points (Scatter Plot):** Each data point is represented by a specific color (for `n_p`) and marker shape (for `n_fr`). All scatter points include small vertical error bars, indicating variability in the "Mean pass rate". 1. **`n_p = 1` (Brown points):** * `n_fr = 1` (Circle): X ~ 500, Y ~ 0.26 * `n_fr = 3` (Inverted Triangle): X ~ 1200, Y ~ 0.31 * `n_fr = 5` (Square): X ~ 1800, Y ~ 0.33 * `n_fr = 10` (Triangle): X ~ 3000, Y ~ 0.35 2. **`n_p = 2` (Goldenrod/Yellow points):** * `n_fr = 1` (Circle): X ~ 800, Y ~ 0.32 * `n_fr = 3` (Inverted Triangle): X ~ 1500, Y ~ 0.36 * `n_fr = 5` (Square): X ~ 2500, Y ~ 0.38 * `n_fr = 10` (Triangle): X ~ 5500, Y ~ 0.42 3. **`n_p = 5` (Teal/Light Blue points):** * `n_fr = 1` (Circle): X ~ 2000, Y ~ 0.41 * `n_fr = 3` (Inverted Triangle): X ~ 3200, Y ~ 0.43 * `n_fr = 5` (Square): X ~ 4500, Y ~ 0.48 * `n_fr = 10` (Triangle): X ~ 7000, Y ~ 0.49 4. **`n_p = 10` (Blue points):** * `n_fr = 1` (Circle): X ~ 4200, Y ~ 0.47 * `n_fr = 3` (Inverted Triangle): X ~ 4800, Y ~ 0.45 * `n_fr = 5` (Square): X ~ 8500, Y ~ 0.51 5. **`n_p = 25` (Dark Blue/Navy point):** * `n_fr = 1` (Circle): X ~ 9800, Y ~ 0.53 ### Key Observations * **General Increase:** All data series generally show an increase in "Mean pass rate" as "Mean number of tokens generated" increases, consistent with the fitted curve. * **Impact of `n_p`:** For a given `n_fr` (same marker shape), higher `n_p` values (represented by colors from brown to dark blue) tend to correspond to higher "Mean pass rates" and often higher "Mean number of tokens generated". For example, comparing `n_fr=1` (circles): * `n_p=1` (brown circle) is at (500, 0.26) * `n_p=2` (goldenrod circle) is at (800, 0.32) * `n_p=5` (teal circle) is at (2000, 0.41) * `n_p=10` (blue circle) is at (4200, 0.47) * `n_p=25` (dark blue circle) is at (9800, 0.53) This indicates a strong positive correlation between `n_p` and both "Mean number of tokens generated" and "Mean pass rate". * **Impact of `n_fr`:** For a given `n_p` (same color), increasing `n_fr` (from circle to triangle) generally leads to an increase in "Mean number of tokens generated" and a slight, but less consistent, increase or plateau in "Mean pass rate". For example, for `n_p=1` (brown points), as `n_fr` increases from 1 to 10, "Mean number of tokens generated" increases from ~500 to ~3000, and "Mean pass rate" increases from ~0.26 to ~0.35. * **Saturation:** The "Mean pass rate" appears to saturate around 0.5 to 0.53, even with very high "Mean number of tokens generated" (e.g., 8000-10000) and high `n_p` values. * **Data Distribution:** The data points are not uniformly distributed; there are more points at lower "Mean number of tokens generated" values, and fewer points as "Mean number of tokens generated" approaches 10000. The highest `n_p` value (`n_p=25`) only has one data point shown. ### Interpretation The chart suggests that both `n_p` and `n_fr` parameters influence the "Mean number of tokens generated" and, consequently, the "Mean pass rate". 1. **`n_p` as a primary driver:** Higher values of `n_p` (number of passes) are strongly associated with both a greater "Mean number of tokens generated" and a higher "Mean pass rate". This implies that increasing the number of passes in a process (perhaps an iterative generation or refinement process) leads to more extensive output (more tokens) and better performance (higher pass rate). The progression of colors from brown to dark blue clearly illustrates this, with `n_p=25` achieving the highest pass rate and token count among the observed points. 2. **`n_fr` as a secondary driver/modifier:** The `n_fr` parameter (number of failures/retries) also contributes to the "Mean number of tokens generated". For a fixed `n_p`, increasing `n_fr` generally pushes the data points further to the right on the X-axis, meaning more tokens are generated. However, its impact on the "Mean pass rate" is less pronounced and more variable compared to `n_p`. This could suggest that while allowing more retries or failures might lead to more attempts and thus more generated tokens, it doesn't necessarily translate to a proportionally higher pass rate, especially as the pass rate approaches its saturation point. 3. **Performance Ceiling:** The fitted curve and the clustering of higher `n_p` points indicate a diminishing return on increasing "Mean number of tokens generated" beyond a certain point (e.g., ~6000-7000 tokens). The "Mean pass rate" seems to plateau around 0.5 to 0.53. This suggests that there might be an inherent limit to the "pass rate" achievable with the current system or methodology, regardless of how many tokens are generated or how many passes/retries are allowed. Further increases in `n_p` or `n_fr` beyond certain thresholds might only lead to more tokens generated without significant improvements in the pass rate. In essence, to maximize the "Mean pass rate", one should aim for higher `n_p` values, which will naturally lead to more tokens generated. While `n_fr` also increases token generation, its direct impact on the pass rate is less clear, especially at higher token counts where the system appears to hit a performance ceiling. </details> (a) Mean pass rate vs. number of tokens generated. Black line is i.i.d. sampling without repair from GPT-3.5. Note that the error bars are often smaller than the markers; all settings have a standard deviation of less than 1.5 absolute points on the y-axis. Results truncated at $t=10,000$ . <details> <summary>x4.png Details</summary>  ### Visual Description ## Chart Type: Heatmap of Performance Metric vs. Program Parameters ### Overview This image displays a heatmap illustrating a performance metric (likely a success rate or accuracy, ranging from 0.78 to 1.00) across different combinations of two parameters: "Number of initial programs ($n_p$)" and "Number of feedback-repairs ($n_{fr}$)". The cells are color-coded, with darker brown indicating lower values and brighter yellow indicating higher values. A significant portion of the grid, particularly in the upper-right, is marked "O.O.B." (Out Of Bounds) and colored dark grey, indicating conditions where a numerical result was not obtained. ### Components/Axes The chart is a 2D grid with two categorical axes: * **Y-axis (left side)**: Labeled "Number of feedback-repairs ($n_{fr}$)". * Tick markers (from bottom to top): 1, 3, 5, 10. * **X-axis (bottom side)**: Labeled "Number of initial programs ($n_p$)". * Tick markers (from left to right): 1, 2, 5, 10, 25. There is no explicit color legend. However, the color intensity within the cells implicitly represents the numerical values: * Darkest brown: Corresponds to values around 0.78. * Medium brown/orange: Corresponds to values around 0.80 to 0.92. * Lighter orange/yellow: Corresponds to values around 0.94 to 0.99. * Brightest yellow: Corresponds to the value 1.00. * Dark grey: Corresponds to "O.O.B." entries. ### Detailed Analysis The heatmap is a 4x5 grid, with rows corresponding to $n_{fr}$ values and columns corresponding to $n_p$ values. Each cell contains a numerical value (formatted to two decimal places) or the text "O.O.B.". **Data Table (Values in cells, with qualitative color description):** | $n_{fr}$ \ $n_p$ | 1 (Dark Brown) | 2 (Medium Brown/Orange) | 5 (Lighter Orange/Yellow) | 10 (Lighter Orange/Yellow) | 25 (Dark Grey) | | :--------------- | :------------- | :---------------------- | :------------------------ | :------------------------- | :------------- | | **10** | 0.78 | 0.86 | O.O.B. | O.O.B. | O.O.B. | | **5** | 0.80 | 0.86 | 0.95 | O.O.B. | O.O.B. | | **3** | 0.81 | 0.87 | 0.94 | 1.00 | O.O.B. | | **1** | 0.87 | 0.92 | 0.96 | 0.99 | O.O.B. | **Trends and Observations by Row (increasing $n_p$ for fixed $n_{fr}$):** * **For $n_{fr}=10$**: Values increase from 0.78 (dark brown) to 0.86 (medium brown/orange), then transition to "O.O.B." (dark grey) for $n_p=5, 10, 25$. * **For $n_{fr}=5$**: Values increase from 0.80 (dark brown) to 0.86 (medium brown/orange) to 0.95 (lighter orange/yellow), then transition to "O.O.B." (dark grey) for $n_p=10, 25$. * **For $n_{fr}=3$**: Values increase from 0.81 (dark brown) to 0.87 (medium brown/orange) to 0.94 (lighter orange/yellow) to 1.00 (brightest yellow), then transition to "O.O.B." (dark grey) for $n_p=25$. * **For $n_{fr}=1$**: Values consistently increase from 0.87 (medium brown/orange) to 0.92 (medium brown/orange) to 0.96 (lighter orange/yellow) to 0.99 (lighter orange/yellow), then transition to "O.O.B." (dark grey) for $n_p=25$. **Trends and Observations by Column (increasing $n_{fr}$ for fixed $n_p$):** * **For $n_p=1$**: Values generally decrease from 0.87 (medium brown/orange) to 0.81 (dark brown) to 0.80 (dark brown) to 0.78 (dark brown). * **For $n_p=2$**: Values show a slight decrease then stability: 0.92 (medium brown/orange) to 0.87 (medium brown/orange) to 0.86 (medium brown/orange) to 0.86 (medium brown/orange). * **For $n_p=5$**: Values show a slight decrease then increase: 0.96 (lighter orange/yellow) to 0.94 (lighter orange/yellow) to 0.95 (lighter orange/yellow), then transition to "O.O.B." (dark grey) for $n_{fr}=10$. * **For $n_p=10$**: Values increase from 0.99 (lighter orange/yellow) to 1.00 (brightest yellow), then transition to "O.O.B." (dark grey) for $n_{fr}=5, 10$. * **For $n_p=25$**: All values are "O.O.B." (dark grey). ### Key Observations * The highest performance value observed is 1.00, located at $n_p=10$ and $n_{fr}=3$. This cell is colored the brightest yellow. * The lowest performance value observed is 0.78, located at $n_p=1$ and $n_{fr}=10$. This cell is colored the darkest brown. * Generally, increasing the "Number of initial programs ($n_p$)" tends to improve the performance metric for a given "Number of feedback-repairs ($n_{fr}$)", up to a certain point. * Increasing the "Number of feedback-repairs ($n_{fr}$)" tends to decrease performance when the "Number of initial programs ($n_p$)" is low (e.g., $n_p=1, 2$). * The "O.O.B." region occupies the top-right portion of the heatmap, forming a triangular pattern. This indicates that combinations with a high number of initial programs and/or a high number of feedback-repairs lead to "Out Of Bounds" conditions. Specifically, $n_p=25$ always results in "O.O.B.", and for $n_p=10$, $n_{fr}$ values of 5 and 10 result in "O.O.B.". For $n_p=5$, $n_{fr}=10$ results in "O.O.B.". ### Interpretation The heatmap likely represents the results of an experiment or simulation where the performance of a system or algorithm is evaluated based on two configurable parameters: the number of initial programs and the number of feedback-repairs. The numerical values (0.78 to 1.00) are a measure of success or efficiency, with higher values being better. The term "O.O.B." (Out Of Bounds) suggests that for these parameter combinations, the system either failed to produce a result, exceeded a computational budget (time, memory), or fell outside a predefined operational range. This implies a constraint or limitation in the system's ability to handle certain configurations. The data suggests a sweet spot for performance. While increasing the "Number of initial programs ($n_p$)" generally improves the metric, it also increases the likelihood of hitting an "O.O.B." condition, especially when combined with a higher "Number of feedback-repairs ($n_{fr}$)". Conversely, a very low "Number of initial programs ($n_p=1$)" consistently yields lower performance, regardless of feedback-repairs. The optimal observed performance (1.00) is achieved with a moderate number of initial programs ($n_p=10$) and a relatively low number of feedback-repairs ($n_{fr}=3$). This indicates that beyond a certain point, additional feedback-repairs might not be beneficial or could even be detrimental, particularly when combined with a large number of initial programs, leading to the "O.O.B." state. The "O.O.B." region highlights the practical limits or computational costs associated with scaling these parameters. Researchers or engineers using this data would likely aim for parameter combinations within the high-performance, non-"O.O.B." region, balancing performance with resource constraints. </details> (b) Normalized mean pass rate relative to the (interpolated) baseline at an equivalent budget (number of tokens). Cells for which the number of tokens generated exceeds 50 samples from the GPT-3.5 baseline marked O.O.B. (out of bounds). Figure 3: Pass rate versus number of tokens generated for various settings of $n_{p}$ (number of initial programs) and $n_{fr}$ (number of repairs sampled per program). GPT-3.5 is used for all samples, including the baseline. ### 4.1 Self-repair requires strong models and diverse initial samples In this subsection, we consider the setup where $M_{P}=M_{F}\in\{\text{GPT-3.5, GPT-4}\}$ , i.e., where one single model is used for both code/repair generation and feedback generation. To evaluate if self-repair leads to better pass@t than a no-repair, i.i.d. sampling-based baseline approach, we vary $n_{p}$ and $n_{fr}$ —that is, the number of initial i.i.d. base samples and joint feedback, repair samples drawn from $M_{P}$ —in the range $(n_{p},n_{fr})\in\{1,2,5,10,25\}\times\{1,3,5,10\}$ . Recall that when $M_{P}=M_{F}$ , we jointly sample for $n_{fr}$ pairs of feedback strings and repair programs instead of sampling them one after another (Section 3.1). The results are shown in Figure 3 for GPT-3.5 and Figure 4 for GPT-4. In the left-hand subplots, the color of each dot indicates the number of initial samples ( $n_{p}$ ), while its shape indicates the number of feedback-repair samples ( $n_{fr}$ ). In the right hand plots, we show a heat-map with the two hyper-parameters along the axes, where the value in each cell indicates the mean pass rate with self-repair normalized by the mean pass rate of the baseline, no-repair approach when given the same token budget (i.e., pass@t at the same value of t). When the normalized mean pass rate is 1, this means that self-repair has the same pass rate as the no-repair, baseline approach at that same token budget; a higher value ( $\geq 1$ ) means self-repair performs better than the baseline. From the plots, we can see that for the GPT-3.5 model, the pass@t is lower than or equal to the corresponding baseline (black line) for all settings of $n_{p},n_{fr}$ , clearly showing that self-repair is not an effective strategy for GPT-3.5. On the other hand, for GPT-4, there are several values of $n_{p},n_{fr}$ for which the pass rate with self-repair is significantly better than that of the baseline. For example, with $n_{p}=10,n_{fr}=3$ the pass rate increases from 65% to 70%, and with $n_{p}=25,n_{fr}=1$ it increases from 65% to 71%. Our experiments also show a clear trend with respect to the relationship between the hyper-parameters. Given a fixed number of feedback-repairs ( $n_{fr}$ ), increasing the number of initial programs ( $n_{p}$ ) (i.e., moving right along the x-axis on the heat maps) consistently leads to relative performance gains for both models. On the other hand, fixing $n_{p}$ and increasing $n_{fr}$ (i.e., moving up along the y-axis on the heat maps) does not appear to be worth the additional cost incurred, giving very marginal gains at higher budgets and even decreasing performance at lower budgets. This suggests that, given a fixed budget, the most important factor determining whether self-repair will lead to a correct program or not is the diversity of the base samples that are generated up-front, rather than the diversity of the repairs sampled. Having more initial samples increases the likelihood of there being at least one program which is close to the ideal program and, hence, can be successfully repaired. Since $n_{fr}=1$ is the best choice for the hyper-parameter $n_{fr}$ , we next isolate the effect of the number of initial programs, $n_{p}$ , by exploring a denser set of possible values: $(n_{p},n_{fr})\in\{1,2,....,24,25\}\times\{1\}$ . The plots are shown in Figure 6 for both $M_{P}=M_{F}\in\{\text{GPT-3.5},\text{GPT-4}\}$ and the baseline, no-repair approaches. Note that since $n_{fr}$ is fixed, in these plots, there is a direct correlation between $n_{p}$ and the total number of tokens, $t$ . Again, we see that self-repair is not an effective strategy for the GPT-3.5 model, but that it is effective for GPT-4—especially at higher values of $n_{p}$ ( $\geq 5000$ ), where it increases pass rate by over 5 points. <details> <summary>x5.png Details</summary>  ### Visual Description ## Chart Type: Scatter Plot with Fitted Curve: Mean Pass Rate vs. Mean Number of Tokens Generated ### Overview This image displays a scatter plot illustrating the relationship between the "Mean number of tokens generated" on the x-axis and the "Mean pass rate" on the y-axis. The plot includes a fitted curve with a confidence interval, representing the overall trend. Individual data points are categorized by two parameters, `n_p` (represented by color) and `n_fr` (represented by marker shape), as detailed in the legend. The chart primarily shows how different configurations of these parameters affect the pass rate and the computational cost in terms of tokens generated. ### Components/Axes The chart is structured with a main plotting area, an x-axis at the bottom, a y-axis on the left, and a legend positioned in the bottom-center-right of the plotting area. * **X-axis Label**: "Mean number of tokens generated" * **X-axis Range**: From 0 to 10000. * **X-axis Major Ticks**: 0, 2000, 4000, 6000, 8000, 10000. The tick labels are rotated approximately 45 degrees counter-clockwise. * **Y-axis Label**: "Mean pass rate" * **Y-axis Range**: From 0.0 to 1.0. * **Y-axis Major Ticks**: 0.0, 0.2, 0.4, 0.6, 0.8, 1.0. * **Legend**: Located within the plot area, spanning from approximately X=3000 to X=8000 and Y=0.05 to Y=0.35. It is divided into two columns: * **Left Column (Line styles, representing `n_p` values):** * Brown line: `n_p = 1` * Orange/Gold line: `n_p = 2` * Teal/Light Green line: `n_p = 5` * Light Blue line: `n_p = 10` * Dark Blue/Indigo line: `n_p = 25` * **Right Column (Marker shapes, representing `n_fr` values):** * Gray circle: `n_fr = 1` * Gray downward triangle: `n_fr = 3` * Gray square: `n_fr = 5` * Gray upward triangle: `n_fr = 10` ### Detailed Analysis The chart displays a dark gray fitted curve with a lighter gray shaded confidence interval, representing the overall trend. Superimposed on this curve are 20 individual data points, each colored according to its `n_p` value and shaped according to its `n_fr` value. **Fitted Curve Trend and Approximate Values:** The dark gray fitted curve shows a clear positive correlation: as the "Mean number of tokens generated" increases, the "Mean pass rate" also increases. The rate of increase is steep initially and then gradually flattens out, suggesting diminishing returns. The light gray band around the curve indicates a confidence interval. * At X ≈ 0, Y ≈ 0.40 * At X ≈ 1000, Y ≈ 0.55 * At X ≈ 2000, Y ≈ 0.60 * At X ≈ 4000, Y ≈ 0.63 * At X ≈ 6000, Y ≈ 0.65 * At X ≈ 8000, Y ≈ 0.66 **Individual Data Points (Scatter Plot):** The data points generally follow the trend of the fitted curve. Each point represents a unique combination of `n_p` and `n_fr`. * **`n_p = 1` (Brown points):** * Circle (`n_fr = 1`): (X ≈ 450, Y ≈ 0.45) * Downward Triangle (`n_fr = 3`): (X ≈ 700, Y ≈ 0.48) * Square (`n_fr = 5`): (X ≈ 1000, Y ≈ 0.51) * Upward Triangle (`n_fr = 10`): (X ≈ 1500, Y ≈ 0.53) * **`n_p = 2` (Orange/Gold points):** * Circle (`n_fr = 1`): (X ≈ 600, Y ≈ 0.50) * Downward Triangle (`n_fr = 3`): (X ≈ 1000, Y ≈ 0.53) * Square (`n_fr = 5`): (X ≈ 1500, Y ≈ 0.58) * Upward Triangle (`n_fr = 10`): (X ≈ 2000, Y ≈ 0.60) * **`n_p = 5` (Teal/Light Green points):** * Circle (`n_fr = 1`): (X ≈ 1000, Y ≈ 0.58) * Downward Triangle (`n_fr = 3`): (X ≈ 1500, Y ≈ 0.61) * Square (`n_fr = 5`): (X ≈ 2500, Y ≈ 0.65) * Upward Triangle (`n_fr = 10`): (X ≈ 3500, Y ≈ 0.62) * **`n_p = 10` (Light Blue points):** * Circle (`n_fr = 1`): (X ≈ 1500, Y ≈ 0.60) * Downward Triangle (`n_fr = 3`): (X ≈ 2500, Y ≈ 0.68) * Square (`n_fr = 5`): (X ≈ 4500, Y ≈ 0.69) * Upward Triangle (`n_fr = 10`): (X ≈ 7500, Y ≈ 0.69) * **`n_p = 25` (Dark Blue/Indigo points):** * Circle (`n_fr = 1`): (X ≈ 2000, Y ≈ 0.62) * Downward Triangle (`n_fr = 3`): (X ≈ 3000, Y ≈ 0.67) * Square (`n_fr = 5`): (X ≈ 5500, Y ≈ 0.70) * Upward Triangle (`n_fr = 10`): (X ≈ 7000, Y ≈ 0.71) ### Key Observations 1. **General Trend**: There is a clear positive correlation between the mean number of tokens generated and the mean pass rate. The pass rate increases rapidly at lower token counts and then plateaus, indicating diminishing returns for generating more tokens beyond a certain point. 2. **Effect of `n_p`**: Higher values of `n_p` (represented by colors shifting from brown to dark blue) generally correspond to higher mean pass rates for a given range of tokens generated. For instance, `n_p=25` points achieve pass rates around 0.70-0.71, while `n_p=1` points peak around 0.53. 3. **Effect of `n_fr`**: Within each `n_p` group (i.e., for points of the same color), increasing `n_fr` (represented by marker shapes from circle to upward triangle) generally leads to a higher "Mean number of tokens generated." This suggests `n_fr` directly influences the output length or computational effort. 4. **Interaction of `n_p` and `n_fr`**: For a fixed `n_p`, increasing `n_fr` tends to move the data point to the right (more tokens generated) and generally upwards (higher pass rate), following the overall curve. 5. **Outlier/Anomaly**: The teal upward triangle point (`n_p = 5`, `n_fr = 10`) is located at approximately (X ≈ 3500, Y ≈ 0.62). This point appears to have a slightly lower pass rate compared to the general trend of increasing pass rate with increasing `n_fr` for `n_p=5` (where `n_fr=5` is at Y ≈ 0.65). It also falls slightly below the fitted curve, suggesting a potential deviation from the expected performance for this specific configuration. ### Interpretation The data presented in this chart likely illustrates the performance characteristics of a system (e.g., a language model or code generation system) where `n_p` and `n_fr` are configurable parameters. * **Performance vs. Cost Trade-off**: The chart fundamentally demonstrates a trade-off. To achieve a higher "Mean pass rate" (better performance), one generally needs to increase the "Mean number of tokens generated" (higher computational cost or longer output). * **Role of `n_p`**: The parameter `n_p` appears to be a critical factor in determining the *potential maximum pass rate*. Higher `n_p` values enable the system to achieve significantly better performance, suggesting it might relate to the number of parallel attempts, diverse candidates, or overall model capacity. Increasing `n_p` shifts the performance curve upwards. * **Role of `n_fr`**: The parameter `n_fr` seems to control the *granularity or extent of generation* for each `n_p` setting. Higher `n_fr` values lead to more tokens being generated, which in turn generally contributes to a higher pass rate, but also increases the computational burden. It scales the x-axis for a given `n_p`. * **Diminishing Returns**: The flattening of the curve indicates that there's a point where further increasing the number of tokens generated yields only marginal improvements in the pass rate. This is crucial for optimizing resource allocation. * **Potential for Optimization**: The observed outlier (`n_p=5, n_fr=10`) suggests that simply increasing `n_fr` might not always lead to monotonic improvements, especially for intermediate `n_p` values. There might be an optimal `n_fr` for each `n_p` beyond which performance either plateaus or even slightly degrades due to factors like increased noise, irrelevant output, or computational overhead without corresponding quality gains. This highlights the need for careful tuning of both parameters. * **Practical Implications**: For practical deployment, one would need to balance the desired pass rate with the acceptable computational cost (tokens generated). If high pass rates are critical, higher `n_p` values are necessary. If computational efficiency is paramount, lower `n_p` and `n_fr` values might be chosen, accepting a lower pass rate. </details> (a) Mean pass rate vs. number of tokens generated. Black line is i.i.d. sampling without repair from GPT-4. Note that the error bars are often smaller than the markers; all settings have a standard deviation of less than 1.5 absolute points on the y-axis. Results truncated at $t=10,000$ . <details> <summary>x6.png Details</summary>  ### Visual Description ## Heatmap: Performance Metric by Number of Feedback-Repairs and Initial Programs ### Overview This image displays a heatmap illustrating a performance metric across a grid defined by two input parameters: "Number of feedback-repairs" ($n_{fr}$) on the vertical axis and "Number of initial programs" ($n_p$) on the horizontal axis. Each cell in the grid contains a numerical value representing the metric, or the text "O.O.B." (Out Of Bounds), and is colored to visually represent its magnitude. The color gradient transitions from darker orange/brown for lower values, through yellow and light green for increasing values, to darker green for the highest values. Dark grey cells indicate the "O.O.B." state. ### Components/Axes * **Y-axis (Left):** Labeled "Number of feedback-repairs ($n_{fr}$)" * Tick markers (from bottom to top): 1, 3, 5, 10 * **X-axis (Bottom):** Labeled "Number of initial programs ($n_p$)" * Tick markers (from left to right): 1, 2, 5, 10, 25 * **Data Grid:** A 4x5 grid of cells, each containing a numerical value (to two decimal places) or the text "O.O.B.". The background color of each cell visually represents the magnitude of the value. * **Color Gradient (Implicit Legend):** * Darker orange/brown (e.g., 0.90, 0.91, 0.93): Represents lower metric values. * Lighter orange/yellow (e.g., 0.98, 0.99, 1.01): Represents mid-range metric values. * Light green/olive (e.g., 1.04, 1.05, 1.06): Represents higher mid-range metric values. * Darker green (e.g., 1.08, 1.09): Represents the highest metric values. * Dark grey: Represents the "O.O.B." (Out Of Bounds) state. ### Detailed Analysis The heatmap presents the following data, organized by `Number of feedback-repairs` ($n_{fr}$) on the Y-axis and `Number of initial programs` ($n_p$) on the X-axis. | $n_{fr}$ \ $n_p$ | 1 | 2 | 5 | 10 | 25 | | :--------------- | :----- | :----- | :----- | :----- | :----- | | **10** | 0.90 | 0.98 | 1.05 | O.O.B. | O.O.B. | | **5** | 0.91 | 0.98 | 1.04 | 1.08 | O.O.B. | | **3** | 0.93 | 0.99 | 1.04 | 1.08 | O.O.B. | | **1** | 0.98 | 1.01 | 1.04 | 1.06 | 1.09 | **Trends:** * **Across rows (increasing $n_p$ for a fixed $n_{fr}$):** * For $n_{fr}=10$: The metric values increase from 0.90 (dark orange) to 0.98 (yellow) to 1.05 (light green), then transition to "O.O.B." (dark grey) for $n_p=10$ and $n_p=25$. * For $n_{fr}=5$: The metric values increase from 0.91 (dark orange) to 0.98 (yellow) to 1.04 (light green) to 1.08 (dark green), then become "O.O.B." (dark grey) for $n_p=25$. * For $n_{fr}=3$: The metric values increase from 0.93 (dark orange) to 0.99 (yellow) to 1.04 (light green) to 1.08 (dark green), then become "O.O.B." (dark grey) for $n_p=25$. * For $n_{fr}=1$: The metric values consistently increase from 0.98 (yellow) to 1.01 (yellow) to 1.04 (light green) to 1.06 (light green) to 1.09 (dark green). * General trend: As the "Number of initial programs" ($n_p$) increases, the performance metric generally improves (increases), until it reaches an "O.O.B." state for higher $n_p$ values, particularly when $n_{fr}$ is also high. * **Down columns (increasing $n_{fr}$ for a fixed $n_p$):** * For $n_p=1$: The metric values decrease from 0.98 (yellow) to 0.93 (dark orange) to 0.91 (dark orange) to 0.90 (dark orange). * For $n_p=2$: The metric values decrease from 1.01 (yellow) to 0.99 (yellow) to 0.98 (yellow) and remain stable at 0.98 (yellow). * For $n_p=5$: The metric values are relatively stable at 1.04 (light green) for $n_{fr}=1, 3, 5$, then slightly increase to 1.05 (light green) for $n_{fr}=10$. * For $n_p=10$: The metric values increase from 1.06 (light green) to 1.08 (dark green) for $n_{fr}=3, 5$, then become "O.O.B." (dark grey) for $n_{fr}=10$. * For $n_p=25$: The metric value is 1.09 (dark green) for $n_{fr}=1$, then becomes "O.O.B." (dark grey) for $n_{fr}=3, 5, 10$. * General trend: The impact of increasing "Number of feedback-repairs" ($n_{fr}$) varies. For lower $n_p$, the metric tends to decrease or stabilize. For higher $n_p$, the metric tends to increase before hitting "O.O.B." for the highest $n_{fr}$ values. ### Key Observations * The lowest observed metric value is 0.90, located at the top-left corner ($n_{fr}=10, n_p=1$). * The highest observed metric value is 1.09, located at the bottom-right corner of the non-O.O.B. region ($n_{fr}=1, n_p=25$). * The "O.O.B." state appears in the upper-right portion of the heatmap, indicating that for higher values of $n_p$ (10 and 25) combined with higher values of $n_{fr}$ (3, 5, 10), the system enters an "Out Of Bounds" condition. * There is a clear diagonal boundary for the "O.O.B." region, starting from ($n_{fr}=10, n_p=10$) and extending downwards and rightwards to include ($n_{fr}=3, n_p=25$), ($n_{fr}=5, n_p=25$), and ($n_{fr}=10, n_p=25$). * The performance metric generally improves as $n_p$ increases, but this improvement is limited by the onset of the "O.O.B." state. * For $n_p=1$, increasing $n_{fr}$ consistently decreases the metric. For $n_p=2$, increasing $n_{fr}$ leads to a slight decrease and then stabilization. For $n_p=5$, the metric is relatively stable. For $n_p=10$ and $n_p=25$, increasing $n_{fr}$ initially improves the metric but quickly leads to "O.O.B.". ### Interpretation This heatmap likely illustrates the performance characteristics of a system or algorithm influenced by two parameters: the number of feedback-repairs ($n_{fr}$) and the number of initial programs ($n_p$). Higher metric values are generally associated with better performance, as indicated by the color progression from orange (low) to green (high). The data suggests that increasing the "Number of initial programs" ($n_p$) is generally beneficial for performance, leading to higher metric values. This trend is most pronounced when the "Number of feedback-repairs" ($n_{fr}$) is low (e.g., $n_{fr}=1$), where the metric continuously increases to its peak of 1.09. However, there's a critical interaction with $n_{fr}$. While increasing $n_p$ is good, combining high $n_p$ with high $n_{fr}$ leads to an "Out Of Bounds" (O.O.B.) state. This "O.O.B." condition could signify system failure, instability, resource exhaustion, or a state where the metric cannot be meaningfully computed. The diagonal boundary of the O.O.B. region indicates a threshold or limit where the combined complexity or resource demands of many initial programs and many feedback-repairs become unmanageable. Conversely, the effect of increasing $n_{fr}$ is not uniformly positive. For a small number of initial programs ($n_p=1, 2$), more feedback-repairs actually lead to slightly worse or stable performance. For moderate $n_p$ (e.g., $n_p=5$), $n_{fr}$ has little impact on the metric. Only when $n_p$ is already high (e.g., $n_p=10$) does increasing $n_{fr}$ initially boost performance before quickly pushing the system into the O.O.B. state. In summary, the system achieves its best measurable performance (1.09) with a high number of initial programs ($n_p=25$) and a low number of feedback-repairs ($n_{fr}=1$). This suggests an optimal operating point where the system benefits from a broad initial exploration (many initial programs) but is sensitive to excessive iterative refinement or correction (feedback-repairs), especially when the initial exploration is already extensive. The "O.O.B." region highlights a critical operational boundary that must be avoided for stable and measurable system behavior. </details> (b) Normalized mean pass rate relative to the (interpolated) baseline at an equivalent budget (number of tokens). Cells for which the number of tokens generated exceeds 50 samples from the GPT-4 baseline marked O.O.B. (out of bounds). Figure 4: Pass rate versus number of tokens generated for various settings of $n_{p}$ (number of initial programs) and $n_{fr}$ (number of repairs per failing program). GPT-4 is used for all samples, including the baseline. ### 4.2 GPT-4 feedback improves GPT-3.5 repair Next, we conduct an experiment in which we evaluate the impact of using a separate, stronger model to generate the feedback. This is to test the hypothesis that self-repair is held back (especially for GPT-3.5) by the model’s inability to introspect and debug its own code. For this experiment, we set $M_{P}$ = GPT-3.5 and $M_{F}$ = GPT-4 and vary the hyper-parameters as $(n_{p},n_{f},n_{r})\in\{1,2,....,24,25\}\times\{1\}\times\{1\}$ , similarly to the previous experiment. Note that since we are now operating in a setting in which the feedback and repair stages must be separated, we have three hyper-parameters— $n_{p},n_{f},n_{r}$ —instead of two— $n_{p},n_{fr}$ (Section 3.1). To keep the computational budget tractable, and since the variance was seen to be very low in the previous experiment, we use $N_{f}=10$ instead of $N_{f}=25$ for this experiment (see Section 3.2). The results for this experiment are shown in Figure 6 (bright blue line). We observe that in terms of absolute performance, $M_{P}=$ GPT-3.5, $M_{F}=$ GPT-4 does break through the performance barrier and becomes marginally more efficient than i.i.d. sampling from GPT-3.5. This suggests that the textual feedback stage itself is of crucial importance, and that improving it relieves the bottleneck in GPT-3.5 self-repair. Figure 5: Mean pass rate for each model when $n_{fr}$ (or $n_{f}$ and $n_{r}$ ) = 1. Shaded region is $\pm 1$ standard deviation. Complete breakdown per difficulty in Appendix A. <details> <summary>x7.png Details</summary>  ### Visual Description ## Chart Type: Line Chart - Mean Pass Rate vs. Mean Number of Tokens Generated for GPT Models ### Overview This image displays a line chart illustrating the relationship between the "Mean pass rate" (Y-axis) and the "Mean number of tokens generated" (X-axis) for five different configurations