# Self-Verifying Reflection Helps Transformers with CoT Reasoning
Abstract
Advanced large language models (LLMs) frequently reflect in reasoning chain-of-thoughts (CoTs), where they self-verify the correctness of current solutions and explore alternatives. However, given recent findings that LLMs detect limited errors in CoTs, how reflection contributes to empirical improvements remains unclear. To analyze this issue, in this paper, we present a minimalistic reasoning framework to support basic self-verifying reflection for small transformers without natural language, which ensures analytic clarity and reduces the cost of comprehensive experiments. Theoretically, we prove that self-verifying reflection guarantees improvements if verification errors are properly bounded. Experimentally, we show that tiny transformers, with only a few million parameters, benefit from self-verification in both training and reflective execution, reaching remarkable LLM-level performance in integer multiplication and Sudoku. Similar to LLM results, we find that reinforcement learning (RL) improves in-distribution performance and incentivizes frequent reflection for tiny transformers, yet RL mainly optimizes shallow statistical patterns without faithfully reducing verification errors. In conclusion, integrating generative transformers with discriminative verification inherently facilitates CoT reasoning, regardless of scaling and natural language.
1 Introduction
Numerous studies have explored the ability of large language models (LLMs) to reason through a chain of thought (CoT), an intermediate sequence leading to the final answer. While simple prompts can elicit CoT reasoning [13], subsequent works have further enhanced CoT quality through reflective thinking [10] and the use of verifiers [4]. Recently, reinforcement learning (RL) [33] has achieved notable success in advanced reasoning models, such as OpenAI-o1 [20] and Deepseek-R1 [5], which show frequent reflective behaviors that self-verify the correctness of current solutions and explore alternatives, integrating generative processes with discriminative inference. However, researchers also report that the ability of these LLMs to detect errors is rather limited, and a large portion of reflection fails to bring correct solutions [11]. Given the weak verification ability, the experimental benefits of reflection and the emergence of high reflection frequency in RL require further explanation.
To address this challenge, we seek to analyze two main questions in this paper: 1) what role self-verifying reflection plays in training and execution of reasoning models, and 2) how reflective reasoning evolves in RL with verifiable outcome rewards [15]. However, the complexity of natural language and the prohibitive training cost of LLMs make it difficult to draw clear conclusions from theoretical abstraction and comprehensive experiments across settings. Inspired by Zeyuan et al. [2], we observe that task-specific reasoning and self-verifying reflection do not necessitate complex language. This allows us to investigate reflective reasoning through tiny transformer models [36], which provide efficient tools to understand self-verifying reflection through massive experiments.
To enable tiny transformers to produce long reflective CoTs and ensure analytic simplicity, we introduce a minimalistic reasoning framework, which supports essential reasoning behaviors that are operable without natural language. In our study, the model self-verifies the correctness of each thought step; then, it may resample incorrect steps or trace back to previous steps. Based on this framework, we theoretically prove that self-verifying reflection improves reasoning accuracy if verification errors are properly bounded, which does not necessitate a strong verifier. Additionally, a trace-back mechanism that allows revisiting previous solutions conditionally improves performance if the problem requires a sufficiently large number of steps.
Our experiments evaluate 1M, 4M, and 16M transformers in solving integer multiplication [7] and Sudoku puzzles [3], which have simple definitions (thus, operable by transformers without language) yet still challenging for even LLM solvers. To maintain relevance to broader LLM research, the tiny transformers are trained from scratch through a pipeline similar to that of training LLM reasoners. Our main findings are listed as follows: 1) Learning to self-verify greatly facilitates the learning of forward reasoning. 2) Reflection improves reasoning accuracy if true correct steps are not excessively verified as incorrect. 3) Resembling the results of DeepSeek-R1 [5], RL can incentivize reflection if the reasoner can effectively explore potential solutions. 4) However, RL fine-tuning increases performance mainly statistically, with limited improvements in generalizable problem-solving skills.
Overall, this paper contributes to the fundamental understanding of reflection in reasoning models by clarifying its effectiveness and synergy with RL. Our findings based on minimal reasoners imply a general benefit of reflection for more advanced models, which operate on a super-set of our simplified reasoning behaviors. In addition, our implementation also provides insights into the development of computationally efficient reasoning models.
2 Related works
CoT reasoning
Pretrained LLMs emerge the ability to produce CoTs from simple prompts [13, 38], which can be explained via the local dependencies [25] and probabilistic distribution [35] of natural-language reasoning. Many recent studies develop models targeted at reasoning, e.g., scaling test-time inference with external verifiers [4, 17, 18, 32] and distilling large general models to smaller specialized models [34, 9]. In this paper, we train tiny transformers from scratch to not only generate CoTs but also self-verify, i.e., detect errors in their own thoughts without external models.
RL fine-tuning for CoT reasoning
RL [33] recently emerges as a key method for CoT reasoning [31, 40]. It optimizes the transformer model by favoring CoTs that yield high cumulated rewards, where PPO [29] and its variant GRPO [31] are two representative approaches. Central to RL fine-tuning are reward models that guide policy optimization: the 1) outcome reward models (ORM) assessing final answers, and the 2) process reward models (PRM) [17] evaluating intermediate reasoning steps. Recent advances in RL with verifiable rewards (RLVR) [5, 41] demonstrate that simple ORM based solely on answer correctness can induce sophisticated reasoning behaviors.
Reflection in LLM reasoning
LLM reflection provides feedback to the generated solutions [19] and may accordingly refine the solutions [10]. Research shows that supervised learning from verbal reflection improves performance, even though the reflective feedback is omitted during execution [42]. Compared to the generative verbal reflection, self-verification uses discriminative labels to indicate the correctness of reasoning steps, which supports reflective execution and is operable without linguistic knowledge. Recently, RL is widely used to develop strong reflective abilities [14, 27, 20]. In particular, DeepSeek-R1 [5] shows that RLVR elicits frequent reflection, and such a result is reproduced in smaller LLMs [24]. In this paper, we further investigate how reflection evolves during RLVR by examining the change of verification errors.
Understanding LLMs through small transformers
Small transformers are helpful tools to understand LLMs, for their architectural consistency with LLMs and low development cost to support massive experiments. For example, transformers smaller than 1B provide insights into how data mixture and data diversity influence LLM training [39, 2]. They also contribute to foundational understanding of CoT reasoning, such as length generalization [12], internalization of thoughts [6], and how CoTs inherently extend the problem-solving ability [8, 16]. In this paper, we further use tiny transformers to better understand reflection in CoT reasoning.
3 Reflective reasoning for transformers
In this section, we develop transformers to perform simple reflective reasoning in long CoTs. Focusing on analytic clarity and broader implications, the design of our framework follows the minimalistic principle, providing only essential reasoning behavior operable without linguistic knowledge. More advanced reasoning frameworks optimized for small-scale models are certainly our next move in future work. In the following, we first introduce the basic formulation of CoT reasoning; then, based on this formulation, we introduce our simple reasoning framework for self-verifying reflection; afterwards, we describe how transformers are trained to reason through this framework.
3.1 Reasoning formulation
<details>
<summary>x1.png Details</summary>
### Visual Description
## Diagram: Model Reasoning Steps and State Transition
### Overview
The image is a diagram illustrating a model's reasoning steps and the corresponding state transitions. It depicts a sequential process, starting with an initial query (Q) and progressing through multiple reasoning steps (R1, R2, ... RT-1) to arrive at a final answer (A). Each reasoning step involves a transition from one state to the next, influenced by the previous state and the current reasoning output.
### Components/Axes
* **Model:** This is the overarching context of the diagram.
* **Reasoning Steps:** This row describes the individual steps in the reasoning process.
* **Q:** Initial query (gray rounded rectangle).
* **R1, R2, ..., RT-1:** Reasoning steps (white rounded rectangles).
* **A:** Final answer (gray rounded rectangle).
* **Rβ ~ Ο(Β· | Sβ):** The reasoning step R1 is sampled from a policy Ο conditioned on the initial state S0.
* **Rβ ~ Ο(Β· | Sβ):** The reasoning step R2 is sampled from a policy Ο conditioned on the state S1.
* **A ~ Ο(Β· | ST-1):** The final answer A is sampled from a policy Ο conditioned on the state ST-1.
* **State Transition:** This row describes how the state changes after each reasoning step.
* **S0 = Q:** The initial state S0 is equal to the query Q.
* **S1 = T(S0, R1):** The state S1 is a function T of the initial state S0 and the reasoning step R1.
* **S2 = T(S1, R2):** The state S2 is a function T of the state S1 and the reasoning step R2.
* **...:** Indicates that the process continues for an unspecified number of steps.
* **ST-1 = T(ST-2, RT-1):** The state ST-1 is a function T of the state ST-2 and the reasoning step RT-1.
* **ST = T(ST-1, A):** The final state ST is a function T of the state ST-1 and the final answer A.
### Detailed Analysis or ### Content Details
The diagram shows a sequence of operations. The initial query `Q` leads to the first reasoning step `R1`. The state transitions from `S0` to `S1` based on the function `T(S0, R1)`. This process repeats until the final reasoning step `RT-1` and the final answer `A` are reached. The dashed line between `R2` and `RT-1` indicates that there can be multiple intermediate reasoning steps.
* **Initial State:** The process begins with the initial state `S0` being equal to the query `Q`.
* **Reasoning Steps:** Each reasoning step `Ri` is sampled from a policy `Ο` conditioned on the previous state `Si-1`.
* **State Transitions:** The state transitions from `Si-1` to `Si` based on the function `T(Si-1, Ri)`.
* **Final State:** The process ends with the final state `ST` being a function of the previous state `ST-1` and the final answer `A`.
### Key Observations
* The diagram illustrates a sequential reasoning process.
* Each reasoning step depends on the previous state.
* The state transitions are determined by a function `T` that takes the previous state and the current reasoning step as input.
* The final state depends on the previous state and the final answer.
### Interpretation
The diagram represents a model's reasoning process as a series of state transitions. The model starts with an initial query and iteratively refines its understanding by performing reasoning steps. Each reasoning step updates the model's internal state, leading to a final answer. The policy `Ο` governs the selection of reasoning steps based on the current state, while the function `T` determines how the state is updated after each step. This framework can be used to model various reasoning processes, such as question answering, problem-solving, and decision-making. The diagram highlights the importance of both the reasoning steps and the state transitions in achieving a final answer.
</details>
Figure 1: The illustration of MTP, where the transformer model $\pi$ reasons the answer $A$ of a query $Q$ through $T-1$ intermediate steps.
<details>
<summary>x2.png Details</summary>
### Visual Description
## Diagram: State Transition Diagram
### Overview
The image is a state transition diagram illustrating a process that transforms an initial state *S<sub>t</sub>* to a subsequent state *S<sub>t+1</sub>* via an intermediate reward *R<sub>t+1</sub>*. The diagram shows calculations performed at each stage.
### Components/Axes
* **Nodes:** Three rounded rectangles representing states and rewards.
* *S<sub>t</sub>*: Initial state.
* *R<sub>t+1</sub>*: Reward at time t+1.
* *S<sub>t+1</sub>*: Subsequent state.
* **Edges:** Arrows indicating transitions between states and rewards.
* Arrow from *S<sub>t</sub>* to *R<sub>t+1</sub>* labeled with "Ο".
* Arrow from *R<sub>t+1</sub>* to *S<sub>t+1</sub>* labeled with "T".
* Arrow from *S<sub>t+1</sub>* back to *S<sub>t</sub>*, forming a loop.
### Detailed Analysis or Content Details
* **Initial State (S<sub>t</sub>):**
* Value: 145
* Operation: multiplied by 340
* Operation: plus 290
* **Transition (Ο):**
* Arrow from *S<sub>t</sub>* to *R<sub>t+1</sub>*.
* **Reward (R<sub>t+1</sub>):**
* 340 (blue) -> 300 (red)
* 145 (blue) x 4 (cyan) = 580 (green)
* 290 (black) + 5800 (green) = 6090 (red)
* **Transition (T):**
* Arrow from *R<sub>t+1</sub>* to *S<sub>t+1</sub>*.
* **Subsequent State (S<sub>t+1</sub>):**
* Value: 145
* Operation: multiplied by 300
* Operation: plus 6090 (red)
* **Loop:**
* Arrow from *S<sub>t+1</sub>* back to *S<sub>t</sub>*.
### Key Observations
* The diagram illustrates a cyclical process.
* Calculations are performed at each state and reward stage.
* The reward *R<sub>t+1</sub>* involves multiple calculations, including a change from 340 to 300, multiplication, and addition.
* The final state *S<sub>t+1</sub>* uses the result from the reward calculation.
### Interpretation
The diagram represents a simplified model of a state transition process, possibly within a reinforcement learning context. The initial state *S<sub>t</sub>* is transformed based on a policy (Ο) to generate a reward *R<sub>t+1</sub>*. This reward, along with a transition function (T), determines the next state *S<sub>t+1</sub>*. The loop suggests that this process is iterative, with the subsequent state feeding back into the initial state for the next iteration. The calculations within each state and reward stage provide specific details on how the values are updated during the transition. The color coding in the reward stage highlights the transformation of values during the process.
</details>
(a) Multiplication
<details>
<summary>x3.png Details</summary>
### Visual Description
## Diagram: Sudoku State Transition
### Overview
The image depicts a state transition diagram for a Sudoku puzzle. It shows an initial state (St), an action/reward step (Rt+1), and the resulting state (St+1) after applying the action. The diagram illustrates how filling in specific cells in the Sudoku grid transforms the puzzle from one state to another.
### Components/Axes
* **St**: Represents the initial state of the Sudoku puzzle. It's a 9x9 grid with some cells filled with numbers.
* **Rt+1**: Represents the reward or action taken. It indicates which cells are filled and with what numbers.
* **St+1**: Represents the resulting state of the Sudoku puzzle after applying the action. It's a 9x9 grid with additional cells filled compared to St.
* **Ο**: Represents the policy or strategy used to select the action.
* **T**: Represents the transition function that maps the current state and action to the next state.
### Detailed Analysis
**Initial State (St):**
The Sudoku grid in the initial state (St) has the following values:
* Row 1: 1, _, 7, _, _, _, 8, _, _
* Row 2: _, _, _, 3, _, 2, _, _, _
* Row 3: _, 2, 3, _, _, 5, _, _, _
* Row 4: _, 9, _, _, _, _, _, _, _
* Row 5: _, 5, _, _, _, 4, 7, _, _
* Row 6: 2, _, 8, 6, _, _, _, _, 9
* Row 7: _, 3, 9, _, 4, 1, _, _, _
* Row 8: _, _, _, _, _, _, 6, _, _
* Row 9: 7, _, _, 8, _, _, _, _, 4
**Action/Reward (Rt+1):**
The action/reward step (Rt+1) indicates that two cells are filled:
* Cell (6,2) is filled with the number 7 (shown in red).
* Cell (7,8) is filled with the number 2 (shown in red).
**Resulting State (St+1):**
The Sudoku grid in the resulting state (St+1) has the following values:
* Row 1: 1, _, 7, _, _, _, 8, _, _
* Row 2: _, _, _, 3, _, 2, _, _, _
* Row 3: _, 2, 3, _, _, 5, _, _, _
* Row 4: _, 9, _, _, _, _, _, _, _
* Row 5: _, 5, _, _, _, 4, 7, _, _
* Row 6: 2, 7, 8, 6, _, _, _, _, 9
* Row 7: _, 3, 9, _, 4, 1, _, 2, _
* Row 8: _, _, _, _, _, _, 6, _, _
* Row 9: 7, _, _, 8, _, _, _, _, 4
### Key Observations
* The diagram shows a single step in solving a Sudoku puzzle.
* The action involves filling two specific cells with numbers.
* The numbers filled in the action/reward step are highlighted in red in the Rt+1 and St+1 states.
* The policy (Ο) and transition function (T) are abstract representations of the Sudoku solving strategy.
### Interpretation
The diagram illustrates a basic reinforcement learning approach to solving Sudoku puzzles. The initial state (St) represents the current puzzle configuration. The policy (Ο) guides the selection of an action, which in this case is filling specific cells. The reward (Rt+1) is associated with the action, and the transition function (T) updates the puzzle state to the next state (St+1). By repeatedly applying this process, the agent (Sudoku solver) aims to reach a solved state. The red numbers highlight the changes made during the transition, emphasizing the impact of the action on the puzzle.
</details>
(b) Sudoku
Figure 2: Example reasoning steps for multiplication and Sudoku, where the core planning is presented in the reasoning step ${R}_{t+1}$ .
CoT Reasoning as a Markov decision process
A general form of CoT reasoning is given as a tuple $({Q},\{{R}\},{A})$ , where ${Q}$ is the input query, $\{{R}\}=({R}_{1},...,{R}_{T-1})$ is the sequence of $T-1$ intermediate steps, and ${A}$ is the final answer. Following Wang [37], we formulate the CoT reasoning as a Markov thought process (MTP). As shown in Figure 1, an MTP follows that [37]:
$$
\displaystyle{R}_{t+1}\sim\pi(\cdot\mid{S}_{t}),\ {S}_{t+1}=\mathcal{T}({S}_{t},{R}_{t+1}), \tag{1}
$$
where ${S}_{t}$ is the $t$ -th reasoning state, $\pi$ is the planning policy (the transformer model), and $\mathcal{T}$ is the (usually deterministic) transition function. The initial state ${S}_{0}:=Q$ is given by the input query. In each reasoning step ${R}_{t+1}$ , the policy $\pi$ plans the next reasoning action that determines the state transition, which is then executed by $\mathcal{T}$ to obtain the next state. The process terminates when the step presents the answer, i.e., $A={R}_{T}$ . For clarity, a table of notations is presented in Appendix A.
An MTP is implemented by specifying the state representations and transition function $\mathcal{T}$ . Since we use tiny transformers that are weak in inferring long contexts, we suggest reducing the length of state representations, so that each state ${S}_{t}$ carries only necessary information for subsequent reasoning. Here, we present two examples to better illustrate how MTPs are designed for tiny transformers.
**Example 1 (An MTP for integer multiplication)**
*As shown in Figure 2(a), to reason the product of two integers $x,yβ₯ 0$ , each state is an expression ${S}_{t}:=[x_{t}Γ y_{t}+z_{t}]$ mathematically equal to $xΓ y$ , initialized as ${S}_{0}=[xΓ y+0]$ . On each step, $\pi$ plans $y_{t+1}$ by eliminate a non-zero digit in $y_{t}$ to $0$ , and it then computes $z_{t+1}=z_{t}+x_{t}(y_{t}-y_{t+1})$ . Consequently, $\mathcal{T}$ updates ${S}_{t+1}$ as $[x_{t+1}Γ y_{t+1}+z_{t+1}]$ with $x_{t+1}=x_{t}$ . Similarly, $\pi$ may also eliminate non-zero digits in $x_{t}$ in a symmetric manner. Finally, $\pi$ yields $A=z_{t}$ as the answer if either $x_{t}$ or $y_{t}$ becomes $0$ .*
**Example 2 (An MTP for Sudoku[3])**
*As shown in Figure 2(b), each Sudoku state is a $9Γ 9$ game board. On each step, the model $\pi$ fills some blank cells to produce a new board, which is exactly the next state. The answer $A$ is a board with no blank cells.*
3.2 The framework of self-verifying reflection
<details>
<summary>x4.png Details</summary>
### Visual Description
## Diagram: State Transition Diagram
### Overview
The image is a state transition diagram illustrating a process that starts with an initial state `Q` and transitions through a series of states based on rules `R`. The diagram shows both successful (green) and unsuccessful (red) transitions, indicated by checkmarks and crosses, respectively.
### Components/Axes
* **Nodes:** Represent states in the process.
* `Q`: Initial state, represented by a gray rounded rectangle. `S0 = Q`
* `R1`: State, represented by a red rounded rectangle with a cross. `S1 = S0`
* `R2`: State, represented by a green rounded rectangle with a checkmark. `S2 = T(S1, R2)`
* `R3`: State, represented by a green rounded rectangle with a checkmark. `S3 = T(S2, R3)`
* `R4`: State, represented by a red rounded rectangle with a cross. `S4 = S3`
* `R5`: State, represented by a red rounded rectangle with a cross. `S5 = S4`
* `R6`: State, represented by a green rounded rectangle with a checkmark. `S6 = T(S5, R6)`
* `A`: State, represented by a green rounded rectangle with a checkmark. `S7 = T(S6, A)`
* **Edges:** Represent transitions between states.
* Green edges indicate successful transitions.
* Red edges indicate unsuccessful transitions.
* **Labels:**
* `Q`: Initial state.
* `R1, R2, R3, R4, R5, R6`: Rules or conditions for transitions.
* `A`: Final state.
* `S0, S1, S2, S3, S4, S5, S6, S7`: State variables.
* `T(x, y)`: A function that determines the next state based on the current state and a rule.
### Detailed Analysis
* The process starts at state `Q` where `S0 = Q`.
* From `Q`, there are two possible transitions:
* A red arrow leads to `R1` where `S1 = S0`. This transition is marked with a cross, indicating failure.
* A green arrow leads to `R2` where `S2 = T(S1, R2)`. This transition is marked with a checkmark, indicating success.
* From `R2`, a green arrow leads to `R3` where `S3 = T(S2, R3)`. This transition is marked with a checkmark, indicating success.
* From `R3`, there are two possible transitions:
* A red arrow leads to `R4` where `S4 = S3`. This transition is marked with a cross, indicating failure.
* A red arrow leads to `R5` where `S5 = S4`. This transition is marked with a cross, indicating failure.
* From `R3`, a green arrow leads to `R6` where `S6 = T(S5, R6)`. This transition is marked with a checkmark, indicating success.
* From `R6`, a green arrow leads to `A` where `S7 = T(S6, A)`. This transition is marked with a checkmark, indicating success.
### Key Observations
* The diagram shows a process that can either succeed or fail at different stages.
* Successful transitions are marked with green arrows and checkmarks, while unsuccessful transitions are marked with red arrows and crosses.
* The state variables `S0` through `S7` represent the state of the process at each stage.
* The function `T(x, y)` determines the next state based on the current state and a rule.
### Interpretation
The state transition diagram illustrates a decision-making process where the system starts in state `Q` and attempts to reach state `A`. The process involves applying rules `R1` through `R6`. Some rules lead to successful transitions (green paths), while others lead to failures (red paths). The diagram highlights the possible paths the system can take and the conditions that determine the outcome of each transition. The function `T(x, y)` represents the logic that governs how the system transitions from one state to another based on the applied rule. The diagram could represent an algorithm, a decision tree, or any process that involves a series of steps with potential success or failure at each step.
</details>
(a) Reflective MTP
<details>
<summary>x5.png Details</summary>
### Visual Description
## Diagram: Decision Tree with Validation
### Overview
The image depicts a decision tree or flow diagram, illustrating a process with multiple branches and validation steps. Each node represents a state or decision point, and the arrows indicate the flow of the process. Some nodes are marked with a checkmark (β), indicating successful validation, while others are marked with an "X", indicating failure. The diagram includes labels for each node and equations describing the transitions between states.
### Components/Axes
* **Nodes:** Represented as rounded rectangles, each labeled with "R" followed by a number (e.g., R1, R2) or "Q" or "A". Each node also contains either a checkmark (β) or an "X".
* **Arrows:** Indicate the flow of the process. Solid arrows represent direct transitions, while dotted arrows represent feedback or alternative paths.
* **Labels:** Each node is labeled with a variable "S" followed by a number (e.g., S0, S1), and an equation that defines the value of that variable.
* **Colors:** The nodes and arrows are color-coded, with shades ranging from gray to orange to red, and green.
### Detailed Analysis
Here's a breakdown of the diagram, starting from the left:
1. **Starting Node:** A gray rounded rectangle labeled "Q" (top-left). The equation below it states "S0 = Q".
2. **First Branch:**
* A solid orange arrow leads from "Q" to a rounded rectangle labeled "R1, β". The equation below it states "S1 = T(S0, R1)".
* A solid green arrow leads from "Q" to a rounded rectangle labeled "R8, β". The equation below it states "S8 = T(S7, R8)".
3. **Second Level (Orange Branch):**
* A solid orange arrow leads from "R1, β" to "R3, β". The equation below it states "S3 = T(S2, R3)".
* A solid orange arrow leads from "R3, β" to "R7, X". The equation below it states "S7 = S0".
4. **Third Level (Orange Branch):**
* A solid red arrow leads from "R1, β" to "R2, X". The equation below it states "S2 = S1".
* A solid red arrow leads from "R3, β" to "R4, β". The equation below it states "S4 = T(S3, R4)".
5. **Fourth Level (Orange Branch):**
* A solid red arrow leads from "R4, β" to "R5, X". The equation below it states "S5 = S4".
* A solid red arrow leads from "R4, β" to "R6, X". The equation below it states "S6 = S3".
6. **Second Level (Green Branch):**
* A solid green arrow leads from "R8, β" to "R9, β". The equation below it states "S9 = T(S8, R9)".
7. **Third Level (Green Branch):**
* A solid green arrow leads from "R9, β" to "A, β". The equation below it states "S10 = T(S9, A)".
8. **Feedback Loops:**
* A dotted red arrow leads from "R2, X" to "R3, β".
* A dotted red arrow leads from "R5, X" to "R4, β".
* A dotted red arrow leads from "R6, X" to "R7, X".
### Key Observations
* The diagram represents a decision-making process with validation steps.
* The process starts with an initial state "Q" and branches into two paths.