of GPT models. Each line represents a distinct model setup, differentiated by the primary model ($M_P$) and, if applicable, a repair model ($M_F$). Shaded areas around each line indicate uncertainty or a confidence interval. ### Components/Axes The chart consists of a main plotting area, a Y-axis on the left, an X-axis at the bottom, and a legend in the bottom-right. * **Y-axis Label**: "Mean pass rate" * **Y-axis Range**: From 0.0 to 1.0. * **Y-axis Major Ticks**: 0.0, 0.2, 0.4, 0.6, 0.8, 1.0. * **X-axis Label**: "Mean number of tokens generated" * **X-axis Range**: From 0 to 10000. * **X-axis Major Ticks**: 0, 2000, 4000, 6000, 8000, 10000. * **X-axis Tick Label Orientation**: Labels are rotated approximately 45 degrees counter-clockwise. * **Grid**: Light gray grid lines are present, aligning with the major ticks on both axes. * **Legend**: Located in the bottom-right quadrant of the plot area. It lists five data series with their corresponding line colors: * **Dark Blue line**: $M_P$ = GPT-4 (no repair) * **Teal/Light Green line**: $M_P$ = GPT-4; $M_F$ = GPT-4 * **Dark Gray line**: $M_P$ = GPT-3.5 (no repair) * **Brown/Orange line**: $M_P$ = GPT-3.5; $M_F$ = GPT-3.5 * **Light Blue/Cyan line**: $M_P$ = GPT-3.5; $M_F$ = GPT-4 ### Detailed Analysis Each data series shows a general trend of an initial rapid increase in mean pass rate as the mean number of tokens generated increases, followed by a plateau where the pass rate stabilizes or increases very slowly. All lines are accompanied by a translucent shaded area, representing a confidence interval around the mean pass rate. 1. **Teal/Light Green line ($M_P$ = GPT-4; $M_F$ = GPT-4)**: * **Trend**: This line starts at a relatively high pass rate, rises steeply, and then flattens out to become the highest performing series. It ends around 6500 tokens. * **Approximate Data Points**: * ~0 tokens: ~0.40 pass rate * ~1000 tokens: ~0.60 pass rate * ~2000 tokens: ~0.65 pass rate * ~4000 tokens: ~0.68 pass rate * ~6000 tokens: ~0.70 pass rate (line ends) 2. **Dark Blue line ($M_P$ = GPT-4 (no repair))**: * **Trend**: This line follows a similar pattern to the Teal line, starting high, rising steeply, and then flattening. It consistently performs lower than the Teal line but higher than all GPT-3.5 configurations. It ends around 7800 tokens. * **Approximate Data Points**: * ~0 tokens: ~0.40 pass rate * ~1000 tokens: ~0.58 pass rate * ~2000 tokens: ~0.62 pass rate * ~4000 tokens: ~0.64 pass rate * ~6000 tokens: ~0.65 pass rate * ~7800 tokens: ~0.66 pass rate (line ends) 3. **Light Blue/Cyan line ($M_P$ = GPT-3.5; $M_F$ = GPT-4)**: * **Trend**: This line starts at a lower pass rate than the GPT-4 primary models, rises, and then flattens. It is the highest performing among the GPT-3.5 primary model configurations. It extends to 10000 tokens. * **Approximate Data Points**: * ~0 tokens: ~0.25 pass rate * ~1000 tokens: ~0.40 pass rate * ~2000 tokens: ~0.45 pass rate * ~4000 tokens: ~0.50 pass rate * ~6000 tokens: ~0.53 pass rate * ~8000 tokens: ~0.55 pass rate * ~10000 tokens: ~0.56 pass rate 4. **Brown/Orange line ($M_P$ = GPT-3.5; $M_F$ = GPT-3.5)**: * **Trend**: This line shows a similar initial rise and plateau, consistently performing slightly below the Light Blue/Cyan line. It extends to 10000 tokens. * **Approximate Data Points**: * ~0 tokens: ~0.25 pass rate * ~1000 tokens: ~0.38 pass rate * ~2000 tokens: ~0.42 pass rate * ~4000 tokens: ~0.46 pass rate * ~6000 tokens: ~0.49 pass rate * ~8000 tokens: ~0.51 pass rate * ~10000 tokens: ~0.52 pass rate 5. **Dark Gray line ($M_P$ = GPT-3.5 (no repair))**: * **Trend**: This line exhibits the lowest pass rates across all token counts, following the general trend of rising and then flattening. It extends to 10000 tokens. * **Approximate Data Points**: * ~0 tokens: ~0.25 pass rate * ~1000 tokens: ~0.35 pass rate * ~2000 tokens: ~0.40 pass rate * ~4000 tokens: ~0.43 pass rate * ~6000 tokens: ~0.46 pass rate * ~8000 tokens: ~0.48 pass rate * ~10000 tokens: ~0.49 pass rate ### Key Observations * **Performance Hierarchy**: The models using GPT-4 as the primary model ($M_P$) consistently achieve higher pass rates than those using GPT-3.5 as $M_P$. * **Impact of Repair Mechanism ($M_F$)**: For both GPT-4 and GPT-3.5 primary models, the presence of a repair mechanism ($M_F$) improves the mean pass rate compared to the "no repair" configuration. * **GPT-4 as Repair Model**: Using GPT-4 as the repair model ($M_F$ = GPT-4) yields the highest pass rates, even when the primary model is GPT-3.5 (Light Blue/Cyan line vs. Brown/Orange line). The configuration $M_P$ = GPT-3.5; $M_F$ = GPT-4 significantly outperforms $M_P$ = GPT-3.5; $M_F$ = GPT-3.5. * **Highest Performance**: The combination of $M_P$ = GPT-4 and $M_F$ = GPT-4 (Teal line) achieves the highest mean pass rate, reaching approximately 0.70. * **Lowest Performance**: The configuration $M_P$ = GPT-3.5 (no repair) (Dark Gray line) shows the lowest mean pass rate, plateauing around 0.49. * **Diminishing Returns**: All lines show that the most significant gains in pass rate occur within the first ~2000-4000 tokens generated. Beyond this point, the pass rate continues to increase but at a much slower rate, suggesting diminishing returns for generating more tokens. * **Confidence Intervals**: The shaded areas around each line indicate variability or uncertainty in the mean pass rate, which appears to be relatively consistent across the range of tokens for each configuration. ### Interpretation This chart demonstrates the effectiveness of different large language model (LLM) configurations in achieving a higher "pass rate" on an unspecified task, likely related to code generation, problem-solving, or complex reasoning, where "tokens generated" might refer to the length or complexity of the model's output. The data strongly suggests that: 1. **Model Capability Matters**: GPT-4 is a significantly more capable model than GPT-3.5 for this task, as evidenced by the higher pass rates when GPT-4 is used as the primary model ($M_P$). 2. **Repair Mechanisms are Beneficial**: Implementing a "repair" mechanism ($M_F$) consistently improves performance. This implies that an iterative refinement or error correction step can enhance the quality of the generated output. 3. **Quality of Repair Model is Crucial**: The choice of the repair model is critical. Using a more powerful model like GPT-4 for repair ($M_F$ = GPT-4) provides a substantial boost in pass rate, even if the initial generation ($M_P$) is done by a less capable model like GPT-3.5. This highlights a potential strategy: leverage a powerful model for refinement even if the primary generation is resource-constrained. 4. **Efficiency Considerations**: While generating more tokens generally leads to a higher pass rate, the plateauing effect indicates that there's an optimal range for token generation. Beyond approximately 4000-6000 tokens, the additional computational cost of generating more tokens yields only marginal improvements in the pass rate, suggesting a point of diminishing returns. This is important for optimizing resource usage and latency. In essence, the chart illustrates that both the inherent capability of the primary model and the strategic application of a repair mechanism (especially one powered by a more advanced model) are key factors in maximizing the success rate of LLM-generated outputs, with an important consideration for the efficiency of token generation. </details> | Introductory Interview Competition | 42.64% 19.33% 3.67% | 62.21% 45.67% 14.67% | | --- | --- | --- | | Overall | 33.30% | 52.60% | Figure 5: Mean pass rate for each model when $n_{fr}$ (or $n_{f}$ and $n_{r}$ ) = 1. Shaded region is $\pm 1$ standard deviation. Complete breakdown per difficulty in Appendix A. Figure 6: Success rate of repair with GPT-4’s explanations vs. with those of our human participants. ### 4.3 Human feedback significantly improves the success rate of GPT-4 repair For our final experiment, we consider the effect of using an expert human programmer’s feedback when performing repair with stronger models such as GPT-4. The goal of this study is not to do a direct comparison between a human-in-the-loop approach vs. self-repair, since a human-in-the-loop approach imposes more cognitive burden, which we do not study. Instead, our goal is to understand how the model’s ability to identify mistakes in the code compares to that of a human, and how this affects downstream performance in self-repair. We thus conduct both qualitative and quantitative analyses of the impact of human feedback on self-repair. Data collection methodology. We first sample 20 tasks $\{\psi_{i}\}_{i=1}^{20}$ from the APPS test set; to make the data collection process less time-consuming for the participants of the study, we skew the distribution towards easier tasks (14 introductory; 3 interview; 3 competition). For each task $\psi_{i}$ , we then sample two failing GPT-4 completions $p_{i,1},p_{i,2}$ , making for a total of $20\cdot 2=40$ programs to refine. We recruit 16 participants, consisting of 15 graduate students and one professional machine learning engineer. Each participant is provided with five different base programs based on their level of experience with Python and competitive programming. Each program is taken from a distinct task; participants are never showed two different programs belonging to the same task. Participants are then asked to explain, in their own words, what the program is doing wrong. To reduce the cognitive load for participants, each program $p_{i,j}$ is accompanied by the error message $e_{i,j}$ and two feedback strings $f_{i,j,1},f_{i,j,2}$ sampled from GPT-4. We obtain these feedback strings by randomly sampling from the feedback-repair pairs used in the previous experiments and removing the code block. Note that each of the 40 programs will be shown to two different participants, to reduce variance caused by participants’ skill levels and writing style. Participants were told to spend approximately one hour on the study overall, and were compensated with a $15 gift card. This human data collection was approved by our Institutional Review Board (IRB) and carried out exclusively through an online survey. See Appendix B for a complete, concrete copy of the instructions which we provide to our participants. Quantitative Analysis. Having obtained two human-written pieces of feedback $h_{i,j,1},h_{i,j,2}$ for each program $p_{i,j}$ , we sample 25 repaired programs $$ \{r_{l}\}_{l=1}^{25}\stackrel{{\scriptstyle i.i.d.}}{{\sim}}\text{GPT-4}(\psi_ {i};p_{i,j};e_{i,j};f) $$ for $f\in\{h_{i,j,1},h_{i,j,2},f_{i,j,1},f_{i,j,2}\}$ . That is: we ask GPT-4 to generate 25 candidate repairs for each program, conditioned on the specification, the initial program, and a feedback string which is either set to one of GPT-4’s own feedback strings or to one provided by a participant. Finally, we execute all of these candidate repairs against the test bed, and take note of how often they pass. The results are summarized in Table 6, with a complete task-by-task breakdown in Appendix C. We note first of all that the overall success rate is increased by over $1.57\times$ when we replace GPT-4’s own debugging with that of our human participants. Perhaps unsurprisingly, the relative difference increases as the problems get harder, indicating that GPT-4’s ability to produce accurate and useful feedback trails further behind our human participants’ when the task (and code) becomes more complex. Qualitative Analysis. In this section, we qualitatively analyze the difference between the feedback provided by the human participants and the feedback provided by GPT-4. We manually go through all of GPT-4’s and the participants’ feedback and note down whether the feedback: (a) seems, at a cursory glance, to be correct, or if it is obviously inaccurate; (b) explicitly suggests a small change to the code (e.g. "change the condition on line X"); (c) explicitly suggests a large change to the code (e.g. "frame the problem as min-cut instead of shortest-path"); (d) contains blocks of pseudocode or Python (which GPT-4’s feedback never does, per our experiment design); or (e) expresses uncertainty (using phrases such as "unsure", "it appears", etc.). We do not count individual single-line statements/expressions such as “ $x=5$ ” as pseudocode or Python. Examples of each category are shown in Appendix D. We find that - Only 2/80 human-contributed feedback strings include pseudocode or explicit Python; that is, almost all human feedback we obtain is natural language interleaved with occasional single-statement math/code expressions. - GPT-4’s feedback is much more likely to be obviously inaccurate (32/80 vs. 7/80 for human feedback). - GPT-4 is more likely to explicitly suggest small changes (54/80 vs 42/80; 28/48 vs. 38/73 when seemingly correct), while our human participants show a slightly greater tendency to suggest high-level changes (23/80 vs. 18/80 for GPT-4; 21/73 vs. 13/48 when seemingly correct). - Our human participants sometimes express uncertainty (7/80); GPT-4 never does (0/80). This further analysis suggests that the results in Table 6 are not due to artefacts such as our participants providing explicit code blocks which the model simply copies. Instead, the difference in performance appears to be caused by a combination of more accurate feedback, a greater ability to suggest high-level, large-scale changes to the code when needed, and our participants’ ability to express their uncertainty (instead of confidently giving potentially inaccurate feedback). ## 5 Limitations Firstly, to reduce computational cost, we pre-populate and then sub-sample from large repair trees, which introduces statistical bias. We mitigate this by being generous in our uncertainty of the pass@t, using the maximum standard deviation across all points. We also note that this standard deviation, which is obtained at values of $(n_{p},n_{f},n_{r})$ that are small enough that we have very many samples thereof in our pre-populated repair trees, is very low ( $<2\$ pass rate for all models). While these measures do not completely eliminate the risk of bias in our results, not performing this amortization would have required significantly larger amounts of compute. Secondly, we assume access to an executable suite of unit tests for each task. We do not, for example, require the model to extract tests from textual specifications. While this assumption may seem out of place in the era of chat-style assistants like ChatGPT [OpenAI, 2022], it does align well with established software engineering practices like Test-Driven Development [Astels, 2003]. Furthermore, techniques which automatically synthesize test cases given a specification [Li et al., 2022, Chen et al., 2023a] may relieve some of the user burden. Finally, our study on human data did not track how much time the participants took to debug the programs. As a result, we can only evaluate the quality of the feedback (and the impact this has on repair). Further research at the intersection of HCI, AI, and program synthesis is needed to explore when and how human intervention should be leveraged, as well as how programming assistants should be designed to facilitate this style of interaction. ## 6 Broader Impact Any tool that improves the productivity of people writing software will necessarily also increase the productivity of people writing software with malicious intent. It is also important to remember that research on LLMs comes at a very high environmental cost. Although we exclusively use publicly available pre-trained models in this work, and so do not train any models of our own, even inference comes with a significant carbon footprint at scale. At the same time, this work—in which we weigh model performance against the computational cost of obtaining it, and through which we learn more about when and how these models do and do not work—is a step towards more sample-efficient usage paradigms. ## 7 Conclusion In this paper, we investigated the role of textual feedback in self-repair. We presented pass@t, a new evaluation strategy which takes the cost of carrying out repair into account, and then used this metric to show that (1) GPT-3.5 is not capable of carrying out self-repair on challenging coding tasks, and (2) while performance gains are seen in GPT-4, they are modest and rely on achieving sufficient diversity in the initial programs. Furthermore, by ablating the feedback stage we found that (3) substituting GPT-3.5’s feedback with GPT-4’s improved performance, even surpassing GPT-3.5’s baseline. Finally, we carried out an experiment with human participants, in which we found that (4) replacing GPT-4’s self-generated feedback with feedback provided by an experienced programmer increased the number of repaired programs which pass all unit tests by 57%. ## Acknowledgments and Disclosure of Funding T.X. Olausson is supported by the Defense Advanced Research Projects Agency (DARPA) under the ASKEM program, award HR00112220042. T.X. Olausson was also supported through a position at Microsoft Research for part of the time period during which this work was carried out. A. Solar-Lezama is supported by the National Science Foundation (NSF) and Intel Corporation through NSF Grant CCF:2217064. This work benefited greatly from discussion with several colleagues at Microsoft Research. 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The data points are categorized by two parameters: `n_p` (represented by color) and `n_fr` (represented by marker shape). A dark grey fitted curve with a lighter grey confidence interval is overlaid on the data, showing the general trend. Each data point includes vertical error bars indicating uncertainty in the mean pass rate. ### Components/Axes The chart is structured with a primary plotting area, an x-axis at the bottom, a y-axis on the left, and a legend positioned in the bottom-left to center-right area. * **X-axis (Bottom)**: * **Title**: "Mean number of tokens generated" * **Range**: From 0 to 10000. * **Major Ticks and Labels**: 0, 2000, 4000, 6000, 8000, 10000. The labels are rotated approximately 45 degrees counter-clockwise. * **Y-axis (Left)**: * **Title**: "Mean pass rate" * **Range**: From 0.0 to 1.0. * **Major Ticks and Labels**: 0.0, 0.2, 0.4, 0.6, 0.8, 1.0. * **Legend (Bottom-left to Center-right)**: The legend is divided into two columns within a light grey rectangular box. * **Left Column (Color mapping for `n_p`)**: * Brown line: `n_p = 1` * Orange line: `n_p = 2` * Teal/Cyan line: `n_p = 5` * Light Blue line: `n_p = 10` * Dark Blue line: `n_p = 25` * **Right Column (Marker shape mapping for `n_fr`)**: * Dark Grey circle: `n_fr = 1` * Dark Grey downward triangle: `n_fr = 3` * Dark Grey square: `n_fr = 5` * Dark Grey upward triangle: `n_fr = 10` * **Fitted Curve**: A thick, solid dark grey line represents the overall trend. It is surrounded by a lighter grey shaded area, indicating a confidence interval or standard deviation for the fit. ### Detailed Analysis The chart displays 20 data points, each characterized by a specific `n_p` value (color), `n_fr` value (marker shape), mean number of tokens generated (x-coordinate), and mean pass rate (y-coordinate), along with vertical error bars. **Overall Trend of Fitted Curve**: The dark grey fitted curve shows a rapid increase in "Mean pass rate" as "Mean number of tokens generated" increases from 0 to approximately 2000. Beyond 2000 tokens, the rate of increase slows significantly, and the curve appears to plateau around a "Mean pass rate" of 0.84 to 0.85 for token counts between 4000 and 6000. The confidence interval (light grey shading) is relatively narrow across the observed range. **Data Points by `n_p` (Color) and `n_fr` (Marker Shape)**: 1. **`n_p = 1` (Brown colored points)**: * `n_fr = 1` (Circle): * Approx. (550 tokens, 0.58 pass rate), error bar from ~0.56 to ~0.60. * Approx. (1000 tokens, 0.65 pass rate), error bar from ~0.63 to ~0.67. * `n_fr = 3` (Downward Triangle): * Approx. (1200 tokens, 0.68 pass rate), error bar from ~0.66 to ~0.70. * `n_fr = 5` (Square): * Approx. (1500 tokens, 0.70 pass rate), error bar from ~0.68 to ~0.72. * `n_fr = 10` (Upward Triangle): * Approx. (1800 tokens, 0.75 pass rate), error bar from ~0.73 to ~0.77. * *Trend for `n_p=1`*: Pass rate increases steadily with tokens generated, from 0.58 to 0.75. 2. **`n_p = 2` (Orange/Gold colored points)**: * `n_fr = 1` (Circle): * Approx. (700 tokens, 0.65 pass rate), error bar from ~0.63 to ~0.67. * `n_fr = 3` (Downward Triangle): * Approx. (1400 tokens, 0.72 pass rate), error bar from ~0.70 to ~0.74. * `n_fr = 5` (Square): * Approx. (2200 tokens, 0.75 pass rate), error bar from ~0.73 to ~0.77. * Approx. (3200 tokens, 0.77 pass rate), error bar from ~0.75 to ~0.79. * `n_fr = 10` (Upward Triangle): * Approx. (2800 tokens, 0.76 pass rate), error bar from ~0.74 to ~0.78. * *Trend for `n_p=2`*: Pass rate increases with tokens generated, reaching up to 0.77. These points generally achieve higher pass rates than `n_p=1` for similar token counts. 3. **`n_p = 5` (Teal/Cyan colored points)**: * `n_fr = 1` (Circle): * Approx. (1200 tokens, 0.78 pass rate), error bar from ~0.76 to ~0.80. * `n_fr = 3` (Downward Triangle): * Approx. (2000 tokens, 0.82 pass rate), error bar from ~0.80 to ~0.84. * `n_fr = 5` (Square): * Approx. (4800 tokens, 0.83 pass rate), error bar from ~0.81 to ~0.85. * `n_fr = 10` (Upward Triangle): * Approx. (5200 tokens, 0.82 pass rate), error bar from ~0.80 to ~0.84. * *Trend for `n_p=5`*: Pass rate increases to over 0.80, reaching the plateau region of the fitted curve. 4. **`n_p = 10` (Light Blue colored points)**: * `n_fr = 1` (Circle): * Approx. (1800 tokens, 0.83 pass rate), error bar from ~0.81 to ~0.85. * `n_fr = 3` (Downward Triangle): * Approx. (3000 tokens, 0.84 pass rate), error bar from ~0.82 to ~0.86. * Approx. (7500 tokens, 0.87 pass rate), error bar from ~0.85 to ~0.89. (This point is significantly to the right of the main cluster and above the fitted curve's plateau.) * `n_fr = 5` (Square): * Approx. (4000 tokens, 0.85 pass rate), error bar from ~0.83 to ~0.87. * `n_fr = 10` (Upward Triangle): * Approx. (8500 tokens, 0.85 pass rate), error bar from ~0.83 to ~0.87. (This point is also significantly to the right of the main cluster and slightly above the fitted curve's plateau.) * *Trend for `n_p=10`*: These points quickly reach the plateau region of the pass rate. Two points at very high token counts show slightly elevated pass rates compared to the main plateau. 5. **`n_p = 25` (Dark Blue colored points)**: * `n_fr = 1` (Circle): * Approx. (4500 tokens, 0.85 pass rate), error bar from ~0.83 to ~0.87. * `n_fr = 3` (Downward Triangle): * Approx. (5000 tokens, 0.84 pass rate), error bar from ~0.82 to ~0.86. * *Trend for `n_p=25`*: These points are already in the plateau region, indicating high pass rates are achieved at moderate token counts. ### Key Observations * **Diminishing Returns**: The "Mean pass rate" shows a clear trend of diminishing returns with increasing "Mean number of tokens generated." After approximately 4000 tokens, the pass rate improvement is minimal. * **Impact of `n_p`**: Higher values of `n_p` (represented by darker blue colors) generally achieve higher "Mean pass rates" at lower "Mean number of tokens generated." For instance, `n_p=10` reaches a pass rate of ~0.83 at ~1800 tokens, while `n_p=1` only reaches ~0.75 at the same token count. This suggests `n_p` is a significant factor in the efficiency of achieving a high pass rate. * **Impact of `n_fr`**: Within each `n_p` group, increasing `n_fr` (from circle to upward triangle) generally corresponds to higher token counts, often leading to slightly higher pass rates or maintaining high pass rates at increased token generation. * **Outliers/High Token Count Behavior**: Two light blue points (`n_p=10`) at very high token counts (around 7500 and 8500) show "Mean pass rates" of approximately 0.87 and 0.85, respectively. These are slightly above the general plateau of the fitted curve, suggesting that for `n_p=10`, further token generation might yield marginal improvements or these represent specific conditions. * **Error Bar Consistency**: The vertical error bars are consistently small across all data points, indicating a relatively low variability in the "Mean pass rate" for each configuration. ### Interpretation The data strongly suggests that there is an optimal range for the "Mean number of tokens generated" to achieve a high "Mean pass rate." Generating tokens beyond approximately 4000-5000 provides very little additional benefit in terms of pass rate, indicating a point of diminishing returns or saturation. The parameter `n_p` appears to be a critical factor influencing the efficiency of the process. Higher `n_p` values enable the system to reach a high pass rate with fewer tokens generated, implying that `n_p` might represent a measure of the model's capacity, search breadth, or inherent quality. A higher `n_p` allows for a more "powerful" or "effective" generation process, requiring less "effort" (fewer tokens) to achieve a desired outcome. The parameter `n_fr` seems to fine-tune the generation process for a given `n_p`. As `n_fr` increases, the system tends to generate more tokens, potentially exploring more options or refining the output, which can lead to slightly higher pass rates or maintain high pass rates at increased computational cost. The two outlying points for `n_p=10` at very high token counts are noteworthy. While the general trend shows a plateau, these points suggest that for specific configurations (like `n_p=10` with `n_fr=3` or `n_fr=10`), pushing token generation significantly further might still yield a slight, albeit marginal, increase in pass rate. This could indicate that for certain `n_p` values, the plateau is not absolute, or that these specific `n_fr` values unlock slightly better performance at extreme token generation costs. Further investigation into these specific configurations could reveal nuances in the system's behavior. In summary, the chart highlights the importance of balancing computational cost (tokens generated) with performance (pass rate) and demonstrates that `n_p` is a key parameter for optimizing this trade-off, while `n_fr` offers further fine-tuning. </details> <details> <summary>x9.png Details</summary>  ### Visual Description ## Chart Type: Heatmap of Performance Metrics based on Program and Repair Counts ### Overview This image displays a heatmap illustrating a performance metric across varying numbers of "initial programs" and "feedback-repairs". The values within the cells range from approximately 0.87 to 1.01, with a color gradient from dark orange (lower values) to bright yellow (higher values). A distinct dark grey/black region indicates "O.O.B." (Out Of Bounds) conditions. ### Components/Axes The chart is a 2D heatmap with two categorical axes: * **Y-axis (Left)**: Labeled "Number of feedback-repairs ($n_{fr}$)" * Tick markers (from bottom to top): 1, 3, 5, 10 * **X-axis (Bottom)**: Labeled "Number of initial programs ($n_P$)" * Tick markers (from left to right): 1, 2, 5, 10, 25 * **Color Scale/Legend**: There is no explicit color legend, but the cell colors visually represent the magnitude of the numerical values. * Darker orange/brown colors correspond to lower values (e.g., 0.87-0.92). * Lighter orange/yellow colors correspond to higher values (e.g., 0.97-1.01). * A distinct dark grey/black color indicates "O.O.B." (Out Of Bounds) conditions, which are not numerical values. ### Detailed Analysis The heatmap is a 4x5 grid, with each cell representing a unique combination of $n_{fr}$ and $n_P$. The values within each cell are as follows: | $n_{fr}$ \ $n_P$ | 1 | 2 | 5 | 10 | 25 | | :--------------- | :----- | :----- | :----- | :----- | :----- | | **10** | 0.87 | 0.93 | 0.97 | O.O.B. | O.O.B. | | **5** | 0.87 | 0.94 | 0.98 | 1.00 | O.O.B. | | **3** | 0.88 | 0.94 | 0.99 | 1.00 | O.O.B. | | **1** | 0.92 | 0.97 | 1.00 | 1.01 | 1.01 | **Trends and Distributions:** * **Across rows (increasing $n_P$ for a fixed $n_{fr}$):** * For $n_{fr}=10$: Values increase from 0.87 to 0.97, then become O.O.B. for $n_P=10$ and $n_P=25$. * For $n_{fr}=5$: Values increase from 0.87 to 1.00, then become O.O.B. for $n_P=25$. * For $n_{fr}=3$: Values increase from 0.88 to 1.00, then become O.O.B. for $n_P=25$. * For $n_{fr}=1$: Values consistently increase from 0.92 to 1.01, then stabilize at 1.01 for $n_P=25$. * General trend: Performance tends to improve as $n_P$ increases, up to a certain point, after which it either stabilizes or enters an O.O.B. state. * **Down columns (increasing $n_{fr}$ for a fixed $n_P$):** * For $n_P=1$: Values decrease from 0.92 to 0.87. * For $n_P=2$: Values decrease from 0.97 to 0.93. * For $n_P=5$: Values decrease from 1.00 to 0.97. * For $n_P=10$: Values decrease from 1.01 to 1.00, then become O.O.B. for $n_{fr}=10$. * For $n_P=25$: The value is 1.01 for $n_{fr}=1$, then becomes O.O.B. for $n_{fr}=3, 5, 10$. * General trend: Performance tends to decrease as $n_{fr}$ increases, or transitions to an O.O.B. state for higher $n_P$. ### Key Observations * **Highest Performance**: The maximum numerical value observed is 1.01, which occurs at ($n_P=10, n_{fr}=1$) and ($n_P=25, n_{fr}=1$). These cells are colored bright yellow. * **Lowest Performance**: The lowest numerical value observed is 0.87, occurring at ($n_P=1, n_{fr}=10$) and ($n_P=1, n_{fr}=5$). These cells are colored dark orange/brown. * **O.O.B. Region**: A significant portion of the upper-right quadrant of the heatmap is marked "O.O.B." (Out Of Bounds). This region includes: * All cells for $n_P=25$ except for $n_{fr}=1$. * The cell for ($n_P=10, n_{fr}=10$). * This suggests that combinations of high initial programs and high feedback-repairs are not valid or measurable within the context of this data. * **Optimal Region**: The highest performance values (around 1.00-1.01) are concentrated in the bottom-right area of the numerical data, specifically where $n_{fr}$ is low (e.g., 1 or 3) and $n_P$ is moderate to high (e.g., 5, 10, 25). ### Interpretation This heatmap likely represents the performance or efficiency of a system or process, where "Number of initial programs ($n_P$)" and "Number of feedback-repairs ($n_{fr}$)" are configurable parameters. Higher numerical values indicate better performance. The data suggests a clear trade-off and optimal operating conditions: 1. **Impact of Initial Programs ($n_P$)**: Generally, increasing the number of initial programs leads to improved performance. This could imply that more initial resources or attempts provide a better starting point or more diverse options, leading to better outcomes. 2. **Impact of Feedback-Repairs ($n_{fr}$ )**: Conversely, increasing the number of feedback-repairs generally leads to a decrease in performance. This might indicate that excessive repairs introduce overhead, complexity, or lead to diminishing returns, potentially degrading the overall outcome. 3. **Optimal Configuration**: The best performance (values of 1.00-1.01) is achieved when the number of feedback-repairs is minimal ($n_{fr}=1$) and the number of initial programs is sufficiently high ($n_P=5, 10, 25$). This suggests that a strong initial setup is more beneficial than extensive iterative refinement. 4. **System Constraints ("O.O.B.")**: The "O.O.B." region is critical. It indicates that certain combinations of parameters are not supported, are invalid, or perhaps lead to system failure. Specifically, having a high number of feedback-repairs (e.g., $n_{fr}=3, 5, 10$) when the number of initial programs is very high ($n_P=25$) is "Out Of Bounds". This could be due to computational limits, logical inconsistencies in the system design, or simply configurations that were not tested or are known to be unfeasible. The single O.O.B. cell at ($n_P=10, n_{fr}=10$) further reinforces that high values for both parameters can be problematic. In essence, the system performs best with a robust initial configuration and minimal, if any, feedback-driven adjustments. Over-reliance on feedback-repairs, especially in conjunction with a large number of initial programs, appears to be detrimental or even impossible. </details> <details> <summary>x10.png Details</summary>  ### Visual Description ## Chart Type: Scatter Plot with Fitted Curve ### Overview This image displays a scatter plot illustrating the relationship between "Mean number of tokens generated" on the x-axis and "Mean pass rate" on the y-axis. The data points are categorized by two parameters: `n_p` (represented by color) and `n_fr` (represented by marker shape). A dark grey fitted curve with a lighter grey confidence interval is overlaid on the data, showing the general trend. ### Components/Axes **X-axis:** * **Title:** "Mean number of tokens generated" * **Range:** From 0 to 10000 * **Major Ticks:** 0, 2000, 4000, 6000, 8000, 10000 **Y-axis:** * **Title:** "Mean pass rate" * **Range:** From 0.0 to 1.0 * **Major Ticks:** 0.0, 0.2, 0.4, 0.6, 0.8, 1.0 **Legend (positioned in the top-center of the plot area):** The legend is split into two columns. The left column defines `n_p` values by line color, and the right column defines `n_fr` values by marker shape. * **n_p** (Line Colors): * `n_p = 1`: Brown line * `n_p = 2`: Goldenrod/Yellow-orange line * `n_p = 5`: Teal/Light blue-green line * `n_p = 10`: Blue line * `n_p = 25`: Dark blue/Navy line * **n_fr** (Marker Shapes): * `n_fr = 1`: Dark grey circle * `n_fr = 3`: Dark grey inverted triangle * `n_fr = 5`: Dark grey square * `n_fr = 10`: Dark grey triangle **Fitted Curve:** * A solid dark grey line represents the central trend. * A lighter grey shaded area surrounds the dark grey line, indicating the confidence interval or uncertainty of the fit. The curve starts near (0, 0.2) and increases, then flattens out, reaching approximately (9500, 0.52). ### Detailed Analysis The plot shows individual data points, each representing a specific combination of `n_p` and `n_fr` values, plotted against the mean number of tokens generated and the mean pass rate. **Overall Trend of Fitted Curve:** The dark grey fitted curve shows a clear positive correlation: as the "Mean number of tokens generated" increases, the "Mean pass rate" generally increases. This increase is initially steep and then gradually flattens out, suggesting a diminishing return on pass rate for very high numbers of tokens generated. The curve starts around a pass rate of 0.2 and approaches a plateau around 0.5 to 0.52. The confidence interval is relatively narrow, indicating a consistent trend. **Data Points by `n_p` and `n_fr`:** * **`n_p = 1` (Brown colored points):** * `n_fr = 1` (Circle): Approximately (1000 tokens, 0.24 pass rate) * `n_fr = 3` (Inverted Triangle): Approximately (1500 tokens, 0.28 pass rate) * `n_fr = 5` (Square): Approximately (2000 tokens, 0.30 pass rate) * `n_fr = 10` (Triangle): Approximately (3000 tokens, 0.32 pass rate) * *Trend:* For `n_p = 1`, increasing `n_fr` generally corresponds to a higher number of tokens generated and a slightly higher pass rate. * **`n_p = 2` (Goldenrod/Yellow-orange colored points):** * `n_fr = 1` (Circle): Approximately (1200 tokens, 0.29 pass rate) * `n_fr = 3` (Inverted Triangle): Approximately (2000 tokens, 0.35 pass rate) * `n_fr = 5` (Square): Approximately (3000 tokens, 0.37 pass rate) * `n_fr = 10` (Triangle): Approximately (5800 tokens, 0.40 pass rate) * *Trend:* Similar to `n_p = 1`, increasing `n_fr` generally leads to more tokens and a higher pass rate. These points are generally above the `n_p = 1` points for similar token counts. * **`n_p = 5` (Teal/Light blue-green colored points):** * `n_fr = 1` (Circle): Approximately (1500 tokens, 0.35 pass rate) * `n_fr = 3` (Inverted Triangle): Approximately (4500 tokens, 0.43 pass rate) * `n_fr = 5` (Square): Approximately (7000 tokens, 0.46 pass rate) * `n_fr = 10` (Triangle): Approximately (4000 tokens, 0.38 pass rate) * *Trend:* These points show a general increase in pass rate with tokens. The `n_fr = 10` point is an outlier in terms of token count compared to the `n_fr = 3` and `n_fr = 5` points for this `n_p` value, having fewer tokens but a slightly lower pass rate than `n_fr = 3`. * **`n_p = 10` (Blue colored points):** * `n_fr = 1` (Circle): Approximately (4200 tokens, 0.46 pass rate) * `n_fr = 3` (Inverted Triangle): Approximately (8800 tokens, 0.50 pass rate) * `n_fr = 5` (Square): Approximately (4800 tokens, 0.45 pass rate) * `n_fr = 10` (Triangle): Approximately (7500 tokens, 0.49 pass rate) * *Trend:* These points generally follow the fitted curve, showing higher pass rates at higher token counts. The `n_fr = 5` point has a slightly lower pass rate than `n_fr = 1` despite more tokens. * **`n_p = 25` (Dark blue/Navy colored points):** * `n_fr = 1` (Circle): Approximately (9500 tokens, 0.52 pass rate) * *Note:* Only one data point is plotted for `n_p = 25`. This point is the highest in both tokens generated and pass rate among all plotted points, and it sits at the upper end of the fitted curve's plateau. ### Key Observations * **General Improvement:** There's a clear positive correlation between the mean number of tokens generated and the mean pass rate, up to a certain point where the pass rate plateaus. * **Impact of `n_p`:** Higher values of `n_p` (represented by colors from brown to dark blue) generally correspond to higher mean pass rates for a given range of tokens generated, or require more tokens to achieve similar pass rates. The `n_p = 25` point achieves the highest pass rate observed. * **Impact of `n_fr`:** For a fixed `n_p`, increasing `n_fr` (represented by marker shapes from circle to triangle) generally leads to an increase in the mean number of tokens generated and often a corresponding increase in the mean pass rate. * **Diminishing Returns:** The fitted curve indicates that beyond approximately 8000 tokens generated, the increase in mean pass rate becomes marginal, suggesting a point of diminishing returns. * **Data Density:** The data points are more densely clustered at lower token counts (below 4000) and become sparser at higher token counts. ### Interpretation The chart suggests that both `n_p` and `n_fr` parameters influence the performance, measured by "Mean pass rate," and the computational cost, measured by "Mean number of tokens generated." Increasing `n_p` appears to be a primary driver for achieving higher pass rates, as evidenced by the `n_p = 25` point reaching the highest pass rate. However, this often comes at the cost of generating more tokens. Similarly, increasing `n_fr` generally pushes the system towards generating more tokens and, consequently, achieving higher pass rates. This implies that `n_fr` might be related to the thoroughness or breadth of the generation process, leading to more tokens and better outcomes. The flattening of the fitted curve is a critical insight. It indicates that there is an upper limit to the "Mean pass rate" that can be achieved, regardless of how many more tokens are generated. This suggests that optimizing for pass rate beyond a certain token generation threshold might be inefficient, as the gains become negligible. Researchers or practitioners might use this information to find an optimal balance between computational cost (tokens generated) and performance (pass rate) by selecting appropriate `n_p` and `n_fr` values that fall within the steep part of the curve before the plateau. For instance, achieving a pass rate of ~0.45-0.46 might be possible with around 4000-5000 tokens (e.g., `n_p = 10, n_fr = 1` or `n_p = 5, n_fr = 3`), while pushing to ~0.50-0.52 requires significantly more tokens (e.g., `n_p = 10, n_fr = 3` or `n_p = 25, n_fr = 1`). </details> <details> <summary>x11.png Details</summary>  ### Visual Description ## Chart Type: Heatmap of Performance Metric vs. Program Parameters ### Overview This image displays a heatmap illustrating the relationship between two parameters, "Number of feedback-repairs ($n_{fr}$)" and "Number of initial programs ($n_P$)", and an associated performance metric. The performance metric is represented by numerical values ranging from approximately 0.73 to 0.98, with higher values indicated by brighter orange/yellow colors and lower values by darker brown colors. Certain parameter combinations are marked as "O.O.B." (Out Of Bounds), shown in dark grey. ### Components/Axes The heatmap is structured with a vertical Y-axis on the left and a horizontal X-axis at the bottom. * **Y-axis (Left)**: * **Title**: "Number of feedback-repairs ($n_{fr}$)" * **Tick Markers**: 10, 5, 3, 1 (from top to bottom, indicating decreasing values). * **X-axis (Bottom)**: * **Title**: "Number of initial programs ($n_P$)" * **Tick Markers**: 1, 2, 5, 10, 25 (from left to right, indicating increasing values). * **Data Grid**: The central area of the chart is a 4x5 grid of cells, each representing a unique combination of $n_{fr}$ and $n_P$. Each cell contains either a numerical value (a performance score) or the text "O.O.B.". * **Color Gradient**: The numerical values are color-coded: * Darkest brown: Approximately 0.73 - 0.77 * Medium brown: Approximately 0.82 - 0.84 * Orange-brown: Approximately 0.89 * Orange: Approximately 0.91 - 0.93 * Bright orange/yellow: Approximately 0.97 - 0.98 * Dark grey: "O.O.B." ### Detailed Analysis The data from the heatmap can be reconstructed into the following table, showing the performance metric for each combination of $n_{fr}$ and $n_P$: | $n_{fr}$ \ $n_P$ | 1 | 2 | 5 | 10 | 25 | | :--------------- | :----- | :----- | :----- | :----- | :----- | | **10** | 0.73 | 0.82 | O.O.B. | O.O.B. | O.O.B. | | **5** | 0.75 | 0.82 | 0.91 | O.O.B. | O.O.B. | | **3** | 0.77 | 0.84 | 0.91 | 0.98 | O.O.B. | | **1** | 0.84 | 0.89 | 0.93 | 0.97 | O.O.B. | **Trends by Row (Constant $n_{fr}$, Increasing $n_P$):** * **$n_{fr}=10$**: The value increases from 0.73 (dark brown) at $n_P=1$ to 0.82 (medium brown) at $n_P=2$. For $n_P \ge 5$, the state is "O.O.B." (dark grey). * **$n_{fr}=5$**: The value increases from 0.75 (dark brown) at $n_P=1$ to 0.82 (medium brown) at $n_P=2$, then to 0.91 (orange) at $n_P=5$. For $n_P \ge 10$, the state is "O.O.B." (dark grey). * **$n_{fr}=3$**: The value increases from 0.77 (dark brown) at $n_P=1$ to 0.84 (medium brown) at $n_P=2$, then to 0.91 (orange) at $n_P=5$, and reaches its peak at 0.98 (bright orange/yellow) at $n_P=10$. For $n_P=25$, the state is "O.O.B." (dark grey). * **$n_{fr}=1$**: The value increases from 0.84 (medium brown) at $n_P=1$ to 0.89 (orange-brown) at $n_P=2$, then to 0.93 (orange) at $n_P=5$, and to 0.97 (bright orange/yellow) at $n_P=10$. For $n_P=25$, the state is "O.O.B." (dark grey). **Trends by Column (Constant $n_P$, Decreasing $n_{fr}$):** * **$n_P=1$**: The value generally increases as $n_{fr}$ decreases: 0.73 ($n_{fr}=10$) -> 0.75 ($n_{fr}=5$) -> 0.77 ($n_{fr}=3$) -> 0.84 ($n_{fr}=1$). * **$n_P=2$**: The value generally increases as $n_{fr}$ decreases: 0.82 ($n_{fr}=10$) -> 0.82 ($n_{fr}=5$) -> 0.84 ($n_{fr}=3$) -> 0.89 ($n_{fr}=1$). * **$n_P=5$**: The value generally increases as $n_{fr}$ decreases: "O.O.B." ($n_{fr}=10$) -> 0.91 ($n_{fr}=5$) -> 0.91 ($n_{fr}=3$) -> 0.93 ($n_{fr}=1$). * **$n_P=10$**: The value generally increases as $n_{fr}$ decreases: "O.O.B." ($n_{fr}=10$) -> "O.O.B." ($n_{fr}=5$) -> 0.98 ($n_{fr}=3$) -> 0.97 ($n_{fr}=1$). Note a slight decrease from 0.98 to 0.97 for $n_{fr}=3$ to $n_{fr}=1$. * **$n_P=25$**: All values are "O.O.B." for all $n_{fr}$ values shown. ### Key Observations * **Performance Gradient**: There is a clear visual gradient indicating that performance generally improves (values increase) as $n_{fr}$ decreases and $n_P$ increases, within the region where data is available. The highest values are found towards the bottom-right of the non-"O.O.B." region. * **Optimal Point**: The highest recorded performance metric is 0.98, achieved at $n_{fr}=3$ and $n_P=10$. * **"O.O.B." Region**: A significant portion of the heatmap, particularly the top-right triangular area, is marked "O.O.B.". This indicates that combinations with a high number of initial programs (e.g., $n_P=25$) or a high number of feedback-repairs combined with moderate to high initial programs (e.g., $n_{fr}=10$ with $n_P \ge 5$) are not valid or were not evaluated. * **Diminishing Returns/Plateau**: While performance generally increases, the change from 0.98 to 0.97 when moving from ($n_{fr}=3, n_P=10$) to ($n_{fr}=1, n_P=10$) suggests that further reducing feedback-repairs beyond a certain point might not yield significant improvements or could even slightly decrease performance. ### Interpretation This heatmap demonstrates the sensitivity of a system's performance to the "Number of feedback-repairs ($n_{fr}$)" and "Number of initial programs ($n_P$)". The data suggests that: 1. **Increasing Initial Programs Generally Helps**: For a given number of feedback-repairs, increasing the number of initial programs ($n_P$) tends to improve performance, up to a certain threshold. For example, with $n_{fr}=1$, performance rises from 0.84 to 0.97 as $n_P$ increases from 1 to 10. 2. **Decreasing Feedback-Repairs Generally Helps**: For a given number of initial programs, decreasing the number of feedback-repairs ($n_{fr}$) also tends to improve performance. For instance, with $n_P=1$, performance increases from 0.73 to 0.84 as $n_{fr}$ decreases from 10 to 1. 3. **Optimal Parameter Balance**: The peak performance of 0.98 is achieved at an intermediate number of feedback-repairs ($n_{fr}=3$) and a relatively high number of initial programs ($n_P=10$). This indicates that there might be an optimal balance between these two parameters, rather than simply maximizing one and minimizing the other. 4. **System Constraints or Infeasibility ("O.O.B.")**: The "O.O.B." region is critical. It implies that certain combinations of parameters are either computationally infeasible, resource-intensive beyond practical limits, or lead to an invalid state for the system. Specifically, very high numbers of initial programs ($n_P=25$) seem to be universally "O.O.B." for all tested $n_{fr}$ values. Similarly, high $n_{fr}$ values (e.g., $n_{fr}=10$) quickly lead to "O.O.B." conditions when $n_P$ increases beyond 2. This suggests that the system might struggle or become unstable when both parameters are high, or when $n_P$ is excessively large. 5. **Practical Implications**: For system designers or users, this heatmap provides guidance on selecting parameters to achieve high performance while avoiding "out of bounds" conditions. Focusing on $n_P$ values around 5-10 and $n_{fr}$ values around 1-3 appears to be the most effective strategy based on this data. </details> <details> <summary>x12.png Details</summary>  ### Visual Description ## Chart: Mean Pass Rate vs. Mean Number of Tokens Generated ### Overview This image displays a 2D scatter plot with error bars, overlaid on a dark gray line with a shaded confidence interval. The chart illustrates the relationship between "Mean number of tokens generated" on the x-axis and "Mean pass rate" on the y-axis. Individual data points are differentiated by color, representing `n_p` values, and shape, representing `n_fr` values, as defined in a dual-column legend. ### Components/Axes **X-axis:** * **Title:** "Mean number of tokens generated" * **Scale:** Ranges from 0 to 10000. * **Major Ticks:** 0, 2000, 4000, 6000, 8000, 10000. * **Labels:** Rotated approximately 45 degrees counter-clockwise for readability. **Y-axis:** * **Title:** "Mean pass rate" * **Scale:** Ranges from 0.0 to 1.0. * **Major Ticks:** 0.0, 0.2, 0.4, 0.6, 0.8, 1.0. **Grid:** * A light gray grid is present, extending from both major X and Y axis ticks across the plotting area. **Legend (Top-right corner):** The legend is divided into two conceptual columns, indicating how `n_p` values are mapped to colors (represented by line segments) and `n_fr` values are mapped to shapes. The legend is positioned within the upper-right quadrant of the plot area. * **Left Column (Colors for `n_p`):** * Brown line segment: `n_p = 1` * Goldenrod/Orange-yellow line segment: `n_p = 2` * Teal/Cyan line segment: `n_p = 5` * Bright Blue line segment: `n_p = 10` * Dark Blue line segment: `n_p = 25` (Note: No data points with this color are visible on the plot.) * **Right Column (Shapes for `n_fr`):** * Dark Gray Circle: `n_fr = 1` * Dark Gray Inverted Triangle: `n_fr = 3` * Dark Gray Square: `n_fr = 5` * Dark Gray Triangle: `n_fr = 10` **Overall Trend Line and Confidence Interval:** * A prominent dark gray line traverses the plot, representing an overall trend. * A semi-transparent gray shaded area surrounds this dark gray line, indicating a confidence interval or standard deviation for the trend. This line and shaded area are not explicitly labeled in the legend. ### Detailed Analysis The chart displays 11 distinct data points, each represented by a specific color (corresponding to `n_p`) and shape (corresponding to `n_fr`). Each point includes horizontal and vertical error bars, indicating uncertainty in both the "Mean number of tokens generated" and "Mean pass rate" respectively. The overall trend, represented by the dark gray line, shows an initial rapid increase in "Mean pass rate" as "Mean number of tokens generated" increases, followed by a flattening or slight increase at higher token counts. Here are the approximate values for each data point, identified by their `n_p` (color) and `n_fr` (shape) values: 1. **`n_p = 1` (Brown), `n_fr = 1` (Circle):** * Mean number of tokens generated: ~550 (horizontal error bar from ~450 to ~650) * Mean pass rate: ~0.04 (vertical error bar from ~0.03 to ~0.05) * This point is located in the lower-left quadrant, close to the origin. 2. **`n_p = 2` (Goldenrod), `n_fr = 3` (Inverted Triangle):** * Mean number of tokens generated: ~1250 (horizontal error bar from ~1150 to ~1350) * Mean pass rate: ~0.07 (vertical error bar from ~0.06 to ~0.08) * This point is to the right and slightly above the first point. 3. **`n_p = 1` (Brown), `n_fr = 5` (Square):** * Mean number of tokens generated: ~1850 (horizontal error bar from ~1750 to ~1950) * Mean pass rate: ~0.08 (vertical error bar from ~0.07 to ~0.09) * This point is further right and slightly above the previous point. 4. **`n_p = 2` (Goldenrod), `n_fr = 10` (Triangle):** * Mean number of tokens generated: ~2550 (horizontal error bar from ~2450 to ~2650) * Mean pass rate: ~0.10 (vertical error bar from ~0.09 to ~0.11) * This point continues the upward trend. 5. **`n_p = 5` (Teal), `n_fr = 1` (Circle):** * Mean number of tokens generated: ~3050 (horizontal error bar from ~2950 to ~3150) * Mean pass rate: ~0.11 (vertical error bar from ~0.10 to ~0.12) * This point is slightly above and to the right of the previous point. 6. **`n_p = 1` (Brown), `n_fr = 3` (Inverted Triangle):** * Mean number of tokens generated: ~3550 (horizontal error bar from ~3450 to ~3650) * Mean pass rate: ~0.10 (vertical error bar from ~0.09 to ~0.11) * This point shows a slight dip in pass rate compared to the previous point, despite more tokens. 7. **`n_p = 2` (Goldenrod), `n_fr = 5` (Square):** * Mean number of tokens generated: ~4050 (horizontal error bar from ~3950 to ~4150) * Mean pass rate: ~0.11 (vertical error bar from ~0.10 to ~0.12) * This point is slightly above the previous point. 8. **`n_p = 5` (Teal), `n_fr = 3` (Inverted Triangle):** * Mean number of tokens generated: ~4550 (horizontal error bar from ~4450 to ~4650) * Mean pass rate: ~0.12 (vertical error bar from ~0.11 to ~0.13) * This point continues the general upward trend. 9. **`n_p = 10` (Bright Blue), `n_fr = 1` (Circle):** * Mean number of tokens generated: ~6250 (horizontal error bar from ~6150 to ~6350) * Mean pass rate: ~0.16 (vertical error bar from ~0.15 to ~0.17) * This point represents a significant jump in tokens generated and a higher pass rate, aligning with the flattening part of the overall trend line. 10. **`n_p = 5` (Teal), `n_fr = 10` (Triangle):** * Mean number of tokens generated: ~6850 (horizontal error bar from ~6750 to ~6950) * Mean pass rate: ~0.16 (vertical error bar from ~0.15 to ~0.17) * This point is very close in pass rate to the previous point but with more tokens generated. 11. **`n_p = 2` (Goldenrod), `n_fr = 10` (Triangle):** * Mean number of tokens generated: ~8350 (horizontal error bar from ~8250 to ~8450) * Mean pass rate: ~0.13 (vertical error bar from ~0.12 to ~0.14) * This point shows a decrease in pass rate compared to the two preceding points, despite a higher number of tokens generated. ### Key Observations * **Overall Trend:** The "Mean pass rate" generally increases with the "Mean number of tokens generated," but the rate of increase diminishes significantly after approximately 4000-5000 tokens, with the curve flattening out. The maximum observed mean pass rate is around 0.16. * **Parameter Influence:** The scatter points suggest that both `n_p` (color) and `n_fr` (shape) influence the mean pass rate and the mean number of tokens generated. Higher `n_p` values (e.g., `n_p=10` in bright blue) tend to be associated with higher token counts and pass rates, while lower `n_p` values (e.g., `n_p=1` in brown) are generally at lower token counts and pass rates. * **`n_fr` Variation:** The shapes representing `n_fr` values are distributed across the range of tokens generated, indicating that `n_fr` might also play a role in the token generation process and pass rate. * **Error Bars:** All data points have relatively small error bars in both dimensions, suggesting that the measured means are fairly stable for each specific configuration of `n_p` and `n_fr`. * **Unused Legend Entry:** The legend includes `n_p = 25` (dark blue line segment), but no data points corresponding to this `n_p` value are plotted on the chart. * **Maximum Pass Rate:** The "Mean pass rate" never exceeds approximately 0.16-0.17, even at the highest token generation counts shown. ### Interpretation This chart likely illustrates the performance of a system or model, where `n_p` and `n_fr` are parameters influencing its output. The "Mean number of tokens generated" could represent the computational effort or output length, while "Mean pass rate" is a measure of success or quality. The overall trend suggests that increasing the number of tokens generated initially improves the pass rate, but there are diminishing returns. Beyond a certain point (around 4000-5000 tokens), generating more tokens does not significantly increase the pass rate, which plateaus at a relatively low value (around 15-16%). This could imply that the system reaches its performance limit or that generating more tokens beyond this threshold does not add value in terms of correctness or quality. The different colored and shaped points indicate that specific combinations of `n_p` and `n_fr` lead to different outcomes. For instance, `n_p=10` (bright blue circle) achieves one of the highest pass rates with a moderate number of tokens, while `n_p=2` (goldenrod triangle) at a very high token count (~8350) results in a lower pass rate than some points with fewer tokens. This suggests a complex interaction between `n_p`, `n_fr`, and the resulting performance. It's not simply "more tokens = better pass rate" across all parameter settings. The absence of data points for `n_p=25` might indicate that experiments for this parameter value were not conducted, or perhaps they yielded results outside the displayed range or were deemed irrelevant. The low overall pass rates (maxing out below 0.2) suggest that the task being evaluated is challenging, or the system's performance is generally limited under these conditions. The confidence interval around the main trend line provides a sense of the variability or uncertainty in the average performance across all observed conditions. </details> <details> <summary>x13.png Details</summary>  ### Visual Description ## Chart Type: Heatmap of Performance Metrics ### Overview This image displays a heatmap illustrating performance metrics across varying numbers of "feedback-repairs" ($n_{fr}$) and "initial programs" ($n_p$). The heatmap uses a color gradient to represent numerical values, with darker browns/oranges indicating lower values and lighter greens/teals indicating higher values. A distinct dark grey color signifies "O.O.B." (Out Of Bounds or Out Of Budget) where data is not available or applicable. ### Components/Axes The chart consists of a central grid representing the data, bounded by a Y-axis on the left and an X-axis at the bottom. * **Y-axis (Left)**: * **Title**: "Number of feedback-repairs ($n_{fr}$)" * **Tick Markers**: 10, 5, 3, 1 (ordered from top to bottom, indicating decreasing values). * **X-axis (Bottom)**: * **Title**: "Number of initial programs ($n_p$)" * **Tick Markers**: 1, 2, 5, 10, 25 (ordered from left to right, indicating increasing values). * **Legend**: No explicit legend is provided. The color mapping is inferred from the values within the cells: * Dark brown: Values around 0.78 - 0.81 * Orange-brown: Values around 0.87 * Orange: Values around 0.91 - 0.93 * Yellow-green: Values around 1.05 * Light green: Values around 1.08 - 1.09 * Teal: Values around 1.13 * Dark grey: "O.O.B." ### Detailed Analysis The heatmap presents a 4x5 grid of values, corresponding to the combinations of $n_{fr}$ and $n_p$. The data can be structured as follows: | $n_{fr}$ \ $n_p$ | 1 | 2 | 5 | 10 | 25 | | :--------------- | :------- | :------- | :------- | :------- | :------- | | **10** | 0.78 | 0.93 | O.O.B. | O.O.B. | O.O.B. | | **5** | 0.79 | 0.91 | 1.09 | O.O.B. | O.O.B. | | **3** | 0.81 | 0.91 | 1.08 | O.O.B. | O.O.B. | | **1** | 0.87 | 0.93 | 1.05 | 1.13 | O.O.B. | ### Key Observations 1. **"O.O.B." Dominance**: A significant portion of the heatmap, particularly towards higher values of $n_p$ and higher values of $n_{fr}$, is marked as "O.O.B.". This indicates that data is not available or the conditions are out of scope for these parameter combinations. Specifically, for $n_p=25$, all $n_{fr}$ values result in "O.O.B.". For $n_p=10$, only $n_{fr}=1$ yields a numerical value (1.13). For $n_p=5$, $n_{fr}=10$ is "O.O.B.". 2. **General Trend with $n_p$**: For a fixed number of feedback-repairs ($n_{fr}$), as the number of initial programs ($n_p$) increases, the performance metric generally tends to increase before hitting the "O.O.B." region. For example, at $n_{fr}=1$, the values are 0.87 ($n_p=1$), 0.93 ($n_p=2$), 1.05 ($n_p=5$), and 1.13 ($n_p=10$). 3. **General Trend with $n_{fr}$**: For a fixed number of initial programs ($n_p$), as the number of feedback-repairs ($n_{fr}$) decreases (moving down the Y-axis): * At $n_p=1$, the values increase from 0.78 ($n_{fr}=10$) to 0.87 ($n_{fr}=1$). * At $n_p=2$, the values are relatively stable, fluctuating slightly around 0.91-0.93. * At $n_p=5$, the values decrease from 1.09 ($n_{fr}=5$) to 1.05 ($n_{fr}=1$), after being "O.O.B." at $n_{fr}=10$. 4. **Highest Value**: The highest observed numerical value is 1.13, occurring at $n_{fr}=1$ and $n_p=10$. This cell is colored teal. 5. **Lowest Value**: The lowest observed numerical value is 0.78, occurring at $n_{fr}=10$ and $n_p=1$. This cell is colored dark brown. 6. **Color Gradient Consistency**: The color gradient consistently maps lower numerical values to darker brown/orange hues and higher numerical values to lighter green/teal hues, with "O.O.B." being distinctly dark grey. ### Interpretation The heatmap suggests that the performance metric is influenced by both the number of feedback-repairs ($n_{fr}$) and the number of initial programs ($n_p$). The "O.O.B." regions are critical. They imply that certain combinations of parameters are either not feasible, not tested, or fall outside the defined operational boundaries of the system being evaluated. Specifically, a high number of initial programs ($n_p=25$) consistently leads to "O.O.B." regardless of feedback-repairs, suggesting a potential limitation or inefficiency at that scale. Similarly, a high number of feedback-repairs ($n_{fr}=10$) combined with a moderate to high number of initial programs ($n_p \ge 5$) also results in "O.O.B.". Within the observable range, there's a general trend where increasing the number of initial programs ($n_p$) tends to improve the performance metric, especially when the number of feedback-repairs ($n_{fr}$) is low. The peak performance (1.13) is achieved with a low number of feedback-repairs ($n_{fr}=1$) and a relatively high number of initial programs ($n_p=10$). Conversely, for a fixed low number of initial programs ($n_p=1$), reducing feedback-repairs ($n_{fr}$) seems to slightly increase the metric. However, for $n_p=2$, the metric remains relatively stable across different $n_{fr}$ values. For $n_p=5$, reducing $n_{fr}$ from 5 to 1 leads to a slight decrease in the metric (1.09 to 1.05). This data could indicate that: * There's an optimal balance between $n_{fr}$ and $n_p$ for maximizing the performance metric, which appears to be around $n_{fr}=1$ and $n_p=10$. * Excessive initial programs ($n_p=25$) or certain combinations involving higher $n_{fr}$ and $n_p$ are not viable or lead to system failures/unmeasurable states ("O.O.B."). * The impact of $n_{fr}$ on performance is less straightforward and depends on the value of $n_p$. For low $n_p$, lower $n_{fr}$ is better; for moderate $n_p$, $n_{fr}$ has less impact; and for higher $n_p$, only very low $n_{fr}$ values yield results. Further investigation into the meaning of "O.O.B." and the nature of the performance metric would provide deeper insights into the system's behavior and limitations. </details> Figure 7: GPT-3.5 results from Figure 3 (Section 4.1) per difficulty (row), from top to bottom: introductory, interview, and competition. <details> <summary>x14.png Details</summary>  ### Visual Description ## Chart Type: Mean Pass Rate vs. Mean Number of Tokens Generated with `n_p` and `n_fr` Parameters ### Overview This image displays a scatter plot with error bars, overlaid with a general trend line and its confidence interval. It illustrates the relationship between the "Mean number of tokens generated" (X-axis) and the "Mean pass rate" (Y-axis) for various combinations of two parameters, `n_p` and `n_fr`. The `n_p` parameter is differentiated by the color of the data points, while the `n_fr` parameter is differentiated by the shape of the data points. ### Components/Axes **X-axis:** * **Title:** "Mean number of tokens generated" * **Range:** From 0 to 10000 * **Major Tick Markers:** 0, 2000, 4000, 6000, 8000, 10000. The tick labels are rotated approximately 45 degrees counter-clockwise. **Y-axis:** * **Title:** "Mean pass rate" * **Range:** From 0.0 to 1.0 * **Major Tick Markers:** 0.0, 0.2, 0.4, 0.6, 0.8, 1.0 **Legend:** The legend is located in the bottom-left quadrant of the plot area, within a light gray box. It is divided into two columns: * **Left Column (Line colors for `n_p` values):** These lines indicate the color mapping for the data points based on the `n_p` parameter. * Brown line: `n_p = 1` * Orange line: `n_p = 2` * Teal/Light Blue line: `n_p = 5` * Blue line: `n_p = 10` * Dark Blue/Navy line: `n_p = 25` * **Right Column (Marker shapes for `n_fr` values):** These gray markers indicate the shape mapping for the data points based on the `n_fr` parameter. * Gray circle: `n_fr = 1` * Gray inverted triangle: `n_fr = 3` * Gray square: `n_fr = 5` * Gray triangle: `n_fr = 10` **Overall Trend Line:** A thick gray line with a lighter gray shaded area (representing a confidence interval or standard deviation) is plotted across the data points. This line represents the general trend of "Mean pass rate" as "Mean number of tokens generated" increases, irrespective of specific `n_p` or `n_fr` values. ### Detailed Analysis **Overall Trend (Gray Line):** The gray trend line starts around a "Mean pass rate" of approximately 0.65 at very low "Mean number of tokens generated" (near 0). It rises steeply, indicating a rapid increase in pass rate with an initial increase in tokens. The slope then gradually decreases, and the line begins to flatten out, reaching a plateau around a "Mean pass rate" of 0.92 to 0.93 for "Mean number of tokens generated" between 3000 and 5500. The shaded gray area around the line suggests a variability or confidence interval of roughly +/- 0.02 to 0.03 in the pass rate. **Individual Data Points (Colored by `n_p`, Shaped by `n_fr`):** Each data point includes vertical error bars, which are generally small, indicating relatively low variability for each specific configuration. The points generally follow the upward-sloping and then plateauing trend of the overall gray line. 1. **`n_p = 1` (Brown colored points):** * These points generally appear at the lower end of the "Mean number of tokens generated" spectrum for a given "Mean pass rate" compared to higher `n_p` values. * **Circles (`n_fr = 1`):** E.g., (~500 tokens, ~0.78 pass rate), (~1000 tokens, ~0.83 pass rate), (~2000 tokens, ~0.88 pass rate), (~3000 tokens, ~0.91 pass rate). * **Inverted Triangles (`n_fr = 3`):** E.g., (~700 tokens, ~0.82 pass rate), (~1500 tokens, ~0.87 pass rate), (~2500 tokens, ~0.90 pass rate). * **Squares (`n_fr = 5`):** E.g., (~900 tokens, ~0.85 pass rate), (~1800 tokens, ~0.89 pass rate), (~3000 tokens, ~0.92 pass rate). * **Triangles (`n_fr = 10`):** E.g., (~1200 tokens, ~0.87 pass rate), (~2200 tokens, ~0.90 pass rate), (~4000 tokens, ~0.93 pass rate). * *Trend:* For `n_p = 1`, increasing `n_fr` generally shifts the points slightly to the right (more tokens) and slightly upwards (higher pass rate). 2. **`n_p = 2` (Orange colored points):** * These points are generally shifted slightly to the right and/or slightly higher in pass rate compared to `n_p = 1` points. * **Circles (`n_fr = 1`):** E.g., (~600 tokens, ~0.82 pass rate), (~1200 tokens, ~0.86 pass rate), (~2200 tokens, ~0.90 pass rate), (~3500 tokens, ~0.92 pass rate). * **Inverted Triangles (`n_fr = 3`):** E.g., (~800 tokens, ~0.85 pass rate), (~1700 tokens, ~0.89 pass rate), (~2800 tokens, ~0.91 pass rate). * **Squares (`n_fr = 5`):** E.g., (~1000 tokens, ~0.87 pass rate), (~2000 tokens, ~0.90 pass rate), (~3200 tokens, ~0.92 pass rate). * **Triangles (`n_fr = 10`):** E.g., (~1400 tokens, ~0.89 pass rate), (~2500 tokens, ~0.91 pass rate), (~4200 tokens, ~0.93 pass rate). 3. **`n_p = 5` (Teal/Light Blue colored points):** * These points continue the trend of shifting further right and/or higher in pass rate. * **Circles (`n_fr = 1`):** E.g., (~700 tokens, ~0.85 pass rate), (~1400 tokens, ~0.89 pass rate), (~2500 tokens, ~0.91 pass rate), (~4000 tokens, ~0.93 pass rate). * **Inverted Triangles (`n_fr = 3`):** E.g., (~900 tokens, ~0.87 pass rate), (~1900 tokens, ~0.90 pass rate), (~3000 tokens, ~0.92 pass rate). * **Squares (`n_fr = 5`):** E.g., (~1100 tokens, ~0.89 pass rate), (~2200 tokens, ~0.91 pass rate), (~3500 tokens, ~0.93 pass rate). * **Triangles (`n_fr = 10`):** E.g., (~1600 tokens, ~0.90 pass rate), (~2800 tokens, ~0.92 pass rate), (~4500 tokens, ~0.94 pass rate). 4. **`n_p = 10` (Blue colored points):** * These points are generally further to the right and higher than `n_p = 5` points. * **Circles (`n_fr = 1`):** E.g., (~800 tokens, ~0.87 pass rate), (~1600 tokens, ~0.90 pass rate), (~2800 tokens, ~0.92 pass rate), (~4500 tokens, ~0.94 pass rate). * **Inverted Triangles (`n_fr = 3`):** E.g., (~1000 tokens, ~0.89 pass rate), (~2100 tokens, ~0.91 pass rate), (~3200 tokens, ~0.93 pass rate). * **Squares (`n_fr = 5`):** E.g., (~1300 tokens, ~0.90 pass rate), (~2400 tokens, ~0.92 pass rate), (~3800 tokens, ~0.94 pass rate). * **Triangles (`n_fr = 10`):** E.g., (~1800 tokens, ~0.91 pass rate), (~3000 tokens, ~0.93 pass rate), (~4800 tokens, ~0.95 pass rate). 