* The orange path involves multiple validation steps, some of which fail (indicated by "X").
* The green path leads to a successful outcome "A, β".
* Feedback loops exist from failed validation states back to earlier stages in the orange path.
### Interpretation
The diagram illustrates a process where multiple attempts may be needed to reach a successful outcome. The orange path represents a more complex and potentially error-prone process, while the green path represents a more direct and successful route. The feedback loops suggest that the process can recover from failures by revisiting earlier stages and adjusting the input. The "T" function likely represents a transformation or processing step that depends on the input from the previous state and the current node. The diagram suggests that the process aims to reach a validated state, and it can do so through different paths, with some paths being more reliable than others.
</details>
(b) Reflective trace-back search (width $m=2$ )
Figure 3: Reflective reasoning based on MTP. β $\checkmark$ β and β $Γ$ β are self-verification labels for positive and negative steps, respectively. The steps that are instantly verified as negative are highlighted in red. In RTBS, the dashed-line arrows back-propagate the negative labels, causing parental steps to be recursively rejected (orange). The green shows the steps that successfully lead to the answer.
Conceptually, reflection provides feedback for the proposed steps and may alter the subsequent reasoning accordingly. Reflection takes flexible forms in natural language (e.g., justifications and comprehensive evaluations), making it extremely costly to analyze. In this work, we propose to equip transformers with the simplest discriminative form of reflection, where the model self-verifies the correctness of each step and is allowed to retry those incorrect attempts. We currently do not consider the high-level revisory behavior that maps incorrect steps to correct ones, as we find learning such a mapping is challenging for tiny models and leads to no significant gain in practice. Specifically, we analyze two basic variants of reflective reasoning in this paper: the reflective MTP and the reflective trace-back search, as described below (see pseudo-code in Appendix D.1).
Reflective MTP (RMTP)
Given any MTP with a policy $\pi$ and transition $\mathcal{T}$ , we use a verifier $\mathcal{V}$ to produce a verification sequence after each reasoning step, denoted as ${V}_{t}\sim\mathcal{V}(Β·|{R}_{t})$ . Such ${V}_{t}$ includes verification label(s): The positive β $\checkmark$ β and negative β $Γ$ " signifying correct and incorrect reasoning of ${R}_{t}$ , respectively. Given the verified step ${\tilde{R}}_{t+1}:=({R}_{t+1},{V}_{t+1})$ that contains verification, we define $\tilde{\mathcal{T}}$ as the reflective transition function that rejects incorrect steps:
$$
{S}_{t+1}=\tilde{\mathcal{T}}({S}_{t},{\tilde{R}}_{t+1})=\tilde{\mathcal{T}}({S}_{t},({R}_{t+1},{V}_{t+1})):=\begin{cases}{S}_{t},&\text{``$\times$''}\in{V}_{t+1};\\
\mathcal{T}({S}_{t},{R}_{t+1}),&\text{otherwise.}\end{cases} \tag{2}
$$
In other words, if $\mathcal{V}$ detects any error (i.e. β $Γ$ ") in ${R}_{t+1}$ , the state remains unchanged so that $\pi$ may re-sample another attempt. Focusing on self-verification, we use a single model called the self-verifying policy $\tilde{\pi}:=\{\pi,\mathcal{V}\}$ to serve simultaneously as the planning policy $\pi$ and the verifier $\mathcal{V}$ . By operating tokens, $\tilde{\pi}$ outputs the verified step ${\tilde{R}}_{t}$ for each input state ${S}_{t}$ . In this way, $\tilde{\mathcal{T}}$ and $\tilde{\pi}$ constitute a new MTP called the RMTP, with illustration in Figure 3(a).
Reflective trace-back search (RTBS)
Though RMTP allows instant rejections of incorrect steps, sometimes the quality of a step can be better determined by actually trying it. For example, a Sudoku solver occasionally makes tentative guesses and traces back if the subsequent reasoning fails. Inspired by o1-journey [26], a trace-back search allowing the reasoner to revisit previous states may be applied to explore solution paths in an MTP. We implement simple RTBS by simulating the depth-first search in the trajectory space. Let $m$ denote the RTBS width, i.e., the maximal number of attempts on each step. As illustrated in Figure 3(b), if $m$ proposed steps are rejected on a state ${S}_{t}$ , the negative label β $Γ$ β will be propagated back to recursively reject the previous step ${R}_{t}$ . As a result, the state traces back to the closest ancestral state that has remaining attempt opportunities.
3.3 Training
<details>
<summary>x6.png Details</summary>
### Visual Description
## Process Diagram: Chain-of-Thought (CoT) Training and Fine-tuning
### Overview
The image is a process diagram illustrating a four-stage approach for training and fine-tuning a model using Chain-of-Thought (CoT) examples. The stages are Pretraining, Non-reflective SFT (Supervised Fine-Tuning), Reflective SFT, and RL (Reinforcement Learning) fine-tuning. The diagram shows the flow of data and the interactions between different components in each stage.
### Components/Axes
* **Stage I: Pretraining (Top-Left, Green Box)**
* Input: "CoT examples (data mixture)" from a "Training Data" cylinder.
* Process: "context windows (randomly drawn)" from the training data.
* Data Representation: A sequence of blocks labeled "Q", "R1", "R2", "A", representing Question, Reasoning Step 1, Reasoning Step 2, and Answer, respectively. These blocks are arranged linearly, with red boxes highlighting example context windows.
* **Stage II: Non-reflective SFT (Center, Blue Box)**
* Input: Policy Ο from Pretraining.
* Process: A series of states (S1, S2, Q) and steps/answers (R1, R2, A) are processed.
* Data Representation: States are represented as "Q", "S1", "S2", and steps/answers are represented as "R1", "R2", "A". Arrows indicate the flow of information.
* **Stage III: Reflective SFT (Bottom-Left, Gray Box)**
* Input: "CoT examples (data mixture)".
* Process: "sampling CoTs through MTP" and "ground-truth verification" using an "Expert Verifier".
* Data Representation: States & steps are represented as "Q", "R1", "S1", "R2", "S2", "R3". These are mapped via Ξ½ to ground-truth verifications "V1", "V2", "V3". The policy Ο is used to transition between Q, Rt, and A, with the state St = T(St-1, Rt).
* **Stage IV: RL fine-tuning (Right, Peach Box)**
* Input: Policy Ο from Non-reflective SFT and Reflective SFT.
* Process: Two parallel processes, MTP (top) and RMTP (bottom), followed by "Reward Model" and "Policy Optimization".
* Data Representation:
* MTP: Transitions between Q, Rt, and A, with the state St = T(St-1, Rt).
* RMTP: Transitions between Q, Rt, Vt, and A, with the state St = T~(St-1, Rt, Vt).
* **Arrows**: Arrows indicate the flow of data and control between stages and components.
* **Labels**: "CoT examples (data mixture)", "Training Data", "context windows (randomly drawn)", "Pretraining", "Non-reflective SFT", "Reflective SFT", "RL fine-tuning", "Expert Verifier", "Reward Model", "Policy Optimization", "MTP", "RMTP".
### Detailed Analysis or ### Content Details
* **Pretraining (I)**: The model is initially trained on a mixture of CoT examples. Context windows are randomly drawn from this data.
* **Non-reflective SFT (II)**: The model is fine-tuned using supervised learning, without explicit reflection on the reasoning process. The policy Ο guides the transitions between states and answers.
* **Reflective SFT (III)**: This stage introduces reflection by sampling CoTs through MTP and verifying them against ground truth using an expert verifier. The policy Ο is used to transition between Q, Rt, and A, with the state St = T(St-1, Rt).
* **RL fine-tuning (IV)**: The model is further fine-tuned using reinforcement learning. Two processes, MTP and RMTP, are used. The reward model provides feedback, and the policy is optimized based on this feedback.
### Key Observations
* The diagram illustrates a multi-stage training process that incorporates both supervised and reinforcement learning.
* The "Reflective SFT" stage introduces a mechanism for verifying the reasoning process against ground truth.
* The "RL fine-tuning" stage uses two parallel processes, MTP and RMTP, which may represent different reward structures or training objectives.
### Interpretation
The diagram presents a comprehensive approach to training and fine-tuning models using Chain-of-Thought reasoning. The process starts with pretraining on a mixture of CoT examples, followed by supervised fine-tuning, reflective fine-tuning, and finally, reinforcement learning fine-tuning. The inclusion of an "Expert Verifier" in the "Reflective SFT" stage suggests an attempt to improve the quality of the reasoning process by comparing it against ground truth. The use of two parallel processes (MTP and RMTP) in the "RL fine-tuning" stage may indicate an exploration of different reward structures or training objectives. Overall, the diagram highlights the importance of both data quality and training methodology in developing effective CoT models.
</details>
Figure 4: The training workflow for transformers to perform CoT reasoning.
As shown in Figure 4, we train the tiny transformers from scratch through consistent techniques of LLM counterparts, such as pretraining, supervised fine-tuning (SFT), and RL fine-tuning. First, we use conventional pipelines to train a baseline model $\pi$ with only the planning ability in MTPs. During (I) pretraining, these CoT examples are treated as a textual corpus, where sequences are randomly drawn to minimize cross-entropy loss of next-token prediction. Then, in (II) non-reflective SFT, the model learns to map each state ${S}_{t}$ to the corresponding step ${R}_{t+1}$ by imitating examples.
Next, we employ (III) reflective SFT to integrate the planning policy $\pi$ with the knowledge of self-verification. To produce ground-truth verification labels, we use $\pi$ to sample non-reflective CoTs, in which the sampled steps are then labeled by an expert verifier (e.g., a rule-based process reward model). Reflective SFT learns to predict these labels from the states and the proposed steps, i.e., $({S}_{t},{R}_{t+1})β{V}_{t+1}$ . To prevent disastrous forgetting, we also mix the same CoT examples as in non-reflective SFT. This converts $\pi$ to a self-verifying policy $\tilde{\pi}$ that can self-verify reasoning steps.
Thus far, we have obtained the planning policy $\pi$ and the self-verifying policy $\tilde{\pi}$ , which can be further strengthened through (IV) RL fine-tuning. As illustrated in Figure 4, RL fine-tuning involves iteratively executing $\pi$ ( $\tilde{\pi}$ ) to collect experience CoTs through an MTP (RMTP), evaluating these CoTs with a reward model, and updating the policy to favor higher-reward solutions. Following the RLVR paradigm [15], we use binary outcome rewards (i.e., $1$ for correct answers and $0$ otherwise) computed by a rule-based answer checker $\operatorname{ORM}(Q,A)$ . When training the self-verifying policy $\tilde{\pi}$ , the RMTP treats verification ${V}_{t}$ as a part of the augmented step ${\tilde{R}}_{t}$ , simulating R1-like training [5] where reflection and solution planning are jointly optimized. We mainly use GRPO [31] as the algorithms to optimize policies. Details of RL fine-tuning are elaborated in Appendix B.
4 Theoretical results
This section establishes theoretical conditions under which self-verifying reflection (RMTP or RTBS in Section 3.2) enhances reasoning accuracy (the probability of deriving correct answers). The general relationship between the verification ability and reasoning accuracy (discussed in Appendix C.1) for any MTP is intractable as the states and transitions can be arbitrarily specified. Therefore, to derive interpretable insights, we discuss a simplified prototype of reasoning that epitomizes the representative principle of CoTs β to incrementally express complex relations by chaining the local relation in each step [25]. Specifically, Given query $Q$ as the initial state, we view a CoT as the step-by-step process that reduces the complexity within states:
- We define $\mathcal{S}_{n}$ as the set of states with a complexity scale of $n$ . For simplicity, we assume that each step, if not rejected by reflection, reduces the complexity scale by $1$ . Therefore, the scale $n$ is the number of effective steps required to derive an answer.
- An answer $A$ is a state with a scale of $0$ , i.e. $Aβ\mathcal{S}_{0}$ . Given an input query $Q$ , the answers $\mathcal{S}_{0}$ are divided into positive (correct) answers $\mathcal{S}_{0}^{+}$ and negative (wrong) answers $\mathcal{S}_{0}^{-}$ .
- States $\mathcal{S}_{n}$ ( $n$ > 0) are divided into 1) positive states $\mathcal{S}_{n}^{+}$ that potentially lead to correct answers and 2) negative states $\mathcal{S}_{n}^{-}$ leading to only incorrect answers through forward transitions.
Consider a self-verifying policy $\tilde{\pi}=\{\pi,\mathcal{V}\}$ to solve this simplified task. We describe its fundamental abilities using the following probabilities (whose meanings will be explained afterwards):
$$
\displaystyle\mu:=p_{{R}\sim\pi}(\mathcal{T}({S},{R})\in\mathcal{S}^{+}_{n-1}\mid{S}\in\mathcal{S}_{n}^{+}) \displaystyle e_{+}:=p_{{R},{V}\sim\tilde{\pi}}(\mathcal{T}({S},{R})\in\mathcal{S}^{-}_{n-1},\text{``$\times$''}\notin{V}\mid{S}\in\mathcal{S}_{n}^{+}), \displaystyle e_{-}:=p_{{R},{V}\sim\tilde{\pi}}(\mathcal{T}({S},{R})\in\mathcal{S}^{+}_{n-1},\text{``$\times$''}\in{V}\mid{S}\in\mathcal{S}_{n}^{+}), \displaystyle f:=p_{{R},{V}\sim\tilde{\pi}}(\text{``$\times$''}\in{V}\mid{S}\in\mathcal{S}_{n}^{-}). \tag{3}
$$
To elaborate, $\mu$ measures the planning ability, defined as the probability that $\pi$ plans a step that leads to a positive next state, given that the current state is positive. For verification abilities, we measure the rates of two types of errors: $e_{+}$ (false positive rate) is the probability of accepting a step that leads to a negative state, and $e_{-}$ (false negative rate) is the probability of rejecting a step that leads to a positive state. Additionally, $f$ is the probability of rejecting any step on negative states, providing the chance of tracing back to previous states. Given these factors, Figure 5 illustrates the state transitions in non-reflective (vanilla MTP) and reflective (RMTB and RTBS) reasoning.
<details>
<summary>x7.png Details</summary>
### Visual Description
## Diagram: State Transition Diagram
### Overview
The image presents a state transition diagram illustrating transitions between two states, denoted as $S_n^-$ and $S_n^+$, and their subsequent states $S_{n-1}^-$ and $S_{n-1}^+$. The diagram shows the probabilities associated with each transition.
### Components/Axes
* **States:**
* $S_n^-$: Represented by a red circle.
* $S_n^+$: Represented by a green circle.
* $S_{n-1}^-$: Represented by a red circle.
* $S_{n-1}^+$: Represented by a green circle.
* **Transitions:**
* From $S_n^-$ to $S_{n-1}^-$: Labeled with probability "1" (red arrow).
* From $S_n^+$ to $S_{n-1}^-$: Labeled with probability "1 - ΞΌ" (red arrow).
* From $S_n^+$ to $S_{n-1}^+$: Labeled with probability "ΞΌ" (green arrow).
### Detailed Analysis
* The diagram shows that state $S_n^-$ transitions directly to state $S_{n-1}^-$ with a probability of 1.
* State $S_n^+$ can transition to either $S_{n-1}^-$ with probability $1 - \mu$ or to $S_{n-1}^+$ with probability $\mu$.
### Key Observations
* The transition from $S_n^-$ is deterministic, always leading to $S_{n-1}^-$.
* The transition from $S_n^+$ is probabilistic, branching to either $S_{n-1}^-$ or $S_{n-1}^+$ based on the value of $\mu$.
### Interpretation
The diagram models a system where the state $S_n^-$ always decrements to $S_{n-1}^-$. The state $S_n^+$ has two possible transitions: it can either decrement to $S_{n-1}^+$ with probability $\mu$, or it can transition to $S_{n-1}^-$ with probability $1 - \mu$. The parameter $\mu$ controls the likelihood of the system staying in the "+" state versus transitioning to the "-" state. This could represent a simplified model of a process where a state can either remain in a positive condition or revert to a negative condition, with $\mu$ representing the probability of remaining in the positive condition.
</details>
(a) Non-reflective reasoning
<details>
<summary>x8.png Details</summary>
### Visual Description
## Diagram: State Transition Diagram for RTBS
### Overview
The image presents a state transition diagram illustrating the behavior of a system, likely related to a retry-based backoff scheme (RTBS). The diagram shows transitions between states denoted as S+ and S-, with subscripts indicating a numerical index. The transitions are labeled with probabilities or rates, representing the likelihood of moving from one state to another. The diagram is divided into two regions: one representing the state after 'm' attempts in RTBS, and the other showing the transitions between states.
### Components/Axes
* **States:**
* S+ (Green circles): Represents a "positive" state.
* S- (Red circles): Represents a "negative" state.
* Subscripts (n+1, n, n-1): Indicate the state's index or position in a sequence.
* **Transitions:** Arrows connecting the states, labeled with probabilities or rates.
* **Parameters:**
* f: Probability of a transition from S-n to S-n.
* ΞΌ: A parameter related to the transition rates.
* e+: A parameter related to the transition rates.
* e-: A parameter related to the transition rates.
* **Equation:** Ξ± := ΞΌΞ΅- + (1 - ΞΌ)(1 - e+)
* **Region:** States S+n+1 and S-n+1 are enclosed in a rounded rectangle, labeled "After m attempts in RTBS".
### Detailed Analysis
* **S- States (Red):**
* S-n transitions to S-n-1 with probability (1-f).
* S-n transitions to S+n with probability f.
* S-n+1 transitions to S-n (dashed grey arrow).
* **S+ States (Green):**
* S+n transitions to S+n-1 with rate ΞΌ(1 - e-).
* S+n transitions to S-n with rate (1 - ΞΌ)e+.
* S+n+1 transitions to S+n (dashed green arrow).
* **Equation:** The equation Ξ± := ΞΌΞ΅- + (1 - ΞΌ)(1 - e+) is given, likely representing a key parameter or probability related to the system's behavior.
### Key Observations
* The diagram shows transitions between "positive" and "negative" states, suggesting a system that can recover from failures or errors.
* The parameters f, ΞΌ, e+, and e- control the transition rates between the states.
* The equation for Ξ± likely represents a crucial aspect of the system's dynamics.
### Interpretation
The state transition diagram models a system that undergoes transitions between positive (S+) and negative (S-) states. The transitions are governed by probabilities and rates, which depend on the parameters f, ΞΌ, e+, and e-. The equation for Ξ± likely represents a key performance metric or probability related to the system's behavior. The diagram is likely used to analyze the system's stability, convergence, or performance under different conditions. The "After m attempts in RTBS" region suggests that the system involves a retry mechanism, where the system attempts to recover from failures by retrying the operation.
</details>
(b) Reflective reasoning through an RMTP or RTBS
Figure 5: The diagram of state transitions starting from scale $n$ in the simplified reasoning, where probabilities are attached to solid lines. In (b) reflective reasoning, the dashed-line arrow presents the trace-back move after $m$ attempts in RTBS.
For input problems with scale $n$ , we use $\rho(n)$ , $\tilde{\rho}(n)$ , and $\tilde{\rho}_{m}(n)$ to respectively denote the reasoning accuracy using no reflection, RMTP, and RTBS (with width $m$ ). Obviously, we have $\rho(n)=\mu^{n}$ . In contrast, the mathematical forms of $\tilde{\rho}(n)$ and $\tilde{\rho}_{m}(n)$ are more complicated and therefore left to Appendix C.2. Our main result provides simple conditions for the above factors $(\mu,e_{-},e_{+},f)$ to ensure an improved accuracy when reasoning through an RMTP or RTBS.
**Theorem 1**
*In the above simplified problem, consider a self-verifying policy $\tilde{\pi}$ where $\mu$ , $e_{-}$ , and $e_{+}$ are non-trivial (i.e. neither $0$ nor $1$ ). Let $\alpha:=\mu e_{-}+(1-\mu)(1-e_{+})$ denote the rejection probability on positive states. Given an infinite computation budget, for $n>0$ we have:
- $\tilde{\rho}(n)β₯\rho(n)$ if and only if $e_{-}+e_{+}β€ 1$ , where equalities hold simultaneously; furthermore, reducing either $e_{-}$ or $e_{+}$ strictly increases $\tilde{\rho}(n)$ .
- $\tilde{\rho}_{m}(n)>\tilde{\rho}(n)$ for a sufficiently large $n$ if and only if $f>\alpha$ and $m>\frac{1}{1-\alpha}$ ; furthermore, such a gap of $\tilde{\rho}_{m}(n)$ over $\tilde{\rho}(n)$ increases strictly with $f$ .*
Does reflection require a strong verifier? Theorem 1 shows that RMTP improves performance over vanilla MTP if the verification errors $e_{+}$ and $e_{-}$ are properly bounded, which does not necessitate a strong verifier. In our simplified setting, this only requires the verifier $\mathcal{V}$ to be better than random guessing (which ensures $e_{-}+e_{+}=1$ ). This also indicates a trivial guarantee of RTBS, as an infinitely large width ( $mβ+β$ ) substantially converts RTBS to RMTB.
When does trace-back search facilitate reflection? Theorem 1 provides the conditions for RTBS to outperform RMTP for a sufficiently large $n$ : 1) The width $m$ is large enough to ensure effective exploration. 2) $f>\alpha$ indicates that negative states are inherently discriminated from positive ones, leading to a higher rejection probability on negative states than on positive states (see Figure 5(b)). In other words, provided $f>\alpha$ , RTBS is ensured to be more effective on complicated queries using a finite $m$ . However, this also implies a risk of over-thought on simple queries that have a small $n$ .
The derivation and additional details of Theorem 1 are provided in Appendix C.3. In addition, we also derive how many steps it costs to find a correct solution in RMTP. The following Proposition 1 (see proof in Appendix C.4) shows that a higher $e_{-}$ causes more steps to be necessarily rejected and increases the solution cost. In contrast, although a higher $e_{+}$ reduces accuracy, it forces successful solutions to rely less on reflection, leading to fewer expected steps. Therefore, a high false negative rate $e_{-}$ is worse than a high $e_{+}$ given the limited computational budget in practice.
**Proposition 1 (RMTP Reasoning Length)**
*For a simplified reasoning problem with scale $n$ , the expected number of steps $\bar{T}$ for $\tilde{\pi}$ to find a correct answer is $\bar{T}=\frac{n}{(1-\mu)e_{+}+\mu(1-e_{-})}$ . Especially, a correct answer will never be found if the denominator is $0$ .*
Appendix C.5 further extends our analysis to more realistic reasoning, where rejected attempts lead to a posterior drop of $\mu$ (or rise of $e_{-}$ ), indicating that the model may not well generalize the current state. In this case, the bound of $e_{-}$ to ensure improvements becomes stricter than that in Theorem 1.
5 Experiments
We conduct comprehensive experiments to examine the reasoning performance of tiny transformers under various settings. We trained simple causal-attention transformers [36] (implemented by LitGPT [1]) with 1M, 4M, and 16M parameters, through the pipelines described in Section 3.3. Details of training data, model architectures, tokenization, and hyperparameters are included in Appendix D. The source code is available at https://github.com/zwyu-ai/self-verifying-reflection-reasoning.
We test tiny transformers in two reasoning tasks: The integer multiplication task (Mult for short) computes the product of two integers $x$ and $y$ ; the Sudoku task fills numbers into blank positions of a $9Γ 9$ matrix, such that each row, column, or $3Γ 3$ block is a permutation of $\{1,...,9\}$ . For both tasks, we divide queries into 3 levels of difficulties: The in-distribution (ID) Easy, ID Hard, and out-of-distribution (OOD) Hard. The models are trained on ID-Easy and ID-Hard problems, while tested additionally on OOD-Hard cases. We define the difficulty of a Mult query by the number $d$ of digits of the greater multiplicand, and that of a Sudoku puzzle is determined by the number $b$ of blanks to be filled. Specifically, we have $1β€ dβ€ 5$ or $9β€ b<36$ for ID Easy, $6β€ dβ€ 8$ or $36β€ b<54$ for ID Hard, and $9β€ dβ€ 10$ or $54β€ b<63$ for OOD Hard.
Our full results are presented in Appendix E. Shown in Appendix E.1, these seemingly simple tasks pose challenges even for some well-known LLMs. Remarkably, through simple self-verifying reflection, our best 4M Sudoku model is as good as OpenAI o3-mini [21], and our best 16M Mult model outperforms DeepSeek-R1 [5] in ID difficulties.
5.1 Results of supervised fine-tuning
First, we conduct (I) pretraining, (II) non-reflective SFT, and (III) reflective SFT as described in Section 3.3. In reflective SFT, we consider learning two types of self-verification: 1) The binary verification includes a single binary label indicating the overall correctness of a planned step; 2) the detailed verification includes a series of binary labels checking the correctness of each meaningful element in the step. The implementation of verification labels is elaborated in Appendix D.2.3. We present our full SFT results in Appendix E.2, which includes training 30 models and executing 54 tests. In the following, we discuss our main findings through visualizing representative results.
<details>
<summary>x9.png Details</summary>
### Visual Description
## Bar Chart: Accuracy vs. Model Size for Different Verification Types
### Overview
The image presents three bar charts comparing the accuracy of different verification types (None, Binary, Detailed) across varying model sizes (1M, 4M, 16M). The charts are grouped by difficulty level: "ID Easy", "ID Hard", and "OOD Hard". Accuracy is measured as a percentage.