5. **`n_p = 25` (Dark Blue/Navy colored points):** * These points represent the highest `n_p` value and are generally located furthest to the right and highest on the plot. * **Circles (`n_fr = 1`):** E.g., (~900 tokens, ~0.89 pass rate), (~1800 tokens, ~0.91 pass rate), (~3000 tokens, ~0.93 pass rate), (~5000 tokens, ~0.95 pass rate). * **Inverted Triangles (`n_fr = 3`):** E.g., (~1100 tokens, ~0.90 pass rate), (~2300 tokens, ~0.92 pass rate), (~3500 tokens, ~0.94 pass rate). * **Squares (`n_fr = 5`):** E.g., (~1500 tokens, ~0.91 pass rate), (~2600 tokens, ~0.93 pass rate), (~4000 tokens, ~0.95 pass rate), (~5500 tokens, ~0.96 pass rate). * **Triangles (`n_fr = 10`):** E.g., (~2000 tokens, ~0.92 pass rate), (~3200 tokens, ~0.94 pass rate), (~5000 tokens, ~0.96 pass rate), and a distinct outlier at (~8200 tokens, ~0.97 pass rate). ### Key Observations * **Diminishing Returns:** The "Mean pass rate" increases with "Mean number of tokens generated" but shows clear diminishing returns, plateauing at a pass rate of approximately 0.92-0.96. * **Influence of `n_p`:** Higher values of `n_p` (represented by darker blue colors) generally correspond to data points that achieve slightly higher "Mean pass rates" but often require a greater "Mean number of tokens generated" to reach the plateau. The points for higher `n_p` values are shifted towards the right side of the plot. * **Influence of `n_fr`:** For a given `n_p` value, increasing `n_fr` (from circle to inverted triangle to square to triangle) tends to shift the data points towards higher "Mean number of tokens generated" for similar or slightly improved "Mean pass rates." This suggests that higher `n_fr` values might involve generating more tokens. * **Outlier Point:** A prominent outlier is the dark blue triangle (`n_p = 25`, `n_fr = 10`) located at approximately 8200 tokens and a 0.97 pass rate. This point achieves the highest observed pass rate but at a significantly higher token generation cost than other configurations that achieve pass rates in the 0.95-0.96 range (which typically require around 4000-5500 tokens). * **Consistent Performance:** The relatively small error bars across all data points suggest that the mean pass rates are consistent and reliable for each tested configuration. ### Interpretation The data suggests a clear trade-off between the "Mean number of tokens generated" and the "Mean pass rate." Initially, increasing token generation significantly improves the pass rate, but this effect quickly diminishes, indicating that there's an optimal range of token generation beyond which further increases yield minimal benefits. The parameters `n_p` and `n_fr` appear to modulate this relationship. Higher `n_p` values seem to enable slightly higher peak pass rates, but they also push the "saturation point" further to the right on the X-axis, meaning more tokens are needed to achieve these higher rates. This could imply that `n_p` relates to the capacity or complexity of the generation process, where higher `n_p` allows for more thorough (and thus token-intensive) exploration, potentially leading to marginal gains in pass rate. Similarly, `n_fr` seems to influence the token generation cost. As `n_fr` increases, more tokens are generally required to achieve comparable or slightly better pass rates for a given `n_p`. This might indicate that `n_fr` controls the breadth or diversity of generated tokens, where higher values lead to more extensive (and thus token-heavy) outputs. The outlier point (`n_p = 25`, `n_fr = 1 </details> <details> <summary>x15.png Details</summary>  ### Visual Description ## Chart Type: Heatmap of Performance Metric ### Overview This image displays a heatmap illustrating a performance metric across varying numbers of "feedback-repairs" ($n_{fr}$) and "initial programs" ($n_p$). The grid cells contain numerical values, likely representing the performance metric, or the label "O.O.B." (Out Of Bounds). The color of the cells indicates the magnitude of the numerical value, with warmer colors (orange) generally corresponding to lower values and cooler/greener colors to higher values. ### Components/Axes **Y-axis (Left Side):** * **Label:** "Number of feedback-repairs ($n_{fr}$)" * **Tick Marks (from bottom to top):** 1, 3, 5, 10 **X-axis (Bottom Side):** * **Label:** "Number of initial programs ($n_p$)" * **Tick Marks (from left to right):** 1, 2, 5, 10, 25 **Data Grid:** The central area of the image is a 4x5 grid of cells. Each cell represents a unique combination of $n_{fr}$ and $n_p$ and contains a value. **Color Gradient (Inferred - No explicit legend):** * **Orange:** Associated with the lowest values (e.g., 0.98, 1.00, 1.01). * **Orange-Yellow:** Associated with intermediate values (e.g., 1.02, 1.03). * **Yellow-Green:** Associated with higher values (e.g., 1.04). * **Green-Yellow:** Associated with the highest value (e.g., 1.05). * **Dark Grey:** Represents "O.O.B." (Out Of Bounds). ### Detailed Analysis The data is presented in a grid format. The rows correspond to the "Number of feedback-repairs ($n_{fr}$)" and the columns correspond to the "Number of initial programs ($n_p$)". **Data Table:** | $n_{fr}$ \ $n_p$ | 1 | 2 | 5 | 10 | 25 | | :--------------- | :----- | :----- | :----- | :----- | :----- | | **10** | 0.98 | 1.01 | 1.02 | 1.03 | O.O.B. | | **5** | 1.00 | 1.02 | 1.03 | 1.03 | O.O.B. | | **3** | 1.02 | 1.03 | 1.04 | 1.04 | 1.04 | | **1** | 1.05 | 1.04 | 1.04 | 1.04 | 1.04 | **Trends and Distributions:** 1. **Row $n_{fr}=10$ (Top Row):** * Values start at 0.98 (orange) for $n_p=1$. * Increase to 1.01 (orange) for $n_p=2$. * Increase to 1.02 (orange-yellow) for $n_p=5$. * Increase to 1.03 (orange-yellow) for $n_p=10$. * The cell for $n_p=25$ is "O.O.B." (dark grey). * *Trend:* Values generally increase with $n_p$ until reaching an "O.O.B." state. 2. **Row $n_{fr}=5$:** * Values start at 1.00 (orange) for $n_p=1$. * Increase to 1.02 (orange-yellow) for $n_p=2$. * Increase to 1.03 (orange-yellow) for $n_p=5$. * Remain at 1.03 (orange-yellow) for $n_p=10$. * The cell for $n_p=25$ is "O.O.B." (dark grey). * *Trend:* Values generally increase or stabilize with $n_p$ until reaching an "O.O.B." state. 3. **Row $n_{fr}=3$:** * Values start at 1.02 (orange-yellow) for $n_p=1$. * Increase to 1.03 (orange-yellow) for $n_p=2$. * Increase to 1.04 (yellow-green) for $n_p=5$. * Remain at 1.04 (yellow-green) for $n_p=10$. * Remain at 1.04 (yellow-green) for $n_p=25$. * *Trend:* Values generally increase with $n_p$ and then stabilize at 1.04. 4. **Row $n_{fr}=1$ (Bottom Row):** * Values start at 1.05 (green-yellow) for $n_p=1$. * Decrease to 1.04 (yellow-green) for $n_p=2$. * Remain at 1.04 (yellow-green) for $n_p=5$. * Remain at 1.04 (yellow-green) for $n_p=10$. * Remain at 1.04 (yellow-green) for $n_p=25$. * *Trend:* Values slightly decrease with $n_p$ and then stabilize at 1.04. **Column-wise Trends:** * **Column $n_p=1$ (Leftmost):** Values increase from 0.98 ($n_{fr}=10$) to 1.05 ($n_{fr}=1$). * **Column $n_p=2$:** Values increase from 1.01 ($n_{fr}=10$) to 1.04 ($n_{fr}=1$). * **Column $n_p=5$:** Values increase from 1.02 ($n_{fr}=10$) to 1.04 ($n_{fr}=1$). * **Column $n_p=10$:** Values increase from 1.03 ($n_{fr}=10$) to 1.04 ($n_{fr}=1$). * **Column $n_p=25$ (Rightmost):** Values are "O.O.B." for $n_{fr}=10$ and $n_{fr}=5$, then become 1.04 for $n_{fr}=3$ and $n_{fr}=1$. ### Key Observations * **Minimum Value:** The lowest performance metric value observed is 0.98, located at $n_{fr}=10, n_p=1$. This cell is distinctly orange. * **Maximum Value:** The highest performance metric value observed is 1.05, located at $n_{fr}=1, n_p=1$. This cell is green-yellow. * **"O.O.B." Region:** The top-right corner of the grid, specifically for $n_p=25$ when $n_{fr}=10$ or $n_{fr}=5$, shows "O.O.B.". This indicates a region where the process or experiment was not applicable, failed, or exceeded some boundary condition. * **Stabilization:** For lower numbers of feedback-repairs ($n_{fr}=3$ and $n_{fr}=1$), the performance metric tends to stabilize at 1.04 as the number of initial programs ($n_p$) increases. * **Color-Value Relationship:** The color scheme suggests that lower numerical values (closer to 0.98) are "better" or more desirable, as they are represented by a distinct orange, while higher values (closer to 1.05) are represented by greener hues. ### Interpretation This heatmap likely represents the outcome of an experiment or simulation where the "Number of feedback-repairs ($n_{fr}$)" and "Number of initial programs ($n_p$)" are input parameters, and the numerical values in the cells are a measure of system performance or efficiency. The inverse relationship between color and value (orange for low, green for high) suggests that the metric being measured is one where lower values are more favorable. For example, it could be an error rate, a cost, or a time taken. The data suggests that: 1. **Optimal Performance:** The best performance (0.98) is achieved when there is a high number of feedback-repairs ($n_{fr}=10$) and a minimal number of initial programs ($n_p=1$). This implies that extensive refinement (feedback-repairs) on a single or very few initial attempts is most effective under these conditions. 2. **Impact of Initial Programs:** Generally, increasing the "Number of initial programs ($n_p$)" tends to degrade performance (increase the metric value) or lead to an "Out Of Bounds" state, especially when combined with a high number of feedback-repairs. This could mean that too many initial programs introduce complexity or redundancy that hinders the process, or that the system struggles to manage a large pool of initial programs with many feedback loops. 3. **Impact of Feedback-Repairs:** Decreasing the "Number of feedback-repairs ($n_{fr}$)" generally leads to worse performance (higher metric values). This highlights the importance of feedback and repair mechanisms for achieving better outcomes. 4. **System Limitations ("O.O.B."):** The "O.O.B." cells in the top-right corner indicate a boundary condition. It's plausible that running a high number of feedback-repairs on a large number of initial programs is computationally too expensive, leads to resource exhaustion, or simply falls outside the defined scope of the experiment. For example, if $n_p$ represents parallel processes and $n_{fr}$ represents iterations, a high combination might exceed available processing power or time limits. 5. **Convergence:** For lower $n_{fr}$ values (1 and 3), the performance metric seems to converge to 1.04 regardless of the number of initial programs (beyond $n_p=2$). This suggests a baseline performance level that cannot be significantly improved by adding more initial programs if feedback-repairs are limited. In essence, the data points to a trade-off: while a high number of feedback-repairs is beneficial, it needs to be balanced with a relatively low number of initial programs to achieve optimal results and avoid "out of bounds" scenarios. The sweet spot for this particular metric appears to be at the lowest $n_p$ and highest $n_{fr}$ tested. </details> <details> <summary>x16.png Details</summary>  ### Visual Description ## Chart: Mean Pass Rate vs. Mean Number of Tokens Generated ### Overview This image displays a 2D scatter plot with error bars, overlaid with a single dark grey line and a lighter grey shaded region representing an overall trend. The chart illustrates the relationship between the "Mean number of tokens generated" (X-axis) and the "Mean pass rate" (Y-axis), with individual data points categorized by two parameters: `n_p` (represented by color) and `n_fr` (represented by shape). ### Components/Axes The chart is structured with a main plotting area, an X-axis at the bottom, a Y-axis on the left, and a legend in the bottom-right quadrant. * **X-axis Label**: "Mean number of tokens generated" * **Range**: From 0 to 10000. * **Major Ticks**: Labeled at 0, 2000, 4000, 6000, 8000, and 10000. The labels are rotated approximately 45 degrees counter-clockwise. * **Minor Ticks**: Not explicitly labeled, but grid lines suggest intermediate divisions. * **Y-axis Label**: "Mean pass rate" * **Range**: From 0.0 to 1.0. * **Major Ticks**: Labeled at 0.0, 0.2, 0.4, 0.6, 0.8, and 1.0. * **Minor Ticks**: Not explicitly labeled, but grid lines suggest intermediate divisions. * **Grid**: A light grey grid is present, aligning with the major tick marks on both axes. * **Legend (bottom-right quadrant)**: The legend is divided into two columns. * **Left Column (`n_p` values, represented by line colors in the legend, which correspond to point colors in the plot)**: * Brown line: `n_p = 1` * Golden line: `n_p = 2` * Teal line: `n_p = 5` * Light blue line: `n_p = 10` * Dark blue line: `n_p = 25` * **Right Column (`n_fr` values, represented by grey shapes in the legend, which correspond to point shapes in the plot)**: * Grey circle: `n_fr = 1` * Grey downward triangle: `n_fr = 3` * Grey square: `n_fr = 5` * Grey upward triangle: `n_fr = 10` * **Overall Trend Line**: A solid dark grey line with a surrounding lighter grey shaded region (likely indicating a confidence interval or standard deviation) is plotted across the chart. ### Detailed Analysis The chart shows a general trend where the "Mean pass rate" increases with the "Mean number of tokens generated" up to a certain point, after which it appears to saturate. The individual data points, each with vertical error bars (approximately +/- 0.02 on the Y-axis), represent specific configurations of `n_p` and `n_fr`. **Overall Trend Line (Dark Grey with Shaded Region):** * Starts around (X=0, Y=0.4). * Steadily increases, passing through approximately (X=1000, Y=0.55), (X=2000, Y=0.6), (X=4000, Y=0.65). * Flattens out, approaching a plateau around Y=0.66 to 0.68 for X values greater than approximately 4000. * The shaded region around the line indicates a narrow band of uncertainty, suggesting a consistent trend. **Individual Data Points (Colored by `n_p`, Shaped by `n_fr`):** 1. **`n_p = 1` (Brown points):** * Brown circle (`n_fr = 1`): Approximately (X=250, Y=0.43). * Brown downward triangle (`n_fr = 3`): Approximately (X=500, Y=0.48). * Brown square (`n_fr = 5`): Approximately (X=1000, Y=0.51). * Brown upward triangle (`n_fr = 10`): Approximately (X=1500, Y=0.52). * **Trend**: For `n_p=1`, increasing `n_fr` from 1 to 10 generally leads to an increase in both tokens generated and mean pass rate, but the pass rate remains relatively low compared to other `n_p` values. 2. **`n_p = 2` (Golden points):** * Golden circle (`n_fr = 1`): Approximately (X=700, Y=0.50). * Golden downward triangle (`n_fr = 3`): Approximately (X=1500, Y=0.57). * Golden square (`n_fr = 5`): Approximately (X=2000, Y=0.59). * Golden upward triangle (`n_fr = 10`): Approximately (X=3500, Y=0.61). * **Trend**: For `n_p=2`, increasing `n_fr` shows a consistent increase in tokens generated and a moderate increase in mean pass rate, reaching up to ~0.61. 3. **`n_p = 5` (Teal points):** * Teal circle (`n_fr = 1`): Approximately (X=1500, Y=0.61). * Teal downward triangle (`n_fr = 3`): Approximately (X=2500, Y=0.65). * Teal square (`n_fr = 5`): Approximately (X=4000, Y=0.66). * Teal upward triangle (`n_fr = 10`): Approximately (X=7500, Y=0.68). * **Trend**: For `n_p=5`, increasing `n_fr` results in a substantial increase in tokens generated and a good improvement in mean pass rate, approaching the saturation level. 4. **`n_p = 10` (Light blue points):** * Light blue circle (`n_fr = 1`): Approximately (X=2000, Y=0.62). * Light blue downward triangle (`n_fr = 3`): Approximately (X=5000, Y=0.69). * Light blue square (`n_fr = 5`): Approximately (X=7000, Y=0.70). * Light blue upward triangle (`n_fr = 10`): Approximately (X=7500, Y=0.69). * **Trend**: For `n_p=10`, increasing `n_fr` generally increases tokens generated and achieves high mean pass rates, with `n_fr=5` reaching the highest point for this `n_p` at ~0.70. The `n_fr=10` point is slightly lower than `n_fr=5` at a similar token count. 5. **`n_p = 25` (Dark blue points):** * Dark blue circle (`n_fr = 1`): Approximately (X=3000, Y=0.66). * Dark blue downward triangle (`n_fr = 3`): Approximately (X=5500, Y=0.69). * Dark blue square (`n_fr = 5`): Approximately (X=6500, Y=0.70). * Dark blue upward triangle (`n_fr = 10`): Approximately (X=7000, Y=0.69). * **Trend**: For `n_p=25`, increasing `n_fr` also increases tokens generated and achieves high mean pass rates, similar to `n_p=10` in the saturation region. The `n_fr=5` point is again the highest at ~0.70, with `n_fr=10` slightly lower. ### Key Observations * **General Improvement**: The mean pass rate generally increases as the mean number of tokens generated increases, up to a point of diminishing returns. * **Impact of `n_p`**: Higher values of `n_p` (represented by colors from brown to dark blue) consistently lead to higher mean pass rates for a given range of tokens generated. They also achieve higher pass rates with fewer tokens compared to lower `n_p` values. For example, `n_p=25` reaches a pass rate of ~0.66 at 3000 tokens, while `n_p=1` only reaches ~0.52 at 1500 tokens. * **Impact of `n_fr`**: For a fixed `n_p`, increasing `n_fr` (represented by shapes from circle to upward triangle) generally corresponds to an increase in the mean number of tokens generated and a corresponding increase in the mean pass rate. This suggests `n_fr` influences the "effort" (tokens) and "reward" (pass rate). * **Saturation**: All series of points appear to converge towards a maximum mean pass rate between approximately 0.68 and 0.70, regardless of further increases in tokens generated beyond ~5000-7000. * **Efficiency**: Higher `n_p` values demonstrate better efficiency, achieving higher pass rates with fewer tokens. For instance, `n_p=25` (dark blue circle) achieves a pass rate of ~0.66 with ~3000 tokens, which is a pass rate that `n_p=5` (teal square) only reaches with ~4000 tokens. * **Error Bars**: The error bars are consistently small and of similar magnitude across all data points, suggesting relatively low variability or high confidence in the mean pass rate measurements. ### Interpretation The data presented in this chart suggests a performance characteristic of a system where `n_p` and `n_fr` are tunable parameters affecting the "Mean pass rate" and the "Mean number of tokens generated." * **`n_p` as a Quality/Capability Parameter**: The parameter `n_p` appears to represent a fundamental capability or quality setting of the system. Higher `n_p` values lead to a better baseline performance, allowing the system to achieve higher pass rates more efficiently (i.e., with fewer tokens). This could imply that `n_p` controls the "power" or "intelligence" of the generation process. * **`n_fr` as a Resource Allocation/Exploration Parameter**: The parameter `n_fr` seems to control the extent of resource allocation or exploration during token generation. Increasing `n_fr` generally pushes the system to generate more tokens, which in turn allows for a higher mean pass rate, especially for higher `n_p` values. This could represent the number of "tries" or "samples" taken during generation. * **Diminishing Returns**: The overall trend and the saturation of individual series indicate that there are diminishing returns to simply generating more tokens. Beyond a certain point (around 5000-7000 tokens), the mean pass rate plateaus, suggesting that the system reaches its maximum potential for a given `n_p` value, and further token generation does not yield significant improvements. * **Optimization Trade-off**: The chart highlights a critical trade-off for system design or operation. If the goal is to maximize the mean pass rate, one would aim for higher `n_p` values (e.g., `n_p=10` or `n_p=25`) and then select an `n_fr` value that pushes the tokens generated into the saturation region (e.g., `n_fr=3` or `n_fr=5` for `n_p=10` or `n_p=25`). However, if minimizing token generation is also a concern, one might choose a configuration like `n_p=10` with `n_fr=3` (5000 tokens, ~0.69 pass rate) over `n_p=25` with `n_fr=5` (6500 tokens, ~0.70 pass rate) if the slight difference in pass rate is acceptable for the token savings. * **System Limits**: The observed plateau suggests an inherent upper limit to the "Mean pass rate" for the system or task being evaluated, regardless of how many tokens are generated, given the current range of `n_p` and `n_fr` values. This limit appears to be around 0.70. </details> <details> <summary>x17.png Details</summary>  ### Visual Description ## Chart Type: Heatmap of Performance Metrics ### Overview This image displays a heatmap or grid table illustrating performance metrics based on two varying parameters: "Number of feedback-repairs ($n_{fr}$)" and "Number of initial programs ($n_p$)". Each cell in the grid contains a numerical value, likely representing a performance score, or the text "O.O.B." (Out Of Bounds/Budget/Bandwidth), and is color-coded to visually represent the magnitude of the numerical values. Lower numerical values are represented by darker orange/brown hues, while higher values transition to yellow-green. "O.O.B." cells are colored dark grey. ### Components/Axes The chart is structured as a 4x5 grid. * **Y-axis (left, vertical):** Labeled "Number of feedback-repairs ($n_{fr}$)", oriented vertically reading upwards. * Tick markers (from bottom to top): 1, 3, 5, 10. * **X-axis (bottom, horizontal):** Labeled "Number of initial programs ($n_p$)", oriented horizontally. * Tick markers (from left to right): 1, 2, 5, 10, 25. The tick labels are slightly rotated counter-clockwise. * **Color Gradient (Implicit Legend):** There is no explicit legend, but the cell colors indicate value ranges: * Dark orange/brown: Represents the lowest numerical values (e.g., 0.88, 0.89, 0.91). * Lighter orange/yellow: Represents intermediate numerical values (e.g., 0.96, 0.97, 0.99). * Yellow-green: Represents the highest numerical values (e.g., 1.03, 1.04, 1.05, 1.06, 1.08, 1.09). * Dark grey: Represents "O.O.B." (Out Of Bounds/Budget/Bandwidth) condition, indicating that a valid numerical result was not obtained for those parameter combinations. ### Detailed Analysis The grid presents the following values for each combination of $n_{fr}$ and $n_p$: | $n_{fr}$ \ $n_p$ | 1 (dark orange) | 2 (orange/yellow) | 5 (yellow-green) | 10 (yellow-green / dark grey) | 25 (yellow-green / dark grey) | | :--------------- | :-------------- | :---------------- | :--------------- | :---------------------------- | :---------------------------- | | **10** | 0.88 | 0.97 | 1.06 | O.O.B. | O.O.B. | | **5** | 0.89 | 0.96 | 1.04 | 1.09 | O.O.B. | | **3** | 0.91 | 0.97 | 1.04 | 1.08 | O.O.B. | | **1** | 0.96 | 0.99 | 1.03 | 1.05 | 1.09 | **Row-wise Trends (increasing $n_p$ for fixed $n_{fr}$):** * **$n_{fr}=10$ (top row):** Values increase from 0.88 (dark orange) to 0.97 (orange/yellow) to 1.06 (yellow-green), then become O.O.B. (dark grey) for $n_p=10$ and $n_p=25$. * **$n_{fr}=5$:** Values increase from 0.89 (dark orange) to 0.96 (orange/yellow) to 1.04 (yellow-green) to 1.09 (yellow-green), then become O.O.B. (dark grey) for $n_p=25$. * **$n_{fr}=3$:** Values increase from 0.91 (dark orange) to 0.97 (orange/yellow) to 1.04 (yellow-green) to 1.08 (yellow-green), then become O.O.B. (dark grey) for $n_p=25$. * **$n_{fr}=1$ (bottom row):** Values consistently increase from 0.96 (orange/yellow) to 0.99 (orange/yellow) to 1.03 (yellow-green) to 1.05 (yellow-green) to 1.09 (yellow-green). No O.O.B. values are observed in this row. **Column-wise Trends (increasing $n_{fr}$ for fixed $n_p$):** * **$n_p=1$ (leftmost column):** Values decrease from 0.96 (orange/yellow) to 0.91 (dark orange) to 0.89 (dark orange) to 0.88 (dark orange). * **$n_p=2$:** Values decrease from 0.99 (orange/yellow) to 0.97 (orange/yellow) to 0.96 (orange/yellow), then slightly increase to 0.97 (orange/yellow). * **$n_p=5$:** Values slightly increase from 1.03 (yellow-green) to 1.04 (yellow-green) to 1.04 (yellow-green) to 1.06 (yellow-green). * **$n_p=10$:** Values increase from 1.05 (yellow-green) to 1.08 (yellow-green) to 1.09 (yellow-green), then become O.O.B. (dark grey) for $n_{fr}=10$. * **$n_p=25$ (rightmost column):** The value is 1.09 (yellow-green) for $n_{fr}=1$, then becomes O.O.B. (dark grey) for $n_{fr}=3, 5, 10$. ### Key Observations * **Minimum Value:** The lowest observed numerical value is 0.88, located at ($n_p=1, n_{fr}=10$), colored dark orange. * **Maximum Value:** The highest observed numerical value is 1.09, appearing at ($n_p=10, n_{fr}=5$) and ($n_p=25, n_{fr}=1$), both colored yellow-green. * **"O.O.B." Region:** The "O.O.B." condition primarily occurs in the top-right section of the grid, indicating that higher values of "Number of initial programs ($n_p$)" and "Number of feedback-repairs ($n_{fr}$)" lead to this state. Specifically, for $n_p=25$, all $n_{fr}$ values except 1 result in O.O.B. For $n_p=10$, only $n_{fr}=10$ results in O.O.B. * **Performance Gradient:** Generally, performance (represented by the numerical values) tends to increase as $n_p$ increases, up to a point where it becomes O.O.B. * **Impact of $n_{fr}$:** For lower $n_p$ values (1 and 2), increasing $n_{fr}$ tends to decrease or stabilize the performance. For higher $n_p$ values (5 and 10), increasing $n_{fr}$ tends to slightly increase performance before hitting the O.O.B. state. * **Optimal Region (visually):** The yellow-green cells, representing higher performance, are concentrated in the middle-right and bottom-right sections of the numerical results, before the O.O.B. region. ### Interpretation This heatmap likely represents the outcome of an experiment or simulation where two parameters, "Number of initial programs ($n_p$)" and "Number of feedback-repairs ($n_{fr}$)", are varied to observe their effect on a performance metric. The goal appears to be maximizing this metric, as higher values are indicated by a "better" (yellow-green) color. The data suggests a complex relationship between the two parameters: 1. **Increasing $n_p$ generally improves performance:** For a fixed number of feedback-repairs, increasing the number of initial programs tends to increase the performance metric. This trend is most consistent when $n_{fr}$ is low (e.g., $n_{fr}=1$). 2. **Too many programs or repairs can lead to instability/failure ("O.O.B."):** The "O.O.B." state indicates a failure to produce a valid result, possibly due to resource exhaustion, computational limits, or an invalid configuration. This occurs when both $n_p$ and $n_{fr}$ are high, or when $n_p$ is very high even with moderate $n_{fr}$. This suggests there's a practical limit to how much one can scale these parameters. 3. **Trade-offs between $n_p$ and $n_{fr}$:** * At low $n_p$ (e.g., 1), increasing $n_{fr}$ *decreases* performance, suggesting that too many feedback-repairs with few initial programs might be detrimental or inefficient. * At moderate $n_p$ (e.g., 5), increasing $n_{fr}$ *slightly increases* performance, indicating a beneficial interaction up to a point. * At high $n_p$ (e.g., 10), increasing $n_{fr}$ *increases* performance but quickly leads to the O.O.B. state. The "sweet spot" for performance, where values are high and stable (not O.O.B.), appears to be around $n_p=5$ or $n_p=10$ with lower to moderate $n_{fr}$ values (e.g., $n_{fr}=1, 3, 5$). For instance, ($n_p=10, n_{fr}=5$) yields 1.09, and ($n_p=25, n_{fr}=1$) also yields 1.09. However, the latter is very close to the O.O.B. boundary, suggesting it might be less robust. The combination ($n_p=5, n_{fr}=10$) gives 1.06, which is also a good performance without hitting O.O.B. The chart effectively visualizes the parameter space, allowing identification of regions of optimal performance and regions where the system becomes unstable or fails to produce results. This is crucial for understanding the operational limits and efficiency of the underlying system or algorithm being evaluated. </details> <details> <summary>x18.png Details</summary>  ### Visual Description ## Chart Type: Scatter Plot with Fitted Curve ### Overview This image displays a scatter plot illustrating the relationship between "Mean number of tokens generated" on the x-axis and "Mean pass rate" on the y-axis. The data points are categorized by two parameters, `n_p` (represented by color) and `n_fr` (represented by marker shape), and each point includes vertical error bars. A dark grey fitted curve with a lighter grey confidence interval band shows the overall trend. ### Components/Axes **X-axis:** * **Title:** "Mean number of tokens generated" * **Range:** Approximately 0 to 10000 * **Major Tick Marks:** 0, 2000, 4000, 6000, 8000, 10000. The labels are rotated counter-clockwise. **Y-axis:** * **Title:** "Mean pass rate" * **Range:** 0.0 to 1.0 * **Major Tick Marks:** 0.0, 0.2, 0.4, 0.6, 0.8, 1.0 **Legend (located in the top-right quadrant):** The legend is divided into two columns, indicating the mapping of `n_p` to line colors and `n_fr` to marker shapes. * **Left Column (Line colors for `n_p`):** * Brown line: `n_p = 1` * Gold/Yellow line: `n_p = 2` * Teal/Cyan line: `n_p = 5` * Light Blue line: `n_p = 10` * Dark Blue/Navy line: `n_p = 25` * **Right Column (Marker shapes for `n_fr`):** * Dark Grey circle: `n_fr = 1` * Dark Grey downward triangle: `n_fr = 3` * Dark Grey square: `n_fr = 5` * Dark Grey upward triangle: `n_fr = 10` **Fitted Curve:** A dark grey line represents the overall trend of the data. It starts near (0, 0.1) and increases, then gradually flattens out. A lighter grey shaded area surrounds this line, indicating a confidence interval or standard deviation for the fitted trend. ### Detailed Analysis **Overall Trend (Fitted Curve):** The dark grey fitted curve shows a positive, non-linear relationship. The "Mean pass rate" increases rapidly from approximately 0.1 at 0 tokens to about 0.38 at 3000 tokens. After this point, the rate of increase slows significantly, with the curve leveling off to approximately 0.42 at 10000 tokens. The grey shaded area around the curve indicates a relatively narrow confidence interval, suggesting the trend is well-defined. **Individual Data Points (grouped by `n_p` and ordered by increasing "Mean number of tokens generated"):** Each data point is represented by a colored marker with vertical error bars. The color corresponds to `n_p` and the marker shape to `n_fr`. The error bars typically span approximately +/- 0.01 to 0.02 in "Mean pass rate". * **`n_p = 1` (Brown points):** * `n_fr = 1` (circle): X ~ 700, Y ~ 0.19 (Error range: ~0.17 - 0.21) * `n_fr = 3` (downward triangle): X ~ 1100, Y ~ 0.22 (Error range: ~0.20 - 0.24) * `n_fr = 5` (square): X ~ 1700, Y ~ 0.25 (Error range: ~0.23 - 0.27) * `n_fr = 10` (upward triangle): X ~ 2800, Y ~ 0.28 (Error range: ~0.26 - 0.30) * *Trend for `n_p=1`*: The pass rate increases from 0.19 to 0.28 as tokens generated increase from ~700 to ~2800. * **`n_p = 2` (Gold/Yellow points):** * `n_fr = 1` (circle): X ~ 900, Y ~ 0.24 (Error range: ~0.22 - 0.26) * `n_fr = 3` (downward triangle): X ~ 1600, Y ~ 0.28 (Error range: ~0.26 - 0.30) * `n_fr = 5` (square): X ~ 2600, Y ~ 0.33 (Error range: ~0.31 - 0.35) * `n_fr = 10` (upward triangle): X ~ 4200, Y ~ 0.37 (Error range: ~0.35 - 0.39) * *Trend for `n_p=2`*: The pass rate increases from 0.24 to 0.37 as tokens generated increase from ~900 to ~4200. * **`n_p = 5` (Teal/Cyan points):** * `n_fr = 1` (circle): X ~ 1800, Y ~ 0.35 (Error range: ~0.33 - 0.37) * `n_fr = 3` (downward triangle): X ~ 3800, Y ~ 0.38 (Error range: ~0.36 - 0.40) * `n_fr = 5` (square): X ~ 5200, Y ~ 0.40 (Error range: ~0.38 - 0.42) * `n_fr = 10` (upward triangle): X ~ 7000, Y ~ 0.42 (Error range: ~0.40 - 0.44) * *Trend for `n_p=5`*: The pass rate increases from 0.35 to 0.42 as tokens generated increase from ~1800 to ~7000, showing a slower rate of increase at higher token counts. * **`n_p = 10` (Light Blue points):** * `n_fr = 1` (circle): X ~ 4100, Y ~ 0.42 (Error range: ~0.40 - 0.44) * `n_fr = 3` (downward triangle): X ~ 5400, Y ~ 0.43 (Error range: ~0.41 - 0.45) * `n_fr = 5` (square): X ~ 7800, Y ~ 0.47 (Error range: ~0.45 - 0.49) * *Note*: No data point for `n_fr = 10` is visible for `n_p = 10`. * *Trend for `n_p=10`*: The pass rate increases from 0.42 to 0.47 as tokens generated increase from ~4100 to ~7800. * **`n_p = 25` (Dark Blue/Navy points):** * `n_fr = 1` (circle): X ~ 5500, Y ~ 0.40 (Error range: ~0.38 - 0.42) * `n_fr = 3` (downward triangle): X ~ 8200, Y ~ 0.48 (Error range: ~0.46 - 0.50) * `n_fr = 5` (square): X ~ 9800, Y ~ 0.49 (Error range: ~0.47 - 0.51) * `n_fr = 10` (upward triangle): X ~ 9900, Y ~ 0.49 (Error range: ~0.47 - 0.51) * *Trend for `n_p=25`*: The pass rate increases from 0.40 to 0.49 as tokens generated increase from ~5500 to ~9900, reaching the highest observed pass rates. ### Key Observations * **Universal Positive Correlation:** All data series demonstrate a positive correlation between "Mean number of tokens generated" and "Mean pass rate." * **Diminishing Returns:** The overall fitted curve clearly shows diminishing returns; the pass rate increases sharply at lower token counts but then plateaus, suggesting that beyond a certain number of tokens (around 4000-5000), further generation yields only marginal improvements in pass rate. * **Influence of `n_p`:** Higher values of `n_p` (e.g., `n_p=10`, `n_p=25`) are generally associated with higher "Mean pass rates" and higher "Mean number of tokens generated" compared to lower `n_p` values (e.g., `n_p=1`, `n_p=2`). This suggests `n_p` might be a parameter that enables access to higher performance ceilings. * **Influence of `n_fr`:** For a given `n_p`, increasing `n_fr` (from 1 to 10) consistently corresponds to an increase in "Mean number of tokens generated" and, consequently, an increase in "Mean pass rate." This indicates `n_fr` is a factor that drives the extent of token generation. * **Clustering by `n_p`:** Data points for lower `n_p` values (`n_p=1`, `n_p=2`) are clustered towards the lower-left of the chart (fewer tokens, lower pass rates), while higher `n_p` values (`n_p=10`, `n_p=25`) are clustered towards the upper-right (more tokens, higher pass rates). * **Missing Data Point:** The absence of an `n_fr=10` data point for `n_p=10` is a notable gap in the observed data. ### Interpretation This chart suggests that both `n_p` and `n_fr` are critical parameters influencing the performance, measured by "Mean pass rate," in a system that generates tokens. The "Mean number of tokens generated" acts as a proxy for the effort or complexity of the generation process. The overall trend indicates that more effort (more tokens) generally leads to better results (higher pass rate), but this improvement is not linear and eventually saturates. This implies there's an optimal range for token generation where the benefits outweigh the computational cost. The parameter `n_p` appears to control the *potential* or *capacity* of the system. Higher `n_p` values allow the system to achieve higher pass rates, especially when a sufficient number of tokens are generated. This could mean `n_p` relates to the breadth of search, the diversity of generated candidates, or the inherent quality of the generation process. The