### Components/Axes
* **X-axis (Model Size):** Categorical axis with three values: 1M, 4M, 16M.
* **Y-axis (Accuracy (%)):** Numerical axis ranging from 0% to 100% for "ID Easy", 0 to 60% for "ID Hard", and 0 to 8 for "OOD Hard", with gridlines at intervals of 25%, 20%, and 2 respectively.
* **Legend (Verification Type):** Located on the top-right of the entire image.
* **None:** Represented by blue bars.
* **Binary:** Represented by orange bars.
* **Detailed:** Represented by green bars.
* **Chart Titles:** "ID Easy", "ID Hard", "OOD Hard" are titles above each respective chart.
### Detailed Analysis
**1. ID Easy Chart:**
* **None (Blue):** Accuracy starts at approximately 23% for 1M, increases to 93% for 4M, and remains at 93% for 16M.
* **Binary (Orange):** Accuracy starts at approximately 96% for 1M, increases to 98% for 4M, and remains at 98% for 16M.
* **Detailed (Green):** Accuracy starts at approximately 23% for 1M, increases to 96% for 4M, and reaches 100% for 16M.
**2. ID Hard Chart:**
* **None (Blue):** Accuracy starts at approximately 2% for 1M, increases to 37% for 4M, and reaches 65% for 16M.
* **Binary (Orange):** Accuracy starts at approximately 53% for 1M, increases to 57% for 4M, and reaches 64% for 16M.
* **Detailed (Green):** Accuracy starts at approximately 2% for 1M, increases to 43% for 4M, and reaches 66% for 16M.
**3. OOD Hard Chart:**
* **None (Blue):** Accuracy starts at approximately 1% for 1M, increases to 2% for 4M, and reaches 2% for 16M.
* **Binary (Orange):** Accuracy starts at approximately 3.7% for 1M, decreases to 3% for 4M, and decreases to 1% for 16M.
* **Detailed (Green):** Accuracy starts at approximately 1% for 1M, increases to 3% for 4M, and reaches 9% for 16M.
### Key Observations
* For "ID Easy", all verification types show high accuracy with larger model sizes (4M and 16M). Binary verification performs consistently well across all model sizes.
* For "ID Hard", accuracy generally increases with model size for all verification types. The "Detailed" verification type shows the most significant improvement.
* For "OOD Hard", the "Detailed" verification type shows the most significant improvement with increasing model size, while "None" and "Binary" verification types remain relatively low.
* The "OOD Hard" chart has a significantly smaller y-axis scale compared to the other two charts, indicating much lower accuracy in this scenario.
### Interpretation
The data suggests that model size has a positive impact on accuracy, especially for "ID Hard" and "OOD Hard" scenarios. The "Detailed" verification type appears to be the most effective in improving accuracy as model size increases, particularly for "OOD Hard" cases. The "Binary" verification type performs well in the "ID Easy" scenario, indicating it might be sufficient for simpler tasks. The "OOD Hard" scenario highlights the challenge of generalizing to out-of-distribution data, where even with larger model sizes, accuracy remains relatively low for "None" and "Binary" verification types. The "Detailed" verification type shows promise in improving accuracy for "OOD Hard" cases, suggesting it might be more robust to distributional shifts.
</details>
Figure 6: The accuracy of non-reflective execution of models in Mult. In each group, we compare training with various types of verification (βNoneβ for no reflective SFT).
Does learning self-verification facilitate learning the planning policy? We compare our models under the non-reflective execution, where self-verification is not actively used in test time. As shown in Figure 6, reflective SFT with binary verification brings remarkable improvements for 1M and 4M in ID-Easy and ID-Hard Mult problems, greatly reducing the gap among model sizes. Although detailed verification does not benefit as much as binary verification in ID problems, it significantly benefits the 16M model in solving OOD-Hard problems. Therefore, learning to self-verify benefits the learning of forward planning, increasing performance even if test-time reflection is not enabled.
Since reflective SFT mixes the same CoT examples as used in non-reflective SFT, an explanation for this phenomenon is that learning to self-verify serves as a regularizer to the planning policy. This substantially improves the quality of hidden embeddings in transformers, which facilitates the learning of CoT examples. Binary verification is inherently a harder target to learn, which produces stronger regularizing effects than detailed verification. However, the complexity (length) of the verification should match the capacity of the model; otherwise, it could severely compromise the benefits of learning self-verification. For instance, learning binary verification and detailed verification fails to improve the 16M model and the 1M model, respectively.
<details>
<summary>x10.png Details</summary>
### Visual Description
## Bar Charts: Accuracy and Error Metrics for Verification Tasks
### Overview
The image presents a set of bar charts comparing the accuracy and error rates of different reflective execution techniques (None, RMTP, RTBS) on two verification tasks (Mult ID-Hard and Sudoku ID-Hard) with two verification types (Binary and Detailed). The charts are organized in a 2x2 grid, with accuracy on the top row and error on the bottom row, and each column representing a different task and verification type. The x-axis represents the model size (1M, 4M, 16M).
### Components/Axes
**Top Row (Accuracy):**
* **Y-axis:** Accuracy (%), ranging from 0 to 80.
* **X-axis:** Model Size (1M, 4M, 16M).
* **Titles (Left to Right):**
* Mult ID-Hard Binary Verification
* Mult ID-Hard Detailed Verification
* Sudoku ID-Hard Binary Verification
* Sudoku ID-Hard Detailed Verification
* **Legend (Top-Right):**
* None (Gray)
* RMTP (Green)
* RTBS (Red)
**Bottom Row (Error):**
* **Y-axis:** Error (%), ranging from 0 to 75.
* **X-axis:** Model Size (1M, 4M, 16M).
* **Titles (Left to Right):** Same as the top row.
* **Legend (Right):**
* RMTP e- (Green with Cross pattern)
* RMTP e+ (Green, empty)
* RTBS e- (Red with Cross pattern)
* RTBS e+ (Red, empty)
### Detailed Analysis
**Mult ID-Hard Binary Verification (Accuracy):**
* **None (Gray):** Accuracy increases with model size, from approximately 53% at 1M to 63% at 16M.
* **RMTP (Green):** Accuracy increases with model size, from approximately 45% at 1M to 78% at 16M.
* **RTBS (Red):** Accuracy increases with model size, from approximately 35% at 1M to 75% at 16M.
**Mult ID-Hard Detailed Verification (Accuracy):**
* **None (Gray):** Accuracy increases with model size, from approximately 3% at 1M to 63% at 16M.
* **RMTP (Green):** Accuracy increases with model size, from approximately 3% at 1M to 78% at 16M.
* **RTBS (Red):** Accuracy increases with model size, from approximately 2% at 1M to 75% at 16M.
**Sudoku ID-Hard Binary Verification (Accuracy):**
* **None (Gray):** Accuracy increases with model size, from approximately 48% at 1M to 53% at 16M.
* **RMTP (Green):** Accuracy increases with model size, from approximately 53% at 1M to 57% at 16M.
* **RTBS (Red):** Accuracy increases with model size, from approximately 55% at 1M to 57% at 16M.
**Sudoku ID-Hard Detailed Verification (Accuracy):**
* **None (Gray):** Accuracy increases with model size, from approximately 3% at 1M to 53% at 16M.
* **RMTP (Green):** Accuracy increases with model size, from approximately 10% at 1M to 60% at 16M.
* **RTBS (Red):** Accuracy increases with model size, from approximately 52% at 1M to 65% at 16M.
**Mult ID-Hard Binary Verification (Error):**
* **RMTP e- (Green with Cross):** Error decreases with model size, from approximately 35% at 1M to 15% at 16M.
* **RMTP e+ (Green, empty):** Error decreases with model size, from approximately 0% at 1M to 0% at 16M.
* **RTBS e- (Red with Cross):** Error decreases with model size, from approximately 30% at 1M to 15% at 16M.
* **RTBS e+ (Red, empty):** Error decreases with model size, from approximately 0% at 1M to 0% at 16M.
**Mult ID-Hard Detailed Verification (Error):**
* **RMTP e- (Green with Cross):** Error decreases with model size, from approximately 10% at 1M to 5% at 16M.
* **RMTP e+ (Green, empty):** Error decreases with model size, from approximately 0% at 1M to 0% at 16M.
* **RTBS e- (Red with Cross):** Error decreases with model size, from approximately 20% at 1M to 5% at 16M.
* **RTBS e+ (Red, empty):** Error decreases with model size, from approximately 0% at 1M to 0% at 16M.
**Sudoku ID-Hard Binary Verification (Error):**
* **RMTP e- (Green with Cross):** Error decreases with model size, from approximately 30% at 1M to 15% at 16M.
* **RMTP e+ (Green, empty):** Error decreases with model size, from approximately 0% at 1M to 0% at 16M.
* **RTBS e- (Red with Cross):** Error decreases with model size, from approximately 40% at 1M to 10% at 16M.
* **RTBS e+ (Red, empty):** Error decreases with model size, from approximately 0% at 1M to 0% at 16M.
**Sudoku ID-Hard Detailed Verification (Error):**
* **RMTP e- (Green with Cross):** Error decreases with model size, from approximately 85% at 1M to 5% at 16M.
* **RMTP e+ (Green, empty):** Error decreases with model size, from approximately 0% at 1M to 0% at 16M.
* **RTBS e- (Red with Cross):** Error decreases with model size, from approximately 80% at 1M to 5% at 16M.
* **RTBS e+ (Red, empty):** Error decreases with model size, from approximately 0% at 1M to 0% at 16M.
### Key Observations
* For both Mult ID-Hard and Sudoku ID-Hard, the accuracy generally increases with model size for all reflective execution techniques.
* RMTP and RTBS generally outperform "None" in terms of accuracy, especially for larger model sizes.
* The error rates generally decrease with increasing model size.
* The "Detailed Verification" task shows a more significant improvement in accuracy with increasing model size compared to "Binary Verification".
* The error rates for Sudoku ID-Hard Detailed Verification are significantly higher at 1M model size compared to other tasks.
### Interpretation
The data suggests that reflective execution techniques (RMTP and RTBS) can improve the accuracy of verification tasks, especially as the model size increases. The "Detailed Verification" task benefits more from larger model sizes and reflective execution compared to "Binary Verification". The high error rates for Sudoku ID-Hard Detailed Verification at smaller model sizes indicate that this task is particularly challenging and requires larger models or further optimization of the verification process. The consistent decrease in error with increasing model size across all tasks and techniques highlights the importance of model size in achieving higher accuracy and reliability in verification tasks. The error metrics 'e-' and 'e+' likely represent different types of errors, with 'e-' being more prevalent.
</details>
Figure 7: Performance of reflective execution methods across different model sizes, including the accuracy (top) and the self-verification errors (bottom).
When do reflective executions improve reasoning accuracy? Figure 7 evaluates the non-reflective, RMTP, and RTBS executions for models in solving ID-Hard problems. Apart from the accuracy, the rates of verification error (i.e., the false positive rate $e_{+}$ and false negative rate $e_{-}$ defined in Section 4) are measured using an oracle verifier. In these results, RMTP reasoning raises the performance over non-reflective reasoning except for the 1M models (which fail in ID-hard Sudoku). Smaller error rates (especially $e_{-}$ ) generally lead to higher improvements, whereas a high $e_{-}$ in binary verification severely compromises the performance of the 1M Mult Model. Overall, reflection improves reasoning if the chance of rejecting correct steps ( $e_{-}$ ) is sufficiently small.
In what task is the trace-back search helpful? As seen in Figure 7, though RTBS shows no advantage against RMTP in Mult, it outperforms RMTP in Sudoku, especially the 4M model with detailed verification. This aligns with Theory 1 β The state of Sudoku (the $9Γ 9$ matrix) is required to comply with explicit verifiable rules, making incorrect states easily discriminated from correct states. However, errors in Mult states can only be checked by recalculating all historical steps. Therefore, we are more likely to have $f>\alpha$ in Sudoku, which grants a higher chance of solving harder problems. This suggests that RTBS can be more helpful than RMTP if incorrect states in the task carry verifiable errors, which validates our theoretical results.
5.2 Results of reinforcement learning
<details>
<summary>x11.png Details</summary>
### Visual Description
## Bar Chart: Accuracy and Error Metrics for Different Verification Types
### Overview
The image presents a set of bar charts comparing the accuracy and error rates of different verification types (None, Binary, Detailed) under various conditions. The charts are organized in a 2x2 grid, with the top row displaying accuracy (%) and the bottom row displaying error (%). The columns represent different scenarios: "Mult ID-Hard (4M)", "Mult OOD-Hard (4M)", "Mult ID-Hard (16M)", and "Mult OOD-Hard (16M)". The charts compare the performance of three reflective execution methods: None, RMTP, and RTBS. Error metrics are further broken down into e- and e+ for RMTP and RTBS.
### Components/Axes
* **Top Row (Accuracy):**
* **Y-axis:** "Accuracy (%)", ranging from 0 to 80 in increments of 20.
* **X-axis:** "Verification Type" with categories "None", "Binary", and "Detailed".
* **Legend (top-right):** "Reflective Execution" with the following mapping:
* Gray: "None"
* Green: "RMTP"
* Dark Red: "RTBS"
* White arrows with circles indicate the range of values for each bar.
* **Bottom Row (Error):**
* **Y-axis:** "Error (%)", ranging from 0 to 75 in increments of 25.
* **X-axis:** "Verification Type" with categories "None", "Binary", and "Detailed".
* **Legend (right):** "Error Metrics" with the following mapping:
* Green with cross pattern: "RMTP e-"
* Green: "RMTP e+"
* Dark Red with cross pattern: "RTBS e-"
* Dark Red: "RTBS e+"
* Black arrows with circles indicate the range of values for each bar.
* **Titles (top of each column):**
* Column 1: "Mult ID-Hard (4M)"
* Column 2: "Mult OOD-Hard (4M)"
* Column 3: "Mult ID-Hard (16M)"
* Column 4: "Mult OOD-Hard (16M)"
### Detailed Analysis
**Accuracy Charts (Top Row):**
* **Mult ID-Hard (4M):**
* "None" verification: Accuracy around 65% for "None", 70% for "RMTP", and 75% for "RTBS".
* "Binary" verification: Accuracy around 65% for "None", 70% for "RMTP", and 75% for "RTBS".
* "Detailed" verification: Accuracy around 65% for "None", 65% for "RMTP", and 65% for "RTBS".
* **Mult OOD-Hard (4M):**
* Accuracy is very low (close to 0%) for all verification types and reflective execution methods.
* **Mult ID-Hard (16M):**
* Accuracy is high (around 75%) for all verification types and reflective execution methods.
* **Mult OOD-Hard (16M):**
* Accuracy is very low (close to 0%) for all verification types and reflective execution methods.
**Error Charts (Bottom Row):**
* **Mult ID-Hard (4M):**
* "None" verification: Error is approximately 0% for all error metrics.
* "Binary" verification: "RMTP e-" is around 5%, "RMTP e+" is around 55%, "RTBS e-" is around 5%, and "RTBS e+" is around 60%.
* "Detailed" verification: "RMTP e-" is around 5%, "RMTP e+" is around 20%, "RTBS e-" is around 5%, and "RTBS e+" is around 25%.
* **Mult OOD-Hard (4M):**
* Error is low (around 0-5%) for all verification types and error metrics.
* **Mult ID-Hard (16M):**
* Error is low (around 0-5%) for all verification types and error metrics.
* **Mult OOD-Hard (16M):**
* "None" verification: Error is approximately 0% for all error metrics.
* "Binary" verification: "RMTP e-" is around 5%, "RMTP e+" is around 75%, "RTBS e-" is around 5%, and "RTBS e+" is around 80%.
* "Detailed" verification: "RMTP e-" is around 5%, "RMTP e+" is around 20%, "RTBS e-" is around 5%, and "RTBS e+" is around 30%.
### Key Observations
* **ID-Hard vs. OOD-Hard:** The "ID-Hard" scenarios (both 4M and 16M) generally show higher accuracy compared to the "OOD-Hard" scenarios, which have very low accuracy.
* **Impact of Verification Type:** For "ID-Hard (4M)", the "Binary" verification type shows a significant difference between e- and e+ error metrics for both RMTP and RTBS. "Detailed" verification reduces the error for e+ metrics.
* **Reflective Execution Methods:** In "ID-Hard (4M)", RTBS generally shows slightly higher accuracy than RMTP.
* **Memory Size (4M vs. 16M):** Increasing memory size from 4M to 16M significantly improves accuracy in the "ID-Hard" scenario, with accuracy reaching approximately 75% regardless of the verification type or reflective execution method.
### Interpretation
The data suggests that the "OOD-Hard" scenarios are significantly more challenging than the "ID-Hard" scenarios, resulting in very low accuracy regardless of the verification type or reflective execution method. For the "ID-Hard (4M)" scenario, using "Binary" verification introduces a large difference between e- and e+ error metrics, which is mitigated by using "Detailed" verification. Increasing the memory size to 16M significantly improves accuracy in the "ID-Hard" scenario, indicating that memory size is a crucial factor for performance in these tasks. The choice of reflective execution method (RMTP vs. RTBS) has a relatively small impact on accuracy compared to the other factors.
</details>
Figure 8: Performance of the 4M and 16M models in Mult after GRPO, including accuracy and the verification error rates. As an ablation, we also include non-reflective models. The vertical arrows start from the baseline accuracy after SFT, presenting the relative change caused by GRPO.
As introduced in Section 3.3, we further apply GRPO to fine-tune the models after SFT. Especially, GRPO based on RMTP allows solution planning and verification to be jointly optimized for self-verifying policies. The full GRPO results are presented in Appendix E.3, and the main findings are presented below. Overall, RL does enable most models to better solve ID problems, yet such improvements arise from a superficial shift in the distribution of known reasoning skills.
How does RL improve reasoning accuracy? Figure 8 presents the performance of 4M and 16M models in Mult after GRPO, where the differences from SFT results are visualized. GRPO effectively enhances accuracy in solving ID-Hard problems, yet the change in OOD performance is marginal. Therefore, RL can optimize ID performance, while failing to generalize to OOD cases.
Does RL truly enhance verification? From the change of verification errors in Figure 8, we find that the false negative rate $e_{-}$ decreases along with an increase in the false positive rate $e_{+}$ . This suggests that models learn an optimistic bias, which avoids rejecting correct steps through a high false positive rate that bypasses verification. In other words, instead of truly improving the verifier (where $e_{-}$ and $e_{+}$ both decrease), RL mainly induces an error-type trade-off, shifting from false negatives ( $e_{+}$ ) to false positives ( $e_{-}$ ).
To explain this, we note that a high $e_{-}$ raises the computational cost (Proposition 1) and thus causes a significant performance loss under the limited budget of RL sampling, making reducing $e_{-}$ more rewarding than maintaining a low $e_{+}$ . Meanwhile, shifting the error type is easy to learn, achievable by adjusting only a few parameters in the output layer of the transformer.
Inspired by DeepSeek-R1 [5], we additionally examine how RL influences the frequency of reflective behavior. To simulate the natural distribution of human reasoning, we train models to perform optional detailed verification by adding examples of empty verification (in the same amount as the full verification) into reflective SFT. This allows the policy to optionally omit self-verification, usually with a higher probability than producing full verification, since empty verification is easier to learn. Consequently, we can measure the reflection frequency by counting the proportion of steps that include non-empty verification. Since models can implicitly omit binary verification by producing false positive labels, we do not explicitly examine the optional binary verification.
When does RL incentivize frequent reflection?
Figure 9 shows reflection frequency in Mult before and after GRPO, comparing exploratory ( $1.25$ ) and exploitative ( $1$ ) temperatures when sampling experience CoTs. With a temperature $1.25$ , GRPO elicits frequent reflection, especially on hard queries. However, reflection frequency remains low if using temperature $1$ . Additional results for other model sizes and Sudoku appear in Appendix E.3.3. In conclusion, RL can adapt reflection frequency to align with the exploratory ability of the planning policy $\pi$ , encouraging more reflection if the policy can potentially explore rewards. This helps explain why RL promotes frequent reflection in LLMs [5], as the flexibility of language naturally fosters exploratory reasoning.
<details>
<summary>x12.png Details</summary>
### Visual Description
## Heatmap Comparison: GRPO Effect on Digit Combinations
### Overview
The image presents three heatmaps comparing the frequency of digit combinations before and after applying the GRPO (presumably an optimization) algorithm. The heatmaps display the frequency of combinations of 'x' digits and 'y' digits, ranging from 1 to 10. The first heatmap shows the distribution "Before GRPO", the second "After GRPO" with a temperature of 1.25, and the third "After GRPO" with a temperature of 1.0. A white dashed L-shape is overlaid on each heatmap, highlighting a specific region.
### Components/Axes
* **X-axis:** "number of x's digits" (values 1 to 10)
* **Y-axis:** "number of y's digits" (values 1 to 10)
* **Heatmap Cell Values:** Frequency of the (x, y) digit combination.
* **Titles:** "Before GRPO", "After GRPO Temperature: 1.25", "After GRPO Temperature: 1.0"
* **Color Scale:** Darker shades represent lower frequencies, while lighter/brighter shades represent higher frequencies.
* **White Dashed L-Shape:** Highlights the region where x + y <= 10.
### Detailed Analysis
#### Heatmap 1: Before GRPO
* **General Trend:** The highest frequencies appear to be concentrated in the top-left corner, indicating that combinations with smaller numbers of digits are more frequent before GRPO.
* **Specific Values:**
* (1, 1): 7
* (1, 2): 23
* (1, 3): 15
* (2, 1): 21
* (2, 2): 23
* (2, 3): 24
* (10, 10): 2
* **Observations:** The frequencies generally decrease as the number of digits increases for both x and y.
#### Heatmap 2: After GRPO (Temperature: 1.25)
* **General Trend:** The frequencies are generally higher across the board compared to the "Before GRPO" heatmap. The distribution is more uniform, with less concentration in the top-left corner.
* **Specific Values:**
* (1, 1): 30
* (1, 2): 43
* (1, 3): 35
* (2, 1): 50
* (2, 2): 56
* (2, 3): 51
* (10, 10): 50
* **Observations:** The GRPO algorithm with a temperature of 1.25 seems to have increased the frequency of combinations with larger numbers of digits.
#### Heatmap 3: After GRPO (Temperature: 1.0)
* **General Trend:** The frequencies are significantly lower than the "After GRPO (Temperature: 1.25)" heatmap, and many combinations have a frequency of 0. The distribution is skewed towards the top-left corner, but less so than the "Before GRPO" heatmap.
* **Specific Values:**
* (1, 1): 0
* (1, 2): 9
* (1, 3): 5
* (2, 1): 19
* (2, 2): 14
* (2, 3): 14
* (10, 10): 2
* **Observations:** The GRPO algorithm with a temperature of 1.0 appears to have reduced the frequency of many combinations, especially those with larger numbers of digits.
### Key Observations
* The GRPO algorithm has a significant impact on the frequency distribution of digit combinations.
* The temperature parameter influences the effect of GRPO. A higher temperature (1.25) leads to a more uniform distribution with higher frequencies, while a lower temperature (1.0) leads to lower frequencies and a distribution skewed towards smaller digit combinations.
* The white dashed L-shape highlights the region where the sum of x and y digits is less than or equal to 10. The frequencies within this region are generally higher than those outside the region, especially in the "Before GRPO" and "After GRPO (Temperature: 1.0)" heatmaps.
### Interpretation
The heatmaps demonstrate the effect of the GRPO algorithm on the frequency of digit combinations. The algorithm aims to optimize the distribution of these combinations, and the temperature parameter controls the degree of optimization.
* **Before GRPO:** The initial distribution favors smaller digit combinations, likely due to a natural bias or prior distribution.
* **After GRPO (Temperature: 1.25):** The algorithm increases the frequency of larger digit combinations, leading to a more uniform distribution. This suggests that the algorithm is exploring a wider range of possibilities.
* **After GRPO (Temperature: 1.0):** The algorithm reduces the frequency of many combinations, potentially focusing on a smaller set of "optimal" combinations. The lower temperature may lead to a more focused search, resulting in a less diverse distribution.
The white dashed L-shape likely represents a constraint or a region of interest. The higher frequencies within this region suggest that the algorithm prioritizes combinations that satisfy this constraint.
In summary, the GRPO algorithm can be used to manipulate the frequency distribution of digit combinations, and the temperature parameter provides control over the exploration-exploitation trade-off. A higher temperature encourages exploration, while a lower temperature encourages exploitation of potentially optimal combinations.
</details>
Figure 9: The hot-maps of reflection frequencies of the 4M transformer in multiplication before and after GRPO using temperatures $1$ and $1.25$ , tested with RMTP execution. The $i$ -th row and $j$ -th column shows the frequency (%) for problems $xΓ y$ where $x$ has $j$ digits and $y$ has $i$ digits.
6 Conclusion and Discussion
In this paper, we provide a foundational analysis of self-verifying reflection in multi-step CoTs using small transformers. Through minimalistic prototypes of reflective reasoning (the RMTP and RTBS), we demonstrate that self-verification benefits both training and execution. Compared to natural-language reasoning based on LLMs, the proposed minimalistic framework performs effective reasoning and reflection using limited computational resources. We also show that RL fine-tuning can enhance the performance in solving in-distribution problems and incentivize reflective thinking for exploratory reasoners. However, the improvements from RL rely on shallow patterns and lack generalizable new skills. Overall, we suggest that self-verifying reflection is inherently beneficial for CoT reasoning, yet its synergy with RL fine-tuning is limited in superficial statistics.
Limitations and future work
Although the current training pipeline enables tiny transformers to reason properly through reflective CoTs, the generalization ability is still low and not improved in RL. Therefore, future work will extend reflection frameworks and explore novel training approaches. Observing the positive effect of learning self-verification, a closer connection between generative and discriminative reasoning may be the key to addressing this challenge. Additionally, how our findings transfer from small transformers to natural-language LLMs needs to be further examined. However, the diversity of natural language and high computational cost pose significant challenges to comprehensive evaluation, and our proposed framework does not sufficiently exploit the emergent linguistic ability of LLMs. To this end, we expect to investigate a more flexible self-verification framework with an efficient evaluator of natural-language reflection in future work.
Acknowledgments and Disclosure of Funding
We gratefully acknowledge Dr. Linyi Yang for providing partial computational resources.
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1. 1 Introduction
1. 2 Related works
1. 3 Reflective reasoning for transformers
1. 3.1 Reasoning formulation
1. 3.2 The framework of self-verifying reflection
1. 3.3 Training
1. 4 Theoretical results
1. 5 Experiments
1. 5.1 Results of supervised fine-tuning
1. 5.2 Results of reinforcement learning
1. 6 Conclusion and Discussion
1. A Notations
1. B Details of reinforcement learning
1. B.1 Proximal policy optimization
1. B.2 Group-reward policy optimization
1. B.3 Technical Implementation
1. C Theory
1. C.1 A general formulation of reasoning performance
1. C.1.1 Bellman equations in RMTP
1. C.1.2 Bellman equations in RTBS
1. C.2 Accuracy derivation in the simplified reasoning task
1. C.3 Derivation of Theorem 1
1. C.3.1 Proof of Proposition 3
1. C.3.2 Proof of Proposition 4
1. C.4 Derivation of RMTP reasoning cost
1. C.5 Considering posterior risks of rejected attempts
1. D Implementation details
1. D.1 Algorithmic descriptions of reflective reasoning
1. D.2 Example CoT data
1. D.2.1 Multiplication CoT
1. D.2.2 Sudoku CoT
1. D.2.3 Verification of reasoning steps
1. D.3 Model architectures and tokenization
1. D.4 Hyperparameters
1. D.5 Computational resources
1. E Supplementary results of experiments
1. E.1 Evaluation of LLMs
1. E.2 Results of supervised fine tuning
1. E.3 Results of GRPO
1. E.3.1 The verification errors after GRPO
1. E.3.2 The planning correctness rate after GRPO
1. E.3.3 Reflection frequency of optional detailed verification
1. E.4 Reflection frequency under controlled verification error rates
1. E.5 Results of PPO
Appendix A Notations
The notations used in the main paper are summarized in Table 1. Notations only appear in the appendix are not included.
Table 1: Notations in the main paper.
| ${Q}$ $\{{R}\}$ ${R}_{t}$ | The query of CoT reasoning The sequence of intermediate reasoning steps The $t$ -th intermediate step in CoT reasoning |
| --- | --- |
| ${A}$ | The answer of CoT reasoning. |
| $T$ | The number of steps (including the final answer) in an CoT |
| $\pi$ | The planning policy in MTP reasoning |
| ${s}_{t}$ | The $t$ -th state in CoT reasoning |
| $\mathcal{T}$ | The transition function in an MTP |
| β $\checkmark$ β | The special token as the positive label of verification. |
| β $Γ$ β | The special token as the negative label of verification |
| ${V}_{t}$ | The verification sequence for the proposed step ${R}_{t}$ . |
| $\mathcal{V}$ | The verifier such that ${V}_{t+1}\sim\mathcal{V}(Β·|{S}_{t},{R}_{t+1})$ |
| $\tilde{{R}_{t}}$ | The verified reasoning step, i.e. $({R}_{t},{V}_{t})$ |
| $\tilde{\mathcal{T}}$ | The reflective transition function in an RMTP |
| $\tilde{\mathcal{\pi}}$ | The self-verifying policy, i.e. $\{\pi,\mathcal{V}\}$ |
| $m$ | The RTBS width, i.e. maximal number of attempts on each state |
| $\mu$ | The probability of proposing a correct step on positive states |
| $e_{-}$ | The probability of instantly rejecting a correct step on positive states |
| $e_{+}$ | The probability of accepting an incorrect step on positive states |
| $f$ | The probability of instantly rejecting any step on negative states |
| $\alpha$ | The shorthand of $\mu e_{-}+(1-\mu)(1-e_{+})$ |
| $\rho(n)$ | The accuracy of non-reflective MTP reasoning |
| $\tilde{\rho}(n)$ | The accuracy of RMTP reasoning for queries with scale $n$ |
| $\tilde{\rho}_{m}(n)$ | The accuracy of RTBS reasoning with width $m$ for queries with scale $n$ |
Appendix B Details of reinforcement learning
This section introduces PPO and GRPO algorithms used in RL fine-tuning. We introduce PPO and GRPO under the context of MTP, which is described in Section 3.1. This also applies to RMTP reasoning in Section 3.2, as RMTP is a special MTP given the self-verifying policy $\tilde{\pi}$ and the reflective transition function $\tilde{\mathcal{T}}$ .
For any sequence ${X}$ of tokens, we additionally define the following notations: ${{X}^{[i]}}$ denotes the $i$ -th token, ${{X}^{[<i]}}$ ( ${{X}^{[β€ i]}}$ ) denotes the former $i-1$ ( $i$ ) tokens, and $|{X}|$ denotes the length (i.e., the number of tokens).
Both PPO and GRPO iteratively update the reasoning policy through online experience. Let $\pi_{\theta}$ to denote a reasoning policy parameterized by $\theta$ . On each iteration, PPO and GRPO use a similar process to update $\theta$ :
1. Randomly draw queries from the task or taring set, and apply the old policy $\pi_{\theta_{old}}$ to sample experience CoTs.
1. Use reward models to assign rewards to the experience CoTs. Let $\operatorname{ORM}$ and $\operatorname{PRM}$ be the outcome reward model and process reward model, respectively. For each CoT $({Q},{R}_{1},...,{R}_{T-1},A)$ , we obtain outcome rewards $r_{o}=\operatorname{ORM}(Q,A)$ and the process rewards $r_{t}=\operatorname{PRM}({S}_{t},{R}_{t+1})$ for $t=0,1,...,T-1$ (where ${R}_{T}=A$ ). In our case, we only use the outcome reward model and thus all process rewards are $0$ .
1. Then, $\theta$ is updated by maximizing an objective function based on the experience CoTs with above rewards. Especially, PPO additionally needs to update an value approximator.
B.1 Proximal policy optimization
PPO [29] is a classic RL algorithm widely used in various applications. It includes a value model $v$ to approximate the value function, namely the expected cumulated rewards:
$$
v({S}_{t},{{R}_{t}^{[<i]}})=\mathbb{E}_{\pi}\left(r_{o}+\sum_{k=t}^{T}r_{k}\right) \tag{7}
$$
Let $q_{t,i}(\theta)=\frac{\pi_{\theta}\left({{R}_{t}^{[i]}}\middle|{S}_{t},{{R}_{t}^{[<i]}}\right)}{\pi_{\theta_{old}}\left({{R}_{t}^{[i]}}\middle|{S}_{t},{{R}_{t}^{[<i]}}\right)}$ be the relative likelihood of the $i$ -th token in the $t$ -th step, and $\pi_{ref}$ be the reference model (e.g., the policy before RL-tuning). Then, the PPO algorithm maximizes
$$
\displaystyle J_{PPO}(\theta)= \displaystyle\mathbb{E}_{{Q}\sim P({Q}),\{{R}\}\sim\pi_{\theta_{old}}}\frac{1}{\sum_{t=1}^{T}|{R}_{t}|}\sum_{t=1}^{T}\sum_{i=1}^{|{R}_{t}|} \displaystyle\left\{\min\left[q_{t,i}(\theta)\hat{A}_{t,i},\operatorname{clip}\left(q_{t,i}(\theta),1-\varepsilon,1+\varepsilon\right)\hat{A}_{t,i}\right]-\beta\mathbb{D}_{KL}\left[\pi_{\theta}\|\pi_{ref}\right]\right\}. \tag{8}
$$
Here, $\hat{A}_{t,i}$ is the advantage of the $i$ -th token in step $t$ , computed using the value model $v$ . For example, $\hat{A}_{t,i}=v({S}_{t},{{R}_{t}^{[<i]}},{{R}_{t}^{[i]}})-v({S}_{t},{{R}_{t}^{[<i]}})$ is a simple way to estimate advantage. In practice, advantages can be estimated using the general advantage estimation (GAE) [28].
The value model $v$ is implemented using the same architecture as the reasoner except for the output layer, which is replaced by a linear function that outputs a scalar value. The value model is initialized using the same parameters as the reasoner, apart from the output layer. Assuming that $v$ is parameterized by $\omega$ , we learn $v$ by minimizing the temporal-difference error:
$$
J_{v}(\omega)=\mathbb{E}_{{Q}\sim P({Q}),{R}\sim\pi_{\theta_{old}}}\sum_{t=1}^{T}\sum_{i=1}^{|{R}_{t}|}\left(v_{\omega}({S}_{t},{{R}_{t}^{[<i]}})-v_{\omega_{old}}({S}_{t+1})\right)^{2}. \tag{9}
$$
Although PPO proves effective in training LLMs [23], we deprecate using it in training tiny transformers due to the difficulty of learning the value function. Since the value model $v$ is also a tiny transformer, its weakness in model capacity severely compromise the precision of value approximation, leading to unreliable advantage estimation.
B.2 Group-reward policy optimization
PPO requires learning an additional value model, which can be expensive and unstable. Alternatively, GRPO [31] directly computes the advantages using the relative rewards from a group of $G$ solutions. For each query ${Q}$ , it samples a group of $G$ solutions:
$$
\{{R}_{g}\}=({R}_{g,1},\ldots,{R}_{g,T_{g}-1},A_{g})\sim\pi_{\theta_{old}},\qquad\text{for}\ g=1,\ldots,G. \tag{10}
$$
In this group, each solution $\{{R}_{g}\}$ contains $T_{g}$ steps, where the answer $A_{g}$ is considered as the final step ${R}_{g,T_{g}}$ . Using the reward models, we obtain process rewards $\boldsymbol{r}_{p}:=\{(r_{g,1},...,r_{g,T_{g}})\}_{g=1}^{G}$ and outcome rewards $\boldsymbol{r}_{o}:=\{r_{g,o}\}_{g=1}^{G}$ . Then, GPRO computes the normalized rewards, given by:
$$
\tilde{r}_{g,t}=\frac{r_{g,t}-\operatorname{mean}\boldsymbol{r}_{p}}{\operatorname{std}\boldsymbol{r}_{p}},\ \tilde{r}_{g,o}=\frac{r_{g,o}-\operatorname{mean}\boldsymbol{r}_{o}}{\operatorname{std}\boldsymbol{r}_{o}} \tag{11}
$$
Afterwards, the advantage of step $t$ in the $g$ -th solution of the group is $\hat{A}_{g,t}=\tilde{r}_{g,o}+\sum_{k=t}^{T_{g}}\tilde{r}_{g,t^{\prime}}$ . Let $q_{g,t,i}(\theta)=\frac{\pi_{\theta}\left({{R}_{g,t}^{[i]}}\middle|{S}_{g,t},{{R}_{g,t}^{[<i]}}\right)}{\pi_{\theta_{old}}\left({{R}_{g,t}^{[i]}}\middle|{S}_{g,t},{{R}_{g,t}^{[<i]}}\right)}$ be the relative likelihood of the $i$ -th token in the $t$ -th step from the $g$ -th solution. Then, the GRPO objective is to maximize the following:
$$
\displaystyle J_{GRPO}(\theta)= \displaystyle\mathbb{E}_{{Q}\sim P({Q}),\{{R}_{g}\}\sim\pi_{old}}\frac{1}{G}\sum_{g=1}^{G}\frac{1}{\sum_{t=1}^{T_{g}}|\tau^{(t)}|}\sum_{t=1}^{T_{g}}\sum_{i=1}^{|{R}_{g,t}|} \displaystyle\left\{\min\left[q_{g,t,i}(\theta)\hat{A}_{g,t},\operatorname{clip}\left(q_{g,t,i}(\theta),1-\varepsilon,1+\varepsilon\right)\hat{A}_{g,t}\right]-\beta\mathbb{D}_{KL}\left[\pi_{\theta}\|\pi_{ref}\right]\right\} \tag{12}
$$
B.3 Technical Implementation
We made two technical modifications that make RL more suitable in our case, described in the following.
First, in RMTP, we mask off the advantage of rejected steps, while the advantage of self-verification labels is reserved. This prevents the algorithm from increasing the likelihood of rejected steps, allowing the planning policy $\pi$ to be properly optimized. In practice, we find this modification facilitates the training of models that perform mandatory detailed verification. Otherwise, RL could make the reasoner excessively rely on reflection, leading to CoTs that are unnecessarily long.
Second, we employ an early-truncating strategy when sampling trajectories in training. If the model has already made a clear error at some step (detected using an oracle process reward model), we truncate the trajectory as it is impossible to find a correct answer. This avoids unnecessarily punishing later steps due to previous deviations, as some later steps may be locally correct in their own context. Empirically, we find this modification reduces the training time required to reach the same performance, while the difference in final performance is marginal.
Appendix C Theory
C.1 A general formulation of reasoning performance
Let $\mathcal{S}$ denote the state space and $\mathcal{A}$ denote the answer space. We use $\mathcal{A}_{{Q}}βeq\mathcal{A}$ to denote the set of correct answers for some input query ${Q}$ . Given any thought state ${S}$ , the accuracy, namely the probability of finding a correct answer, is denoted as
$$
\rho_{{Q}}({S})=p_{({R}_{t+1},{R}_{t+2},\ldots,{A})\sim\pi}({A}\in\mathcal{A}_{{Q}}\mid{S}_{t}={S}) \tag{13}
$$
C.1.1 Bellman equations in RMTP
By considering the reasoning correctness as the binary outcome reward, we may use Bellman equations [33] to provide a general formulation of the reasoning performance for arbitrary MTPs (RMTP). For simplicity, we use ${S}$ , ${R}$ , and ${S}^{\prime}$ to respectively denote the state, step, and next state in a transition.
Initially, in the absence of a trace-back mechanism, the accuracy $\rho_{{Q}}(s)$ can be interpreted as the value function when considering the MTP as a goal-directed decision process. For simplicity, we denote the transition probability drawn from the reasoning dynamics $\mathcal{T}$ as $p({S^{\prime}}\mid{S},{R})$ . In non-reflective reasoning, the state transition probability $p({S^{\prime}}\mid{S})$ can be expressed as:
$$
p({S^{\prime}}|{S})=\sum_{{R}}p({S^{\prime}}|{S},{R})\pi({R}|{S}) \tag{14}
$$
When using RMTP execution, assuming that $\xi({S},{R}):=p_{{V}\sim\mathcal{V}}(\text{``$Γ$''}β{V}\mid{S},{R})$ represents the probability of rejecting the step ${R}$ , we have:
$$
p({S^{\prime}}|{S})=\begin{cases}\sum_{R}\pi({R}|{S})(1-\xi({S},{R}))p({S^{\prime}}|{S},{R}),&\text{if }{S^{\prime}}\neq{S}\\
\sum_{R}\pi({R}|{S})\left((1-\xi({S},{R}))p({S^{\prime}}|{S},{R})+\xi({S},{R})\right),&\text{if }{S^{\prime}}={S}\end{cases} \tag{15}
$$
Consequently, the Bellman equation follows:
$$
\rho_{{Q}}({S})=\begin{cases}1,&\text{if }{S}\in\mathcal{A}_{{Q}}\\
0,&\text{if }{S}\in\mathcal{A}\setminus\mathcal{A}_{{Q}}\\
\sum_{{S^{\prime}}}\rho_{{Q}}({S^{\prime}})p({S^{\prime}}\mid{S}),&\text{if }s\in\mathcal{S}\setminus\mathcal{A}\end{cases} \tag{16}
$$
C.1.2 Bellman equations in RTBS
Let $m$ denote the number of attempts at each state, and let $\phi({S})$ represent the failure probability (i.e., the probability of tracing back after $m$ rejected steps) at state ${S}$ . The probability of needing to retry a proposed step due to instant rejection or recursive rejection is given by:
$$
\epsilon({S})=\sum_{{R}}\pi({R}|{S})\left(\xi({S},{R})+\sum_{{S^{\prime}}}(1-\xi({S},{R}))p({S^{\prime}}|{S},{R})\phi({S^{\prime}})\right) \tag{17}
$$
The failure probability is then given by $\phi({S})=\epsilon^{m}({S})$ . When there are $k$ attempts remaining at the current state ${S}$ , we denote the accuracy as $\rho_{x}({S},k)$ , given by:
$$
\rho_{Q}({S},k)=\begin{cases}\epsilon({S})\rho_{x}({S},k-1)+\sum_{{R}}\pi({R}|{S})(1-\xi({S},{R}))\sum_{{S^{\prime}}}p({S^{\prime}}|{S},{R})\rho_{Q}({S^{\prime}}),&k>0\\
0,&k=0\end{cases} \tag{18}
$$
It follows that $\rho_{x}({S})=\rho_{x}({S},m)$ . This leads to a recursive formulation that ultimately results in the following equations for each $sβ\mathcal{S}$ :
$$
\displaystyle\epsilon({S})=\sum_{{R}}\pi({R}|{S})\left(\xi({S},{R})+\sum_{{S^{\prime}}}(1-\xi({S},{R}))p({S^{\prime}}|{S},{R})\epsilon^{m}({S^{\prime}})\right), \displaystyle\rho_{x}({S})=\frac{1-\epsilon^{m}({S})}{1-\epsilon({S})}\sum_{{S^{\prime}}}\rho_{Q}({S^{\prime}})\pi({R}|{S})(1-\xi({S},{R}))p({S^{\prime}}|{S},{R}). \tag{19}
$$
C.2 Accuracy derivation in the simplified reasoning task
In the following, we derive the accuracy of reflective reasoning with and without the trace-back search, given the simplified reasoning task in Section 4. For each proposed step on a correct state, we define several probabilities to simplify notations: $\alpha:=\mu e_{-}+(1-\mu)(1-e_{+})$ is the probability of being instantly rejected; $\beta=\mu(1-e_{-})$ is the probability of being correct and accepted; $\gamma=(1-\mu)e_{+}$ is the probability of being incorrect but accepted. Note that $\beta$ here no longer refers to the KL-divergence factor in Appendix B.
**Proposition 2**
*The RTMP accuracy $\tilde{\rho}(n)$ for problems with a scale of $n$ is
$$
\tilde{\rho}(n)=\left(\frac{\beta}{1-\alpha}\right)^{n} \tag{21}
$$
Let $m$ be the width of RTBS. Let $\delta_{m}(n)$ and $\epsilon_{m}(n)$ be the probability of a proposed step being rejected (either instantly or recursively) on a correct state and incorrect state of scale $n$ , respectively. We have $\sigma_{m}(0)=\epsilon_{m}(0)=0$ and the following recursive equations for $n>0$ :
$$
\displaystyle\delta_{m}(n)=\alpha+\beta(\delta_{m}(n-1))^{m}+\gamma(\epsilon_{m}(n-1))^{m} \displaystyle\epsilon_{m}(n)=f+(1-f)\left(\epsilon_{m}(n-1)\right)^{m} \tag{22}
$$
Then, the RTBS accuracy $\tilde{\rho}_{m}(n)$ for problems with a scale of $n$ is given by
$$
\tilde{\rho}_{m}(n)=\prod_{t=1}^{n}\sigma_{m}(t),\qquad\text{where }\sigma_{m}(t)=\beta\sum_{i=0}^{m-1}(\delta_{m}(t))^{i}=\frac{1-(\delta_{m}(t))^{m}}{1-\delta_{m}(t)}\beta. \tag{24}
$$
In addition, $\delta_{m}(n)$ , $\epsilon_{m}(n)$ and $\sigma_{m}(n)$ all motonously increase and converge in relation to $n$ .*
* Proof*
We first consider reasoning through RTBS. Let $\phi_{m}(n)$ and $\psi_{m}(n)$ denote the probabilities of failure (reaching the maximum number of attempts) in correct and incorrect states, respectively. Let $\tilde{\rho}_{i|m}(n)$ indicate the accuracy after the $i$ attempts at the current sub-problem of scale $n$ . Therefore, we have $\tilde{\rho}_{m}(n)=\tilde{\rho}_{0|m}(n)$ and $\tilde{\rho}_{m|m}(n)=0$ . At a correct state, we have the following possible cases: - A correct step is proposed and instantly accepted with probability $\beta=\mu(1-e_{-})$ . In this case, the next state has a scale of $n-1$ , which is correctly solved with probability $\rho_{0|m}(n-1)$ and fails (i.e., is recursively rejected) with probability $\phi_{m}(n-1)$ .
- A correct step is proposed and instantly rejected with probability $\mu e_{-}$ .
- An incorrect step is proposed and instantly accepted with probability $\gamma=(1-\mu)e_{+}$ . In this scenario, the next state has a scale of $n-1$ , which fails with probability $\psi_{m}(n-1)$ .
- An incorrect step is proposed and instantly rejected with probability $\beta=(1-\mu)(1-e_{+})$ . Thus, we have a probability of $\alpha=\mu e_{-}+(1-\mu)(1-e_{+})$ to instantly reject the step, and a probability of $\beta\phi_{m}(n-1)+\gamma\psi_{m}(n-1)$ to recursively reject the step. Therefore, the overall probability of rejecting an attempt on correct states is:
$$
\delta_{m}(n)=\alpha+\beta\phi_{m}(n-1)+\gamma\psi_{m}(n-1). \tag{25}
$$
Since failure occurs after $m$ rejections, we have:
$$
\phi_{m}(n)=\left(\delta_{m}(n)\right)^{m} \tag{26}
$$ At an incorrect state, we have a probability $f$ to instantly reject a step. Otherwise, we accept the step, and the next state fails with probability $\psi_{m}(n-1)$ . Therefore, the overall probability of rejecting an attempt for incorrect states is:
$$
\epsilon_{m}(n)=f+(1-f)\psi_{m}(n-1). \tag{27}
$$
Similarly, we obtain:
$$
\psi_{m}(n)=\left(\epsilon_{m}(n)\right)^{m} \tag{28}
$$ By substituting Equations (26) and (28) into Equations (25) and (27), we obtain Equations (22) and (23). If an attempt is rejected (either instantly or recursively), we initiate another attempt which solves the problem with a probability of $\rho_{i+1|m}(n)$ . Therefore, we have the recursive form of the accuracy, given by:
$$
\tilde{\rho}_{i|m}(n)=\beta\tilde{\rho}_{0|m}(n-1)+\delta(n,m)\tilde{\rho}_{i+1|m}(n) \tag{29}
$$
Thus, we can expand $\tilde{\rho}_{m}(n)$ as:
$$
\displaystyle\tilde{\rho}_{m}(n) \displaystyle=\tilde{\rho}_{0|m}(n) \displaystyle=\beta\tilde{\rho}_{m}(n-1)+\delta_{m}(n)\tilde{\rho}_{1|m}(n) \displaystyle\cdots \displaystyle=(\beta+\delta_{m}(n)\beta+\delta_{m}^{2}(n)\beta+\cdots+\delta_{m}^{m}(n)\beta)\tilde{\rho}_{m}(n-1) \displaystyle=\sigma_{m}(n)\tilde{\rho}_{m}(n-1) \tag{30}
$$ Note that $n=0$ indicates that the state is exactly the outcome, which means $\tilde{\rho}_{m}(0)=1$ . Then, Equation (24) is evident given the recursive form in Equation (31). For reflective reasoning without trace-back, we can simply replace $\delta_{m}(n)$ with $\alpha$ in $\sigma_{m}(n)$ , as only instant rejections are allowed. We then set $mββ$ , leading to Equation (21).
Monotonicity
We first prove the monotonic increase of $\epsilon_{m}(n)$ . Equation (23) gives $\epsilon_{m}(n)=f+(\epsilon_{m}(n-1))^{m}$ and $\epsilon_{m}(n+1)=f+(\epsilon_{m}(n))^{m}$ for each $n>1$ . Therefore, if $\epsilon_{m}(n)β₯\epsilon_{m}(n-1)$ , we have:
$$
\epsilon_{m}(n+1)=f+(\epsilon_{m}(n))^{m}\geq f+(\epsilon_{m}(n-1))^{m}=\epsilon_{m}(n). \tag{32}
$$
Additionally, it is clear that $\epsilon_{m}(1)=fβ₯ 0=\epsilon_{m}(0)$ . Using mathematical induction, we conclude that $\epsilon_{m}(n+1)>\epsilon_{m}(n)$ for all $nβ₯ 0$ . The monotonicity of $\delta_{m}(n)$ can be proven similarly, and the monotonicity of $\sigma_{m}(n)$ is evident from that of $\delta_{m}(n)$ . Since $\delta_{m}(n)$ and $\epsilon_{m}(n)$ are probabilities, they are bounded in $[0,1]$ and thus converge monotonically. β
Illustration of accuracy curves
Using the recursive formulae in Proposition 2, we are able to implement a program to compute the reasoning accuracy in the simplified reasoning problem in Section 4 and thereby visualize the accuracy curves of various reasoning algorithms. For example, Figure 10 presents the reasoning curves given $\mu=0.8$ , $e_{-}=0.3$ , $e_{+}=0.2$ , and $f=0.8$ , which lead to $\alpha=0.4<f$ . For this example, we may observe the following patterns: 1) An overly small width $m$ in RTBS leads to poor performance; and 2) by choosing $m$ properly, $\tilde{\rho}_{m}(n)$ remains stable when $nββ$ . These observations are formally described and proved in Appendix C.3.