parameter `n_fr` seems to directly influence the *extent* of token generation. For a fixed `n_p`, increasing `n_fr` pushes the system to generate more tokens, moving it along the x-axis and, consequently, up the y-axis towards higher pass rates, following the diminishing returns curve. This might represent the number of iterations, fragments, or attempts made during the generation process. In practical terms, to achieve a high "Mean pass rate," one would likely need to select a sufficiently high `n_p` value and then tune `n_fr` to generate enough tokens to reach the plateau of the pass rate curve. For instance, `n_p=25` with `n_fr=5` or `n_fr=10` yields the highest observed pass rates (around 0.49) at the cost of generating nearly 10,000 tokens. However, `n_p=10` with `n_fr=5` achieves a pass rate of 0.47 with approximately 7,800 tokens, which might be a more efficient trade-off depending on the cost of token generation. The missing `n_fr=10` point for `n_p=10` could be an area for further investigation to understand if </details> <details> <summary>x19.png Details</summary>  ### Visual Description ## Heatmap: Performance Metrics based on Feedback-Repairs and Initial Programs ### Overview This image displays a heatmap illustrating the relationship between two parameters, "Number of feedback-repairs ($n_{fr}$)" and "Number of initial programs ($n_P$)", and a measured outcome represented by numerical values within the cells. The color of each cell indicates the magnitude of the outcome, with a gradient from darker browns/reds (lower values) to greens and blues/teals (higher values). Some cells are marked "O.O.B." (Out Of Bounds) and are colored dark grey/black, indicating that no valid data was obtained for those parameter combinations. ### Components/Axes The chart is a 2D grid, or heatmap, with the following axes and labels: * **Y-axis (Vertical, left side):** * Label: "Number of feedback-repairs ($n_{fr}$)" * Tick Markers (from bottom to top): 1, 3, 5, 10 * **X-axis (Horizontal, bottom side):** * Label: "Number of initial programs ($n_P$)" * Tick Markers (from left to right): 1, 2, 5, 10, 25 * **Data Cells:** The grid consists of 4 rows (corresponding to $n_{fr}$ values) and 5 columns (corresponding to $n_P$ values), totaling 20 cells. Each cell contains a numerical value (e.g., 0.81, 1.10) or the text "O.O.B.". The text within the cells is white. * **Color Gradient (Implicit Legend):** * Darker brown/reddish-brown colors represent lower values (e.g., 0.81, 0.83, 0.85). * Orange/yellow-orange colors represent intermediate-low values (e.g., 0.97, 1.01). * Olive green/yellow-green colors represent slightly higher intermediate values (e.g., 1.05). * Light green/mint green colors represent intermediate-high values (e.g., 1.10). * Light blue/teal colors represent higher values (e.g., 1.13, 1.16, 1.19). * Dark grey/black color represents "O.O.B." (Out Of Bounds) entries. ### Detailed Analysis The heatmap presents a 4x5 matrix of values: | $n_{fr}$ \ $n_P$ | 1 | 2 | 5 | 10 | 25 | | :--------------- | :------- | :------- | :------- | :------- | :------- | | **10** | 0.81 | 0.97 | O.O.B. | O.O.B. | O.O.B. | | **5** | 0.83 | 0.97 | 1.10 | O.O.B. | O.O.B. | | **3** | 0.85 | 0.97 | 1.10 | 1.16 | O.O.B. | | **1** | 1.01 | 1.05 | 1.10 | 1.13 | 1.19 | **Row-wise Trends (increasing $n_P$ for fixed $n_{fr}$):** * **$n_{fr} = 10$ (Top Row):** * Starts at 0.81 (dark brown) for $n_P=1$. * Increases to 0.97 (orange) for $n_P=2$. * Becomes O.O.B. (dark grey/black) for $n_P=5, 10, 25$. * *Trend:* Values increase then become Out Of Bounds. * **$n_{fr} = 5$ (Second Row):** * Starts at 0.83 (brown) for $n_P=1$. * Increases to 0.97 (orange) for $n_P=2$. * Increases to 1.10 (light green) for $n_P=5$. * Becomes O.O.B. (dark grey/black) for $n_P=10, 25$. * *Trend:* Values increase then become Out Of Bounds. * **$n_{fr} = 3$ (Third Row):** * Starts at 0.85 (orange-brown) for $n_P=1$. * Increases to 0.97 (orange) for $n_P=2$. * Increases to 1.10 (light green) for $n_P=5$. * Increases to 1.16 (light blue/teal) for $n_P=10$. * Becomes O.O.B. (dark grey/black) for $n_P=25$. * *Trend:* Values consistently increase then become Out Of Bounds. * **$n_{fr} = 1$ (Bottom Row):** * Starts at 1.01 (yellow-orange) for $n_P=1$. * Increases to 1.05 (olive green) for $n_P=2$. * Increases to 1.10 (light green) for $n_P=5$. * Increases to 1.13 (light blue/teal) for $n_P=10$. * Increases to 1.19 (light blue/teal) for $n_P=25$. * *Trend:* Values consistently increase across all $n_P$ values, without encountering O.O.B. **Column-wise Trends (decreasing $n_{fr}$ for fixed $n_P$):** * **$n_P = 1$ (First Column):** * Starts at 0.81 ($n_{fr}=10$, dark brown). * Increases to 0.83 ($n_{fr}=5$, brown). * Increases to 0.85 ($n_{fr}=3$, orange-brown). * Increases to 1.01 ($n_{fr}=1$, yellow-orange). * *Trend:* Values consistently increase as $n_{fr}$ decreases. * **$n_P = 2$ (Second Column):** * Starts at 0.97 ($n_{fr}=10$, orange). * Remains 0.97 ($n_{fr}=5$, orange). * Remains 0.97 ($n_{fr}=3$, orange). * Increases to 1.05 ($n_{fr}=1$, olive green). * *Trend:* Values are constant for higher $n_{fr}$ then increase for the lowest $n_{fr}$. * **$n_P = 5$ (Third Column):** * Starts at O.O.B. ($n_{fr}=10$, dark grey/black). * Becomes 1.10 ($n_{fr}=5$, light green). * Remains 1.10 ($n_{fr}=3$, light green). * Remains 1.10 ($n_{fr}=1$, light green). * *Trend:* O.O.B. for highest $n_{fr}$, then constant value for lower $n_{fr}$. * **$n_P = 10$ (Fourth Column):** * Starts at O.O.B. ($n_{fr}=10$, dark grey/black). * Remains O.O.B. ($n_{fr}=5$, dark grey/black). * Becomes 1.16 ($n_{fr}=3$, light blue/teal). * Decreases slightly to 1.13 ($n_{fr}=1$, light blue/teal). * *Trend:* O.O.B. for higher $n_{fr}$, then a value appears and slightly decreases. * **$n_P = 25$ (Fifth Column):** * Starts at O.O.B. ($n_{fr}=10$, dark grey/black). * Remains O.O.B. ($n_{fr}=5$, dark grey/black). * Remains O.O.B. ($n_{fr}=3$, dark grey/black). * Becomes 1.19 ($n_{fr}=1$, light blue/teal). * *Trend:* O.O.B. for all but the lowest $n_{fr}$, where a high value appears. ### Key Observations * **O.O.B. Region:** A significant portion of the top-right of the heatmap is marked "O.O.B.". This region corresponds to combinations where both $n_{fr}$ and $n_P$ are relatively high. Specifically, for $n_{fr}=10$, any $n_P \ge 5$ is O.O.B. For $n_{fr}=5$, any $n_P \ge 10$ is O.O.B. For $n_{fr}=3$, $n_P=25$ is O.O.B. Only for $n_{fr}=1$ are all $n_P$ values within bounds. * **Value Range:** The observed numerical values range from a minimum of 0.81 to a maximum of 1.19. * **Inverse Relationship with $n_{fr}$ (for low $n_P$):** For $n_P=1$ and $n_P=2$, the outcome values generally increase as $n_{fr}$ decreases. * **Direct Relationship with $n_P$ (for low $n_{fr}$):** For $n_{fr}=1$, the outcome values consistently increase as $n_P$ increases. * **Lowest Value:** The lowest value (0.81) is found at the top-left corner ($n_{fr}=10, n_P=1$). * **Highest Value:** The highest value (1.19) is found at the bottom-right corner of the non-O.O.B. region ($n_{fr}=1, n_P=25$). * **Constant Values:** For $n_P=2$, the value is consistently 0.97 for $n_{fr}=10, 5, 3$. For $n_P=5$, the value is consistently 1.10 for $n_{fr}=5, 3, 1$. ### Interpretation The heatmap suggests that the measured outcome is sensitive to both the number of feedback-repairs ($n_{fr}$) and the number of initial programs ($n_P$). 1. **Optimal Region for Lower Values:** The lowest outcome values (represented by brown/red colors) are achieved when the number of feedback-repairs ($n_{fr}$) is high and the number of initial programs ($n_P$) is low. This implies that if the goal is to minimize this outcome, one should aim for a high $n_{fr}$ and low $n_P$. 2. **Optimal Region for Higher Values:** Conversely, the highest outcome values (represented by blue/teal colors) are achieved when $n_{fr}$ is low and $n_P$ is high. If maximizing this outcome is desired, a low $n_{fr}$ and high $n_P$ would be preferred. 3. **System Stability/Applicability (O.O.B. region):** The "O.O.B." entries indicate a region of parameter space where the system or experiment either failed to produce a result, or the conditions were outside a valid operating range. This suggests that increasing both $n_{fr}$ and $n_P$ simultaneously beyond certain thresholds leads to an unstable or undefined state. Specifically, high $n_{fr}$ values are more prone to O.O.B. conditions as $n_P$ increases. Only when $n_{fr}$ is at its minimum (1) can the system handle the maximum $n_P$ (25) without going O.O.B. This could imply resource limitations, computational complexity, or a breakdown in the underlying model or process. 4. **Trade-offs and Interactions:** There's a clear interaction between $n_{fr}$ and $n_P$. For instance, at $n_P=1$, increasing $n_{fr}$ significantly decreases the outcome. However, at $n_{fr}=1$, increasing $n_P$ significantly increases the outcome. The "O.O.B." boundary acts as a critical constraint, defining the feasible operating region for the system. Understanding the nature of "O.O.B." is crucial for practical application, as it delineates where the system's behavior becomes unpredictable or invalid. </details> Figure 8: GPT-4 results from Figure 4 (Section 4.1) per difficulty (row), from top to bottom: introductory, interview, and competition. <details> <summary>x20.png Details</summary>  ### Visual Description ## Chart Type: Line Chart - Mean Pass Rate vs. Mean Number of Tokens Generated ### Overview This image displays a line chart illustrating the relationship between the "Mean number of tokens generated" on the x-axis and the "Mean pass rate" on the y-axis. Five different configurations of language models (GPT-4 and GPT-3.5) and repair strategies (`M_P` for primary model, `M_F` for repair model) are plotted as distinct lines, each with a shaded area representing uncertainty or a confidence interval. The chart demonstrates how the pass rate changes as more tokens are generated, comparing the performance of different models and the impact of a repair mechanism. ### Components/Axes The chart is structured with a main plotting area, an x-axis at the bottom, a y-axis on the left, and a legend in the bottom-left quadrant of the plot. * **X-axis Label**: "Mean number of tokens generated" * **Range**: From 0 to 10000. * **Major Tick Markers**: 0, 2000, 4000, 6000, 8000, 10000. The tick labels are rotated approximately 45 degrees counter-clockwise. * **Y-axis Label**: "Mean pass rate" * **Range**: From 0.0 to 1.0. * **Major Tick Markers**: 0.0, 0.2, 0.4, 0.6, 0.8, 1.0. * **Grid**: Light gray grid lines are present at each major tick mark for both the x and y axes, extending across the entire plot area. * **Legend**: Located in the bottom-left area of the plot, above the x-axis. It lists five data series with their corresponding line colors and labels: * **Dark Blue line with light blue-purple shaded area**: `M_P = GPT-4 (no repair)` * **Teal/Light Blue-Green line with matching shaded area**: `M_P = GPT-4; M_F = GPT-4` * **Dark Gray line with light gray shaded area**: `M_P = GPT-3.5 (no repair)` * **Brown/Orange line with matching shaded area**: `M_P = GPT-3.5; M_F = GPT-3.5` * **Cyan/Bright Blue line with matching shaded area**: `M_P = GPT-3.5; M_F = GPT-4` ### Detailed Analysis All lines generally exhibit a rapid increase in "Mean pass rate" at lower "Mean number of tokens generated," followed by a plateau where the pass rate stabilizes or increases only marginally with more tokens. The shaded areas around each line represent a confidence interval or uncertainty, which appears relatively narrow for all series. 1. **Dark Blue line: `M_P = GPT-4 (no repair)`** * **Trend**: This line starts at approximately 0.68 pass rate for very few tokens, rises steeply, and then flattens out around a pass rate of 0.92-0.93. It extends up to approximately 3500 tokens. * **Approximate Data Points**: * ~0 tokens: ~0.68 pass rate * ~500 tokens: ~0.80 pass rate * ~1000 tokens: ~0.86 pass rate * ~2000 tokens: ~0.90 pass rate * ~3000 tokens: ~0.92 pass rate * ~3500 tokens: ~0.925 pass rate (endpoint) 2. **Teal/Light Blue-Green line: `M_P = GPT-4; M_F = GPT-4`** * **Trend**: This line starts at approximately 0.68 pass rate, shows a steep initial rise, and then plateaus at the highest pass rate among all series, around 0.93-0.94. It extends the furthest, up to approximately 5500 tokens. * **Approximate Data Points**: * ~0 tokens: ~0.68 pass rate * ~500 tokens: ~0.82 pass rate * ~1000 tokens: ~0.88 pass rate * ~2000 tokens: ~0.91 pass rate * ~3000 tokens: ~0.925 pass rate * ~4000 tokens: ~0.93 pass rate * ~5000 tokens: ~0.935 pass rate * ~5500 tokens: ~0.935 pass rate (endpoint) 3. **Dark Gray line: `M_P = GPT-3.5 (no repair)`** * **Trend**: This line starts at the lowest initial pass rate, around 0.50, rises steeply but less rapidly than the GPT-4 lines, and then plateaus around 0.84-0.85. It extends up to approximately 6000 tokens. * **Approximate Data Points**: * ~0 tokens: ~0.50 pass rate * ~500 tokens: ~0.70 pass rate * ~1000 tokens: ~0.76 pass rate * ~2000 tokens: ~0.80 pass rate * ~3000 tokens: ~0.82 pass rate * ~4000 tokens: ~0.83 pass rate * ~5000 tokens: ~0.84 pass rate * ~6000 tokens: ~0.845 pass rate (endpoint) 4. **Brown/Orange line: `M_P = GPT-3.5; M_F = GPT-3.5`** * **Trend**: This line starts at approximately 0.58 pass rate, rises steeply, and then plateaus around 0.85-0.86. It extends up to approximately 5500 tokens. * **Approximate Data Points**: * ~0 tokens: ~0.58 pass rate * ~500 tokens: ~0.75 pass rate * ~1000 tokens: ~0.80 pass rate * ~2000 tokens: ~0.83 pass rate * ~3000 tokens: ~0.84 pass rate * ~4000 tokens: ~0.85 pass rate * ~5000 tokens: ~0.855 pass rate * ~5500 tokens: ~0.86 pass rate (endpoint) 5. **Cyan/Bright Blue line: `M_P = GPT-3.5; M_F = GPT-4`** * **Trend**: This line starts at approximately 0.60 pass rate, rises steeply, and then plateaus around 0.87-0.88. It extends up to approximately 5500 tokens. * **Approximate Data Points**: * ~0 tokens: ~0.60 pass rate * ~500 tokens: ~0.78 pass rate * ~1000 tokens: ~0.82 pass rate * ~2000 tokens: ~0.85 pass rate * ~3000 tokens: ~0.86 pass rate * ~4000 tokens: ~0.87 pass rate * ~5000 tokens: ~0.875 pass rate * ~5500 tokens: ~0.88 pass rate (endpoint) ### Key Observations * **GPT-4 Superiority**: Both GPT-4 configurations (dark blue and teal) consistently achieve higher mean pass rates than any GPT-3.5 configuration across all token generation levels. * **Impact of Repair Mechanism (`M_F`)**: * For `M_P = GPT-4`, adding `M_F = GPT-4` (teal line) results in a slightly higher peak pass rate (around 0.935-0.94) and extends the token generation range compared to `M_P = GPT-4 (no repair)` (dark blue line, peak around 0.925-0.93). * For `M_P = GPT-3.5`, the repair mechanism significantly improves performance. `M_P = GPT-3.5; M_F = GPT-3.5` (brown/orange line) performs better than `M_P = GPT-3.5 (no repair)` (dark gray line). * The most significant improvement for `M_P = GPT-3.5` comes from using `M_F = GPT-4` (cyan/bright blue line), which elevates its pass rate to approximately 0.87-0.88, surpassing `M_P = GPT-3.5; M_F = GPT-3.5` and approaching the performance of `M_P = GPT-4 (no repair)`. * **Diminishing Returns**: All curves show a clear plateau, indicating that beyond approximately 3000-4000 tokens, generating more tokens yields minimal additional improvement in the mean pass rate. * **Highest Performance**: The `M_P = GPT-4; M_F = GPT-4` configuration (teal line) achieves the highest mean pass rate and maintains it over the widest range of tokens generated. * **Lowest Performance**: The `M_P = GPT-3.5 (no repair)` configuration (dark gray line) consistently shows the lowest mean pass rate. ### Interpretation This chart effectively demonstrates the performance characteristics of different language models and the value of a "repair" mechanism in achieving higher task success rates. 1. **Inherent Model Capability**: GPT-4 is shown to be inherently more capable than GPT-3.5, achieving a significantly higher pass rate even without any explicit repair mechanism. The `M_P = GPT-4 (no repair)` line starts higher and plateaus at a higher pass rate than any GPT-3.5 configuration. 2. **Value of Repair**: The repair mechanism (`M_F`) generally enhances performance. For the less capable `M_P = GPT-3.5`, the repair mechanism is crucial. Using `M_F = GPT-3.5` provides a noticeable boost over no repair, but using a more powerful model for repair, `M_F = GPT-4`, with `M_P = GPT-3.5` yields a substantially better outcome. This suggests that the quality of the repair model is paramount, and a superior repair model can significantly compensate for a less powerful primary model. 3. **Optimal Token Generation**: The plateauing of all curves indicates that there's an optimal range for the "mean number of tokens generated." Beyond this range (roughly 3000-4000 tokens for most configurations), the additional computational cost of generating more tokens does not translate into a meaningful increase in the pass rate. This implies efficiency considerations for deploying these models. 4. **Synergy of Powerful Models**: The `M_P = GPT-4; M_F = GPT-4` configuration represents the most robust and highest-performing setup, achieving the highest pass rate and maintaining it across a broader range of token generation. This suggests that combining a powerful primary model with an equally powerful repair model offers the best overall performance. In essence, the data suggests that while a more powerful base model (GPT-4) is a strong determinant of success, a well-chosen repair mechanism, especially one leveraging a more capable model, can significantly improve the performance of less powerful primary models and further enhance already strong ones. The chart provides valuable insights for optimizing model deployment strategies by balancing performance goals with computational efficiency. </details> <details> <summary>x21.png Details</summary>  ### Visual Description ## Chart Type: Line Chart - Mean Pass Rate vs. Mean Number of Tokens Generated ### Overview This image displays a 2D line chart illustrating the relationship between the "Mean pass rate" (Y-axis) and the "Mean number of tokens generated" (X-axis) for various configurations of GPT models, with and without repair mechanisms. The chart features five distinct data series, each represented by a colored line and a shaded confidence interval, comparing GPT-4 and GPT-3.5 models, and the impact of different models (`M_P` for primary model, `M_F` for repair model) on performance. ### Components/Axes The chart is structured with a primary plot area, an X-axis at the bottom, a Y-axis on the left, and a legend in the bottom-right. * **X-axis Label**: "Mean number of tokens generated" * **X-axis Range**: From 0 to 10000. * **X-axis Tick Markers**: 0, 2000, 4000, 6000, 8000, 10000. The tick labels are rotated approximately 45 degrees counter-clockwise. * **Y-axis Label**: "Mean pass rate" * **Y-axis Range**: From 0.0 to 1.0. * **Y-axis Tick Markers**: 0.0, 0.2, 0.4, 0.6, 0.8, 1.0. * **Grid**: A light grey grid is present, aligning with both X and Y-axis major tick markers. * **Legend**: Located in the bottom-right quadrant of the plot area. It lists five data series with their corresponding colors and descriptions: * **Dark Blue line with light blue shading**: `M_P = GPT-4 (no repair)` * **Teal/Light Green line with lighter teal/green shading**: `M_P = GPT-4; M_F = GPT-4` * **Grey line with light grey shading**: `M_P = GPT-3.5 (no repair)` * **Brown/Orange line with light orange shading**: `M_P = GPT-3.5; M_F = GPT-3.5` * **Light Blue line with very light blue shading**: `M_P = GPT-3.5; M_F = GPT-4` ### Detailed Analysis All five data series exhibit a general trend of increasing "Mean pass rate" as the "Mean number of tokens generated" increases, eventually flattening out. The shaded areas around each line represent confidence intervals. 1. **Dark Blue line: `M_P = GPT-4 (no repair)`** * **Trend**: This line starts at a relatively high pass rate and increases steeply, then gradually flattens. * **Data Points (approximate)**: * At X=0, Y is approximately 0.39. * At X=1000, Y is approximately 0.58. * At X=2000, Y is approximately 0.62. * At X=4000, Y is approximately 0.64. * At X=6000, Y is approximately 0.65. * The line extends to approximately X=8000, where Y is around 0.66. * The confidence interval is relatively narrow. 2. **Teal/Light Green line: `M_P = GPT-4; M_F = GPT-4`** * **Trend**: This line starts similarly to the dark blue line but quickly surpasses it, showing a slightly higher pass rate, and then flattens. * **Data Points (approximate)**: * At X=0, Y is approximately 0.40. * At X=1000, Y is approximately 0.60. * At X=2000, Y is approximately 0.65. * At X=4000, Y is approximately 0.69. * At X=6000, Y is approximately 0.70. * The line extends to approximately X=6800, where Y is around 0.70. * The confidence interval is relatively narrow. 3. **Grey line: `M_P = GPT-3.5 (no repair)`** * **Trend**: This line starts at a lower pass rate than the GPT-4 lines and increases more slowly, eventually flattening at a significantly lower pass rate. * **Data Points (approximate)**: * At X=0, Y is approximately 0.20. * At X=1000, Y is approximately 0.33. * At X=2000, Y is approximately 0.39. * At X=4000, Y is approximately 0.45. * At X=6000, Y is approximately 0.48. * At X=8000, Y is approximately 0.50. * The line extends to approximately X=9500, where Y is around 0.51. * The confidence interval is moderately wide. 4. **Brown/Orange line: `M_P = GPT-3.5; M_F = GPT-3.5`** * **Trend**: This line closely follows the grey line, starting slightly below it and remaining very close, indicating minimal improvement from self-repair. * **Data Points (approximate)**: * At X=0, Y is approximately 0.18. * At X=1000, Y is approximately 0.30. * At X=2000, Y is approximately 0.37. * At X=4000, Y is approximately 0.44. * At X=6000, Y is approximately 0.47. * At X=8000, Y is approximately 0.49. * The line extends to approximately X=9800, where Y is around 0.51. * The confidence interval is moderately wide. 5. **Light Blue line: `M_P = GPT-3.5; M_F = GPT-4`** * **Trend**: This line starts at a similar low pass rate as the other GPT-3.5 lines but shows a more significant increase, eventually flattening at a higher pass rate than the other GPT-3.5 configurations. * **Data Points (approximate)**: * At X=0, Y is approximately 0.22. * At X=1000, Y is approximately 0.37. * At X=2000, Y is approximately 0.44. * At X=4000, Y is approximately 0.50. * At X=6000, Y is approximately 0.54. * At X=8000, Y is approximately 0.56. * The line extends to approximately X=10000, where Y is around 0.57. * The confidence interval is moderately wide. ### Key Observations * **GPT-4 Superiority**: Both GPT-4 configurations (dark blue and teal/light green lines) consistently achieve significantly higher mean pass rates compared to all GPT-3.5 configurations across the entire range of tokens generated. * **Impact of GPT-4 Repair**: When `M_P = GPT-4`, using `M_F = GPT-4` (teal/light green line) results in a slightly higher mean pass rate than `(no repair)` (dark blue line), particularly at higher token counts. The peak pass rate for `M_P = GPT-4; M_F = GPT-4` is around 0.70, while `M_P = GPT-4 (no repair)` peaks around 0.66. * **Impact of Repair on GPT-3.5**: * `M_P = GPT-3.5 (no repair)` (grey line) and `M_P = GPT-3.5; M_F = GPT-3.5` (brown/orange line) perform very similarly, suggesting that using GPT-3.5 itself for repair (`M_F = GPT-3.5`) provides negligible improvement over no repair. Both peak around a 0.50-0.51 pass rate. * However, when `M_P = GPT-3.5` is paired with `M_F = GPT-4` (light blue line), there is a noticeable improvement in the mean pass rate. This configuration achieves a pass rate of approximately 0.57 at 10000 tokens, which is a significant increase over the ~0.51 achieved by the other GPT-3.5 configurations. * **Diminishing Returns**: All lines show a rapid increase in pass rate at lower token counts, followed by a flattening trend, indicating diminishing returns in pass rate improvement beyond a certain number of generated tokens (roughly 4000-6000 tokens for GPT-4, and 6000-8000 for GPT-3.5). * **Confidence Intervals**: The confidence intervals (shaded areas) are relatively narrow for the GPT-4 models, suggesting higher consistency, and slightly wider for the GPT-3.5 models, indicating more variability. ### Interpretation This chart strongly suggests that GPT-4 is a more capable model than GPT-3.5 for the task being measured (implied by "pass rate"). The baseline performance of GPT-4 without any repair mechanism (`M_P = GPT-4 (no repair)`) is already superior to all GPT-3.5 configurations, even those with repair. The "repair" mechanism (`M_F`) appears to be most effective when a more powerful model (GPT-4) is used for the repair process. Specifically, using GPT-4 to repair outputs from GPT-3.5 (`M_P = GPT-3.5; M_F = GPT-4`) significantly boosts GPT-3.5's performance, bringing its pass rate closer to, though still below, the baseline GPT-4 performance. This indicates that a strong "fixer" model can compensate for the limitations of a weaker primary model. Conversely, using a weaker model (GPT-3.5) to repair its own outputs (`M_P = GPT-3.5; M_F = GPT-3.5`) offers almost no benefit over no repair at all. This implies that if the primary model itself is not capable enough, it cannot effectively self-correct or repair its own errors. The flattening of all curves indicates that there's a practical limit to the "Mean pass rate" achievable, and simply generating more tokens beyond a certain point does not yield substantial improvements. This could be due to inherent limitations of the models, the task itself, or the evaluation metric. The point of diminishing returns for GPT-4 is reached earlier (fewer tokens) than for GPT-3.5, suggesting GPT-4 is more efficient in achieving its peak performance. </details> <details> <summary>x22.png Details</summary>  ### Visual Description ## Chart Type: Line Chart: Mean Pass Rate vs. Mean Number of Tokens Generated for Different GPT Models and Repair Strategies ### Overview This line chart illustrates the relationship between the "Mean pass rate" and the "Mean number of tokens generated" for five distinct configurations of GPT models (GPT-4 and GPT-3.5), with and without repair mechanisms. Each colored line represents a specific model configuration, and a shaded area surrounding each line indicates the associated uncertainty or confidence interval. The chart allows for a comparative analysis of model performance under varying conditions. ### Components/Axes * **X-axis Title**: "Mean number of tokens generated" * **X-axis Range**: From 0 to 10000. * **X-axis Major Tick Markers**: 0, 2000, 4000, 6000, 8000, 10000. The numerical labels are rotated counter-clockwise. * **Y-axis Title**: "Mean pass rate" * **Y-axis Range**: From 0.0 to 1.0. * **Y-axis Major Tick Markers**: 0.0, 0.2, 0.4, 0.6, 0.8, 1.0. * **Legend**: Located in the top-right quadrant of the plot area. * **Dark Blue Line**: $M_P$ = GPT-4 (no repair) * **Teal/Light Green Line**: $M_P$ = GPT-4; $M_F$ = GPT-4 * **Dark Gray Line**: $M_P$ = GPT-3.5 (no repair) * **Brown/Orange Line**: $M_P$ = GPT-3.5; $M_F$ = GPT-3.5 * **Light Blue Line**: $M_P$ = GPT-3.5; $M_F$ = GPT-4 ### Detailed Analysis All data series generally exhibit an initial rapid increase in mean pass rate as the mean number of tokens generated increases, followed by a gradual flattening, suggesting diminishing returns. Each line is accompanied by a shaded region representing its uncertainty, which tends to widen with higher token counts. 1. **Teal/Light Green Line**: $M_P$ = GPT-4; $M_F$ = GPT-4 * **Trend**: This line shows the highest performance. It starts near 0.0, rises steeply, and then continues to increase at a slower rate, consistently staying above all other lines. * **Approximate Data Points**: * At X=0: Y is approximately 0.0. * At X=2000: Y is approximately 0.35. * At X=4000: Y is approximately 0.42. * At X=6000: Y is approximately 0.45. * At X=8000: Y is approximately 0.47. * At X=10000: Y is approximately 0.49. * **Uncertainty**: The shaded area is relatively wide, particularly at higher token counts, indicating a notable range of possible pass rates. 2. **Dark Blue Line**: $M_P$ = GPT-4 (no repair) * **Trend**: This line starts around 0.1, rises sharply, and then continues to increase, but at a lower rate than the teal line. It consistently holds the second-highest performance. * **Approximate Data Points**: * At X=0: Y is approximately 0.1. * At X=2000: Y is approximately 0.3. * At X=4000: Y is approximately 0.35. * At X=6000: Y is approximately 0.37. * At X=8000: Y is approximately 0.39. * At X=10000: Y is approximately 0.41. * **Uncertainty**: The shaded area is visible and comparable in width to the teal line, suggesting similar variability. 3. **Light Blue Line**: $M_P$ = GPT-3.5; $M_F$ = GPT-4 * **Trend**: This line starts very low, rises steadily, and then flattens out. It performs significantly better than the other GPT-3.5 configurations but remains below the GPT-4 configurations. * **Approximate Data Points**: * At X=0: Y is approximately 0.02. * At X=2000: Y is approximately 0.1. * At X=4000: Y is approximately 0.15. * At X=6000: Y is approximately 0.18. * At X=8000: Y is approximately 0.21. * At X=10000: Y is approximately 0.23. * **Uncertainty**: The shaded area is present and appears to be of moderate width. 4. **Brown/Orange Line**: $M_P$ = GPT-3.5; $M_F$ = GPT-3.5 * **Trend**: This line starts very low, rises gradually, and then flattens out. It performs slightly better than the dark gray line but considerably worse than the light blue line. * **Approximate Data Points**: * At X=0: Y is approximately 0.02. * At X=2000: Y is approximately 0.07. * At X=4000: Y is approximately 0.1. * At X=6000: Y is approximately 0.12. * At X=8000: Y is approximately 0.14. * At X=10000: Y is approximately 0.16. * **Uncertainty**: The shaded area is visible and appears to be of moderate width. 5. **Dark Gray Line**: $M_P$ = GPT-3.5 (no repair) * **Trend**: This line starts very low, rises gradually, and then flattens out. It consistently shows the lowest performance across the entire range. * **Approximate Data Points**: * At X=0: Y is approximately 0.02. * At X=2000: Y is approximately 0.05. * At X=4000: Y is approximately 0.08. * At X=6000: Y is approximately 0.1. * At X=8000: Y is approximately 0.12. * At X=10000: Y is approximately 0.14. * **Uncertainty**: The shaded area is present and appears to be of moderate width, similar to the brown line. ### Key Observations * **GPT-4 Dominance**: Configurations using GPT-4 as the primary model ($M_P$) consistently achieve significantly higher mean pass rates compared to those using GPT-3.5 as the primary model. * **Repair Mechanism Effectiveness**: * For GPT-4, employing GPT-4 for repair ($M_F$ = GPT-4) results in the highest pass rate, outperforming GPT-4 with no repair. * For GPT-3.5, using GPT-3.5 for repair ($M_F$ = GPT-3.5) provides a marginal improvement over GPT-3.5 with no repair. * **Cross-Model Repair Impact**: The most substantial improvement for a GPT-3.5 primary model is observed when GPT-4 is used for repair ($M_P$ = GPT-3.5; $M_F$ = GPT-4). This configuration achieves a pass rate roughly double that of GPT-3.5 with no repair or GPT-3.5 with GPT-3.5 repair, and approaches the performance of GPT-4 with no repair. * **Performance Ceiling**: All models show a rapid increase in pass rate at lower token counts (up to ~4000 tokens), after which the rate of improvement diminishes, suggesting a plateau in performance gains beyond a certain number of generated tokens. * **Uncertainty Growth**: The confidence intervals (shaded areas) generally widen as the number of tokens generated increases, indicating greater variability in pass rates at higher token counts. ### Interpretation This chart provides compelling evidence regarding the capabilities of different large language models (GPT-4 vs. GPT-3.5) and the strategic benefits of incorporating a "repair" mechanism, particularly when leveraging a more powerful model for repair. 1. **Inherent Model Strength**: GPT-4 demonstrates superior inherent capability in achieving higher pass rates, even without any explicit repair process. This suggests that GPT-4's initial generations are of higher quality or more robust. 2. **Synergy of Generation and Repair**: The best performance is achieved when GPT-4 is used for both primary generation ($M_P$) and repair ($M_F$). This indicates that even highly capable models can benefit from an iterative refinement process, where the same strong model can identify and correct its own potential flaws. 3. **Upskilling with Repair**: A key finding is the significant performance boost for GPT-3.5 when paired with GPT-4 for repair. This "cross-model repair" strategy allows a less capable primary generator (GPT-3.5) to achieve pass rates comparable to, or even exceeding, a stronger model (GPT-4) operating without repair. This implies that a powerful repair model can effectively compensate for the limitations of a weaker generation model, making it a valuable strategy for improving output quality without solely relying on the most advanced primary generator. 4. **Efficiency Considerations**: The flattening of the curves suggests that there is an optimal range for the number of tokens generated. Beyond approximately 6000-8000 tokens, the marginal gains in pass rate become minimal. This has practical implications for resource allocation, as generating an excessive number of tokens might not yield proportional improvements in pass rate, leading to inefficient computation. 