<details>
<summary>x13.png Details</summary>
### Visual Description
## Line Chart: Reasoning Accuracy vs. Problem Scale
### Overview
The image is a line chart comparing the reasoning accuracy (Ο) of different algorithms (RTBS with varying 'm' values, RMTP, and 'no reflection') against the problem scale (n). The chart shows how the accuracy of each algorithm changes as the problem scale increases.
### Components/Axes
* **X-axis:** problem scale (n), ranging from 0 to 50 in increments of 10.
* **Y-axis:** reasoning accuracy (Ο), ranging from 0.0 to 1.0 in increments of 0.2.
* **Legend:** Located on the right side of the chart, it identifies each line by algorithm:
* Blue: RTBS m=1
* Orange: RTBS m=2
* Green: RTBS m=3
* Red: RTBS m=4
* Purple: RTBS m=5
* Brown: RTBS m=6
* Black: RMTP
* Dashed Black: no reflection
### Detailed Analysis
* **RTBS m=1 (Blue):** Starts at approximately 0.95 and rapidly decreases to nearly 0 around n=10, remaining close to 0 for the rest of the scale.
* **RTBS m=2 (Orange):** Starts at approximately 0.95 and decreases to around 0.35 by n=50. The rate of decrease slows down as n increases.
* **RTBS m=3 (Green):** Starts at approximately 0.95 and decreases to around 0.73 by n=50. The rate of decrease slows down as n increases.
* **RTBS m=4 (Red):** Starts at approximately 0.95 and remains relatively constant at around 0.75 across the entire problem scale.
* **RTBS m=5 (Purple):** Starts at approximately 0.95 and decreases to around 0.65 by n=50. The rate of decrease slows down as n increases.
* **RTBS m=6 (Brown):** Starts at approximately 0.95 and decreases to around 0.25 by n=50. The rate of decrease slows down as n increases.
* **RMTP (Black):** Starts at 1.0 and decreases to approximately 0.32 by n=50. The rate of decrease slows down as n increases.
* **No reflection (Dashed Black):** Starts at approximately 0.95 and rapidly decreases to nearly 0 around n=20, remaining close to 0 for the rest of the scale.
### Key Observations
* RTBS with m=1 and 'no reflection' show the most significant drop in reasoning accuracy as the problem scale increases.
* RTBS with m=4 maintains the most consistent reasoning accuracy across all problem scales.
* The reasoning accuracy of RMTP decreases more gradually than RTBS with m=1 and 'no reflection'.
* As 'm' increases in RTBS, the reasoning accuracy tends to be higher for larger problem scales.
### Interpretation
The chart illustrates the impact of problem scale on the reasoning accuracy of different algorithms. RTBS with lower 'm' values and 'no reflection' are highly susceptible to decreasing accuracy as the problem scale increases, suggesting they are less robust for larger problems. RMTP and RTBS with higher 'm' values demonstrate better performance and maintain higher accuracy, indicating they are more suitable for handling larger problem scales. The consistent performance of RTBS with m=4 suggests it may be a good choice when a stable reasoning accuracy is desired, regardless of the problem scale. The data suggests that the choice of algorithm and its parameters significantly affects the ability to maintain reasoning accuracy as problem complexity grows.
</details>
Figure 10: The accuracy curves of non-reflective MTP $\rho(n)$ , RMTP $\tilde{\rho}(n)$ , and RTBS $\tilde{\rho}_{m}(n)$ , using $\mu=0.8$ , $e_{-}=0.3$ , $e_{+}=0.2$ , and $f=0.8$ .
Furthermore, in Figure 10 we see that a small $m$ stabilizes the drop of $\tilde{\rho}_{m}(n)$ when $n$ is large, yet it also makes $\tilde{\rho}_{m}(n)$ drop sharply in the area where $n$ is small. This indicates the potential of using an adaptive width in RTBS, where $m$ is set small when the current subproblem (state) requires a large number $n$ of steps to solve, and $m$ increases when $n$ is reduced by previous reasoning steps. Since this paper currently focuses on the minimalistic reflection framework, we expect to explore such an extension in future work.
C.3 Derivation of Theorem 1
Theorem 1 is obtained by merging the following Proposition 3 and Proposition 4, which also provide supplementary details on the non-trivial assumptions of factors $\mu$ , $e_{-}$ , and $e_{+}$ . Additionally, Proposition 4 also shows that there exists an ideal range of RTBS width $m$ such that stabilizes the drop of $\tilde{\rho}_{m}(n)$ when $nββ$ .
**Proposition 3 (RMTP Validity conditions)**
*For all $nβ₯ 0$ , we have $\tilde{\rho}(n)β₯\rho(n)\iff e_{-}+e_{+}β€ 1$ . Additionally, if $\mu>0$ and $e_{-}<1$ , then for all $nβ₯ 1$ we have that $\tilde{\rho}(n)=\rho(n)\iff e_{-}+e_{+}=1$ and $\tilde{\rho}(n)$ decreases strictly with either $e_{-}$ or $e_{+}$ .*
**Proposition 4 (RTBS Validity Condition)**
*Assuming $0<\mu<1$ , $e_{-}<1$ , and $e_{+}>0$ , then
$$
\lim_{n\to\infty}\frac{\tilde{\rho}_{m}(n)}{\tilde{\rho}(n)}>1\iff\left(m>\frac{1}{1-\alpha}\ \text{and}\ f>\alpha\right). \tag{33}
$$
Furthermore, we have
- $\lim_{nββ}\frac{\tilde{\rho}_{m}(n)}{\tilde{\rho}_{m}(n-1)}=1$ if $mβ[\frac{1}{\mu(1-e_{-})},\frac{1}{1-f}]$ .
- $\tilde{\rho}_{m}(n)$ increases strictly with $f$ for all $nβ₯ 2$ .*
The proof of propositions 3 and 4 is given in Appendix C.3.1 and Appendix C.3.2, which are based on the previous derivation in Appendix C.2.
C.3.1 Proof of Proposition 3
In any case, we have $\tilde{\rho}(0)=\rho(0)=1$ .
If $\mu=0$ or $e_{-}=1$ , we clearly have $\tilde{\rho}(n)=\rho(n)=0$ for $nβ₯ 1$ .
If $0>\mu$ and $e_{-}<1$ , we can transform $\tilde{\rho}(n)$ (given in Proposition 2) as:
$$
\tilde{\rho}(n)=\left(\frac{1}{1+\frac{e_{+}}{1-e_{-}}(\mu^{-1}-1)}\right)^{n}=\left(\frac{\mu(1-e_{-})}{\mu(1-e_{+}-e_{-})+e_{+}}\right)^{n}. \tag{34}
$$
This shows that $\rho(n)$ strictly decreases with both $e_{+}$ and $e_{-}$ , and $\tilde{\rho}(n)=\mu^{n}\iff e_{+}+e_{-}=1$ . Therefore, we also have $\tilde{\rho}(n)>\mu^{n}\iff e_{+}+e_{-}<1$ .
The Proposition is proved by combining all the above cases.
C.3.2 Proof of Proposition 4
The assumptions $0<\mu<1$ , $e_{-}<1$ , and $e_{+}>0$ ensure that $\beta>0$ and $\gamma>0$ . Proposition 2 suggests the monotonous convergence of $\delta_{m}(n)$ , $\epsilon_{m}(n)$ , and $\sigma_{m}(n)$ . For simplicity, we denote $\delta:=\lim_{nββ}\delta_{m}(n)$ , $\epsilon:=\lim_{nββ}\epsilon_{m}(n)$ , and $\sigma:=\lim_{nββ}\sigma_{m}(n)$ . From Equations (22) and (23), we have:
$$
\displaystyle\delta \displaystyle=\alpha+\beta\delta^{m}+\gamma\epsilon^{m} \displaystyle\epsilon \displaystyle=f+(1-f)\epsilon^{m} \tag{35}
$$
Note that $\epsilon=\delta=1$ gives the trivial solution of the above equations. However, there may exist another solution (if any) such that $\delta<1$ or $\epsilon<1$ under certain circumstances. Since $\epsilon_{m}(0)=0$ and $\delta_{m}(0)=0$ , the limits $\epsilon$ and $\delta$ take the smaller solution within $[0,1]$ . In the following, we first discuss when another non-trivial solution exists.
**Lemma 1**
*For any $mβ₯ 1$ , if $0β€ p<\frac{m-1}{m}$ , then $x=p+(1-p)x^{m}$ has a unique solution $x_{*}β[p,1)$ , which strictly increases with $p$ . Otherwise, if $\frac{m-1}{m}β€ pβ€ 1$ , the only solution in $[0,1]$ is $x_{*}=1$ .*
* Proof*
Define $F(x):=p+(1-p)x^{m}-x$ . We find:
$$
F^{\prime}(x)=m(1-p)x^{m-1}-1.
$$
It is observed that $F^{\prime}(x)$ increases monotonically with $x$ . Additionally, we have $F^{\prime}(0)=-1<0$ , $F^{\prime}(1)=m(1-p)-1$ , and $F(1)=0$ . We only consider the scenario where $p>0$ , since $x=0β[0,1)$ is obviously the unique solution. If $0β€ p<\frac{m-1}{m}$ , we have $1-p>\frac{1}{m}$ . This implies $F^{\prime}(1)>0$ . Combining $F^{\prime}(0)<0$ , there exists $\xiβ(0,1)$ such that $F^{\prime}(\xi)=0$ . As a result, $F(x)$ strictly decreases in $[0,\xi]$ and increases in $[\xi,1)$ . Therefore, we have $F(\xi)<F(1)=0$ . Since $F(p)=(1-p)p^{m}>0$ , we know that there exists a unique $x_{*}β[p,\xi)$ such that $F(x_{*})=0$ . If $\frac{m-1}{m}β€ pβ€ 1$ , we have $1-pβ€\frac{1}{m}$ and $F^{\prime}(1)β€ 0$ . In this case, $F^{\prime}(x)<0$ in $[0,1)$ due to the monotonicity of $F^{\prime}(x)$ . Thus, $F(x)>F(1)=0$ for all $xβ[0,1)$ . Therefore, $x_{*}=1$ is the only solution within $[0,1]$ . Now, we prove the monotonic increase of $x_{*}$ when $0β€ p<\frac{m-1}{m}$ . We have:
$$
\displaystyle\frac{\mathrm{d}x_{*}}{\mathrm{d}p}=1+m(1-p)x_{*}^{m-1}\frac{\mathrm{d}x_{*}}{\mathrm{d}p}-x_{*}^{m} \displaystyle\frac{\mathrm{d}x_{*}}{\mathrm{d}p}=\frac{1-x_{*}^{m}}{1-m(1-p)x_{*}^{m-1}}=\frac{1-x_{*}^{m}}{-F^{\prime}(x_{*})} \tag{37}
$$
The previous discussion shows that with $x_{*}<[p,\xi)$ for some $\xi$ such that $F^{\prime}(\xi)=0$ . Given that $F^{\prime}(x)$ increases monotonically, we have $F^{\prime}(x_{*})<0$ and thus $\frac{\mathrm{d}x_{*}}{\mathrm{d}p}>0$ . β
**Lemma 2**
*Assume $pβ₯ 0$ , $q>0$ , and $p+qβ€ 1$ . Then, the equation $x=p+qx^{m}$ has a unique solution $x_{*}β[0,1)$ , which increases monotonically with $pβ[0,1-q]$ .*
* Proof*
Define $F(x):=p+qx^{m}-x$ . Since $F(0)β₯ 0$ and $F(1)<0$ , there exists a solution $x_{*}β[0,1)$ . Since $F$ is convex, we know there is at most one other solution. Clearly, the other solution appears in $(1,+β)$ , since $F(+β)>0$ . Therefore, $F(x)=0$ must have a unique solution $x_{*}$ in $[0,1)$ . Additionally, $x_{*}$ must appear to the left of the minimum of $F$ , which yields $F^{\prime}(x_{*})<0$ . Using the Implicit Function Theorem, we write:
$$
\frac{\mathrm{d}x_{*}}{\mathrm{d}{p}}=\frac{1}{1-mqx_{*}^{m-1}}=-\frac{1}{F^{\prime}(x_{*})}>0 \tag{39}
$$
Thus, we conclude that $x_{*}$ increases monotonically with $p$ . β
Applying Lemmas 1 and 2 to Equation (36), we find that $\epsilon=1$ if and only if $fβ₯\frac{m-1}{m}$ ; otherwise, $\epsilon$ strictly increases with $p$ . Therefore, $f<\frac{m-1}{m}$ indicates that $\epsilon<1$ , leading to $(\alpha+\gamma\epsilon)+\gamma<1$ . Using Lemma 2 again in Equation (35), we have $f<\frac{m-1}{m}\implies\delta<1$ . Conversely, $fβ₯\frac{m-1}{m}$ yields $\epsilon=1$ . and thus $f,\alpha+\gammaβ₯\frac{m-1}{m}\implies\delta=1$ .
First, we consider the special case where $\delta=1$ , which occurs if both $f,\alpha+\gammaβ₯\frac{m-1}{m}$ , namely $mβ€\min\{\frac{1}{1-f},\frac{1}{\beta}\}$ . In this case, we write $\sigma=(1+\delta+Β·s+\delta^{m-1})\beta=m\beta$ . Therefore, we have:
| | $\displaystyle\lim_{nββ}\frac{{\tilde{\rho}(n)}}{\rho(n)}>1$ | $\displaystyle\iff\sigma>\frac{\beta}{1-\alpha}$ | |
| --- | --- | --- | --- |
This leads to the validity condition that $\frac{1}{1-\alpha}<mβ€\min\{\frac{1}{1-f},\frac{1}{\beta}\}$ .
Next, we consider the case where $\delta<1$ , which occurs when $f<\frac{m-1}{m}$ or $\alpha+\gamma<\frac{m-1}{m}$ . This leads to $\betaβ₯\frac{1}{m}>0$ , and we can write:
$$
\displaystyle\delta^{m}=\frac{1}{\beta}\left(\delta-\alpha-\gamma\epsilon^{m}\right), \displaystyle\sigma=\frac{1-\delta^{m}}{1-\delta}=\frac{(1-\delta^{m})\beta}{(1-\alpha)-(\beta\delta^{m}+\gamma\epsilon^{m})}. \tag{40}
$$
Then, we can derive:
| | $\displaystyle\lim_{nββ}\frac{{\tilde{\rho}(n)}}{\rho(n)}<1$ | $\displaystyle\iff\sigma>\frac{\beta}{1-\alpha}$ | |
| --- | --- | --- | --- |
Since we have assumed $\delta<1$ , we have $\epsilon=1>\delta$ if $fβ₯\frac{m-1}{m}$ ; otherwise if $f=\alpha<\frac{m-1}{m}$ , then $\epsilon=\delta$ leads to $\delta=\epsilon$ being a solution of Equation (35). Additionally, from Lemmas 1 and 2, we know that a higher $\alpha$ would increase $(\alpha+\gamma\epsilon)$ , which eventually raises $\delta$ above $\epsilon$ ; conversely, a lower $\alpha$ causes $\delta$ to drop below $\epsilon$ . To summarize, we have the following conditions when $\delta<1$ :
$$
\displaystyle 1\geq\epsilon>\delta \displaystyle\iff\left(\alpha+\gamma<\frac{m-1}{m}\leq f\right)\text{ or }\left(\alpha<f<\frac{m-1}{m}\right) \displaystyle\iff\left(\frac{1}{\beta}<m\leq\frac{1}{1-f}\right)\text{ or }\left(\alpha<f\text{ and }m>\frac{1}{1-f}\right) \tag{42}
$$
Combining the conditions when $\delta=1$ , we have:
$$
\displaystyle\lim_{n\to\infty}\frac{{\tilde{\rho}(n)}}{\rho(n)}>1 \displaystyle\iff \displaystyle\left(\frac{1}{1-\alpha}<m\leq\min\left\{\frac{1}{1-f},\frac{1}{\beta}\right\}\right)\text{ or }\left(\frac{1}{\beta}<m\leq\frac{1}{1-f}\right)\text{ or }\left(\alpha<f\text{ and }m>\frac{1}{1-f}\right) \displaystyle\iff \displaystyle m>\frac{1}{1-\alpha}\text{ and }f>\alpha \tag{44}
$$
Thus far, we have obtained Equation (33). Now, we start proving the two additional statements in Proposition 4.
First, we prove that $\frac{1}{\beta}β€ mβ€\frac{1}{1-f}$ ensures that $\sigma=1$ . First, if $\delta=1$ , we have $mβ€\frac{1}{\beta}β€\frac{1}{1-f}$ . In this case, $\sigma=m\beta$ , and thus $\sigma=1$ when $m=\frac{1}{\beta}$ . Alternatively, if $\delta<1$ , we have $m>\min\{\frac{1}{\beta},\frac{1}{1-f}\}$ . We can express that:
$$
\sigma=\frac{1-\delta^{m}}{1-\delta}\beta=\frac{\beta-(\delta-\alpha-\gamma\epsilon^{m})}{1-\delta}=1-\gamma\frac{1-\epsilon}{(1-\delta)(1-f)} \tag{46}
$$
Using Lemma 2, we know that $\delta$ increases with $\alpha+\gamma\epsilon$ , which increases with $\epsilon$ . Therefore, we have $\frac{\mathrm{d}\delta}{\mathrm{d}\epsilon}>0$ . Then, we obtain
$$
\mathrm{d}\sigma/\mathrm{d}\epsilon=\sum_{i=1}^{m}i\delta^{m-1}\beta\frac{\mathrm{d}\delta}{\mathrm{d}\epsilon}>0, \tag{47}
$$
$$
\displaystyle\sigma \displaystyle=\frac{1-\delta^{m}}{1-\delta}\beta=\frac{\beta-(\delta-\alpha-\gamma\epsilon^{m})}{1-\delta}=\frac{\alpha+\beta+\gamma-(1-\epsilon^{m})\gamma-\delta}{1-\delta} \displaystyle=\frac{1-\delta-\gamma(1-\frac{\epsilon-f}{1-f})}{1-\delta} \tag{48}
$$
Therefore, $\sigma$ increases with $\epsilon$ and reaches its maximum value of $1$ when $\epsilon=1$ . As a result, we conclude that $\frac{1}{\beta}β€ mβ€\frac{1}{1-f}$ ensures that $\sigma=1$ . Combining $\beta=\mu(1-e_{-})$ and $\sigma=\lim_{nββ}\sigma_{m}(n)=\lim_{nββ}\frac{\tilde{\rho}_{m}(n)}{\tilde{\rho}_{m}(n-1)}$ , we have proved that $\lim_{nββ}\frac{\tilde{\rho}_{m}(n)}{\tilde{\rho}_{m}(n-1)}=1$ if $mβ[\frac{1}{\mu(1-e_{-})},\frac{1}{1-f}]$ .
Next, we prove the monotonicity of $\tilde{\rho}_{m}(n)$ with respect to $f$ . To prove this, we first prove the monotonicity of $\epsilon_{m}(t)$ for all $t$ with respect to $f$ .
**Lemma 3**
*For $n>0$ , $\epsilon_{m}(t)$ as defined in Equation 23 increases strictly with $f$ .*
* Proof*
We regard
$$
\epsilon_{m}(0;f)\equiv 0,\qquad\epsilon_{m}(t;f)=f+(1-f)\bigl[\epsilon_{m}(t-1;f)\bigr]^{m} \tag{49}
$$
as a function of $f$ on $[0,1]$ . When $t=1$ we have
$$
\epsilon_{m}(1;f)=f+(1-f)\bigl[\epsilon_{m}(0;f)\bigr]^{m}=f, \tag{50}
$$
so $\frac{β\epsilon_{m}(1;f)}{β f}=1>0$ . Thus $\epsilon_{m}(1;f)$ is strictly increasing with $f$ . Further, assume for some $kβ₯ 1$ that
$$
0\leq\epsilon_{m}(k;f)<1\quad\text{and}\quad\frac{\partial\epsilon_{m}(k;f)}{\partial f}>0\quad\forall f\in[0,1]. \tag{51}
$$
Differentiate the recursion for $t=k+1$ :
$$
\epsilon_{m}(k+1;f)=f+(1-f)\bigl[\epsilon_{m}(k;f)\bigr]^{m}, \tag{52}
$$
$$
\frac{\partial\epsilon_{m}(k+1;f)}{\partial f}=1-\bigl[\epsilon_{m}(k;f)\bigr]^{m}+(1-f)\,m\bigl[\epsilon_{m}(k;f)\bigr]^{m-1}\,\frac{\partial\epsilon_{m}(k;f)}{\partial f}. \tag{53}
$$
By the inductive hypothesis, $\epsilon_{m}(k;f)<1$ implies $\bigl[\epsilon_{m}(k;f)\bigr]^{m}<1$ . Therefore, we have
$$
1-\bigl[\epsilon_{m}(k;f)\bigr]^{m}>0. \tag{54}
$$
Since $1-fβ₯ 0$ , $mβ₯ 1$ , $\bigl[\epsilon_{m}(k;f)\bigr]^{m-1}β₯ 0$ , and $\frac{β\epsilon_{m}(k;f)}{β f}>0$ , the second term is also positive. Hence
$$
\frac{\partial\epsilon_{m}(k+1;f)}{\partial f}>0, \tag{55}
$$
showing that $\epsilon_{m}(k+1;f)$ is strictly increasing. This completes the induction. β
Given Lemma 3, we can also prove the monotonicity of $\delta_{m}(t)$ using mathematical induction: It is easy to write that $\delta_{m}(2)=\alpha+\beta\alpha^{m}+\gamma f^{m}$ , showing that $\delta_{m}(2)$ increases strictly with $f$ . Then, for $t>2$ , assuming that $\delta_{m}(t-1)$ increases strictly with $f$ and using Given Lemma 3, we know that $\delta_{m}(t)$ increases strictly with $f$ from Equation (22).
Therefore, we have $\delta_{m}(1)=\alpha$ and that $\delta_{m}(t)$ strictly increases with $f$ for $nβ₯ 2$ . According to Equation (24), from the above monotonicity of $\delta_{m}(t)$ , it is obvious that $\sigma_{m}(t)$ increases with respect to $f$ for all $t$ . This gives the corollary that $\tilde{\rho}_{m}(n)$ increases with $f$ for all $n$ .
C.4 Derivation of RMTP reasoning cost
In this section, we derive how many steps it costs to find a correct solution in RMTP and thereby prove Proposition 1.
* Proposition1*
The probability of accepting the correct step after the $i$ -th attempt is given by $\alpha^{i-1}\beta$ . Assuming a maximum number of attempts $m$ , the number of attempts consumed at each step satisfies:
$$
\Pr(i\ \text{attempts}\mid\text{correct})=\frac{\alpha^{i-1}\beta}{\beta+\alpha\beta+\cdots+\alpha^{m-1}\beta}=\frac{(1-\alpha)\alpha^{i-1}}{1-\alpha^{m}} \tag{56}
$$
Therefore, the average number of attempts required for a correct reasoning step is given by
$$
A_{m}=\sum_{i=1}^{m}i\cdot\frac{(1-\alpha)\alpha^{i}}{1-\alpha^{m}}=\frac{1-\alpha}{1-\alpha^{m}}\sum_{i=1}^{m}i\alpha^{i-1} \tag{57}
$$ To simplify the summation expression $\sum_{i=1}^{m}i\alpha^{i-1}$ (where $0<\alpha<1$ ), we can use the method of telescoping series. Let $S=\sum_{i=1}^{m}i\alpha^{i-1}$ . We calculate $\alpha S$ :
$$
\alpha S=\sum_{i=1}^{m}i\alpha^{i} \tag{58}
$$
Thus,
$$
S-\alpha S=\sum_{i=1}^{m}i\alpha^{i-1}-\sum_{i=1}^{m}i\alpha^{i} \tag{59}
$$
This gives us
$$
(1-\alpha)S=1+\alpha+\alpha^{2}+\cdots+\alpha^{m-1}-m\alpha^{m}=\frac{1-\alpha^{m}}{1-\alpha}-m\alpha^{m} \tag{60}
$$
Rearranging, we have
$$
\sum_{i=1}^{m}i\alpha^{i-1}=\frac{1-\alpha^{m}-(1-\alpha)m\alpha^{m}}{(1-\alpha)^{2}} \tag{61}
$$
Thus, the average number of attempts can be further expressed as: $$
\displaystyle A_{m} \displaystyle=\frac{1-\alpha}{1-\alpha^{m}}\cdot\frac{1-\alpha^{m}-(1-\alpha)m\alpha^{m}}{(1-\alpha)^{2}} \displaystyle=\frac{1-\alpha^{m}-(1-\alpha)m\alpha^{m}}{(1-\alpha)(1-\alpha^{m})} \displaystyle=\frac{1}{1-\alpha}-m\frac{\alpha^{m}}{1-\alpha^{m}} \tag{62}
$$
Considering the limit as $mββ$ , it can be shown using limit properties that $\lim_{mββ}m\frac{\alpha^{m}}{1-\alpha^{m}}=0$ . If the correct solution is obtained (i.e., correct steps are accepted at each step), the average number of steps taken is given by
$$
\bar{T}=nA_{\infty}=\frac{n}{1-\alpha}=\frac{n}{1-\mu e_{-}-(1-\mu)(1-e_{+})}=\frac{n}{(1-\mu)e_{+}+\mu(1-e_{-})} \tag{66}
$$
β
C.5 Considering posterior risks of rejected attempts
Our previous analysis is based on a coarse binary partition of states (correct and incorrect) for each scale, which enhances clarity yet does not apply to real-world complexity. Therefore, we can introduce stronger constraints by taking into account the posterior distribution of states in $\mathcal{S}_{n}^{+}$ after multiple attempts. For example, if the state has produced several incorrect attempts on state ${S}$ (or rejected several correct attempts), it is more likely that the current state has not been well generalized by the model. Consequently, the chances of making subsequent errors increase. In this case, $\mu$ is likely to decrease with the number of attempts, while $e_{-}$ increases with the number of attempts. Thus, the probability of accepting the correct action will decrease as the number of attempts increases.