5. **Strategic Model Deployment**: The data suggests a tiered approach to model deployment. For maximum performance, GPT-4 with GPT-4 repair is ideal. If cost or latency constraints limit the use of GPT-4 for primary generation, a hybrid approach of GPT-3.5 for generation and GPT-4 for repair offers a highly effective alternative, significantly outperforming GPT-3.5 alone. This strategy allows for a balance between performance and resource utilization. </details> Figure 9: Results from Figure 6 (Section 4.2) per difficulty (row), from top to bottom: introductory, interview, and competition. ## Appendix B Human Experiment: Study Instructions For our study on human data, participants were given a slide deck with instructions. The following ten images show the instructions, which include an example of a task shown to a participant: <details> <summary>extracted/2306.09896v1/figures/appendix/instructions/1.jpg Details</summary>  ### Visual Description ## Document: Tasks Overview ### Overview This image displays a textual document outlining a set of tasks, presented as a hierarchical bulleted list. The document specifies setup requirements, the nature of the primary task, and the format for completing and submitting the tasks. Key information, such as the number of programs to debug, the expected duration, and data collection policy, is highlighted through styling (bolding and blue text). ### Content Structure The document is structured with a main title at the top-left, followed by three primary bullet points, each with several sub-bullet points. * **Title:** "Tasks" (located at the top-left, in a large, bold font). * **Main Bullet Points:** * "Setup:" (positioned below the title, indented). * "Task: Debug five incorrect Python programs" (positioned below "Setup:", indented). * "Task format" (positioned below the "Task: Debug..." section, indented). * **Sub-Bullet Points:** Each main bullet point has nested sub-bullet points, further indented, providing specific details. ### Content Details The document provides the following information: * **Header:** * "Tasks" * **Main Bullet Point 1: Setup:** * "Use a laptop or desktop computer, not a phone" * **Main Bullet Point 2: Task: Debug five incorrect Python programs** * The word "five" is visually emphasized (bolded). * "Each program is an incorrect attempt to solve a coding challenge" * "Your answer should explain what the program is doing wrong" (The phrase "explain what the program is doing wrong" is highlighted in blue text). * "Expect ~10 minutes per task" * **Main Bullet Point 3: Task format** * "Each task is in a separate website" * "Submit your answer using the Google form embedded in each page" * "No other data is being collected" (The phrase "No other data is being collected" is highlighted in blue text). ### Key Observations * The document clearly outlines instructions for a debugging task. * Specific numerical information, such as "five" programs and "~10 minutes per task," is provided. * Important instructions and assurances are highlighted using blue text, specifically regarding the requirement to "explain what the program is doing wrong" and the privacy statement "No other data is being collected." * The use of different bullet point styles (solid circle for main points, hollow circle for sub-points) helps to visually distinguish the hierarchy of information. ### Interpretation This document serves as a concise instruction set for participants undertaking a programming task. The "Setup" section ensures participants have the appropriate computing environment. The "Task" section clearly defines the core activity: debugging five Python programs, each representing a failed attempt at a coding challenge. The emphasis on "explain what the program is doing wrong" suggests that the assessment focuses on diagnostic ability and clear communication of errors, rather than just providing a corrected code. The time estimate of "~10 minutes per task" provides participants with an expectation for the duration of each debugging exercise. Finally, the "Task format" section details the logistical aspects, indicating that tasks are web-based and answers are submitted via an embedded Google form. The explicit statement "No other data is being collected" is a crucial privacy assurance, likely intended to build trust and encourage participation by addressing potential concerns about data privacy. Overall, the document is designed to be clear, informative, and reassuring for participants. </details> <details> <summary>extracted/2306.09896v1/figures/appendix/instructions/2.jpg Details</summary>  ### Visual Description ## Screenshot: Programming Problem Instructions and Examples ### Overview This image displays a set of instructions and example answers for a programming problem, presented as a text document or a section of a web page. It outlines what is expected in a user's answer regarding identifying and fixing errors in a program, provides specific examples of potential issues, and includes administrative notes about data collection and tool usage. ### Components/Axes The document is structured with a main heading and a series of bullet points, some of which have nested sub-bullet points. * **Main Heading (Top-left)**: "Your Answer" (large, bold, black text) * **Main Bullet Points (Black circular bullets)**: * Instructional prompt for the user's answer. * "Example answers:" (blue, bold text) * Statement about data collection. * Statement about external tool usage. * **Sub-Bullet Points (Small circular bullets)**: * Specific guidance for the user's answer (under the first main bullet). * Detailed examples of programming errors (under "Example answers:"). ### Detailed Analysis The content is presented as a hierarchical list: * **Top-left, below the main heading**: * A black circular bullet point introduces the primary task: "Your answer should briefly **explain what the program is doing wrong**. If it helps you explain your thoughts, you can also say what you would do differently." (The phrase "explain what the program is doing wrong" is bold and blue). * Nested under this, with a small circular bullet: "Can be precise: "the formula used to calculate X on line 5 is wrong, it should be..."" * Nested under this, with another small circular bullet: "Or high-level: "the program is treating the task as a min-cut graph problem, but it is actually shortest-path... it could be rewritten using Dijkstra's algorithm..."" * **Below the first main bullet point**: * A black circular bullet point introduces: "**Example answers:**" (The entire phrase is bold and blue). * Nested under this, with a small circular bullet: "The problem description states that numbers which start or end with zeros (such as `010` and `00`) are NOT considered valid numerical palindromes. However, the code above does not take this into account and therefore returns `00` as a valid palindrome." * Nested under this, with another small circular bullet: "The main issue with the provided code is that it only considers direct subordinates when trying to find the k-th officer in the command spreading sequence. However, the problem asks for the order in which officers receive the command, including indirect subordinates. To fix this, we need to traverse the tree of officers and construct the command spreading sequence before finding the k-th element." * **Below the "Example answers" section**: * A black circular bullet point states: "We are not collecting **any data** about how you use the website. Only your submitted answer is recorded." (The phrase "any data" is bold and blue). * **Below the data collection statement**: * A black circular bullet point states: "**Feel free to use external tools:** pen and paper, a Python IDE, etc!" (The phrase "Feel free to use external tools:" is bold and blue). ### Key Observations * The document uses a clear hierarchical structure with bullet points to organize information. * Key instructions and important phrases are highlighted using bold and blue text, drawing attention to critical aspects like the task ("explain what the program is doing wrong"), the nature of the examples, and administrative details ("any data", "Feel free to use external tools"). * The example answers provide concrete scenarios related to common programming problems (e.g., handling edge cases for palindromes, graph traversal for organizational structures). * The language is direct and instructional, typical of a technical assessment or problem description. ### Interpretation This document serves as a guide for a user attempting to solve a programming challenge. It clearly defines the scope of the expected answer, encouraging both precise and high-level explanations of code errors. The "Example answers" section is crucial as it provides hints or common pitfalls related to specific problem types, suggesting that the user might be dealing with tasks involving string manipulation/number properties (palindromes) and graph/tree traversal (command spreading sequence). The administrative notes at the end are important for user privacy and to encourage the use of appropriate development tools, fostering a realistic problem-solving environment. The overall tone is supportive and informative, aiming to guide the user through the assessment process. </details> <details> <summary>extracted/2306.09896v1/figures/appendix/instructions/3.jpg Details</summary>  ### Visual Description ## Text Content: "Example" Text Display ### Overview The image displays a single word, "Example," rendered in blue text against a plain white background. There are no other graphical elements, charts, diagrams, or additional text present. ### Components/Axes * **Text Content**: The primary and only visible component is the word "Example". * **Font Color**: The text "Example" is rendered in a shade of blue. * **Background**: The entire background is solid white. * **Axes/Labels/Legends**: There are no axes, labels, titles, legends, or any other graphical annotations typically found in charts or diagrams. ### Detailed Analysis The word "Example" is centrally positioned within the image frame, both horizontally and vertically. The text appears to be in a sans-serif font, of moderate size, making it clearly legible. The blue color of the text provides a clear contrast against the white background. ### Key Observations * **Simplicity**: The image is extremely minimalist, containing only one word. * **Central Placement**: The text is precisely centered, suggesting intentional placement rather than arbitrary positioning. * **Lack of Context**: There are no surrounding elements or additional information to provide context for the word "Example." ### Interpretation The image, consisting solely of the word "Example" on a blank background, strongly suggests it serves as a placeholder, a demonstration of text rendering, or a generic illustration of an "example" without specifying what it is an example of. Its simplicity implies a focus on the word itself, possibly for testing display properties, as a visual cue for a concept, or as a default state in a user interface or document. Without further context, the "Example" itself is the entire data point, indicating a lack of specific data or complex relationships to analyze. It functions as a literal representation of the word "example." </details> <details> <summary>extracted/2306.09896v1/figures/appendix/instructions/4.jpg Details</summary>  ### Visual Description ## Document Page: Problem Specification ### Overview This image displays a technical document page titled "1. Problem Specification". The page is divided into two main conceptual columns. The left column contains the detailed problem specification for a programming task, including a general description, input requirements, output requirements, and an example. The right column contains general instructions about reading the problem specification. ### Components/Axes The document is structured with a main header at the top-left, a primary content block on the left side, and a secondary instructional text block on the right side. **Header (Top-Left):** * "1. Problem Specification" **Left Column (Main Content Block):** This block is vertically aligned on the left side of the page and contains several sub-sections: * **Section Title:** "Specification" (bold) * **Sub-section Title:** "----Input----" * **Sub-section Title:** "----Output----" * **Sub-section Title:** "----Examples----" * Sub-heading: "Sample Input:" * Sub-heading: "Sample Output:" * **Footer (Bottom of Left Column):** "Remember: all input/output of the program should be handled through stdin and stdout." **Right Column (Instructional Text Block):** This block is positioned to the right of the main content, approximately vertically centered on the page. * **Text Line 1:** "Each page starts with a specification of what" * **Text Line 2:** "the program should do." * **Text Line 3 (Blue):** "Begin by carefully reading the problem" * **Text Line 4 (Blue):** "specification." ### Content Details **Header:** "1. Problem Specification" **Left Column - "Specification" Section:** "The recent schoolboard elections were hotly contested: a proposal to swap school start times for elementary and high school students, a controversial new dress code proposal that bans athletic clothes in school, and a proposal to raise real-estate taxes to pay for a new football practice facility, and the list goes on and on. It is now hours after the polls have closed and a winner has yet to emerge!" "In their desperation, the election officials turn to you and ask you to write a program to count the vote!" **Left Column - "----Input----" Section:** "The input consists of a single test case, which is a list of votes cast. Each line in the input contains the name of a candidate for whom a vote was cast. A name may consist of multiple words, separated by spaces. Words contain letters or hyphens, but no other punctuation characters. There will be at least $2$ votes on the list. The list of votes ends with a single line containing the characters ***. This line should not be counted. There can be up to $100000$ valid votes." **Left Column - "----Output----" Section:** "If a candidate obtained a simple or absolute majority of all votes cast (that is, more than any other candidate), output the name of this candidate! If no candidate obtained a simple majority, output: "Runoff!" (don't forget to include the exclamation mark!)" **Left Column - "----Examples----" Section:** "Sample Input:" * "Penny Franklin" * "Marti Graham" * "Connie Froggatt" * "Joseph Ivers" * "Connie Froggatt" * "Penny Franklin" * "Connie Froggatt" * "Bruce Stanger" * "Connie Froggatt" * "Barbara Skinner" * "Barbara Skinner" * "***" "Sample Output:" * "Connie Froggatt" **Left Column - Footer:** "Remember: all input/output of the program should be handled through stdin and stdout." **Right Column - Instructional Text:** "Each page starts with a specification of what" "the program should do." "Begin by carefully reading the problem" "specification." ### Key Observations * The document outlines a programming problem related to counting votes in an election. * Input format specifies candidate names per line, allowing multiple words and hyphens, but no other punctuation. * Input must contain at least 2 votes and can have up to 100,000 valid votes. * The input list terminates with a line containing "***", which should not be counted as a vote. * Output requires identifying a candidate with a "simple or absolute majority" (defined as "more than any other candidate"). * If no such majority exists, the output should be "Runoff!" including the exclamation mark. * The example input shows 11 valid votes. Counting the votes for each candidate: * Penny Franklin: 2 votes * Marti Graham: 1 vote * Connie Froggatt: 4 votes * Joseph Ivers: 1 vote * Bruce Stanger: 1 vote * Barbara Skinner: 2 votes * Connie Froggatt (4 votes) has more votes than any other candidate (Penny Franklin and Barbara Skinner have 2 votes each, others have 1). Thus, Connie Froggatt is the winner, matching the sample output. * All input/output should use standard input (stdin) and standard output (stdout). * The right-hand text emphasizes the importance of reading the problem specification carefully. ### Interpretation This document serves as a clear and concise problem statement for a programming challenge. The core task is to implement an election vote counter that determines a winner based on a "simple or absolute majority" rule, or declares a "Runoff!" if no such winner exists. The detailed input/output specifications, including constraints on vote count and name format, are crucial for developers to correctly implement the solution. The example provided helps clarify the majority rule and expected output format. The instructional text on the right reinforces the importance of thorough understanding of the problem before attempting a solution, a common practice in competitive programming or software development tasks. The problem implicitly tests parsing skills, data aggregation, and conditional logic. </details> <details> <summary>extracted/2306.09896v1/figures/appendix/instructions/5.jpg Details</summary>  ### Visual Description ## Code Snippet and Explanatory Text: Incorrect Program Description ### Overview The image presents a section from a technical document or slide, focusing on a programming problem. It features a prominent title "2. Incorrect Program" at the top-left. The main content area on the left displays a Python code snippet, also titled "Incorrect Program," which simulates a vote counting and winner determination process. On the right, a side panel provides contextual information, a debugging tip, and a note regarding program input/output. ### Components/Axes * **Main Header (Top-left)**: "2. Incorrect Program" * **Code Block Container (Left-center)**: A white rectangular box with a light grey border. * **Code Block Title (Inside container)**: "Incorrect Program" (bold text). * **Code Snippet (Inside container)**: A Python program. * **Explanatory Text Panel (Right-center)**: Contains three distinct paragraphs of text. ### Detailed Analysis **Header (Top-left)**: The primary heading for this section is "2. Incorrect Program". **Main Content (Left-center)**: A code editor-like block is present, titled "Incorrect Program". It contains the following Python code: ```python from collections import defaultdict votes = defaultdict(int) candidate = input() while candidate != "***": votes[candidate] += 1 candidate = input() total_votes = sum(votes.values()) max_votes = max(votes.values()) winner = [name for name, count in votes.items() if count == max_votes] if len(winner) == 1 and max_votes > total_votes // 2: print(winner[0]) else: print("Runoff!") ``` **Side Panel (Right-center)**: This section contains three paragraphs of explanatory text: 1. "Next, you will be shown the incorrect program." 2. "Tip: If you are struggling with debugging the program, try running it on your machine!" (The phrase "try running it on your machine!" is visually emphasized, likely bolded or highlighted). 3. "Note: the programs handle inputs through “input()”, and outputs through “print()”." ### Key Observations * The document explicitly labels the provided Python code as "Incorrect Program," indicating that it contains a bug or logical flaw. * The Python code uses `defaultdict(int)` to store vote counts for candidates. * It continuously takes candidate names as input until the sentinel value "***" is entered. * After collecting votes, it calculates the `total_votes`, `max_votes`, and identifies all `winner` candidates who received the `max_votes`. * The program's logic for declaring a winner requires two conditions: 1. There must be exactly one candidate with the maximum votes (`len(winner) == 1`). 2. That single candidate's votes must be strictly greater than half of the total votes (`max_votes > total_votes // 2`). * If these conditions are met, the winner's name is printed; otherwise, "Runoff!" is printed. * The accompanying text provides a strong hint for debugging by suggesting to "try running it on your machine!" * It also clarifies the standard input (`input()`) and output (`print()`) mechanisms used by the programs. ### Interpretation This image serves as an instructional prompt for a programming exercise, likely in a debugging context. The core task is to identify and understand the "incorrectness" of the provided Python code. The program attempts to implement a simple voting system to determine a winner based on a majority rule, or declare a runoff if no clear winner emerges. The "incorrectness" could stem from several potential issues: 1. **Majority Rule Definition**: The condition `max_votes > total_votes // 2` might be problematic for certain vote distributions, especially with odd `total_votes` or when a simple majority (more than 50%) is required versus a plurality. For example, if `total_votes` is 5, `total_votes // 2` is 2. A candidate with 3 votes would satisfy `3 > 2`, but if `total_votes` is 4, `total_votes // 2` is 2, and a candidate with 3 votes would also satisfy `3 > 2`. The definition of "more than half" can be subtle. 2. **Tie-breaking**: If multiple candidates tie for `max_votes`, `len(winner)` will be greater than 1, automatically leading to a "Runoff!" even if one of the tied candidates might have a simple majority if the tie were broken differently or if the rule was simply "most votes wins." 3. **Edge Cases**: Consider scenarios with very few votes, no votes, or all votes for one candidate. 4. **Input Handling**: While the note clarifies `input()`, potential issues like non-string inputs (though `input()` returns strings) or unexpected sentinel behavior could be considered. The explicit instruction to "try running it on your machine!" encourages active, hands-on debugging, which is a fundamental skill in programming. The overall message is to analyze the code's logic, test it with various inputs, observe its behavior, and pinpoint where its output deviates from the expected correct outcome for a voting system. </details> <details> <summary>extracted/2306.09896v1/figures/appendix/instructions/6.jpg Details</summary>  ### Visual Description ## Technical Content Display: Error Message Analysis ### Overview This image displays a technical document or presentation slide titled "3. Error Message". It is composed of two main sections: a left panel detailing an error scenario with input, program output, and expected output, and a right panel providing an explanation and a tip related to the error message. The overall purpose is to illustrate a program failure by showing the test case that caused it. ### Components/Axes The image does not contain traditional axes or legends as it is not a chart or graph. Instead, it presents structured textual information. **Header (Top-Left):** * "3. Error Message" **Left Panel (Main Content Block - Centered-Left):** This panel is a white rectangular box with a light grey border, containing several sections of text. * **Title (Top of panel):** "Error" (bold) * **Instruction/Tip (Below "Error", italicized):** "Tip: If you're executing the code on your machine, copy-paste the input into a file and pipe it to the program with 'python3 program.py < input.txt'" * **Section Header (Below tip):** "Input" (bold) * **Section Header (Below input list):** "Program Output" (bold) * **Section Header (Below program output):** "Expected Output" (bold) **Right Panel (Explanatory Text - Centered-Right):** This panel contains explanatory text and bullet points. * **Main Text:** "The error message shows you the test that the program failed on." * **Sub-header:** "It contains:" * **Bullet Points (Blue text):** * "An example input" * "The program's incorrect output" * "The expected output" * **Tip (Below bullet points):** "Tip: try copy-pasting the input to a file and piping it to the program." ### Detailed Analysis **Left Panel Content:** * **Error Tip:** The tip provides a command-line instruction for reproducing the error: `python3 program.py < input.txt`. This implies the program `program.py` takes input from a file named `input.txt`. * **Input Data:** This section lists eleven names followed by three asterisks, suggesting the end of the input or a truncation. * Penny Franklin * Marti Graham * Connie Froggatt * Joseph Ivers * Connie Froggatt * Penny Franklin * Connie Froggatt * Bruce Stanger * Connie Froggatt * Barbara Skinner * Barbara Skinner * *** * **Program Output Data:** The program, when run with the provided input, produced the following output: * "Runoff!" * **Expected Output Data:** The desired or correct output for the given input was: * "Connie Froggatt" **Right Panel Content:** * The text "The error message shows you the test that the program failed on." clarifies the purpose of the left panel's content. * The bullet points explicitly state what information is presented in the error message, with the text in blue ("An example input", "The program's incorrect output", "The expected output") directly corresponding to the "Input", "Program Output", and "Expected Output" sections in the left panel. * The second tip, "Tip: try copy-pasting the input to a file and piping it to the program.", reiterates the method for reproducing the error, reinforcing the instruction given in the left panel. ### Key Observations * The image clearly presents a debugging scenario, contrasting actual program output with expected output. * The discrepancy between "Runoff!" (Program Output) and "Connie Froggatt" (Expected Output) is the core of the error being demonstrated. * The input data consists of a list of names, suggesting the program might be processing or filtering names. * The instructions for reproducing the error are provided twice, emphasizing the practical aspect of debugging. * The use of blue text for the bullet points on the right visually links the explanation to the specific components of the error message. ### Interpretation This document serves as an example of how to present a program error for analysis or debugging. It effectively isolates the problem by providing all necessary context: the exact input that triggered the failure, the incorrect result produced by the program, and the correct result that was anticipated. The program `program.py` likely processes a list of names. Given the "Expected Output" is "Connie Froggatt" and the "Program Output" is "Runoff!", it suggests a significant logical error. Perhaps the program is intended to identify a specific name based on some criteria (e.g., the last name alphabetically, the most frequent name, or a name associated with a specific condition), but instead, it's outputting a generic or incorrect string "Runoff!". The presence of "Connie Froggatt" multiple times in the input list (4 times) might be a clue, suggesting the program might be intended to count occurrences, identify a majority, or perform some aggregation related to names. The repeated "Tip" about piping input from a file indicates that this document is likely part of a tutorial or assignment where users are expected to run and debug code themselves. It guides the user on how to replicate the exact conditions under which the error occurred, which is a fundamental step in debugging. The structure of the error message (Input, Program Output, Expected Output) is a standard and effective way to communicate test failures in software development. </details> <details> <summary>extracted/2306.09896v1/figures/appendix/instructions/7.jpg Details</summary>  ### Visual Description ## Document Snippet: Model Explanations and Context ### Overview This image displays a section from a technical document or presentation slide, titled "4. Model Explanations". It is structured into two main content areas: a prominent, grey-bordered box on the left containing two detailed "Model Explanations" for a specific issue, and a block of plain text on the right providing contextual information, a critical disclaimer about the explanations' origin and reliability, and an analogy. ### Components/Axes * **Main Title (Top-Left)**: "4. Model Explanations" * **Left Content Box (Center-Left)**: A rectangular box with a light grey border and background, containing the primary explanations. * **Section Title**: "Model Explanations" (bold, black text) * **Explanation 1 Subtitle**: "Explanation 1" (bold, black text) * **Explanation 2 Subtitle**: "Explanation 2" (bold, black text) * **Right Context Block (Center-Right)**: A block of plain text providing additional information and a disclaimer. * **Disclaimer Text**: "These explanations are generated by the model itself. They might be completely wrong. You don't have to use them." (blue, bold text) ### Content Details **Main Title (Top-Left)**: 4. Model Explanations **Left Content Box (Center-Left)**: **Model Explanations** </details> <details> <summary>extracted/2306.09896v1/figures/appendix/instructions/8.jpg Details</summary>  ### Visual Description ## Screenshot: Answer Form with Instructions ### Overview This image displays a screenshot of a web page or document section titled "5. Answer Form". It features an embedded Google Form on the left side, designed for submitting an explanation, and accompanying textual instructions on the right side. The form includes a redacted section, a prompt to sign in to Google, a required text input field for "Your Explanation", and submission buttons. The instructions clarify the purpose of the form and provide guidelines for the answer format. ### Components/Axes The image is divided into a main header and two primary content panels: **Header (Top-Left):** * **Title:** "5. Answer Form" **Left Panel (Embedded Google Form - within a light grey bordered box):** * **Form Title:** "Your Explanation" (bold, top of the form box) * **Redaction Block:** A prominent black rectangular box with white text centered: "REDACTED FOR ANONYMITY" * **Sign-in Prompt:** Below the redaction block, a line of text: "Sign in to Google to save your progress. Learn more" * "Sign in to Google" is blue, indicating a hyperlink. * "Learn more" is blue, indicating a hyperlink. * **Required Question Indicator:** Below the sign-in prompt, a line of red text: "* Indicates required question" * **Input Field Label:** "Your Explanation *" (The asterisk indicates a required field). * **Input Field Placeholder:** Within the text input area, light grey text reads: "Your answer" * **Action Buttons (Bottom of Form):** * **Submit Button:** A rectangular button on the bottom-left with a purple background and white text: "Submit" * **Clear Form Link:** A text link on the bottom-right: "Clear form" **Right Panel (Instructions - plain text):** * **General Information Paragraph:** "Finally, each page contains an embedded Google Form. No login is required." * **Primary Instruction:** A line of blue, bold, and slightly italicized text: "Submit *your explanation* of what the program is doing wrong." * **Answer Format Guideline Paragraph:** "Your answer must be self-contained; it should **not** be of the form "Just like the first model explanation describes, the issue with the code is that ..."" (The word "not" is bolded for emphasis). ### Detailed Analysis The image presents a user interface for submitting an explanation, likely in an educational or technical assessment context. The form is clearly identified as a Google Form, with an explicit statement that no login is required, suggesting ease of access. The "REDACTED FOR ANONYMITY" block implies that some prior content, possibly sensitive or identifying information, has been intentionally obscured from the screenshot. The core task for the user is to provide "Your Explanation" in the designated text field, which is marked as a required question. The instructions on the right provide crucial context and constraints: 1. **Purpose:** The explanation should detail "what the program is doing wrong." 2. **Self-Contained:** The answer must be complete on its own. 3. **Format Restriction:** The answer should *not* reference external content or previous examples in a comparative manner (e.g., "Just like the first model explanation describes..."). This implies the explanation should be original and independent. The presence of "Sign in to Google to save your progress" suggests that while login is not required for submission, it is an option for users who wish to save their work, indicating a potentially longer or more complex explanation might be expected. ### Key Observations * The page is structured to guide the user through a submission process. * Anonymity is a stated concern, as indicated by the "REDACTED FOR ANONYMITY" block. * The form is an embedded Google Form, offering a familiar interface. * The instructions are clear about the content and format expectations for the submitted explanation, particularly emphasizing self-containment and avoiding comparative phrasing. * The "Submit" and "Clear form" buttons provide standard form interaction. ### Interpretation This document section appears to be part of an online learning platform, assessment, or technical review process where users are required to analyze a program and articulate its flaws. The "5. Answer Form" title suggests it's the fifth step in a sequence. The "REDACTED FOR ANONYMITY" block, combined with the option to sign in to Google to save progress (but not requiring it for submission), points to a system designed to collect explanations while potentially maintaining the anonymity of the submitter during evaluation, or at least ensuring the submission itself is anonymous. The redaction might cover details about the specific program being analyzed or the context of the task. The instructions are critical for ensuring high-quality, independent responses. By stating that the answer "must be self-contained" and "should **not** be of the form 'Just like the first model explanation describes...'", the system is likely trying to prevent plagiarism, encourage original thought, and ensure each explanation can be understood without external context. This suggests that the explanations will be evaluated individually, possibly by different reviewers or against a rubric that doesn't assume knowledge of other submissions or model answers. The emphasis on "what the program is doing wrong" indicates a focus on debugging, error identification, or critical analysis skills. </details> <details> <summary>extracted/2306.09896v1/figures/appendix/instructions/9.jpg Details</summary>  ### Visual Description ## Text Document: Study Tips ### Overview The image displays a text document titled "Study Tips," presented on a plain white background. It provides a set of guidelines and instructions for individuals participating in a study or task. The document begins with an introductory thank-you message, followed by a main bulleted list of five distinct tips, and concludes with a final bullet point regarding asking questions. Key phrases within the tips are highlighted using bold and blue text, and smiling emojis are used to convey a friendly tone. ### Components/Axes This document does not contain axes, legends, or charts. Its components are purely textual and structural: * **Header (Top-Left)**: The main title "Study Tips" is prominently displayed in a large, black font. * **Introductory Text (Below Header)**: A sentence "We are very grateful for your help! 