Therefore, we consider the scenario where the probabilities of error increase while the correctness rate $\mu$ drops after each attempt. We define $e_{i+},e_{i-},\mu_{i}$ , etc., to represent the probabilities related to the $i$ -th attempt, corresponding to the calculations of $\alpha_{i},\beta_{i},\gamma_{i}$ , etc. We have that $\beta_{i}=\mu_{i}(1-e_{i-})$ is monotonically decreasing, and $\gamma_{i}=(1-\mu_{i})e_{i+}$ is monotonically increasing with the index $i$ of the attempt. In this case, the derivation is similar to that of Proposition 2. Therefore, we skip all unnecessary details and present the results directly.
**Proposition 5**
*Given the above posterior risks, the RTMP accuracy $\tilde{\rho}(n)$ for problems with a scale of $n$ is
$$
\tilde{\rho}(n)=\left(\beta_{1}+\sum_{i=2}^{\infty}\beta_{i}\prod_{j=1}^{i-1}\alpha_{i}\right)^{n} \tag{67}
$$
Let $m$ be the width of RTBS. Let $\delta_{i|m}(n)$ denote the probability of a proposed step being rejected (either instantly or recursively) at the $i$ -th attempt on a correct state, and $\epsilon_{m}(n)$ follows the same definition as in Proposition 2. We have $\delta_{i|m}(0)=\epsilon_{m}(0)=0$ and the following recursive equations for $n>0$ :
$$
\displaystyle\delta_{i|m}(n)=\alpha_{i}+\beta_{i}\prod_{j=1}^{m}\delta_{j|m}(n-1)+\gamma_{i}(\epsilon_{m}(n-1))^{m},\qquad i=1,\cdots,m \displaystyle\epsilon_{m}(n)=f+(1-f)\left(\epsilon_{m}(n-1)\right)^{m} \tag{68}
$$
Then, the RTBS accuracy $\tilde{\rho}_{m}(n)$ for problems of scale $n$ is given by:
$$
\tilde{\rho}_{m}(n)=\prod_{t=1}^{n}\sigma_{m}(t),\qquad\text{where }\sigma_{m}(t)=\beta_{1}+\sum_{j=2}^{m}\beta_{j}\prod_{i=1}^{j-1}\delta_{i|m}(t,m)\beta. \tag{70}
$$
In addition, $\delta_{i|m}(n)$ , $\epsilon_{m}(n)$ , and $\sigma_{m}(n)$ all monotonically increase and converge with respect to $n$ .*
The theoretical result in this new setting becomes much more challenging to derive an exact validity condition that remains clear and understandable. However, it is still useful to derive a bound that sufficiently guarantees the effectiveness of reflection. In the following, we show that a sufficient condition becomes much stricter than that in Proposition 3.
**Proposition 6**
*Assume $\mu_{1}<1$ and $k=βf_{i}\frac{\beta_{i+1}}{\beta_{i}}$ is the lower bound of the decay rate of the probability of accepting the correct step in multiple attempts. Then, a sufficient condition for $\tilde{\rho}(n)>\rho(n)$ is:
$$
\frac{e_{1-}}{k(1-\mu_{1})}+\sup_{i}e_{i+}<1 \tag{71}
$$*
* Proof*
Considering $\alpha_{i}=\mu_{i}e_{i-}+(1-\mu_{i})(1-\max_{i}e_{i+})$ , let $\underline{\alpha}=(1-\mu_{1})(1-\sup_{i}e_{i+})$ be its lower bound. It can be seen that
$$
\displaystyle\beta_{1}+\alpha_{1}\beta_{2}+\alpha_{1}\alpha_{2}\beta_{3}+\cdots+\beta_{m}\prod_{i=1}^{m-1}\alpha_{i}\geq\sum_{j=1}^{m}(\underline{\alpha}k)^{m-1}\beta_{1}=\beta_{1}\frac{1-(\underline{\alpha}k)^{m}}{1-\underline{\alpha}k} \tag{72}
$$
As $mββ$ , the sufficient condition for reflection validity is:
$$
\displaystyle\frac{\beta_{1}}{1-\underline{\alpha}k}>\mu_{1} \displaystyle\iff \displaystyle\frac{1-e_{1-}}{1-k(1-\mu_{1})(1-\sup_{i}e_{i+})}>1 \displaystyle\iff \displaystyle(1-\mu_{1})(1-\sup_{i}e_{i+})>\frac{e_{-}}{k} \displaystyle\iff \displaystyle\frac{e_{1-}}{k}-\sup_{i}e_{i+}(1-\mu_{1})<1 \displaystyle\iff \displaystyle\frac{e_{1-}}{k(1-\mu_{1})}+\sup_{i}e_{i+}<1 \tag{73}
$$
β
Appendix D Implementation details
This section describes the details of the training datasets, model architectures, and hyper-parameters used in experiments. Our implementation derives the models architectures, pretraining, and SFT from LitGPT [1] (version 0.4.12) under Apache License 2.0.
D.1 Algorithmic descriptions of reflective reasoning
Algorithms 1 and 2 presents the pseudo-code of reasoning execution through RMTP and RTBS, respectively. In practice, we introduce a reflection budget $M$ to avoid infinite iteration. If reflective reasoning fails to find a solution within $M$ steps, the algorithm retrogrades to non-reflective reasoning.
To implement RTBS, we maintain a stack to store the reversed list of parental states, allowing them to be restored if needed. Different from our theoretical analysis, our practical implementation does not limit the number of attempts on the input query $Q$ (as long as the total budget $M$ is not used up) as $Q$ has no parent (i.e. the stack is empty).
Algorithm 1 Reflective reasoning through RMTP
0: the query ${Q}$ , the augmented policy $\tilde{\pi}=\{\pi,\mathcal{V}\}$ , transition function $\mathcal{T}$ , and reflective reasoning budget $M$
1: $tβ 0,{S}_{t}β Q$
2: repeat
3: Infer $R_{t+1}\sim\pi(Β·\mid{S}_{t})$
4: $Rejectβ\text{False}$
5: if $tβ€ M$ then
6: Infer ${V}_{t+1}\sim\mathcal{V}(Β·\mid{S}_{t},{R}_{t+1})$
7: $Rejectβ\text{True}$ if β $Γ$ β $β{V}_{t+1}$
8: if $Reject=\text{True}$ then
9: ${S}_{t+1}β{S}_{t}$
10: else
11: ${S}_{t+1}β\mathcal{T}({S}_{t},{R}_{t+1})$
12: if ${R}_{t+1}$ is the answer then
13: $Tβ t+1,{A}β{R}_{t+1}$
14: else
15: $tβ t+1$
16: until The answer $A$ is produced
17: return $A$
Algorithm 2 Reflective trace-back search
0: the query ${Q}$ , the augmented policy $\tilde{\pi}=\{\pi,\mathcal{V}\}$ , transition function $\mathcal{T}$ , search width $m$ , and reflective reasoning budget $M$
1: $tβ 0,{S}_{t}β Q$
2: $iβ 0$ {The index of attempts}
3: Initialize an empty stack $L$
4: repeat
5: Infer $R_{t+1}\sim\pi(Β·\mid{S}_{t})$
6: $iβ i+1$
7: $Rejectβ\text{False}$
8: if $tβ€ M$ then
9: Infer ${V}_{t+1}\sim\mathcal{V}(Β·\mid{S}_{t},{R}_{t+1})$
10: $Rejectβ\text{True}$ if β $Γ$ β $β{V}_{t+1}$
11: if $Reject=\text{True}$ then
12: if $i<m$ then
13: ${S}_{t+1}β{S}_{t}$
14: else
15: {Find a parent state that has remaining number of attempts.}
16: repeat
17: Pop $(s_{k},j)$ from $L$
18: ${S}_{t+1}β s_{k},iβ j$
19: until $i<m$ or $L$ is empty
20: else
21: Push $({S}_{t+1},i)$ into $L$
22: ${S}_{t+1}β\mathcal{T}({S}_{t},{R}_{t+1}),iβ 0$
23: if ${R}_{t+1}$ is the answer then
24: $Tβ t+1,{A}β{R}_{t+1}$
25: else
26: $tβ t+1$
27: until the answer $A$ is produced
28: return $A$
D.2 Example CoT data
We implement predefined programs to generate examples of CoTs and self-verification. Figure 11 presents the example reasoning steps (correct) for non-reflective training and the example detailed verification for reflective training. In our practical implementations, the reasoning steps include additional tokens, such as preprocessing and formatting, to assist planning and transition. To simplify transition function $\mathcal{T}$ , the example steps also include exactly how the states are supposed to be updated, which removes the task-specific prior in $\mathcal{T}$ .
<details>
<summary>x14.png Details</summary>
### Visual Description
## Diagram: State, Step, and Verification Process
### Overview
The image depicts a diagram illustrating a process involving a state, a step, and verification, within a context window. The process is divided into three main phases: "state St", "step Rt+1" (further divided into "planning" and "update"), and "verification Vt+1 (detailed)". Each phase contains specific actions or data transformations.
### Components/Axes
* **Horizontal Axis:** Represents the "Context window" and implies a temporal progression from left to right.
* **Vertical Sections:**
* **State St (Leftmost, Blue):** Represents the initial state, containing numerical data.
* **Step Rt+1 (Middle, Orange and Light Blue):** Divided into "planning" (Orange) and "update" (Light Blue) phases, showing actions and state modifications.
* **Verification Vt+1 (Rightmost, Pink):** Represents the verification phase, containing checks and reflections.
### Detailed Analysis
**1. State St (Blue Section):**
* Contains the following text within `<state>` tags:
* `145*86093`
* `+101500`
**2. Step Rt+1 (Orange and Light Blue Sections):**
* **Planning (Orange Section):**
* Contains the following text within `<action>` tags:
* `left * 8:`
* `- 40`
* `-32+4=36`
* `-8+3=11`
* `-1`
* `cumulate 1 1600000:`
* `-0+0=0`
* `-0+0=0`
* `-5+0=5`
* `-1+0=1`
* `-0+0=0`
* `-1+6=7`
* `-1`
* `-1`
* `get 11701500.`
* **Update (Light Blue Section):**
* Contains the following text within `<state>` tags:
* `145*6093`
* `+11701500`
**3. Verification Vt+1 (Pink Section):**
* Contains the following text within `<reflect>` tags:
* `<action>`
* `β:-β-β-β-β`
* `cumulate β:`
* `-β-β-β-β-β-β-β-β`
* `</action>`
* `<state>`
* `β*β+β`
* `</reflect>`
### Key Observations
* The diagram illustrates a sequential process, moving from an initial state through planning and update steps to a final verification stage.
* The "step" phase is divided into "planning" and "update," suggesting a two-stage process for modifying the state.
* The verification phase uses checkmarks (β) and other symbols to indicate the results of the verification process.
### Interpretation
The diagram likely represents a simplified model of a computational or decision-making process. The "state" represents the current data or situation, the "step" represents actions taken to modify the state, and the "verification" represents checks to ensure the actions were valid or successful. The "context window" suggests that this process is part of a larger sequence or system. The specific numerical values and operations within each phase would need further context to fully understand their meaning. The use of `<state>`, `<action>`, and `<reflect>` tags suggests a structured data format or programming paradigm.
</details>
(a) Mult
<details>
<summary>x15.png Details</summary>
### Visual Description
## Diagram: State Transition with Verification
### Overview
The image depicts a state transition diagram, showing the evolution of a "state" through a "step" involving preprocessing, planning & update, and finally a verification stage. The diagram illustrates how the state is transformed and validated within a "context window."
### Components/Axes
* **Horizontal Axis:** Represents the progression of the process, labeled as "Context window" with an arrow indicating the direction of flow.
* **State S<sub>t</sub>:** The initial state, represented as a 10x9 grid of numbers enclosed within `<state>` and `</state>` tags.
* **Step R<sub>t+1</sub>:** The processing step, divided into "preprocessing" and "planning & update" stages.
* **Preprocessing:** Lists operations like "rows," "cols," "blocks," and "reduce" along with associated numerical sequences.
* **Planning & Update:** Represents the updated state, again as a 10x9 grid of numbers enclosed within `<state>` and `</state>` tags.
* **Verification V<sub>t+1</sub> (detailed):** The verification stage, represented by a series of checkmarks arranged in a grid, enclosed within `<reflect>` and `</reflect>` tags.
### Detailed Analysis
**1. State S<sub>t</sub> (Leftmost, Light Blue):**
* The initial state is represented as a grid of numbers. The grid is 10 rows by 9 columns.
* The state is enclosed within XML-like tags: `<state>` and `</state>`.
* The numbers within the grid are:
* 3 5 7 8 2 0 4 1 9
* 0 0 1 7 9 0 0 0 6
* 8 6 9 1 5 4 0 7 2
* 0 8 2 4 1 7 5 0 0
* 5 1 3 9 6 8 7 2 4
* 9 7 4 0 0 5 1 0 8
* 2 3 0 5 4 1 9 8 0
* 0 4 5 6 8 0 2 3 1
* 1 9 8 3 7 2 0 4 5
**2. Step R<sub>t+1</sub> (Center, Divided into Light Blue and Light Orange):**
* This section represents the transformation of the state. It is divided into two sub-sections: "preprocessing" (light blue) and "planning & update" (light orange).
* **Preprocessing (Light Blue):**
* Lists operations performed on the state.
* "rows": Followed by "12345789 1679 12456789"
* "cols": Followed by "123589 13456789 12345789"
* "blocks": Followed by "1356789 1245789 124679"
* "reduce": Followed by "12345789 1679 12456789"
* **Planning & Update (Light Orange):**
* Represents the updated state after preprocessing.
* Enclosed within XML-like tags: `<state>` and `</state>`.
* The numbers within the grid are:
* 3 5 7 8 2 6 4 1 9
* 4 2 1 7 9 3 0 5 6
* 8 6 9 1 5 4 3 7 2
* 6 8 2 4 1 7 5 0 3
* 5 1 3 9 6 8 7 2 4
* 9 7 4 2 3 5 1 6 8
* 2 3 6 5 4 1 9 8 7
* 7 4 5 6 8 9 2 3 1
* 1 9 8 3 7 2 6 4 5
**3. Verification V<sub>t+1</sub> (Rightmost, Light Purple):**
* Represents the verification of the updated state.
* Consists of a grid of checkmarks. The grid is 10 rows by 9 columns.
* Enclosed within XML-like tags: `<reflect>` and `</reflect>`.
### Key Observations
* The diagram illustrates a sequential process: State -> Preprocessing -> Planning & Update -> Verification.
* The state is represented numerically, likely indicating a matrix or grid-based representation.
* The preprocessing step involves operations on "rows," "cols," and "blocks," suggesting a spatial or structural analysis of the state.
* The verification step uses checkmarks, indicating a binary pass/fail assessment of the updated state.
### Interpretation
The diagram likely represents a step in an algorithm or system that processes and validates states. The "state" could represent a variety of data structures, such as an image, a game board, or a set of parameters. The preprocessing step suggests feature extraction or transformation of the state. The planning & update step likely involves applying a model or rule set to modify the state. The verification step ensures that the updated state meets certain criteria or constraints. The "context window" suggests that this process is part of a larger sequence or iterative loop. The use of XML-like tags (`<state>`, `</state>`, `<reflect>`, `</reflect>`) suggests that this diagram is part of a larger system that uses a structured data format. The checkmarks in the verification stage indicate a binary assessment, suggesting a boolean outcome for each element of the state.
</details>
(b) Sudoku
Figure 11: Example reasoning steps with detailed verfication for integer multiplication and Sudoku.
D.2.1 Multiplication CoT
Each state is an expression $x_{t}Γ y_{t}+r_{t}$ , where $x_{t}$ and $y_{t}$ are the remaining values of two multiplicands, and $r_{t}$ is the cumulative result. For an input query $xΓ y$ , the expert reasoner assigns $x_{1}=x$ , $y_{1}=y$ , and $r_{1}=0$ .
For each step, the reasoner plans a number $uβ\{1,...,9\}$ to eliminate in $x_{t}$ (or $y_{t}$ ). Specifically, it computes $\delta=uΓ y_{t}$ or ( $\delta=uΓ x_{t}$ ). Next, it finds the digits in $x_{t}$ (or $y_{t}$ ) that are equal to $u$ and set them to $0$ in $x_{t+1}$ (or $y_{t+1}$ ). For each digit set to $0$ , the reasoner cumulates $\deltaΓ 10^{i}$ to $r_{t}$ , where $i$ is the position of the digit (starting from $0$ for the unit digit). An example of a reasoning step is shown in Figure 11(a). Such steps are repeated until either $x_{T}$ or $y_{T}$ becomes $0$ , then the answer is $r_{T}$ .
D.2.2 Sudoku CoT
Each state is a $9Γ 9$ matrix representing the partial solution, where blank numbers are represented by $0$ . On each step, the reasoner preprocesses the state by listing the determined numbers of each row, columns, and blocks. Given these information, the model reduces the blank positions that has only one valid candidate. If no blank can be reduced, the model randomly guess a blank position that has the fewest candidates. Such process continues until there exist no blanks (i.e., zeros) in the matrix.
An example of a reasoning step is shown in the right of Figure 11(b). The planned updates (i.e., which positions are filled with which numbers) is intrinsically included in a new puzzle, which is directly taken as the next state by the transition function $\mathcal{T}$ .
D.2.3 Verification of reasoning steps
Binary Verification
The Binary verification labels are generated using a rule-based checker of each reasoning step. In Multiplication, it simply checks whether the next state $x_{t+1}Γ y_{t+1}+r_{t+1}$ is equal to the current state $x_{t}Γ y_{t}+r_{t}$ . In Sudoku, it checks whether existing numbers in the old matrix are modified and whether the new matrix has duplicated numbers in any row, column, and block.
Detailed Verification
In Multiplication, we output a label for each elemental computation β addition or unit-pair product β is computed correctly. In Sudoku, we output a label for each position in the new matrix, signifying whether the number violates the rule of Sudoku (i.e. conflicts with other numbers in the same row, column, or block) or is inconsistent with the previous matrix. These labels are organized using a consistent format as the CoT data. Examples of detailed reflection for correct steps is in Figure 11(b). If the step contains errors, we replace the corresponding β $\checkmark$ β with β $Γ$ β.
D.3 Model architectures and tokenization
Table 2: The model architectures of models for the transitional implementation.
| Vocabulary size | 128 | | | | | |
| --- | --- | --- | --- | --- | --- | --- |
| Embedding size | 128 | 256 | 512 | 128 | 256 | 512 |
| Number of layers | 5 | | | | | |
| Number of attention Heads | 4 | 8 | 8 | 4 | 8 | 8 |
Our models architectures uses multi-head causal attention with LayerNorm [36] with implementation provided by LitGPT [1]. Table 2 specifies the architecture settings of transformer models with 1M, 4M, 16M parameters.
Tokenizers
We employ the byte-pair encoding algorithm [30] to train tokenizers on reasoning CoT examples for tiny transformers. Special tokens for reflection and reasoning structure (e.g., identifiers for the beginning and ending positions of states and reasoning steps) are manually added to the vocabulary. Since the vocabulary size is small (128 in our experiments), the learned vocabulary is limited to elemental characters and the high-frequency words for formatting.
D.4 Hyperparameters
Table 3 presents the hyper-parameters used in training and testing the tiny-transformer models. In the following sections, we describe how these parameters are involved in our implementation.
Table 3: The main hyper-parameters used in this work.
| Task | Mult | Sudoku | | | | |
| --- | --- | --- | --- | --- | --- | --- |
| Model size | 1M | 4M | 16M | 1M | 4M | 16M |
| Training CoT examples: $N_{CoT}$ | 32K | 36K | | | | |
| Total pretraining tokens: $N_{pre\_tok}$ | 1B | | | | | |
| Pretraining batch size: $B_{pre}$ | 128 | | | | | |
| Pretraining learning rate: $\eta_{pre}$ | 0.001 $β$ 0.00006 | | | | | |
| SFT batch size: $B_{SFT}$ | 128 | | | | | |
| SFT learning rate: $\eta_{SFT}$ | 0.001 $β$ 0.00006 | | | | | |
| Non-reflective SFT epochs: $E_{SFT}$ | 5 | | | | | |
| Reflective sampling temperature: Solving $\tau_{refl:s}$ | 0.75 | | | | | |
| Reflective sampling temperature: Proposing $\tau_{refl:p}$ | 1 | 1.25 | 1.5 | 1 | 1.25 | 1.5 |
| Reflective SFT epochs: $E_{RSFT}$ | 3 | | | | | |
| PPO replay buffer size: $N_{PPO:buf}$ | 512 | | | | | |
| GRPO replay buffer size: $N_{GRPO:buf}$ | 1024 | | | | | |
| RL sampling interval: $E_{RL:int}$ | 4 | | | | | |
| RL sampling temperature: Planning $\tau_{RL:\pi}$ | 1.25 | 1 | 1.25 | 1.25 | | |
| RL sampling temperature: Feedback $\tau_{RL:\pi_{f}}$ | 1 | | | | | |
| RL clipping factor: $\varepsilon$ | 0.1 | | | | | |
| RL KL-divergence factor: $\beta$ | 0.1 | | | | | |
| GRPO group size: $G$ | 8 | | | | | |
| RL total epochs: $E_{RL}$ | 512 | | | | | |
| RL learning rate: $\eta_{RL}$ | 0.00005 $β$ 0.00001 | | | | | |
| PPO warm-up epochs: $E_{PPO:warmup}$ | 64 | | | | | |
| Testing first-attempt temperature: $\tau_{\pi:first}$ | 0 | 1 | | | | |
| Testing revision temperature: $\tau_{\pi:rev}$ | 1 | | | | | |
| Testing verification temperature: $\tau_{\pi_{f}}$ | 0 | | | | | |
| Testing non-reflective steps $T$ : | 32 | | | | | |
| Testing reflective steps $\tilde{T}$ : | 64 | | | | | |
| RTBS width: $m$ | 4 | | | | | |
Non-reflective training
The pretraining and SFT utilize a dataset $N_{CoT}$ of CoT examples generated by an expert reasoning program. Pretraining treats these CoT examples as plain text and minimizes the cross-entropy loss for next-token prediction, using the batch size $B_{pre}$ and the learning rate $\eta_{pre}$ . The pretraining process terminates after predicting a total of $N_{pre\_tok}$ tokens. The non-reflective SFT uses the same dataset as that used in pretraining. It maximizes the likelihood of predicting example outputs (reasoning steps) from prompts (reasoning states), using the batch size $B_{SFT}$ and the learning rate $\eta_{SFT}$ . The total number of non-reflective SFT epochs is $E_{SFT}$ .
Reflective SFT
To perform non-reflective SFT, we use the model after non-reflective training to sample trajectories for each input query in the training set. The reflective sampling involves two decoding temperatures: the lower solving temperature $\tau_{refl:s}$ is used to walk through the solution path, while a higher proposing temperature $\tau_{refl:p}$ is used to generate diverse steps, which are fed into the reflective dataset. Then, the verification examples, which include binary or detailed labels, are generated by an expert verifier program. The reflective SFT includes $E_{RSFT}$ epochs, using the same batch size and learning rate as the non-reflective SFT.
Reinforcement learning
We use online RL algorithms as described in Appendix B, including PPO and GRPO. These algorithms include an experience replay buffer to store $N_{PPO:buf}$ and $N_{GRPO:buf}$ example trajectories, respectively. After every $E_{RL:int}$ epochs trained on the buffer, the buffer is updated by sampling new trajectories, using the temperature $\tau_{RL:\pi}$ for planning steps and the temperature $\tau_{RL:\tilde{\pi}}$ for reflective feedback. According to Equations 8 and 12, the hyper-parameters in both the PPO and GRPO objectives include the clipping factor $\varepsilon$ and the KL-Divergence factor $\beta$ . Additionally, GRPO defines $G$ as the number of trajectories in a group. We run RL algorithms for $E_{RL}$ epochs, using the learning rate $\eta_{RL}$ . PPO involves $E_{PPO:warmup}$ warm-up epochs at the beginning of training, during which only the value model is optimized.