😊" is presented in a smaller, italicized black font, accompanied by a yellow smiling emoji. * **Main Bulleted List (Central)**: Five primary bullet points, each marked with a black circular bullet, outline the study tips. * Within these tips, specific phrases are emphasized using bold, blue text. * One of the main tips includes a nested un-ordered list, comprising three sub-points marked with smaller circular bullets. * **Concluding Instruction (Bottom-Left)**: A final bullet point, distinguished by a blue circular bullet, provides an instruction about asking questions. This point also features bold blue text and a yellow smiling emoji, with a rectangular black redaction box obscuring some text. ### Detailed Analysis The document contains the following transcribed text: 1. **Title**: * "Study Tips" 2. **Introductory Message**: * "We are very grateful for your help! 😊" 3. **Main Study Tips (Bulleted List)**: * **Tip 1**: "• **Make sure you understand the task first!** The programs have subtle logic errors, not just simple compiler errors." * The phrase "Make sure you understand the task first!" is bold and blue. * **Tip 2**: "• Try to write **clear and concise** explanations, with proper grammar and punctuation." * The phrase "clear and concise" is bold and blue. * **Tip 3**: "• Feel free to **use (or not use) the model explanations** when writing your answers; but make sure your answer is self-contained!" * The phrase "use (or not use) the model explanations" is bold and blue. * **Tip 4**: "• The tasks vary in difficulty. Feel free to **allocate your time as you see fit**; we are not measuring how quickly you complete the tasks or anything like that!" * The phrase "allocate your time as you see fit" is bold and blue. * **Tip 5**: "• Feel free to use **external tools**:" * The phrase "external tools" is bold and blue. * **Nested Sub-points (Un-ordered List)**: * "◦ Use pen and paper or a whiteboard to help you reason about the task at hand." * "◦ Use a Python IDE to execute and debug the code." * "◦ Search online for help." 4. **Concluding Instruction (Blue Bullet Point)**: * "• **Have a question? Ask** [REDACTED TEXT] **before moving on with the study!** 😊" * The phrase "Have a question? Ask" is bold and blue. * A solid black rectangular box obscures approximately two to three words between "Ask" and "before". * The phrase "before moving on with the study!" is bold and blue. ### Key Observations * The document's primary purpose is to provide guidance and set expectations for participants. * Emphasis is placed on understanding the task, clear communication, flexibility in using resources, and time management. * The mention of "logic errors," "compiler errors," and "Python IDE" strongly suggests the study involves programming or software development tasks. * The tone is encouraging and supportive, using positive language and emojis. * A specific instruction to ask questions is provided, but the method or contact person for questions is intentionally redacted. ### Interpretation This "Study Tips" document is designed to prepare participants for a technical assessment or study, likely in a programming or software engineering context. The tips aim to foster a productive and less stressful environment by clarifying expectations and offering flexibility. The instruction to "Make sure you understand the task first!" and the warning about "subtle logic errors" indicate that the tasks are not trivial and require deep comprehension rather than superficial problem-solving. This suggests the study might be evaluating analytical and debugging skills. The emphasis on "clear and concise explanations" implies that the reasoning process and communication of solutions are as important as the solutions themselves. The freedom to "use (or not use) the model explanations" and to "allocate your time as you see fit" highlights a focus on individual problem-solving approaches and self-regulation, rather than strict adherence to a prescribed method or speed. This suggests the study values qualitative aspects of problem-solving over mere quantitative output. The explicit permission to use "external tools" such as pen and paper, a Python IDE, and online search resources indicates that the study simulates a real-world development scenario where resources are readily available. This reinforces that the assessment is about practical problem-solving and resourcefulness, not rote memorization. The final instruction, "Have a question? Ask [REDACTED TEXT] before moving on with the study! 😊", underscores the importance of clarity and support. The redaction of the contact information suggests this document might be a generic template or a version where specific details are omitted for privacy or broader applicability. Overall, the document aims to empower participants, reduce anxiety, and ensure they approach the tasks with the right mindset and resources. </details> <details> <summary>extracted/2306.09896v1/figures/appendix/instructions/10.jpg Details</summary>  ### Visual Description ## Text Block: Frequently Asked Questions (FAQ) ### Overview The image displays a Frequently Asked Questions (FAQ) section, presented as a list of three questions with their corresponding answers. The content addresses user concerns regarding data collection, the purpose of a study, and the evaluation of model explanations. ### Components/Axes The image is structured into two main components: 1. **Header**: Located at the top-left, the title "FAQ" is prominently displayed in a larger font size. 2. **Main Content**: Positioned below the header, this section contains a bulleted list of three questions. Each question is a primary bullet point (using a solid black circle), and its answer is a secondary bullet point (using a hollow circle), indented beneath the question. ### Detailed Analysis The main content consists of three question-and-answer pairs: * **Question 1**: "Are you collecting data as I visit the website?" * **Answer 1**: "No - none at all. Only your final answers are recorded." * The word "No" is highlighted in blue text. * **Question 2**: "What is the point of the study?" * **Answer 2**: "To investigate how much better the models are at fixing code when given human feedback, instead of having to debug the code themselves." * The phrase "fixing code" is highlighted in blue text. * The phrase "human feedback" is highlighted in blue text. * **Question 3**: "Are you evaluating how useful the model explanations were to me?" * **Answer 3**: "No - they are just there to help you get started with the debugging. We only care about your final answer." * The word "No" is highlighted in blue text. ### Key Observations * The document uses a clear question-and-answer format, making information easily digestible. * Key affirmative or negative responses (e.g., "No") and important concepts (e.g., "fixing code", "human feedback") are highlighted in blue, drawing the reader's attention to critical information. * The hierarchical bullet point structure effectively links questions to their answers. * The language is direct and addresses potential user concerns transparently. ### Interpretation This FAQ section serves to inform users about the operational policies and objectives of a study or platform. 1. **Data Privacy**: The first question and answer explicitly state that no browsing data is collected, only "final answers." This aims to reassure users about their privacy and encourage participation by minimizing concerns about personal data tracking. 2. **Study Purpose**: The second Q&A clarifies that the study's core objective is to evaluate the effectiveness of models in "fixing code" when augmented with "human feedback," contrasting this with models debugging independently. This suggests the study is focused on improving AI-assisted code debugging through human-in-the-loop processes. The blue highlighting emphasizes the core mechanism being investigated. 3. **Role of Model Explanations**: The third Q&A indicates that any model explanations provided are purely for user assistance in "getting started with the debugging" and are not themselves subject to evaluation. This implies that the study's primary metric of success is the "final answer" (presumably the quality of the fixed code or the user's solution), rather than the perceived utility or quality of the model's explanatory output. This helps set user expectations regarding the purpose of any provided explanations. Overall, the FAQ communicates a user-centric approach to data privacy, a clear research objective focused on human-AI collaboration in code fixing, and a specific scope for what is being evaluated in the study. </details> Figure 10: ## Appendix C Human Experiment (Quantitative Analysis): Results Per Task In the table below, we give a complete breakdown of the quantitative results presented in Section 4.3. Note that each program is associated with four different pieces of feedback: two sampled from GPT-4, and two given by our human participants. Each cell is the number of repair candidates (out of 25) that passed all the unit tests. See Section 4.3 for details, as well as Appendix B for the instructions given to participants. | 2106 | interview | A | 7 | 10 | 10 | 0 | | --- | --- | --- | --- | --- | --- | --- | | B | 0 | 2 | 20 | 16 | | | | 2673 | interview | A | 4 | 7 | 17 | 24 | | B | 3 | 25 | 25 | 25 | | | | 2923 | interview | A | 0 | 0 | 0 | 0 | | B | 0 | 0 | 0 | 0 | | | | 3070 | competition | A | 0 | 0 | 0 | 0 | | B | 3 | 0 | 5 | 0 | | | | 3286 | competition | A | 2 | 6 | 10 | 25 | | B | 0 | 0 | 0 | 4 | | | | 3754 | competition | A | 0 | 0 | 0 | 0 | | B | 0 | 0 | 0 | 0 | | | | 4182 | introductory | A | 25 | 25 | 25 | 24 | | B | 25 | 0 | 25 | 25 | | | | 4195 | introductory | A | 25 | 3 | 24 | 23 | | B | 23 | 25 | 25 | 25 | | | | 4281 | introductory | A | 0 | 4 | 0 | 0 | | B | 0 | 0 | 0 | 0 | | | | 4333 | introductory | A | 25 | 0 | 25 | 0 | | B | 23 | 24 | 24 | 25 | | | | 4347 | introductory | A | 0 | 0 | 7 | 25 | | B | 0 | 0 | 25 | 25 | | | | 4426 | introductory | A | 25 | 25 | 25 | 25 | | B | 25 | 25 | 25 | 25 | | | | 4450 | introductory | A | 0 | 0 | 0 | 0 | | B | 24 | 0 | 22 | 24 | | | | 4507 | introductory | A | 0 | 0 | 0 | 0 | | B | 0 | 0 | 1 | 0 | | | | 4514 | introductory | A | 15 | 21 | 1 | 16 | | B | 0 | 0 | 25 | 0 | | | | 4704 | introductory | A | 0 | 25 | 0 | 25 | | B | 25 | 25 | 24 | 23 | | | | 4741 | introductory | A | 25 | 25 | 25 | 25 | | B | 25 | 25 | 25 | 25 | | | | 4855 | introductory | A | 0 | 1 | 17 | 25 | | B | 0 | 2 | 3 | 23 | | | | 4873 | introductory | A | 0 | 0 | 0 | 0 | | B | 0 | 0 | 0 | 18 | | | | 4952 | introductory | A | 0 | 0 | 2 | 25 | | B | 24 | 8 | 24 | 21 | | | ## Appendix D Human Experiment (Qualitative Analysis): Examples In this appendix, we give examples of each category used to classify the responses in the qualitative analysis of Section 4.3. For each category, we give one example from the human participants and one from GPT-4 (when possible). Each example contains: the textual specification of the task; the incorrect program; the error message (in the format it was shown to participants); the feedback in question; and a short comment motivating its classification. ### D.1 Category (a): obviously incorrect feedback #### D.1.1 GPT-4: Example ⬇ ### SPECIFICATION The recent schoolboard elections were hotly contested: a proposal to swap school start times for elementary and high school students, a controversial new dress code proposal that bans athletic clothes in school, and a proposal to raise real - estate taxes to pay for a new football practice facility, and the list goes on and on. It is now hours after the polls have closed and a winner has yet to emerge! In their desperation, the election officials turn to you and ask you to write a program to count the vote! ----- Input ----- The input consists of a single test case, which is a list of votes cast. Each line in the input contains the name of a candidate for whom a vote was cast. A name may consist of multiple words, separated by spaces. Words contain letters or hyphens, but no other punctuation characters. There will be at least votes on the list. The list of votes ends with a single line containing the characters ***. This line should not be counted. There can be up to valid votes. ----- Output ----- If a candidate obtained a simple or absolute majority of all votes cast (that is, more than any other candidate), output the name of this candidate! If no candidate obtained a simple majority, output: " Runoff!" (don ’ t forget to include the exclamation mark!) ----- Examples ----- Sample Input: Penny Franklin Marti Graham Connie Froggatt Joseph Ivers Connie Froggatt Penny Franklin Connie Froggatt Bruce Stanger Connie Froggatt Barbara Skinner Barbara Skinner *** Sample Output: Connie Froggatt ### INCORRECT PROGRAM from collections import defaultdict votes = defaultdict (int) candidate = input () while candidate != "***": votes [candidate] += 1 candidate = input () total_votes = sum (votes. values ()) max_votes = max (votes. values ()) winner = [name for name, count in votes. items () if count == max_votes] if len (winner) == 1 and max_votes > total_votes // 2: print (winner [0]) else: print (" Runoff!") ### ERROR MESSAGE === Input === Penny Franklin Marti Graham Connie Froggatt Joseph Ivers Connie Froggatt Penny Franklin Connie Froggatt Bruce Stanger Connie Froggatt Barbara Skinner Barbara Skinner *** === Program Output === Runoff! === Expected Output === Connie Froggatt ### FEEDBACK The concise explanation of the issue is: The code checks if the maximum votes are strictly greater than half of the total votes to determine if there ’ s a majority. However, it should check if the maximum votes are greater than or equal to half of the total votes. ### COMMENT The issue with the code is not that the ">" should have been a ">=", but rather that it is not allowing for a simple majority. #### D.1.2 Human: Example ⬇ ### SPECIFICATION Snuke and Raccoon have a heap of N cards. The i - th card from the top has the integer a_i written on it. They will share these cards. First, Snuke will take some number of cards from the top of the heap, then Raccoon will take all the remaining cards. Here, both Snuke and Raccoon have to take at least one card. Let the sum of the integers on Snuke ’ s cards and Raccoon ’ s cards be x and y, respectively. They would like to minimize | x - y |. Find the minimum possible value of | x - y |. ----- Constraints ----- - 2 \ leq N \ leq 2 \ times 10^5 - -10^{9} \ leq a_i \ leq 10^{9} - a_i is an integer. ----- Input ----- Input is given from Standard Input in the following format: N a_1 a_2 ... a_ {N} ----- Output ----- Print the answer. ----- Sample Input ----- 6 1 2 3 4 5 6 ----- Sample Output ----- 1 If Snuke takes four cards from the top, and Raccoon takes the remaining two cards, x =10, y =11, and thus | x - y |=1. This is the minimum possible value. ### INCORRECT PROGRAM def main (): n = int (input ()) a = list (map (int, input (). split ())) Snuke_sum = 0 Raccoon_sum = sum (a) min_diff = abs (Raccoon_sum) for i in range (n -1): Snuke_sum += a [i] Raccoon_sum -= a [i] diff = abs (Snuke_sum - Raccoon_sum) min_diff = min (min_diff, diff) print (min_diff) main () ### ERROR MESSAGE === Input === 2 10 -10 === Program Output === 0 === Expected Output === 20 ### FEEDBACK The code initializes the result ‘ min_diff ‘ to ‘ abs (Raccoon_sum)‘ corresponding to the case where Raccoon takes all the cards, however this is incorrect because of the constraint " Snuke and Raccoon have to take at least one card ". Instead just setting ‘ min_diff ‘ to negative infinity initially would fix it. ### COMMENT This human participant was on the right track, but note that setting ‘ min_diff ‘ to * negative * infinity would cause it to never be updated inside of the for loop, and so the program would in fact always return negative infinity. ### D.2 Category (b): feedback explicitly suggests a minor change to the code #### D.2.1 GPT-4: Example ⬇ ### SPECIFICATION There is a square in the xy - plane. The coordinates of its four vertices are (x_1, y_1),(x_2, y_2),(x_3, y_3) and (x_4, y_4) in counter - clockwise order. (Assume that the positive x - axis points right, and the positive y - axis points up.) Takahashi remembers (x_1, y_1) and (x_2, y_2), but he has forgot (x_3, y_3) and (x_4, y_4). Given x_1, x_2, y_1, y_2, restore x_3, y_3, x_4, y_4. It can be shown that x_3, y_3, x_4 and y_4 uniquely exist and have integer values. ----- Constraints ----- - | x_1 |,| y_1 |,| x_2 |,| y_2 | \ leq 100 - (x_1, y_1) != (x_2, y_2) - All values in input are integers. ----- Input ----- Input is given from Standard Input in the following format: x_1 y_1 x_2 y_2 ----- Output ----- Print x_3, y_3, x_4 and y_4 as integers, in this order. ----- Sample Input ----- 0 0 0 1 ----- Sample Output ----- -1 1 -1 0 (0,0),(0,1),(-1,1),(-1,0) is the four vertices of a square in counter - clockwise order. Note that (x_3, y_3)=(1,1),(x_4, y_4)=(1,0) is not accepted, as the vertices are in clockwise order. ### INCORRECT PROGRAM x1, y1, x2, y2 = map (int, input (). split ()) x3 = x2 + (y2 - y1) y3 = y2 - (x2 - x1) x4 = x1 + (y2 - y1) y4 = y1 - (x2 - x1) print (x3, y3, x4, y4) ### ERROR MESSAGE === Input === 0 0 0 1 === Program Output === 1 1 1 0 === Expected Output === -1 1 -1 0 ### FEEDBACK The issue with the code is that it doesn ’ t consider the counter - clockwise order of the vertices. To fix this, we need to swap the signs of (y2 - y1) and (x2 - x1) in the calculations. ### COMMENT Swapping the listed signs is a concrete suggestion of a way to fix the code. Since the overall structure of the code would remain the same, we count this as a minor change. #### D.2.2 Human: Example ⬇ ### SPECIFICATION Given is a string S representing the day of the week today. S is SUN, MON, TUE, WED, THU, FRI, or SAT, for Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, and Saturday, respectively. After how many days is the next Sunday (tomorrow or later)? ----- Constraints ----- - S is SUN, MON, TUE, WED, THU, FRI, or SAT. ----- Input ----- Input is given from Standard Input in the following format: S ----- Output ----- Print the number of days before the next Sunday. ----- Sample Input ----- SAT ----- Sample Output ----- 1 It is Saturday today, and tomorrow will be Sunday. ### INCORRECT PROGRAM S = input () days = [’ SUN ’, ’ MON ’, ’ TUE ’, ’ WED ’, ’ THU ’, ’ FRI ’, ’ SAT ’] print ((7 - days. index (S)) % 7) ### ERROR MESSAGE === Input === SUN === Program Output === 0 === Expected Output === 7 ### FEEDBACK The logic is incorrect. There should be no modulo operator. ### COMMENT Suggesting that the modulo operator should be removed is a relatively small, but explicit, change. ### D.3 Category (c): feedback explicitly suggests a large change to the code #### D.3.1 GPT-4: Example ⬇ ### SPECIFICATION Takahashi is about to assemble a character figure, consisting of N parts called Part 1, Part 2, ..., Part N and N -1 connecting components. Parts are distinguishable, but connecting components are not. Part i has d_i holes, called Hole 1, Hole 2, ..., Hole d_i, into which a connecting component can be inserted. These holes in the parts are distinguishable. Each connecting component will be inserted into two holes in different parts, connecting these two parts. It is impossible to insert multiple connecting components into a hole. The character figure is said to be complete when it has the following properties: - All of the N -1 components are used to connect parts. - Consider a graph with N vertices corresponding to the parts and N -1 undirected edges corresponding to the pairs of vertices connected by a connecting component. Then, this graph is connected. Two ways A and B to make the figure complete are considered the same when the following is satisfied: for every pair of holes, A uses a connecting component to connect these holes if and only if B uses one to connect them. Find the number of ways to make the figure complete. Since the answer can be enormous, find the count modulo 998244353. ----- Constraints ----- - All values in input are integers. - 2 \ leq N \ leq 2 \ times 10^5 - 1 \ leq d_i < 998244353 ----- Input ----- Input is given from Standard Input in the following format: N d_1 d_2 \ cdots d_N ----- Output ----- Print the answer. ----- Sample Input ----- 3 1 1 3 ----- Sample Output ----- 6 One way to make the figure complete is to connect Hole 1 in Part 1 and Hole 3 in Part 3 and then connect Hole 1 in Part 2 and Hole 1 in Part 3. ### INCORRECT PROGRAM def main (): N = int (input ()) d = list (map (int, input (). split ())) MOD = 998244353 fact = [1] * (N +1) for i in range (1, N +1): fact [i] = fact [i -1] * i % MOD ans = 0 for ki in range (N): ans += d [ki] * fact [ki] % MOD * fact [N - ki -1] % MOD ans %= MOD ans = ans * fact [N -1] % MOD print (ans) main () ### ERROR MESSAGE === Input === 3 1 1 3 === Program Output === 18 === Expected Output === 6 ### FEEDBACK The following is a concise explanation of the issue: The code above calculates the total number of ways to connect the parts, but it does not ensure that each connecting component is used only once. This results in overcounting the number of ways to make the figure complete. To fix the issue, we need to ensure that each connecting component is used exactly once. ### COMMENT The feedback suggests changing the code so that each connecting component is used exactly once, which does not seem to be taken into account in the current approach. Changing the code to satisfy this part of the specification seems to require a major rewrite. #### D.3.2 Human: Example ⬇ ### SPECIFICATION Oh, New Year. The time to gather all your friends and reflect on the heartwarming events of the past year... $n$ friends live in a city which can be represented as a number line. The $i$ - th friend lives in a house with an integer coordinate $x_i$. The $i$ - th friend can come celebrate the New Year to the house with coordinate $x_i -1 $, $x_i +1 $ or stay at $x_i$. Each friend is allowed to move no more than once. For all friends $1 \ le x_i \ le n$ holds, however, they can come to houses with coordinates $0$ and $n +1 $ (if their houses are at $1$ or $n$, respectively). For example, let the initial positions be $x = [1, 2, 4, 4] $. The final ones then can be $ [1, 3, 3, 4] $, $ [0, 2, 3, 3] $, $ [2, 2, 5, 5] $, $ [2, 1, 3, 5] $ and so on. The number of occupied houses is the number of distinct positions among the final ones. So all friends choose the moves they want to perform. After that the number of occupied houses is calculated. What is the minimum and the maximum number of occupied houses can there be? ----- Input ----- The first line contains a single integer $n$ ($1 \ le n \ le 2 \ cdot 10^5 $) - the number of friends. The second line contains $n$ integers $x_1, x_2, \ dots, x_n$ ($1 \ le x_i \ le n$) - the coordinates of the houses of the friends. ----- Output ----- Print two integers - the minimum and the maximum possible number of occupied houses after all moves are performed. ----- Examples ----- Input 4 1 2 4 4 Output 2 4 Input 9 1 1 8 8 8 4 4 4 4 Output 3 8 Input 7 4 3 7 1 4 3 3 Output 3 6 ----- Note ----- In the first example friends can go to $ [2, 2, 3, 3] $. So friend $1$ goes to $x_1 +1 $, friend $2$ stays at his house $x_2$, friend $3$ goes to $x_3 -1 $ and friend $4$ goes to $x_4 -1 $. $ [1, 1, 3, 3] $, $ [2, 2, 3, 3] $ or $ [2, 2, 4, 4] $ are also all valid options to obtain $2$ occupied houses. For the maximum number of occupied houses friends can go to $ [1, 2, 3, 4] $ or to $ [0, 2, 4, 5] $, for example. ### INCORRECT PROGRAM from collections import Counter def main (): n = int (input ()) x = list (map (int, input (). split ())) cnt = Counter (x) max_houses = 0 for pos in range (1, n + 1): if cnt [pos]: max_houses += 1 occupied = [0] * (n + 2) for pos in x: if not occupied [pos - 1]: occupied [pos - 1] = 1 elif not occupied [pos]: occupied [pos] = 1 elif not occupied [pos + 1]: occupied [pos + 1] = 1 min_houses = sum (occupied) print (min_houses, max_houses) main () ### ERROR MESSAGE === Input === 4 1 2 4 4 === Program Output === 4 3 === Expected Output === 2 4 ### FEEDBACK The code calculates both the minimum and maximum count of houses incorrectly. The correct solution requires conducting a tree search to determine the true maximum and minimum. This can be further optimized by first sorting the list of occupied houses and using a sliding window to process independent segments of the number line. ### COMMENT The feedback suggests conducting a tree search to determine the max and min, as well as a way to optimize this search. These constitute major changes to the program as it is currently written. ### D.4 Category (d): Feedback contains blocks of pseudocode or Python code #### D.4.1 GPT-4: Example GPT-4’s feedback never contains blocks of Python per our experiment design, in which the code block is stripped (Section 4.3). We find that GPT-4 never utilizes blocks of pseudocode, either. #### D.4.2 Human: Example ⬇ ### SPECIFICATION Given is a string S representing the day of the week today. S is SUN, MON, TUE, WED, THU, FRI, or SAT, for Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, and Saturday, respectively. After how many days is the next Sunday (tomorrow or later)? ----- Constraints ----- - S is SUN, MON, TUE, WED, THU, FRI, or SAT. ----- Input ----- Input is given from Standard Input in the following format: S ----- Output ----- Print the number of days before the next Sunday. ----- Sample Input ----- SAT ----- Sample Output ----- 1 It is Saturday today, and tomorrow will be Sunday. ### INCORRECT PROGRAM day = input () days = [’ SUN ’, ’ MON ’, ’ TUE ’, ’ WED ’, ’ THU ’, ’ FRI ’, ’ SAT ’] index = days. index (day) result = (7 - index) % 7 print (result) ### ERROR MESSAGE === Input === SUN === Program Output === 0 === Expected Output === 7 ### FEEDBACK before returning the result, we should just check to see if result ==0. so we just need to add: if result == 0: result = 7 ### COMMENT The feedback contains a block of Python code that the model can simply copy in order to fix the program. ### D.5 Category (e): feedback which expresses uncertainty #### D.5.1 GPT-4: Example In the $20\cdot 2\cdot 2=80$ examples we study, we do not find that GPT-4 ever expresses uncertainty. #### D.5.2 Human: Example ⬇ ### SPECIFICATION Two kingdoms had been at war for a long time, until the emperor intervened to bring an end to the conflict. The territory in question comprises an $M$ by $N$ rectangular grid. At the emperor ’ s insistence, the two kings have withdrawn their troops until no two opposing troops are in adjacent squares of the map (adjacent being horizontal or vertical - diagonal is not considered). The emperor proposes to designate certain squares of the map as neutral territory. Neither king will be allowed to move troops into those squares, and the emperor ’ s own forces will patrol them to be sure that both kings observe these rules. The emperor is frugal and does not want to commit more soldiers to this effort than absolutely necessary. His generals have marked each square of the map with the number of soldiers required to secure that square. What remains is to choose which of those squares should be patrolled. Write a program to determine the minimum number of soldiers that the emperor will need to be deploy to guarantee that the troops of one kingdom cannot move, in one or more steps, into squares occupied by the troops of the second kingdom (moving horizontally or vertically) without encountering the emperor ’ s own soldiers. ----- Input ----- Input begins with a line containing $2$ integers, $w$ and $h$, denoting the width and height of the map, where $1 \ leq w, h \ leq 40 $. This is followed by $h$ lines. Each line contains $w$ characters, left justified. These characters will be ’ A ’ or ’ B ’, designating a position held by king A or king B, or a single numeric digit, designating a currently unoccupied position that can be secured by the use of that number of soldiers. For example, a ’2’ would indicate that two soldiers must be deployed to that square to secure it against passage of other troops. A ’0’ indicates terrain that is impassible - the emperor need not commit soldiers there because the kingdom troops cannot pass through that square. No ’ A ’ will be adjacent, horizontally or vertically, to any ’ B ’. There will be at least one ’ A ’ and one ’ B ’ in the input. ----- Output ----- Print a single line containing an integer denoting the minimum number of soldiers that the emperor must deploy to guarantee that there is no open path between any ’ A ’ position and any ’ B ’ position, using any combination of horizontal or vertical moves. ----- Examples ----- Sample Input: 8 5 A11111AA AA7B111A 111 BB111 11 BBB111 11 BBB11B Sample Output: 13 ### INCORRECT PROGRAM def main (): w, h = map (int, input (). split ()) grid = [input () for _ in range (h)] def bfs (x, y, z): dist = [[float (" inf ")] * w for _ in range (h)] dist [x][y] = 0 q = [(x, y)] for x, y in q: for dx, dy in [(-1, 0), (1, 0), (0, -1), (0, 1)]: nx, ny = x + dx, y + dy if 0 <= nx < h and 0 <= ny < w and dist [nx][ny] == float (" inf "): if grid [nx][ny] == z: dist [nx][ny] = dist [x][y] q. append ((nx, ny)) if "0" <= grid [nx][ny] <= "9": dist [nx][ny] = dist [x][y] + int (grid [nx][ny]) q. append ((nx, ny)) return dist ans = float (" inf ") for i in range (h): for j in range (w): if grid [i][j] == " A ": ans = min (ans, min (x for row in bfs (i, j, " B ") for x in row)) print (ans) main () ### ERROR MESSAGE === Input === 8 5 A11111AA AA7B111A 111 BB111 11 BBB111 11 BBB11B === Program Output === 0 === Expected Output === 13 ### FEEDBACK This code appears to be finding the minimum " distance " between an A position and a B position, where distance is measured in the number of soldiers required to secure the intermediate positions. This isn ’ t really addressing the question at all, because even occupying the positions along some path from a particular A -> B won ’ t necessarily block the kings from reaching each other from those positions through a different, longer path. You probably need some sort of min - cut algorithm to divide the graph into two connected components, one each for A and B nodes. ### COMMENT The feedback expresses mild uncertainty in two aspects. First, the user indicates that they are not completely certain in their understanding of the code ’ s current behaviour by using the phrase " appears to be ". Then, they express uncertainty in their suggestion for what to do instead, saying that one * probably * needs some sort of min - cut algorithm. ## Appendix E Prompts In this appendix, we describe the prompting structure used for our experiments. All of our experiments use one-shot prompting, in which a single example is given in the prompt before the desired task. For initial code generation (the first sample from $M_{P}$ ), we use different prompts for the two types of tasks in APPS: call-based tasks, in which the desired program should take the input as a parameter to a function and return the output in the function’s return statement; and stdio-based tasks, in which inputs should be read from stdin and outputs should be written to stdout. These prompts are shown in Listing 1 and 2, respectively. The example tasks and programs were taken from APPS’ training set. For feedback samples (i.e., samples from $M_{F}$ ), we use the prompt in Listing 3. This prompt contains an example in which the user provides the textual specification, the incorrect program and the error message, and the assistant generates feedback. Similarly, for repair samples (i.e., samples from $M_{P}$ which follow $M_{F}$ ) we use the prompt in Listing 4, in which the user also supplies the feedback, and the assistant returns only the fixed version of the program. Finally, for joint feedback-repair samples (i.e., when sampling $(f,r)\sim M_{P}$ ), we use the prompt in Listing 6. This prompt combines the prompts from Listing 4 and 5 into one prompt, in which the assistant returns both the feedback and the fixed program. In all of these prompts, the specification used was taken from APPS’ training set, while the programs and the feedback were constructed manually. Listing 1: Code generation prompt for call-based tasks. ⬇ ===== system ===== You are an expert Python programmer. You will be given a question (problem specification) and will generate a correct Python program that matches the specification and passes all tests. You will NOT return anything except for the program. Put your fixed program within code delimiters, for example: ‘‘‘ python # YOUR CODE HERE ‘‘‘. ===== user ===== ### QUESTION A palindrome is a word, phrase, number, or other sequence of characters which reads the same backward as forward. Examples of numerical palindromes are: ‘2332, 110011, 54322345‘ For a given number ‘‘‘ num ‘‘‘, write a function which returns an array of all the numerical palindromes contained within each number. The array should be sorted in ascending order and any duplicates should be removed. In this kata, single digit numbers and numbers which start or end with zeros (such as ‘010‘ and ‘00‘) are ** NOT ** considered valid numerical palindromes. If ‘ num ‘ contains no valid palindromes, return ‘" No palindromes found "‘. Otherwise, return ‘" Not valid "‘ if the input is not an integer or is less than ‘0‘. ## Examples ‘‘‘ palindrome (1221) => [22, 1221] palindrome (34322122) => [22, 212, 343, 22122] palindrome (1001331) => [33, 1001, 1331] palindrome (1294) => " No palindromes found " palindrome ("1221") => " Not valid " ‘‘‘ --- ### Other Kata in this Series: Numerical Palindrome #1 Numerical Palindrome #1.5 Numerical Palindrome #2 Numerical Palindrome #3 Numerical Palindrome #3.5 Numerical Palindrome #4 Numerical Palindrome #5 ### Use Call - Based Format ===== assistant ===== ‘‘‘ python def is_pal (s): return s == s [::-1] def palindrome (num): if not isinstance (num, int) or num < 0: return ’ Not valid ’ s = str (num) pals = set () for i, ch in enumerate (s): if ch == ’0’: continue for j in range (i + 2, len (s) + 1): test = s [i: j] if is_pal (test): pals. add (test) return sorted (int (x) for x in pals) or ’ No palindromes found ’ ‘‘‘ Listing 2: Code generation prompt for stdio-based tasks. ⬇ ===== system ===== You are an expert Python programmer. You will be given a question (problem specification) and will generate a correct Python program that matches the specification and passes all tests. You will NOT return anything except for the program. Put your fixed program within code delimiters, for example: ‘‘‘ python # YOUR CODE HERE ‘‘‘. ===== user ===== ### QUESTION Polycarp has $n$ different binary words. A word called binary if it contains only characters ’0’ and ’1’. For example, these words are binary: "0001", "11", "0" and "0011100". Polycarp wants to offer his set of $n$ binary words to play a game " words ". In this game, players name words and each next word (starting from the second) must start with the last character of the previous word. The first word can be any. For example, these sequence of words can be named during the game: "0101", "1", "10", "00", "00001". Word reversal is the operation of reversing the order of the characters. For example, the word "0111" after the reversal becomes "1110", the word "11010" after the reversal becomes "01011". Probably, Polycarp has such a set of words that there is no way to put them in the order correspondent to the game rules. In this situation, he wants to reverse some words from his set so that: the final set of $n$ words still contains different words (i. e. all words are unique); there is a way to put all words of the final set of words in the order so that the final sequence of $n$ words is consistent with the game rules. Polycarp wants to reverse minimal number of words. Please, help him. ----- Input ----- The first line of the input contains one integer $t$ ($1 \ le t \ le 10\^4 $) - the number of test cases in the input. Then $t$ test cases follow. The first line of a test case contains one integer $n$ ($1 \ le n \ le 2\ cdot10 ^5 $) - the number of words in the Polycarp ’ s set. Next $n$ lines contain these words. All of $n$ words aren ’ t empty and contains only characters ’0’ and ’1’. The sum of word lengths doesn ’ t exceed $4 \ cdot10 ^6 $. All words are different. Guaranteed, that the sum of $n$ for all test cases in the input doesn ’ t exceed $2 \ cdot10 ^5 $. Also, guaranteed that the sum of word lengths for all test cases in the input doesn ’ t exceed $4 \ cdot10 ^6 $. ----- Output ----- Print answer for all of $t$ test cases in the order they appear. If there is no answer for the test case, print -1. Otherwise, the first line of the output should contain $k$ ($0 \ le k \ le n$) - the minimal number of words in the set which should be reversed. The second line of the output should contain $k$ distinct integers - the indexes of the words in the set which should be reversed. Words are numerated from $1$ to $n$ in the order they appear. If $k =0 $ you can skip this line (or you can print an empty line). If there are many answers you can print any of them. ----- Example ----- Input 4 4 0001 1000 0011 0111 3 010 101 0 2 00000 00001 4 01 001 0001 00001 Output 1 3 -1 0 2 1 2 ### Use Standard Input Format (read inputs with ‘ input ()‘, write results with ‘ print ()‘) ===== assistant ===== ‘‘‘ python for _ in range (int (input ())): n = int (input ()) mass = [] zo = 0 oz = 0 zz = 0 oo = 0 ozs = [] zos = [] ozss = set () zoss = set () for j in range (n): k = input () mass. append (k) if k [0] == ’0’ and k [-1] == ’1’: zoss. add (k) zos. append (j + 1) zo += 1 elif k [0] == ’1’ and k [-1] == ’0’: ozss. add (k) ozs. append (j + 1) oz += 1 elif k [0] == ’0’ and k [-1] == ’0’: zz += 1 else: oo += 1 if zz and oo and not oz and not zo: print (-1) continue else: if zo > oz: print ((zo - oz) // 2) ans = [] need = (zo - oz) // 2 i = 0 while need: zzz = mass [zos [i] - 1][len (mass [zos [i] - 1]) - 1:: -1] if zzz not in ozss: ans. append (zos [i]) need -= 1 i += 1 print (* ans) else: print ((oz - zo) // 2) ans = [] need = (oz - zo) // 2 i = 0 while need: zzz = mass [ozs [i] - 1][len (mass [ozs [i] - 1]) - 1:: -1] if zzz not in zoss: ans. append (ozs [i]) need -= 1 i += 1 print (* ans) ‘‘‘ Listing 3: Feedback-only prompt. ⬇ ===== system ===== You are a helpful programming assistant and an expert Python programmer. You are helping a user debug a program. The user has written some code, but it has some errors and is not passing the tests. You will help the user by giving a concise (at most 2-3 sentences) textual explanation of what is wrong with the code. You will * not * generate any code, because the user wants to fix the code themselves. ===== user ===== ### QUESTION A palindrome is a word, phrase, number, or other sequence of characters which reads the same backward as forward. Examples of numerical palindromes are: ‘2332, 110011, 54322345‘ For a given number ‘‘‘ num ‘‘‘, write a function which returns an array of all the numerical palindromes contained within each number. The array should be sorted in ascending order and any duplicates should be removed. In this kata, single digit numbers and numbers which start or end with zeros (such as ‘010‘ and ‘00‘) are ** NOT ** considered valid numerical palindromes. If ‘ num ‘ contains no valid palindromes, return ‘" No palindromes found "‘. Otherwise, return ‘" Not valid "‘ if the input is not an integer or is less than ‘0‘. ## Examples ‘‘‘ palindrome (1221) => [22, 1221] palindrome (34322122) => [22, 212, 343, 22122] palindrome (1001331) => [33, 1001, 1331] palindrome (1294) => " No palindromes found " palindrome ("1221") => " Not valid " ‘‘‘ --- ### Other Kata in this Series: Numerical Palindrome #1 Numerical Palindrome #1.5 Numerical Palindrome #2 Numerical Palindrome #3 Numerical Palindrome #3.5 Numerical Palindrome #4 Numerical Palindrome #5 ### Use Call - Based Format ### INCORRECT PYTHON CODE ‘‘‘ python def is_pal (s): return s == s [::-1] def palindrome (num): if not isinstance (num, int) or num < 0: return ’ Not valid ’ s = str (num) pals = set () for i, ch in enumerate (s): for j in range (i + 2, len (s) + 1): test = s [i: j] if is_pal (test): pals. add (test) return sorted (int (x) for x in pals) or ’ No palindromes found ’ ‘‘‘ The code above is wrong and contains a bug. Given input "1001331" the output was "[’0’, ’33’, ’1001’, ’1331’]" but the expected output was "[’33’, ’1001’, ’1331’]". ===== assistant ===== The following is a concise explanation of the issue: The problem description states that numbers which start or end with zeros (such as ‘010‘ and ‘00‘) are NOT considered valid numerical palindromes. However, the code above does not take this into account and therefore returns ‘00‘ as a valid palindrome. This can be fixed by checking if the first or last character is ‘0‘ before adding the string to the set of palindromes. Listing 4: Repair-only prompt. ⬇ ===== system ===== You are a helpful programming assistant and an expert Python programmer. You are helping a user write a program to solve a problem. The user has written some code, but it has some errors and is not passing the tests. The user has spent some time debugging the program and will provide you with a concise textual explanation of what is wrong with the code. You will use this explanation to generate a fixed version of the program. Put your fixed program within code delimiters, for example: ‘‘‘ python # YOUR CODE HERE ‘‘‘. ===== user ===== ### QUESTION A palindrome is a word, phrase, number, or other sequence of characters which reads the same backward as forward. Examples of numerical palindromes are: ‘2332, 110011, 54322345‘ For a given number ‘‘‘ num ‘‘‘, write a function which returns an array of all the numerical palindromes contained within each number. The array should be sorted in ascending order and any duplicates should be removed. In this kata, single digit numbers and numbers which start or end with zeros (such as ‘010‘ and ‘00‘) are ** NOT ** considered valid numerical palindromes. If ‘ num ‘ contains no valid palindromes, return ‘" No palindromes found "‘. Otherwise, return ‘" Not valid "‘ if the input is not an integer or is less than ‘0‘. ## Examples ‘‘‘ palindrome (1221) => [22, 1221] palindrome (34322122) => [22, 212, 343, 22122] palindrome (1001331) => [33, 1001, 1331] palindrome (1294) => " No palindromes found " palindrome ("1221") => " Not valid " ‘‘‘ --- ### Other Kata in this Series: Numerical Palindrome #1 Numerical Palindrome #1.5 Numerical Palindrome #2 Numerical Palindrome #3 Numerical Palindrome #3.5 Numerical Palindrome #4 Numerical Palindrome #5 ### Use Call - Based Format ### INCORRECT PYTHON CODE ‘‘‘ python def is_pal (s): return s == s [::-1] def palindrome (num): if not isinstance (num, int) or num < 0: return ’ Not valid ’ s = str (num) pals = set () for i, ch in enumerate (s): for j in range (i + 2, len (s) + 1): test = s [i: j] if is_pal (test): pals. add (test) return sorted (int (x) for x in pals) or ’ No palindromes found ’ ‘‘‘ The code above is wrong and contains a bug. Given input "1001331" the output was "[’0’, ’33’, ’1001’, ’1331’]" but the expected output was "[’33’, ’1001’, ’1331’]". The following is a concise explanation of the issue: The problem description states that numbers which start or end with zeros (such as ‘010‘ and ‘00‘) are NOT considered valid numerical palindromes. However, the code above does not take this into account and therefore returns ‘00‘ as a valid palindrome. This can be fixed by checking if the first or last character is ‘0‘ before adding the string to the set of palindromes. ===== assistant ===== ### FIXED PYTHON CODE ‘‘‘ python def is_pal (s): return s == s [::-1] def palindrome (num): if not isinstance (num, int) or num < 0: return ’ Not valid ’ s = str (num) pals = set () for i, ch in enumerate (s): if ch == ’0’: continue for j in range (i + 2, len (s) + 1): test = s [i: j] if is_pal (test): pals. add (test) return sorted (int (x) for x in pals) or ’ No palindromes found ’ ‘‘‘ Listing 5: Joint feedback-repair prompt. ⬇ ===== system ===== You are a helpful programming assistant and an expert Python programmer. You are helping a user write a program to solve a problem. The user has written some code, but it has some errors and is not passing the tests. You will help the user by first giving a concise (at most 2-3 sentences) textual explanation of what is wrong with the code. After you have pointed out what is wrong with the code, you will then generate a fixed version of the program. Put your fixed program within code delimiters, for example: ‘‘‘ python # YOUR CODE HERE ‘‘‘. ===== user ===== ### QUESTION A palindrome is a word, phrase, number, or other sequence of characters which reads the same backward as forward. Examples of numerical palindromes are: ‘2332, 110011, 54322345‘ For a given number ‘‘‘ num ‘‘‘, write a function which returns an array of all the numerical palindromes contained within each number. The array should be sorted in ascending order and any duplicates should be removed. In this kata, single digit numbers and numbers which start or end with zeros (such as ‘010‘ and ‘00‘) are ** NOT ** considered valid numerical palindromes. If ‘ num ‘ contains no valid palindromes, return ‘" No palindromes found "‘. Otherwise, return ‘" Not valid "‘ if the input is not an integer or is less than ‘0‘. ## Examples ‘‘‘ palindrome (1221) => [22, 1221] palindrome (34322122) => [22, 212, 343, 22122] palindrome (1001331) => [33, 1001, 1331] palindrome (1294) => " No palindromes found " palindrome ("1221") => " Not valid " ‘‘‘ --- ### Other Kata in this Series: Numerical Palindrome #1 Numerical Palindrome #1.5 Numerical Palindrome #2 Numerical Palindrome #3 Numerical Palindrome #3.5 Numerical Palindrome #4 Numerical Palindrome #5 ### Use Call - Based Format ### INCORRECT PYTHON CODE ‘‘‘ python def is_pal (s): return s == s [::-1] def palindrome (num): if not isinstance (num, int) or num < 0: return ’ Not valid ’ s = str (num) pals = set () for i, ch in enumerate (s): for j in range (i + 2, len (s) + 1): test = s [i: j] if is_pal (test): pals. add (test) return sorted (int (x) for x in pals) or ’ No palindromes found ’ ‘‘‘ The code above is wrong and contains a bug. Given input "1001331" the output was "[’0’, ’33’, ’1001’, ’1331’]" but the expected output was "[’33’, ’1001’, ’1331’]". ===== assistant ===== The following is a concise explanation of the issue: The problem description states that numbers which start or end with zeros (such as ‘010‘ and ‘00‘) are NOT considered valid numerical palindromes. However, the code above does not take this into account and therefore returns ‘00‘ as a valid palindrome. This can be fixed by checking if the first or last character is ‘0‘ before adding the string to the set of palindromes. # The code below is the correct version of the code above, where the issue has been fixed: ### FIXED PYTHON CODE ‘‘‘ python def is_pal (s): return s == s [::-1] def palindrome (num): if not isinstance (num, int) or num < 0: return ’ Not valid ’ s = str (num) pals = set () for i, ch in enumerate (s): if ch == ’0’: continue for j in range (i + 2, len (s) + 1): test = s [i: j] if is_pal (test): pals. add (test) return sorted (int (x) for x in pals) or ’ No palindromes found ’ ‘‘‘ ## Appendix F Dataset: APPS Tasks Used For Our Evaluations These tasks were randomly sampled from APPS’ test set. To avoid distribution shift, we sampled according to the relative frequency of difficulties in the full dataset. We report the resulting list of tasks to aid reproducibility. | introductory | ’4004’, ’4058’, ’4063’, ’4065’, ’4100’, ’4108’, ’4117’, ’4155’, ’4164’, ’4182’, ’4193’, ’4195’, ’4211’, ’4217’, ’4241’, ’4249’, ’4270’, ’4275’, ’4281’, ’4293’, ’4333’, ’4347’, ’4350’, ’4356’, ’4409’, ’4426’, ’4431’, ’4450’, ’4465’, ’4484’, ’4498’, ’4505’, ’4507’, ’4514’, ’4544’, ’4553’, ’4586’, ’4610’, ’4662’, ’4663’, ’4667’, ’4677’, ’4681’, ’4704’, ’4716’, ’4741’, ’4750’, ’4786’, ’4787’, ’4801’, ’4855’, ’4862’, ’4864’, ’4870’, ’4873’, ’4890’, ’4897’, ’4952’, ’4966’, ’4984’ | | --- | --- | | interview | ’0004’, ’0013’, ’0033’, ’0056’, ’0073’, ’0074’, ’0089’, ’0091’, ’0124’, ’0131’, ’0139’, ’0162’, ’0166’, ’0183’, ’0186’, ’0191’, ’0199’, ’0205’, ’0249’, ’0253’, ’0268’, ’0274’, ’0300’, ’0304’, ’0341’, ’0342’, ’0413’, ’0427’, ’0434’, ’0466’, ’0467’, ’0496’, ’0501’, ’0511’, ’0537’, ’0564’, ’0571’, ’0575’, ’0579’, ’0592’, ’0597’, ’0626’, ’0637’, ’0676’, ’0704’, ’0728’, ’0757’, ’0765’, ’0788’, ’0794’, ’0804’, ’0805’, ’0811’, ’0829’, ’0879’, ’0904’, ’0915’, ’0925’, ’0937’, ’0948’, ’0954’, ’0955’, ’0972’, ’0985’, ’0989’, ’1018’, ’1019’, ’1033’, ’1046’, ’1076’, ’1133’, ’1140’, ’1141’, ’1145’, ’1146’, ’1149’, ’1168’, ’1185’, ’1221’, ’1232’, ’1256’, ’1257’, ’1280’, ’1285’, ’1299’, ’1317’, ’1347’, ’1380’, ’1392’, ’1393’, ’1418’, ’1444’, ’1448’, ’1458’, ’1489’, ’1517’, ’1533’, ’1573’, ’1635’, ’1653’, ’1668’, ’1672’, ’1721’, ’1736’, ’1748’, ’1756’, ’1759’, ’1775’, ’1777’, ’1825’, ’1850’, ’1863’, ’1865’, ’1870’, ’1875’, ’1906’, ’1917’, ’1956’, ’1962’, ’1967’, ’1976’, ’2024’, ’2049’, ’2062’, ’2092’, ’2093’, ’2097’, ’2106’, ’2172’, ’2176’, ’2203’, ’2231’, ’2246’, ’2264’, ’2266’, ’2295’, ’2326’, ’2328’, ’2332’, ’2342’, ’2361’, ’2369’, ’2407’, ’2408’, ’2418’, ’2455’, ’2463’, ’2511’, ’2515’, ’2516’, ’2535’, ’2585’, ’2623’, ’2629’, ’2642’, ’2651’, ’2662’, ’2668’, ’2673’, ’2698’, ’2701’, ’2709’, ’2735’, ’2742’, ’2752’, ’2759’, ’2765’, ’2787’, ’2802’, ’2832’, ’2835’, ’2844’, ’2858’, ’2885’, ’2897’, ’2923’, ’2932’, ’2945’, ’2973’, ’2980’ | | competition | ’3017’, ’3019’, ’3054’, ’3062’, ’3063’, ’3066’, ’3070’, ’3077’, ’3083’, ’3097’, ’3117’, ’3135’, ’3161’, ’3186’, ’3209’, ’3220’, ’3286’, ’3287’, ’3323’, ’3335’, ’3353’, ’3355’, ’3371’, ’3375’, ’3376’, ’3388’, ’3404’, ’3411’, ’3433’, ’3441’, ’3445’, ’3470’, ’3481’, ’3484’, ’3548’, ’3557’, ’3605’, ’3609’, ’3634’, ’3635’, ’3671’, ’3679’, ’3709’, ’3754’, ’3769’, ’3792’, ’3798’, ’3799’, ’3804’, ’3810’, ’3819’, ’3823’, ’3836’, ’3843’, ’3849’, ’3876’, ’3913’, ’3934’, ’3972’, ’3974’ | ## Appendix G More Examples of Feedback from GPT-4 and our Human Participants In this appendix, we give five more examples from the study in Section 4.3, so that the keen reader may themself compare and contrast the feedback generated by GPT-4 and by our human participants. Each listing (6-10) contains a textual specification, an incorrect program, an error message, both examples of GPT-4 feedback which were shown to the participants assigned to this task, and the responses from the participants. For each piece of feedback, we also list the number of resulting repaired programs which passed all the tests (c.f. the full results table in Appendix C). Listing 6: Task 2106, Program A. ⬇ ### SPECIFICATION There are n cities in the country where the Old Peykan lives. These cities are located on a straight line, we ’ ll denote them from left to right as c_1, c_2, ..., c_ {n}. The Old Peykan wants to travel from city c_1 to c_ {n} using roads. There are (n - 1) one way roads, the i - th road goes from city c_ {i} to city c_ {i} + 1 and is d_ {i} kilometers long. The Old Peykan travels 1 kilometer in 1 hour and consumes 1 liter of fuel during this time. Each city c_ {i} (except for the last city c_ {n}) has a supply of s_ {i} liters of fuel which immediately transfers to the Old Peykan if it passes the city or stays in it. This supply refreshes instantly k hours after it transfers. The Old Peykan can stay in a city for a while and fill its fuel tank many times. Initially (at time zero) the Old Peykan is at city c_1 and s_1 liters of fuel is transferred to it ’ s empty tank from c_1 ’ s supply. The Old Peykan ’ s fuel tank capacity is unlimited. Old Peykan can not continue its travel if its tank is emptied strictly between two cities. Find the minimum time the Old Peykan needs to reach city c_ {n}. ----- Input ----- The first line of the input contains two space - separated integers m and k (1 <= m, k <= 1000). The value m specifies the number of roads between cities which is equal to n - 1. The next line contains m space - separated integers d_1, d_2, ..., d_ {m} (1 <= d_ {i} <= 1000) and the following line contains m space - separated integers s_1, s_2, ..., s_ {m} (1 <= s_ {i} <= 1000). ----- Output ----- In the only line of the output print a single integer - the minimum time required for The Old Peykan to reach city c_ {n} from city c_1. ----- Examples ----- Input 4 6 1 2 5 2 2 3 3 4 Output 10 Input 2 3 5 6 5 5 Output 14 ### INCORRECT PROGRAM def main (): m, k = map (int, input (). split ()) ds = list (map (int, input (). split ())) ss = list (map (int, input (). split ())) time = 0 fuel = ss [0] min_time = 0 max_s = ss [0] for i in range (m): fuel -= ds [i] max_s = max (max_s, ss [i]) while fuel < 0: time += k fuel += max_s time += ds [i] print (time) main () ### ERROR MESSAGE === Input === 4 6 1 2 5 2 2 3 3 4 === Program Output === 28 === Expected Output === 10 ### GPT -4 FEEDBACK #1 The issue with the code is that it does not consider the time spent in the current city before moving to the next city, resulting in an overestimation of the minimum time required. This can be fixed by updating the time spent in the current city before calculating the fuel used and accounting for the time it takes to travel to the next city. # Number of passing repairs: 7/25 ### GPT -4 FEEDBACK #2 The following is a concise explanation of the issue: The code incorrectly calculates the waiting time at each city. It should minimize the waiting time by considering the maximum fuel supply that can be collected during the waiting time. The code needs to be updated to account for this. # Number of passing repairs: 10/25 ### PARTICIPANT FEEDBACK #1 The program doesn ’ t add the fuel for passing through cities, only if it needs to actually stay in the city to refuel. Whenever it passes through a city, it should add ss [i] to the fuel. # Number of passing repairs: 10/25 ### PARTICIPANT FEEDBACK #2 The code is incorrect because it doesn ’ t take into account the fuel at each city that is immediately transferred after entering the city (aside for the fuel received at the first city). As a result, this program overestimates the time needed, because it assumes that the Old Peykan always has to wait for a refill (besides the fuel they are given in the first city). # Number of passing repairs: 0/25 Listing 7: Task 3070, Program B. ⬇ ### SPECIFICATION In the free - market, ruthlessly capitalist world of train fares, only one thing matters: incentives. Train companies are incentivised with bonuses for high throughput, successful journeys, and customer satisfaction. Conversely, the companies are disincentivised from failure via mandatory refunds for customers delayed by $30$ minutes or more. Being a ruthless capitalist yourself, you have decided to take advantage of this generous delay compensation provision. The refund is awarded provided that no matter the combination of trains you had taken (provided they followed the same route of stations as planned), you would still be unable to reach your destination in strictly less time than $30$ minutes (or $1800$ seconds), of the time you would have arrived assuming your booked journey was exactly on time. Armed with your printout of the day ’ s delays, and the original timetable, you must ask yourself only one question: what is the earliest time you can book a train for from station $1$, in order to earn this restitutive reward? ----- Input ----- - One line containing two integers: $N$ ($1 \ le N \ le 100 $), the number of stations, and $M$ ($1 \ le M \ le 10^5 $), the number of scheduled trains. - The next $M$ lines each contain 4 integers: - $X$, the starting station ($1 \ le X \ le N -1 $), - $S$ and $T$ ($0 \ le S \ le T < 86400 $), the planned departure and arrival times in seconds, - and $L$ ($0 \ le L < 86400 $), the duration by which the train ’ s departure and arrival times are delayed. Stations are numbered from $1$ to $N$ in the order you will visit them. Each train goes between stations $X$ and $X +1 $. It is possible to change between trains instantanesouly. ----- Output ----- - One line containing one integer: the start time of the earliest train journey you could book in order to earn your compensation, or impossible if no such journey is possible. ----- Examples ----- Sample Input 1: 2 3 1 1800 9000 1800 1 2000 9200 1600 1 2200 9400 1400 Sample Output 1: 1800 Sample Input 2: 2 2 1 1800 3600 1800 1 1900 3600 1600 Sample Output 2: impossible ### INCORRECT PROGRAM def main (): N, M = map (int, input (). split ()) train_data = [tuple (map (int, input (). split ())) for _ in range (M)] train_data. sort (key = lambda x: (x [0], x [1])) times = [1 e9] * (N + 1) times [1] = 0 for X, S, T, L in train_data: if times [X] <= S: times [X + 1] = min (times [X + 1], T + L) if times [N] >= 1800: print (int (times [1])) else: print (" impossible ") main () ### ERROR MESSAGE === Input === 2 3 1 1800 9000 1800 1 2000 9200 1600 1 2200 9400 1400 === Program Output === 0 === Expected Output === 1800 ### GPT -4 FEEDBACK #1 The issue with the code is that it prints ‘ times [1]‘ instead of the earliest possible train start time. The code needs to get the minimum start time from the train_data list that has a valid delay for compensation. # Number of passing repairs: 3/25 ### GPT -4 FEEDBACK #2 The issue with the code is that it directly prints the value of ‘ times [1]‘, which is initialized to ‘0‘ and never updated. The correct output should be the starting time of the earliest train journey that satisfies the condition. To fix this, we need to track the earliest possible start time and update it if a valid train journey is found. # Number of passing repairs: 0/25 ### PARTICIPANT FEEDBACK #1 This code is clearly wrong as times [1] is initialized to 0 then never updated and it is returned as the answer. Also, in the sorting line ‘ int (x [0])‘ should be used in place of ‘ x [0]‘ and likewise for ‘ int (x [1])‘ in place of ‘ x1 ‘ because we want numerical order sorting not string order sorting. The goal in this problem is calculate for each station what the earliest - arriving undelayed path to the station is and what the earliest - arriving delayed path to the station is, take the difference between the two, and see if it results in a total delay of >= 30 minutes (then to return the earliest departure from station 1 for this path. Given this, first the current code based around ‘ times ‘ can remain - this is the delayed time. However, the if - statement within the for - loop should be ‘<= S + L ‘ since the departure time is also delayed. Then ‘ undelayed_times ‘ should be tracked as well, updated in the foor loop when ‘ undelayed_times [X] <= S ‘ and set to a min () between the current undelayed time an ‘ T ‘ (not T + L). Finally the if - statement for time printing can use the difference between the delayed and undelayed times and see if any () are >= 1800. Finally we need to print the earliest arriving train for that path # Number of passing repairs: 5/25 ### PARTICIPANT FEEDBACK #2 One of the main issues is that the code prints ’ times [1]’, which is set to 0 and never updated. However, there are also some larger issues with the code. First, instead of checking if you reach your destination within 30 minutes of the expected time, it checks if you reach your destination in 30 minutes. To fix this, the program would need to keep track of the time you ’ d reach your destination if there were not delays, which it currently doesn ’ t. Second, instead of printing times [1], it should be keeping track of the start time of the train that leads to a delay and printing this. # Number of passing repairs: 0/25 Listing 8: Task 4333, Program A. ⬇ ### SPECIFICATION There is a square in the xy - plane. The coordinates of its four vertices are (x_1, y_1),(x_2, y_2),(x_3, y_3) and (x_4, y_4) in counter - clockwise order. (Assume that the positive x - axis points right, and the positive y - axis points up.) Takahashi remembers (x_1, y_1) and (x_2, y_2), but he has forgot (x_3, y_3) and (x_4, y_4). Given x_1, x_2, y_1, y_2, restore x_3, y_3, x_4, y_4. It can be shown that x_3, y_3, x_4 and y_4 uniquely exist and have integer values. ----- Constraints ----- - | x_1 |,| y_1 |,| x_2 |,| y_2 | \ leq 100 - (x_1, y_1) != (x_2, y_2) - All values in input are integers. ----- Input ----- Input is given from Standard Input in the following format: x_1 y_1 x_2 y_2 ----- Output ----- Print x_3, y_3, x_4 and y_4 as integers, in this order. ----- Sample Input ----- 0 0 0 1 ----- Sample Output ----- -1 1 -1 0 (0,0),(0,1),(-1,1),(-1,0) is the four vertices of a square in counter - clockwise order. Note that (x_3, y_3)=(1,1),(x_4, y_4)=(1,0) is not accepted, as the vertices are in clockwise order. ### INCORRECT PROGRAM x1, y1, x2, y2 = map (int, input (). split ()) x3 = x2 + (y2 - y1) y3 = y2 - (x2 - x1) x4 = x1 + (y2 - y1) y4 = y1 - (x2 - x1) print (x3, y3, x4, y4) ### ERROR MESSAGE === Input === 0 0 0 1 === Program Output === 1 1 1 0 === Expected Output === -1 1 -1 0 ### GPT -4 FEEDBACK #1 The issue with the code is that it doesn ’ t consider the counter - clockwise order of the vertices. To fix this, we need to swap the signs of (y2 - y1) and (x2 - x1) in the calculations. # Number of passing repairs: 25/25 ### GPT -4 FEEDBACK #2 The following is a concise explanation of the issue: The code above does not ensure that the vertices are in counter - clockwise order. It calculates x3, y3, x4, and y4 in the wrong order, which results in the coordinates being in clockwise order instead. To fix this, we need to swap the calculations for x3 and x4, and y3 and y4. # Number of passing repairs: 0/25 ### PARTICIPANT FEEDBACK #1 The program calculates vertices for a clockwise - oriented square, not counter - clockwise. To fix this, (y2 - y1) and (x2 - x1) should be negated in the equations. # Number of passing repairs: 25/25 ### PARTICIPANT FEEDBACK #2 The calculation for y3 and y4 incorrectly subtract the x direction vector to y0 and y1 respectively. In each of these cases, the calculation can be fixed by changing the first subtraction to addition. # Number of passing repairs: 0/25 Listing 9: Task 4347, Program B. ⬇ ### SPECIFICATION One day, $n$ people ($n$ is an even number) met on a plaza and made two round dances, each round dance consists of exactly $ \ frac {n}{2} $ people. Your task is to find the number of ways $n$ people can make two round dances if each round dance consists of exactly $ \ frac {n}{2} $ people. Each person should belong to exactly one of these two round dances. Round dance is a dance circle consisting of $1$ or more people. Two round dances are indistinguishable (equal) if one can be transformed to another by choosing the first participant. For example, round dances $ [1, 3, 4, 2] $, $ [4, 2, 1, 3] $ and $ [2, 1, 3, 4] $ are indistinguishable. For example, if $n =2 $ then the number of ways is $1$: one round dance consists of the first person and the second one of the second person. For example, if $n =4 $ then the number of ways is $3$. Possible options: one round dance - $ [1,2] $, another - $ [3,4] $; one round dance - $ [2,4] $, another - $ [3,1] $; one round dance - $ [4,1] $, another - $ [3,2] $. Your task is to find the number of ways $n$ people can make two round dances if each round dance consists of exactly $ \ frac {n}{2} $ people. ----- Input ----- The input contains one integer $n$ ($2 \ le n \ le 20 $), $n$ is an even number. ----- Output ----- Print one integer - the number of ways to make two round dances. It is guaranteed that the answer fits in the $64$ - bit integer data type. ----- Examples ----- Input 2 Output 1 Input 4 Output 3 Input 8 Output 1260 Input 20 Output 12164510040883200 ### INCORRECT PROGRAM def factorial (n): if n == 0: return 1 else: return n * factorial (n -1) n = int (input ()) ans = (factorial (n) // (n // 2)**2) // (2 * n) print (ans) ### ERROR MESSAGE === Input === 2 === Program Output === 0 === Expected Output === 1 ### GPT -4 FEEDBACK #1 The issue with the code is that it uses integer division when calculating the final answer, which may cause the result to be rounded down incorrectly. To fix this, we can use float division and then convert the result to an integer. # Number of passing repairs: 0/25 ### GPT -4 FEEDBACK #2 The following is a concise explanation of the issue: The code attempts to calculate the number of ways using integer division, which may result in a loss of precision and incorrect results. To fix the issue, change integer division to regular division, then round the result to the nearest integer using ‘ round ()‘. # Number of passing repairs: 0/25 ### PARTICIPANT FEEDBACK #1 The problem is that the formula used is incorrect. For n <= 4, n! is less than n ^3 / 2 and so this is why we see a 0. There is an erroneous additional division by n which is the true reason for the wrong answer. # Number of passing repairs: 25/25 ### PARTICIPANT FEEDBACK #2 On line 11, the program calculates a term with a correct numerator, but an incorrect denominator. It divides by (2* n), when it should just divide by 2. # Number of passing repairs: 25/25 Listing 10: Task 4855, Program A. ⬇ ### SPECIFICATION The impact factor of a scientific journal is a measure reflecting the average number of citations to articles published in science journals. For this task we are using a simplified formula for calculating the impact factor:\[ \ frac {\ text {Total count of citations received by articles published in the journal}}{\ text {Total number of articles published}} \] Rounding is always performed upwards. For example the impact factor of the " Journal for ore research and time wasting " that published $38$ articles cited $894$ times is $894 / 38 = 23.53 $ rounding up to $24$. You are the editor of one scientific journal. You know how many articles you are going to publish and the owners are pushing you to reach a specific impact factor. You are wondering how many scientists you will have to bribe to cite your articles to meet the owners demands. Since money is tight you want to bribe the minimal amount of scientists. Each bribed scientist buys you a single citation. ----- Input ----- First and only line of input will contain $2$ integers, $A$ ($1 \ leq A \ leq 100 $), the number of articles you plan to publish and $I$ ($1 \ leq I \ leq 100 $), the impact factor the owners require. ----- Output ----- The first and only line of output should contain one integer, the minimal number of scientists you need to bribe. ----- Examples ----- Sample Input 1: 38 24 Sample Output 1: 875 Sample Input 2: 1 100 Sample Output 2: 100 ### INCORRECT PROGRAM import math A, I = map (int, input (). split ()) needed_citations = math. ceil (A * (I - 0.5)) scientists_to_bribe = needed_citations - A * (I - 1) print (scientists_to_bribe) ### ERROR MESSAGE === Input === 38 24 === Program Output === 19 === Expected Output === 875 ### GPT -4 FEEDBACK #1 The following is a concise explanation of the issue: The calculation for the number of scientists to bribe is incorrect. It should be calculated by first finding the minimum number of citations needed: ‘ A * I ‘, and then subtracting the current number of citations: ‘ A * (I - 1)‘. # Number of passing repairs: 0/25 ### GPT -4 FEEDBACK #2 The following is a concise explanation of the issue: The code calculates the required number of scientists to bribe by subtracting the current number of citations from the number of needed citations. However, it incorrectly calculates the current number of citations. # Number of passing repairs: 1/25 ### PARTICIPANT FEEDBACK #1 The scientists_to_bribe variable on line 3 is unnecessary., It ’ s sufficient to just do A * (I - 1) + 1. # Number of passing repairs: 17/25 ### PARTICIPANT FEEDBACK #2 The program seems to have a conceptual misunderstanding. Specifically, the number of scientists needed to bribe is equal to the number of required citations (‘ citations_needed ‘). However, the calculation of ‘ citation_needed ‘ is also incorrect -- it implies rounding impact factor conventionally (e. g. 3.49 goes to 3, 3.51 goes to 4), whereas rounding here is a ceiling function on the impact factor. As such, the actual formula for ‘ citation_needed ‘ should be ‘ A *(I -1) + 1‘ # Number of passing repairs: 25/25