Testing
During testing, we execute the reasoner using three decoding temperatures: $\tau_{\pi:first}$ for the first planning attempt, $\tau_{\pi:rev}$ for the revised planning attempt after being rejected, and $\tau_{\pi_{f}}$ for self-verification. We use low temperatures to improve accuracy for more deterministic decisions, such as self-verifying feedback and the first attempt in Mult. We use higher temperatures for exploratory decisions, such as planning in Sudoku and revised attempts in Mult. We set the non-reflective reasoning budget to $T$ steps and the reflective reasoning budget to $\tilde{T}$ steps. If the reflective budget is exhausted, the reasoner reverts to non-reflective reasoning. We set the search width of RTBS to $m$ .
D.5 Computational resources
Since our models are very small, it is entirely feasible to reproduce all our results on any PC (even laptops) that has a standard NVIDIA GPU. Using our hyper-parameters, the maximum GPU memory used for training the 1M, 4M, and 16M models is approximately 4GB, 12GB, and 16GB, respectively, which can be easily reduced by using smaller batch sizes. To run multiple experiments simultaneously, we utilize cloud servers with a total of 5 GPUs (one NVIDIA RTX-3090 GPU and four NVIDIA A10 GPUs).
For each model size and task, a complete pipeline (non-reflective training, reflective training, and RL) takes about two days on a single GPU. This includes 1-2 hours for non-reflective training, 8-12 hours for data collection for reflective training, 1-2 hours for reflective SFT, 6-12 hours for RL, and 6-12 hours for testing.
Appendix E Supplementary results of experiments
In this section, we present supplementary results from our experiments: 1) we assess the reasoning accuracy of various large language models on integer multiplication and Sudoku tasks; 2) we report the accuracy outcomes of models after implementing different supervised fine-tuning strategies; 3) we provide full results of reasoning accuracy after GRPO; 4) we additionally provide the results of PPO, which is weaker than GRPO in reflective reasoning.
E.1 Evaluation of LLMs
In this section, we provide the reasoning accuracy of LLMs on Mult and Sudoku, including GPT-4o [22], OpenAI o3-mini [21], and DeepSeek-R1 [5]. Since GPT-4o is not a CoT-specialized model, we use the magic prompt βletβs think step-by-stepβ [13] to elicit CoT reasoning. For o3-mini and DeepSeek-R1, we only prompt with the natural description of the queries. As shown in Table 4, among these LLMs, OpenAI o3-mini produces the highest accuracy in both tasks.
To illustrate how well tiny transformers can do in these tasks, we also present the best performance (results selected from Tables 5 and 7) of our 1M, 4M, and 16M models for each difficulty level, respectively, showing a performance close to or even better than some of the LLM reasoners. For example, according to our GRPO results (see Table 7), our best 4M Sudoku reasoner performs (RTBS through optional detailed verification) equally well to OpenAI o3-mini, and our best 16M Mult reasoner (through binary verification) outperforms DeepSeek-R1 in ID difficulties. Note that this paper mainly focuses on fundamental analysis and does not intend to compete with the general-purpose LLM reasoners, which can certainly gain better accuracy if specially trained on our tasks. Such a comparison is inherently unfair due to the massive gap in resource costs and data scale. The purpose of these results is to show how challenging these tasks can be, providing a conceptual notion of how well a tiny model can perform.
Table 4: The accuracy (%) of GRT-4o, DeepSeek-R1, and OpenAI o3-mini in integer multiplication and Sudoku, compared with the best performance of our 1M (1M*), 4M (4M*), and 16M (16M*) transformers. The βOOD-Hardβ for LLMs only refers to the same difficulty as used in testing our tiny transformers, as OOD-Hard questions may have been seen in the training of LLMs.
| Mult ID-Hard OOD-Hard | ID-Easy 32.6 18.6 | 73.2 97.2 96.4 | 100 77.0 61.4 | 96.8 52.7 3.7 | 96.2 77.0 5.8 | 98.7 81.1 9.4 | 99.7 |
| --- | --- | --- | --- | --- | --- | --- | --- |
| Sudoku | ID-Easy | 40.7 | 99.6 | 90.4 | 33.9 | 97.2 | 99.8 |
| ID-Hard | 2.8 | 52.8 | 4.4 | 0.4 | 58.1 | 72.2 | |
| OOD-Hard | 0.0 | 0.0 | 0.0 | 0.0 | 6.9 | 14.4 | |
E.2 Results of supervised fine tuning
Table 5 includes our complete results of reasoning accuracy after non-reflective and reflective SFT. As discussed in Section 3.1, our implementation uses Reduced states that maintain only useful information for tiny transformers. To justify this, we also test the vanilla Complete implementation, where each state ${S}_{t}=({Q}_{t},{R}_{1}...,{R}_{t-1})$ includes the full history of past reasoning steps. As a baseline, the Direct thought without intermediate steps is also presented.
Table 5: The reasoning accuracy (%) for 1M, 4M, and 16M transformers after SFT.
| 1M | Mult | ID Easy | 21.8 | 10.6 | 23.6 | 95.8 | 94.5 | 93.4 | 22.0 | 33.4 | 24.2 |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| ID Hard | 3.0 | 1.4 | 2.0 | 52.7 | 44.6 | 35.5 | 2.2 | 4.8 | 2.8 | | |
| OOD Hard | 1.4 | 0.3 | 1.0 | 3.7 | 2.2 | 1.2 | 1.0 | 0.8 | 0.4 | | |
| Sudoku | ID Easy | 2.8 | 0 | 1.4 | 33.0 | 32.4 | 2.4 | 17.4 | 18.7 | 19.4 | |
| ID Hard | 0 | 0 | 0 | 0.3 | 0.1 | 0 | 0.1 | 0 | 0 | | |
| OOD Hard | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | | |
| 4M | Mult | ID Easy | 15.6 | 17.2 | 92.0 | 97.7 | 97.6 | 97.3 | 94.5 | 93.8 | 93.3 |
| ID Hard | 1.7 | 1.9 | 37.3 | 56.9 | 62.2 | 53.0 | 43.4 | 47.6 | 42.4 | | |
| OOD Hard | 1.2 | 1.0 | 2.2 | 2.9 | 1.8 | 1.1 | 3.7 | 3.3 | 2.7 | | |
| Sudoku | ID Easy | 13.0 | 3.9 | 52.2 | 92.1 | 96.8 | 96.0 | 54.4 | 81.9 | 88.5 | |
| ID Hard | 0.1 | 0 | 3.3 | 40.9 | 46.3 | 53.3 | 5.2 | 16.9 | 45.7 | | |
| OOD Hard | 0 | 0 | 0 | 0.4 | 4.0 | 6.7 | 0.0 | 1.1 | 2.0 | | |
| 16M | Mult | ID Easy | 15.1 | 59.2 | 99.2 | 98.8 | 98.9 | 98.8 | 99.2 | 99.5 | 98.5 |
| ID Hard | 1.6 | 9.6 | 65.9 | 65.2 | 76.7 | 74.9 | 65.9 | 76.4 | 73.5 | | |
| OOD Hard | 1.2 | 1.0 | 2.5 | 1.1 | 1.3 | 1.3 | 9.2 | 9.4 | 7.2 | | |
| Sudoku | ID Easy | 35.8 | 15.9 | 95.7 | 97.1 | 97.9 | 92.5 | 93.0 | 99.0 | 99.7 | |
| ID Hard | 0.4 | 0 | 48.8 | 50.1 | 53.1 | 54.8 | 46.9 | 57.9 | 70.7 | | |
| OOD Hard | 0 | 0 | 0.4 | 0.9 | 4.4 | 6.0 | 0.7 | 8.2 | 14.4 | | |
Reducing the redundancy of states in long CoTs benefits tiny transformers. The left three columns in Table 5 compare the above thought implementations for non-reflective models. We see that both direct and complete thoughts fail to provide an acceptable performance even in ID-Easy difficulty. This proves the importance of avoiding long-context inference by reducing redundancy in representing states. Considering the huge performance gap, we exclude the complete and direct implementations from our main discussion.
Estimated errors of self-verification
For RMTP and RTBS executions, we employ the oracle verifiers to maintain test-time statistics of the average $e_{-}$ and $e_{+}$ (see definition in Section 4) of reasoning states. The results are shown in Table 6, where we also present the difference in how much reflective reasoning raises the performance over non-reflective reasoning. We only count the errors in the first attempts on reasoning states to avoid positive bias, as the reasoner may be trapped in some state and repeat the same error for many steps.
Table 6: The percentage (%) of test-time verification errors (i.e., $e_{-}$ and $e_{+}$ ) after reflective SFT. Additionally, we compute $\Delta$ as the difference of how much reflective reasoning raises the performance over non-reflective reasoning, i.e. RMTP (RTBS) accuracy minus non-reflective accuracy.
| 1M ID Hard OOD Hard | Mult 3.8 16.4 | ID Easy 37.6 32.9 | 19.3 $-8.1$ $-1.5$ | 4.4 3.6 6.0 | $-1.3$ 33.0 22.5 | 14.9 $-17.2$ $-2.5$ | 4.9 0.9 13.6 | $-2.4$ 6.9 2.2 | 10.2 $+2.6$ $-0.2$ | 18.3 14.5 13.2 | $+11.4$ 7.4 2.4 | 24.4 $+0.6$ $-0.6$ | 19.4 | $+2.2$ |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| Sudoku | ID Easy | 9.9 | 35.2 | $-0.6$ | 31.1 | 43.9 | $-30.6$ | 87.1 | 0.1 | $+1.3$ | 85.1 | 0.1 | $+2$ | |
| ID Hard | 21.1 | 31.0 | $-0.2$ | 33.1 | 28.6 | $-0.3$ | 82.8 | 0 | $-0.1$ | 79.4 | 0 | $-0.1$ | | |
| OOD Hard | 60.3 | 7.5 | $0$ | 60.2 | 13.4 | $0$ | 87.9 | 0 | $0$ | 84.5 | 0 | $0$ | | |
| 4M | Mult | ID Easy | 25.1 | 5.9 | $-0.1$ | 58.1 | 8.9 | $-0.4$ | 30.4 | 3.7 | $-0.7$ | 28.7 | 7.5 | $-1.2$ |
| ID Hard | 2.4 | 23.6 | $+5.3$ | 26.0 | 30.8 | $-3.9$ | 3.3 | 25.1 | $+4.2$ | 10.0 | 29.3 | $-1.0$ | | |
| OOD Hard | 7.5 | 42.9 | $-1.1$ | 18.0 | 61.7 | $-1.8$ | 5.9 | 28.1 | $-0.4$ | 10.9 | 28.2 | $-1.0$ | | |
| Sudoku | ID Easy | 39.5 | 9.5 | $+4.7$ | 40.4 | 11.5 | $+3.9$ | 23.8 | 0.1 | $+27.5$ | 46.7 | 0.3 | $+34.1$ | |
| ID Hard | 41.3 | 1.9 | $+5.4$ | 56.0 | 6.7 | $+12.4$ | 17.3 | 0.2 | $+11.7$ | 22.1 | 0.3 | $+40.5$ | | |
| OOD Hard | 78.5 | 0.8 | $+3.6$ | 70.6 | 0.6 | $+6.3$ | 31.5 | 0.1 | $+1.1$ | 35.9 | 0.1 | $+2$ | | |
| 16M | Mult | ID Easy | 11.3 | 8.6 | $+0.1$ | 6.1 | 9.4 | $+0.0$ | 15.7 | 2.1 | $+0.3$ | 3.8 | 2.9 | $-0.7$ |
| ID Hard | 1.4 | 13.9 | $+11.5$ | 1.8 | 16.9 | $+9.7$ | 2.5 | 7.0 | $+10.5$ | 4.4 | 7.2 | $+7.6$ | | |
| OOD Hard | 1.3 | 86.4 | $+0.2$ | 1.5 | 88.2 | $+0.2$ | 8.5 | 18.3 | $+0.2$ | 11.7 | 19.7 | $-2$ | | |
| Sudoku | ID Easy | 40.1 | 3.3 | $+0.8$ | 10.1 | 4.7 | $-4.6$ | 6.6 | 1.7 | $+6$ | 9.1 | 6.4 | $+6.7$ | |
| ID Hard | 50.5 | 4.3 | $+3$ | 37.2 | 9.4 | $+4.7$ | 15.4 | 0.1 | $+11.0$ | 10.6 | 0.6 | $+23.8$ | | |
| OOD Hard | 75.2 | 4.2 | $+3.5$ | 65.0 | 3.1 | $+5.1$ | 28.3 | 0.1 | $+7.5$ | 24.8 | 0.0 | $+13.7$ | | |
Our full results provide more evidence for the findings discussed in Section 5.1:
- Learning to self-verify enhances non-reflective execution for 9 out of 12 models (2 verification types, 3 model sizes, and 2 tasks), such that accuracy does not decrease in any difficulty and increases in at least one difficulty.
- RMTP improves performance over non-reflective execution for all 4M and 16M models. However, RMTP based on binary verification fails to benefit the 1M models, which suffer from a high $e_{-}$ .
- 4M and 16M Sudoku models greatly benefit from RTBS, especially using detailed verification.
E.3 Results of GRPO
The complete results of models after GRPO are given in Table 7. To have a convenient comparison, Table 8 presents the difference of accuracy across Table 5 and Table 7, showing that the difference of accuracy is caused by GRPO.
Table 7: The accuracy (%) of the 1M, 4M, and 16M transformers after GRPO.
| Verification Type | None | Binary | Detailed | Optional Detailed | | | | | | | | |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| Reflective Execution | None | None | RMTP | RTBS | None | RMTP | RTBS | None | RMTP | RTBS | | |
| 1M | Mult | ID Easy | 52.6 | 96.2 | 95.9 | 95.7 | 53.0 | 49.5 | 45.1 | 48.6 | 47.7 | 48.8 |
| ID Hard | 11.6 | 50.0 | 44.0 | 42.0 | 11.4 | 9.7 | 8.1 | 12.2 | 12.7 | 12.6 | | |
| OOD Hard | 1.1 | 2.5 | 1.9 | 1.6 | 1.0 | 0.9 | 0.4 | 1.2 | 1.3 | 1.2 | | |
| Sudoku | ID Easy | 1.3 | 33.9 | 29.2 | 4.5 | 17.6 | 20.7 | 18.7 | 23.0 | 23.0 | 22.6 | |
| ID Hard | 0 | 0.4 | 0 | 0.2 | 0 | 0.1 | 0 | 0.1 | 0.1 | 0 | | |
| OOD Hard | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | | |
| 4M | Mult | ID Easy | 98.0 | 98.6 | 98.7 | 98.8 | 98.2 | 98.0 | 98.4 | 98.2 | 98.4 | 98.6 |
| ID Hard | 65.6 | 73.6 | 77.0 | 76.7 | 63.0 | 64.3 | 63.2 | 63.9 | 66.8 | 66.1 | | |
| OOD Hard | 2.3 | 2.7 | 2.7 | 2.3 | 5.8 | 5.3 | 5.3 | 3.3 | 3.2 | 3.3 | | |
| Sudoku | ID Easy | 58.7 | 93.8 | 97.2 | 96.7 | 57.8 | 85.3 | 92.2 | 77.0 | 94 | 98.2 | |
| ID Hard | 3.2 | 43.9 | 53.8 | 58.1 | 5.6 | 24.7 | 47.7 | 21.4 | 37.7 | 61.3 | | |
| OOD Hard | 0 | 0.4 | 4.9 | 6.9 | 0 | 0.4 | 2.0 | 0 | 1.8 | 4.2 | | |
| 16M | Mult | ID Easy | 99.8 | 99.2 | 99.2 | 99.1 | 99.7 | 99.6 | 99.4 | 99.2 | 99.4 | 99.3 |
| ID Hard | 77.2 | 75.2 | 81.1 | 79.6 | 76.3 | 77.8 | 77.6 | 75.9 | 78.4 | 77.7 | | |
| OOD Hard | 1.8 | 1.3 | 1.8 | 1.8 | 8.4 | 8.2 | 7.4 | 6.0 | 5.5 | 5.6 | | |
| Sudoku | ID Easy | 96.3 | 97.6 | 98.8 | 94.6 | 93.3 | 98.8 | 99.8 | 88.7 | 97.6 | 99.0 | |
| ID Hard | 51.3 | 51.7 | 58.0 | 62.3 | 46.7 | 60.4 | 72.2 | 42.2 | 57.3 | 70.9 | | |
| OOD Hard | 0.7 | 0.7 | 6.0 | 7.8 | 0.2 | 6.7 | 12.0 | 0.2 | 6.7 | 11.1 | | |
Table 8: The difference of accuracy (%) of the 1M, 4M, and 16M transformers after GRPO. Positive values mean that GRPO raises the accuracy of the models above SFT.
| 1M | Mult | ID Easy | $+29.0$ | $+0.4$ | $+1.4$ | $+2.3$ | $+31.0$ | $+16.1$ | $+20.9$ |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| ID Hard | $+9.6$ | $-2.7$ | $-0.6$ | $+6.5$ | $+9.2$ | $+4.9$ | $+5.3$ | | |
| OOD Hard | $+0.1$ | $-1.2$ | $-0.3$ | $+0.4$ | $0.0$ | $+0.1$ | $0.0$ | | |
| Sudoku | ID Easy | $-0.1$ | $+0.9$ | $-3.2$ | $+2.1$ | $+0.2$ | $+2.0$ | $-0.7$ | |
| ID Hard | $0.0$ | $+0.1$ | $-0.1$ | $+0.2$ | $-0.1$ | $+0.1$ | $0.0$ | | |
| OOD Hard | $0.0$ | $0.0$ | $0.0$ | $0.0$ | $0.0$ | $0.0$ | $0.0$ | | |
| 4M | Mult | ID Easy | $+6.0$ | $+0.9$ | $+1.1$ | $+1.5$ | $+3.7$ | $+4.2$ | $+5.1$ |
| ID Hard | $+28.3$ | $+16.7$ | $+14.8$ | $+23.7$ | $+19.6$ | $+16.7$ | $+20.8$ | | |
| OOD Hard | $0.1$ | $-0.2$ | $+0.9$ | $+1.2$ | $+2.1$ | $+2.0$ | $+2.6$ | | |
| Sudoku | ID Easy | $+6.5$ | $+1.7$ | $+0.4$ | $+0.7$ | $+3.4$ | $+3.4$ | $+3.7$ | |
| ID Hard | $-0.1$ | $+3.0$ | $+7.5$ | $+4.8$ | $+0.4$ | $+7.8$ | $+2.0$ | | |
| OOD Hard | $0.0$ | $+0.4$ | $+4.9$ | $+6.9$ | $0$ | $-0.7$ | $0$ | | |
| 16M | Mult | ID Easy | $+0.6$ | $+0.4$ | $+0.3$ | $+0.3$ | $+0.5$ | $+0.1$ | $+0.9$ |
| ID Hard | $+11.3$ | $+10.0$ | $+4.4$ | $+4.7$ | $+10.4$ | $+1.4$ | $+4.1$ | | |
| OOD Hard | $-0.7$ | $+0.2$ | $+0.5$ | $+0.5$ | $-0.8$ | $-1.2$ | $+0.2$ | | |
| Sudoku | ID Easy | $+0.6$ | $+0.5$ | $+0.9$ | $+2.1$ | $+0.3$ | $-0.2$ | $+0.1$ | |
| ID Hard | $+2.5$ | $+1.6$ | $+4.9$ | $+7.5$ | $-0.2$ | $+2.5$ | $+1.5$ | | |
| OOD Hard | $+0.3$ | $-0.2$ | $+1.6$ | $+1.8$ | $-0.5$ | $-1.5$ | $-2.4$ | | |
Reflection usually extends the limit of RL. For reflective models, GRPO samples experience CoTs through RMTP, where self-verification $\mathcal{V}$ and the forward policy $\pi$ are jointly optimized in the form of a self-verifying policy $\tilde{\pi}$ . By comparing the RMTP results (columns 3, 6, and 9) with the non-reflective model (the first column) in Table 7, we find that GRPO usually converges to higher accuracy solving ID-Hard problems in RMTP. This shows that having reflection in long CoTs extends the limit of RL, compared to only exploiting a planning policy.
Interestingly, optional detailed verification generally demonstrates higher performance after GRPO than mandatory verification. A probable explanation is that a mandatory verification may cause the reasoner to overly rely on reflection, which stagnates the learning of the planning policy.
Overall, our full results provide more evidence to better support our findings discussed in Section 5.2:
- RL enhances 24 out of 42 ID-Hard results in Table 8 by no less than 3% (measured in absolute difference). However, only 8 out of 42 OOD-Hard results are improved by no less than 1%.
- In table 9, an increase of $e_{+}$ is observed in 20 out of 25 cases where $e_{-}$ decreases by more than 5% (measured in absolute difference).
E.3.1 The verification errors after GRPO
Furthermore, we also present the estimated errors of verification after GRPO in Table 9, in order to investigate how self-verification evolves during RL. Our main observation is that if a model has a high $e_{-}$ before GPRO, then GRPO tends to reduce $e_{-}$ and also increases $e_{+}$ . This change in verification errors is a rather superficial (lazy) way to obtain improvements. If the model faithfully improves verification through RL, both types of errors should simultaneously decrease β such a case occurs only in the ID-Easy difficulty or when $e_{-}$ is already low after SFT. This highlights a potential retrograde of self-verification ability after RL.
Table 9: The percentage (%) of test-time verification errors (i.e., $e_{-}$ and $e_{+}$ ) after GRPO. The arrows βββ (increase) and βββ (decrease) present the change compared to the results in SFT (Table 6).
| 1M ID Hard OOD Hard | Mult 16.5β 20.3 41.2β 57.6 | ID Easy 17.6β 20.0 1.6β 31.3 | 6.8β 12.5 7.0β 10.6 40.2β 46.2 | 3.3β 1.1 13.7β 19.3 1.5β 24.0 | 5.0β 9.9 54.6β 55.5 53.9β 67.5 | 3.5β 1.4 4.6β 2.3 16.4β 18.6 | 12.4β 22.6 42.1β 56.6 57.3β 70.5 | 17.2β 1.1 5.7β 1.7 19.1β 21.5 | 3.5β 27.9 | 17.6β 1.8 |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| Sudoku | ID Easy | 1.2β 11.1 | 1.7β 36.9 | 13.1β 18.0 | 4.0β 47.9 | 0.1β 87.2 | 0.0β 0.0 | 0.0β 87.1 | 0.4β 0.5 | |
| ID Hard | 2.9β 24.0 | 0.9β 30.1 | 5.5β 27.6 | 0.4β 29.0 | 0.1β 83.7 | 0.0β 0.0 | 0.1β 80.3 | 0.0β 0.0 | | |
| OOD Hard | 3.1β 63.4 | 0.8β 8.3 | 4.5β 55.7 | 0.9β 14.3 | 0.5β 88.4 | 0.0β 0.0 | 0.6β 85.1 | 0.0β 0.0 | | |
| 4M | Mult | ID Easy | 2.5β 22.6 | 4.8β 1.1 | 30.2β 27.9 | 7.1β 1.8 | 21.1β 9.3 | 3.0β 0.7 | 5.0β 23.7 | 6.8β 0.7 |
| ID Hard | 53.1β 55.5 | 21.8β 1.8 | 30.6β 56.6 | 28.5β 2.3 | 20.9β 24.2 | 20.1β 5.0 | 22.0β 32.0 | 24.8β 4.5 | | |
| OOD Hard | 60.0β 67.5 | 24.3β 18.6 | 52.5β 70.5 | 40.2β 21.5 | 55.7β 61.6 | 20.9β 7.2 | 53.4β 64.3 | 22.9β 5.3 | | |
| Sudoku | ID Easy | 7.9β 31.6 | 3.2β 6.3 | 2.8β 43.2 | 7.7β 3.8 | 11.5β 12.3 | 1.3β 1.4 | 27.1β 19.6 | 1.9β 2.2 | |
| ID Hard | 28.7β 70.0 | 0.5β 1.4 | 2.2β 58.2 | 4.7β 2.0 | 4.5β 12.8 | 1.6β 1.8 | 9.2β 12.9 | 0.1β 0.2 | | |
| OOD Hard | 6.9β 85.4 | 0.7β 0.1 | 11.0β 81.6 | 0.2β 0.4 | 2.1β 29.4 | 0β 0.1 | 5.9β 30.0 | 0.1β 0.2 | | |
| 16M | Mult | ID Easy | 4.2β 7.1 | 7.2β 1.4 | 0.4β 5.7 | 7.8β 1.6 | 8.1β 7.6 | 1.9β 0.2 | 7.8β 11.6 | 2.7β 0.2 |
| ID Hard | 7.9β 9.3 | 12.8β 1.1 | 6.7β 8.5 | 15.6β 1.3 | 22.7β 25.2 | 3.9β 3.1 | 18.6β 23.0 | 4.3β 2.9 | | |
| OOD Hard | 79.0β 80.3 | 47.1β 39.3 | 89.2β 90.7 | 46.4β 41.8 | 46.0β 54.5 | 12.5β 5.8 | 46.4β 58.1 | 14.7β 5.0 | | |
| Sudoku | ID Easy | 24.5β 64.6 | 6.2β 9.5 | 25.6β 35.7 | 8.5β 13.2 | 2.1β 8.7 | 0.6β 1.1 | 3.3β 5.8 | 5.4β 1.0 | |
| ID Hard | 25.3β 75.8 | 0.0β 4.3 | 16.4β 53.6 | 2.0β 7.4 | 0.5β 15.9 | 0.0β 0.1 | 2.4β 13.0 | 0.4β 1.0 | | |
| OOD Hard | 7.9β 83.1 | 3.5β 0.7 | 12.7β 77.7 | 2.1β 1.0 | 7.8β 36.1 | 0.0β 0.1 | 6.8β 31.6 | 0.1β 0.1 | | |
E.3.2 The planning correctness rate after GRPO
Table 10: The planning correctness rate ( $\mu$ ) before and after GRPO. Each result is reported by $\mu_{\text{SFT}}β\mu_{\text{GRPO}}$ .
| Mult 4M 16M | Detailed $98.3β 99.5$ $99.7β 99.9$ | 1M $68.4β 79.1$ $80.0β 85.9$ | $70.2β 81.7$ $35.0β 38.0$ $47.9β 43.4$ | $54.4β 59.5$ | $42.9β 41.9$ |
| --- | --- | --- | --- | --- | --- |
| Binary | 1M | $98.8β 99.1$ | $81.2β 80.3$ | $42.7β 38.6$ | |
| 4M | $99.3β 99.7$ | $77.6β 89.9$ | $57.1β 48.1$ | | |
| 16M | $99.4β 99.8$ | $79.6β 85.1$ | $75.2β 44.8$ | | |
| Sudoku | Detailed | 1M | $34.1β 33.0$ | $13.2β 12.4$ | $9.0β 8.6$ |
| 4M | $85.0β 86.8$ | $65.2β 72.0$ | $70.1β 70.3$ | | |
| 16M | $98.6β 98.1$ | $92.5β 94.0$ | $84.9β 83.9$ | | |
| Binary | 1M | $59.1β 60.3$ | $36.6β 36.1$ | $19.5β 19.9$ | |
| 4M | $97.3β 97.8$ | $80.2β 81.4$ | $74.5β 70.9$ | | |
| 16M | $99.0β 99.2$ | $88.5β 85.1$ | $68.4β 64.6$ | | |
We also report how GRPO influences the step-wise planning ability, measured by $\mu$ (defined in Section 4), across various tasks, verification types, and model sizes. Shown in Table 10, GRPO increases the planning correctness rate $\mu$ in most ID cases, except for the Sudoku models with binary verification. This indicates that the proposed steps are more likely to be correct and further reduces the overall penalties of false positive verification, making an optimistic verification bias (a high $e_{+}$ in exchange for a low $e_{-}$ ) even more rewarding. In particular, the planning ability shows almost no improvement in OOD problems.
E.3.3 Reflection frequency of optional detailed verification
To show how GRPO adapts the reflection frequency for optional detailed verification, Figure 12 shows the reflection frequency of 1M and 16M transformers before and after GRPO, and the reflection frequency of the 4M model is previously shown in Section 5.2. Similarly, Figure 13 shows the reflection frequency for 1M, 4M, and 16M models in Sudoku.
According to results in Table 5, reflective execution does not improve performance for the 1M model, implying its weakness in exploring correct solutions. Therefore, GRPO does not much incentivize reflection for the 1M model. Contrarily, it greatly encourages reflection for 4M and 16M models, for they explore more effectively than the 1M model. These results align with the discussion in Section 5.2 that RL adapts the reflection frequency based on how well the proposing policy can explore higher rewards.
<details>
<summary>x16.png Details</summary>
### Visual Description
## Heatmaps: Before and After GRPO
### Overview
The image presents two heatmaps side-by-side, titled "Before GRPO" and "After GRPO". Each heatmap visualizes data related to the number of digits in 'x' and 'y', with the color intensity representing the magnitude of the data at each coordinate. A dashed white line is overlaid on each heatmap, forming an L-shape.
### Components/Axes
* **Titles:** "Before GRPO" (left), "After GRPO" (right)
* **X-axis:** "number of x's digits", with ticks labeled 1 to 10.
* **Y-axis:** "number of y's digits", with ticks labeled 1 to 10.
* **Data Representation:** Color intensity, with warmer colors (red, orange) indicating higher values and cooler colors (dark red, black) indicating lower values. The exact numerical value is printed in cyan on each cell.
* **L-shaped Overlay:** A dashed white line forming an L-shape on each heatmap. The L-shape starts at (9,1) and extends to (9,8) and (1,8).
### Detailed Analysis or ### Content Details
**Heatmap: Before GRPO**
| | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| :---- | :- | :- | :- | :- | :- | :- | :- | :- | :- | :- |
| **1** | 38 | 46 | 28 | 22 | 22 | 28 | 23 | 22 | 14 | 11 |
| **2** | 33 | 28 | 27 | 25 | 23 | 17 | 20 | 23 | 18 | 14 |
| **3** | 41 | 27 | 20 | 22 | 19 | 14 | 15 | 17 | 17 | 16 |
| **4** | 30 | 20 | 24 | 20 | 18 | 19 | 15 | 17 | 18 | 17 |
| **5** | 30 | 32 | 25 | 22 | 21 | 16 | 15 | 20 | 18 | 19 |
| **6** | 41 | 34 | 28 | 21 | 20 | 14 | 19 | 19 | 17 | 17 |
| **7** | 38 | 30 | 25 | 17 | 16 | 21 | 18 | 14 | 14 | 12 |
| **8** | 32 | 22 | 15 | 18 | 23 | 18 | 14 | 13 | 12 | 12 |
| **9** | 23 | 16 | 17 | 20 | 17 | 14 | 14 | 11 | 10 | 8 |
| **10**| 17 | 14 | 16 | 16 | 14 | 14 | 12 | 9 | 7 | 7 |
**Heatmap: After GRPO**
| | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| :---- | :- | :- | :- | :- | :- | :- | :- | :- | :- | :- |
| **1** | 34 | 52 | 46 | 36 | 32 | 35 | 42 | 42 | 39 | 32 |
| **2** | 31 | 44 | 36 | 28 | 25 | 25 | 32 | 43 | 40 | 44 |
| **3** | 27 | 34 | 20 | 14 | 13 | 26 | 33 | 38 | 42 | 42 |
| **4** | 14 | 19 | 17 | 20 | 26 | 33 | 33 | 44 | 48 | 38 |
| **5** | 8 | 22 | 24 | 26 | 32 | 36 | 36 | 39 | 42 | 37 |
| **6** | 21 | 30 | 36 | 34 | 33 | 34 | 44 | 49 | 38 | 36 |
| **7** | 26 | 38 | 38 | 36 | 38 | 43 | 39 | 45 | 34 | 32 |
| **8** | 26 | 26 | 34 | 30 | 40 | 44 | 40 | 31 | 30 | 24 |
| **9** | 23 | 29 | 33 | 30 | 34 | 34 | 34 | 33 | 22 | 30 |
| **10**| 25 | 29 | 37 | 36 | 33 | 29 | 38 | 22 | 17 | 23 |
### Key Observations
* **Highest Values:** Before GRPO, the highest value is 46 at (1,2). After GRPO, the highest value is 52 at (1,2).
* **L-Shaped Region:** The L-shaped region defined by the dashed white line appears to contain generally lower values in the "Before GRPO" heatmap compared to the "After GRPO" heatmap.
* **Value Changes:** Many individual cell values change significantly between the "Before" and "After" heatmaps.
### Interpretation
The heatmaps compare a certain metric before and after the application of "GRPO" (likely an optimization or processing technique). The x and y axes represent the number of digits in two input numbers, and the color intensity/numerical value represents some performance metric related to these inputs.
The L-shaped region likely represents a specific constraint or condition being applied. The change in values within this region after GRPO suggests that the technique has a targeted effect on this subset of inputs.
The overall shift in values and color intensities indicates that GRPO has a significant impact on the performance metric being visualized. The specific nature of this impact (improvement or degradation) would require further context about the metric itself. The increase in values in the L-shaped region after GRPO suggests that GRPO improves the metric for inputs within this constraint.
</details>
(a) 1M
<details>
<summary>x17.png Details</summary>
### Visual Description
## Heatmap: Comparison of Values Before and After GRPO
### Overview
The image presents two heatmaps side-by-side, visualizing numerical values before and after the application of a process labeled "GRPO". The heatmaps share the same axes, representing the "number of x's digits" and "number of y's digits," both ranging from 1 to 10. The color intensity in each cell corresponds to the magnitude of the numerical value within that cell. A white dashed line is drawn on each heatmap, forming an L-shape.
### Components/Axes
* **Titles:** "Before GRPO" (left heatmap), "After GRPO" (right heatmap)
* **Y-axis Label:** "number of y's digits" (vertical, on both heatmaps)
* **X-axis Label:** "number of x's digits" (horizontal, on both heatmaps)
* **X-axis Markers:** 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 (on both heatmaps)
* **Y-axis Markers:** 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 (on both heatmaps)
* **Data Values:** Numerical values are displayed within each cell of the heatmaps. The color of the cells varies based on the value, ranging from dark red/black (low values) to yellow/white (high values).
### Detailed Analysis
**Left Heatmap (Before GRPO):**
* **General Trend:** The values are generally lower compared to the right heatmap. The highest values are concentrated in the top-left corner.
* **Specific Values:**
* (1,1): 38
* (1,2): 33
* (1,3): 29
* (1,4): 19
* (1,5): 15
* (1,6): 12
* (1,7): 9
* (1,8): 6
* (1,9): 4
* (2,1): 19
* (2,2): 15
* (2,3): 4
* (2,4): 5
* (2,5): 5
* (2,6): 4
* (2,7): 5
* (2,8): 3
* (2,9): 1
* (2,10): 1
* (3,1): 6
* (3,2): 7
* (3,3): 4
* (3,4): 9
* (3,5): 6
* (3,6): 4
* (3,7): 6
* (3,8): 4
* (3,9): 2
* (3,10): 5
* (4,1): 5
* (4,2): 5
* (4,3): 6
* (4,4): 5
* (4,5): 4
* (4,6): 5
* (4,7): 5
* (4,8): 4
* (4,9): 5
* (4,10): 6
* (5,1): 6
* (5,2): 8
* (5,3): 5
* (5,4): 4
* (5,5): 3
* (5,6): 3
* (5,7): 5
* (5,8): 4
* (5,9): 6
* (5,10): 4
* (6,1): 6
* (6,2): 7
* (6,3): 4
* (6,4): 4
* (6,5): 4
* (6,6): 3
* (6,7): 4
* (6,8): 4
* (6,9): 5
* (6,10): 7
* (7,1): 6
* (7,2): 4
* (7,3): 4
* (7,4): 3
* (7,5): 4
* (7,6): 5
* (7,7): 5
* (7,8): 6
* (7,9): 7
* (7,10): 6
* (8,1): 5
* (8,2): 7
* (8,3): 5
* (8,4): 6
* (8,5): 5
* (8,6): 7
* (8,7): 10
* (8,8): 9
* (8,9): 8
* (8,10): 6
* (9,1): 6
* (9,2): 7
* (9,3): 8
* (9,4): 8
* (9,5): 10
* (9,6): 14
* (9,7): 10
* (9,8): 8
* (9,9): 5
* (9,10): 7
* (10,1): 5
* (10,2): 6
* (10,3): 9
* (10,4): 10
* (10,5): 10
* (10,6): 9
* (10,7): 9
* (10,8): 7
* (10,9): 8
* (10,10): 6
**Right Heatmap (After GRPO):**
* **General Trend:** The values are significantly higher compared to the left heatmap. The heatmap is predominantly yellow and white, indicating high values across the board.
* **Specific Values:**
* (1,1): 75
* (1,2): 67
* (1,3): 69
* (1,4): 70
* (1,5): 67
* (1,6): 67
* (1,7): 67
* (1,8): 56
* (1,9): 54
* (1,10): 51
* (2,1): 72
* (2,2): 72
* (2,3): 71
* (2,4): 70
* (2,5): 68
* (2,6): 70
* (2,7): 70
* (2,8): 67
* (2,9): 67
* (2,10): 75
* (3,1): 64
* (3,2): 73
* (3,3): 74
* (3,4): 75
* (3,5): 76
* (3,6): 72
* (3,7): 75
* (3,8): 72
* (3,9): 78
* (3,10): 82
* (4,1): 61
* (4,2): 71
* (4,3): 80
* (4,4): 79
* (4,5): 78
* (4,6): 79
* (4,7): 81
* (4,8): 82
* (4,9): 88
* (4,10): 88
* (5,1): 64
* (5,2): 74
* (5,3): 80
* (5,4): 80
* (5,5): 81
* (5,6): 83
* (5,7): 85
* (5,8): 88
* (5,9): 92
* (5,10): 93
* (6,1): 60
* (6,2): 71
* (6,3): 78
* (6,4): 83
* (6,5): 85
* (6,6): 85
* (6,7): 89
* (6,8): 93
* (6,9): 95
* (6,10): 96
* (7,1): 53
* (7,2): 67
* (7,3): 81
* (7,4): 84
* (7,5): 86
* (7,6): 90
* (7,7): 94
* (7,8): 95
* (7,9): 96
* (7,10): 96
* (8,1): 52
* (8,2): 68
* (8,3): 81
* (8,4): 87
* (8,5): 89
* (8,6): 94
* (8,7): 96
* (8,8): 96
* (8,9): 96
* (8,10): 98
* (9,1): 52
* (9,2): 72
* (9,3): 84
* (9,4): 88
* (9,5): 92
* (9,6): 96
* (9,7): 96
* (9,8): 98
* (9,9): 97
* (9,10): 96
* (10,1): 61
* (10,2): 77
* (10,3): 86
* (10,4): 90
* (10,5): 93
* (10,6): 96
* (10,7): 97
* (10,8): 97
* (10,9): 96
* (10,10): 95
### Key Observations
* The "GRPO" process significantly increases the values across all combinations of x and y digits.
* The white dashed line seems to separate regions of different performance or behavior, but without further context, its exact meaning is unclear.
* The top-left corner shows the most significant improvement after applying "GRPO".
### Interpretation
The heatmaps demonstrate the effectiveness of the "GRPO" process in increasing the measured values, whatever they represent. The axes "number of x's digits" and "number of y's digits" likely refer to input parameters or characteristics of a system being evaluated. The "GRPO" process appears to optimize or enhance the system's performance across various combinations of these parameters. The white dashed line might indicate a threshold or boundary related to the application of "GRPO," or it could delineate regions where the process has varying degrees of impact. Further context is needed to fully understand the meaning of the values and the significance of the dashed line.
</details>
(b) 16M
Figure 12: The hot-maps of reflection frequency (%) of 1M and 16M multiplication models before and after GRPO, which uses a sampling temperature of $1.25$ . All models are tested using RMTP execution.
<details>
<summary>x18.png Details</summary>
### Visual Description
## Bar Chart: Reflection Frequency Before and After GRPO
### Overview
The image presents two bar charts side-by-side, comparing the reflection frequency (%) against the number of blanks before and after a process called GRPO. Both charts display the same x and y axis scales, with a vertical dashed red line at x=54. The charts show the distribution of reflection frequencies across different numbers of blanks.
### Components/Axes
* **Titles:**
* Left Chart: "Before GRPO"
* Right Chart: "After GRPO"
* **Y-axis (both charts):**
* Label: "reflection frequency (%)"
* Scale: 0.0 to 1.0, with increments of 0.2 (0.0, 0.2, 0.4, 0.6, 0.8, 1.0)
* **X-axis (both charts):**
* Label: "number of blanks"
* Scale: 9 to 54, with increments of 9 (9, 18, 27, 36, 45, 54)
* **Vertical Line:** Dashed red line at x=54 in both charts.
### Detailed Analysis
**Left Chart: Before GRPO**
* The bars are blue.
* The reflection frequency generally fluctuates between 0.0 and 0.2.
* The bars are generally between 0.0 and 0.1 for number of blanks between 9 and 27.
* The bars are generally between 0.1 and 0.2 for number of blanks between 27 and 54.
**Right Chart: After GRPO**
* The bars are blue.
* The reflection frequency generally fluctuates between 0.0 and 0.2.
* The bars are generally between 0.0 and 0.1 for number of blanks between 9 and 27.
* The bars are generally between 0.1 and 0.2 for number of blanks between 27 and 54.
### Key Observations
* Both charts show a similar distribution of reflection frequencies.
* The reflection frequencies are generally low, mostly below 0.2.
* The dashed red line at x=54 might indicate a threshold or a specific point of interest.
* There is no significant visual difference between the "Before GRPO" and "After GRPO" charts.
### Interpretation
The charts compare the reflection frequency distribution before and after the GRPO process. The data suggests that the GRPO process does not significantly alter the reflection frequency distribution across the number of blanks. The reflection frequencies remain generally low in both cases. The vertical line at x=54 could represent a critical number of blanks, but without further context, its significance is unclear. The similarity between the two charts implies that GRPO, as applied here, has little to no impact on the reflection frequency distribution.
</details>
(a) 1M
<details>
<summary>x19.png Details</summary>
### Visual Description
## Bar Chart: Reflection Frequency Before and After GRPO
### Overview
The image presents two bar charts side-by-side, comparing the reflection frequency as a function of the number of blanks "Before GRPO" and "After GRPO". The x-axis represents the number of blanks, ranging from approximately 9 to 54. The y-axis represents the reflection frequency in percentage, ranging from 0.0 to 1.0. A vertical dashed red line is present at x=54 in both charts.
### Components/Axes
* **Titles:**
* Left Chart: "Before GRPO"
* Right Chart: "After GRPO"
* **X-axis:** "number of blanks" with tick marks at approximately 9, 18, 27, 36, 45, and 54.
* **Y-axis:** "reflection frequency (%)" with tick marks at 0.0, 0.2, 0.4, 0.6, 0.8, and 1.0.
* **Bars:** Blue bars representing the reflection frequency for each number of blanks.
* **Vertical Line:** Dashed red line at x=54 in both charts.
### Detailed Analysis
**Left Chart: Before GRPO**
* **Trend:** The reflection frequency fluctuates randomly between 0.0 and 0.25 across the number of blanks. There is no clear trend.
* **Data Points:**
* At x=9, reflection frequency is approximately 0.15.
* At x=18, reflection frequency is approximately 0.22.
* At x=27, reflection frequency is approximately 0.12.
* At x=36, reflection frequency is approximately 0.15.
* At x=45, reflection frequency is approximately 0.10.
* At x=54, reflection frequency is approximately 0.12.
**Right Chart: After GRPO**
* **Trend:** The reflection frequency generally increases with the number of blanks.
* **Data Points:**
* At x=9, reflection frequency is approximately 0.72.
* At x=18, reflection frequency is approximately 0.75.
* At x=27, reflection frequency is approximately 0.80.
* At x=36, reflection frequency is approximately 0.85.
* At x=45, reflection frequency is approximately 0.90.
* At x=54, reflection frequency is approximately 0.98.
### Key Observations
* The reflection frequency is significantly higher "After GRPO" compared to "Before GRPO".
* "Before GRPO", the reflection frequency is relatively constant and low.
* "After GRPO", the reflection frequency increases with the number of blanks, approaching 1.0.
* The vertical red line at x=54 is present in both charts, possibly indicating a significant point or limit.
### Interpretation
The charts demonstrate the effect of GRPO (likely an optimization or processing step) on the reflection frequency as a function of the number of blanks. "Before GRPO", the reflection frequency is low and inconsistent, suggesting a less efficient or less controlled process. "After GRPO", the reflection frequency is significantly improved and shows a positive correlation with the number of blanks, indicating a more efficient and controlled process. The GRPO process appears to have significantly enhanced the reflection properties, especially as the number of blanks increases. The vertical line at x=54 might represent a design limit or a target value for the number of blanks.
</details>
(b) 4M
<details>
<summary>x20.png Details</summary>
### Visual Description
## Bar Chart: Reflection Frequency Before and After GRPO
### Overview
The image presents two bar charts side-by-side, comparing the reflection frequency (%) against the number of blanks "Before GRPO" and "After GRPO". Both charts share the same x and y axes. A vertical dashed red line is present at x=54 on both charts. The "Before GRPO" chart shows low reflection frequencies, while the "After GRPO" chart shows significantly higher reflection frequencies.
### Components/Axes
* **Titles:**
* Left Chart: "Before GRPO"
* Right Chart: "After GRPO"
* **Y-axis (Reflection Frequency (%)):**
* Label: "reflection frequency (%)"
* Scale: 0.0 to 1.0, with increments of 0.2 (0.0, 0.2, 0.4, 0.6, 0.8, 1.0)
* **X-axis (Number of Blanks):**
* Label: "number of blanks"
* Scale: 9 to 54, with increments of 9 (9, 18, 27, 36, 45, 54)
* **Bars:** Blue bars represent the reflection frequency for each number of blanks.
* **Vertical Line:** A dashed red vertical line is present at the x=54 position on both charts.
### Detailed Analysis
**Left Chart: Before GRPO**
* The reflection frequency is generally low, mostly below 0.2.
* The bars fluctuate, indicating some variation in reflection frequency across different numbers of blanks.
* Specific values are difficult to extract precisely due to the bar chart format, but the reflection frequency appears to range from approximately 0.05 to 0.15.
**Right Chart: After GRPO**
* The reflection frequency is significantly higher compared to the "Before GRPO" chart, mostly above 0.9.
* The bars are consistently high, indicating a more uniform reflection frequency across different numbers of blanks.
* Specific values are difficult to extract precisely, but the reflection frequency appears to range from approximately 0.95 to 1.0.
### Key Observations
* **Significant Increase:** The GRPO process leads to a substantial increase in reflection frequency across all numbers of blanks.
* **Uniformity:** The "After GRPO" chart shows a more uniform reflection frequency compared to the "Before GRPO" chart.
* **Vertical Line:** The dashed red line at x=54 is a reference point, possibly indicating a threshold or a specific number of blanks of interest.
### Interpretation
The charts demonstrate the impact of the GRPO process on reflection frequency. Before GRPO, the reflection frequency is low and variable. After GRPO, the reflection frequency is significantly increased and becomes more uniform across different numbers of blanks. This suggests that GRPO is an effective process for enhancing reflection properties, leading to more consistent and higher reflection frequencies. The vertical line at x=54 may indicate a critical number of blanks where this enhancement is particularly important.
</details>
(c) 16M
Figure 13: The histograms of reflection frequency of 1M, 4M, and 16M Sudoku models before and after GRPO, which uses a sampling temperature of $1.25$ . All models are tested using RMTP execution.
E.4 Reflection frequency under controlled verification error rates
To investigate how verification error rates ( $e_{-}$ and $e_{+}$ ) influence the reflection frequency in GRPO, we ran a controlled experiment in which the error rates were fixed by intervening with expert verifications. After each time the transformer generated a non-empty verification, we replaced the verification sequence with the expert verification, where randomized noise is injected to achieve the prescribed false-negative rate $e_{-}$ and false-positive rate $e_{+}$ .
We used the 4M Mult model and ran GRPO (sampling temperature = 1.25) for 25 epochs in the in-distribution setting. We measured the fraction of steps at which the model invoked non-empty reflection (βreflection frequencyβ) after 25 epochs. Especially, we are interested in how reflection frequency changes, given a low $e_{-}=0.1$ or a high $e_{-}=0.4$ . In both cases, we set $e_{+}=0.1$ . The results are as follows:
- Using a low $e_{-}=0.1$ , the reflection frequency increases to $59.8\%$ after 25 GRPO epochs.
- Using a high $e_{-}=0.4$ , the reflection frequency drops to $0.0\%$ after 25 GRPO epochs. That is, the model learns to completely disuse reflection.
Discussion.
When the verifier rejects many correct steps (high $e_{-}$ ), the model learns to avoid invoking reflection, driving the observed reflection frequency to nearly $0\%$ . Conversely, when $e_{-}$ is low (with the same $e_{+}$ ), reflection becomes beneficial and the model increases reflection usage (here to $60\%$ ). Intuitively, reducing excessive false negatives shortens CoT lengths and makes reflection more rewarding; when $e_{-}$ is large, the model can trade off reflection for a no-reflection policy (which corresponds to the extreme $e_{-}=0,e_{+}=1$ ), thereby avoiding costly rejections. This experiment demonstrates that the model learns to reduce $e_{-}$ by strategically bypassing verification.
E.5 Results of PPO
As discussed in Appendix B.1, we prefer GRPO over PPO for tiny transformers, as the value model in PPO increases computational cost and introduces additional approximation bias in computing advantages.
Table 11 presents the reasoning accuracy after PPO, and Table 12 gives the difference compared to the SFT results in Table 5. Our results show that PPO is much weaker than GRPO. Although PPO effectively improves the non-reflective models, the performance of reflective reasoning deteriorates after PPO. To explain this, self-verification in reasoning steps causes a higher complexity of the value function, which may obfuscate tiny transformers. Overall, we suggest that GRPO is a more suitable algorithm to optimize reflective reasoning for tiny transformers.
Table 11: The accuracy (%) of the 1M, 4M, and 16M transformers after PPO.
| Verification Type | None | Binary | Detailed | Optional Detailed | | | | | | | | |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| Reflective Execution | None | None | RMTP | RTBS | None | RMTP | RTBS | None | RMTP | RTBS | | |
| 1M | Mult | ID Easy | 39.6 | 96.5 | 94.1 | 90.6 | 28.3 | 30.1 | 27.2 | 37.9 | 49.0 | 44.4 |
| ID Hard | 7.8 | 49.6 | 43.7 | 32.2 | 2.4 | 3.1 | 2.4 | 5.9 | 9.6 | 7.3 | | |
| OOD Hard | 1.1 | 2.6 | 1.8 | 1.2 | 0.7 | 0.8 | 0.7 | 1.0 | 1.0 | 0.8 | | |
| Sudoku | ID Easy | 1.7 | 36.1 | 33.7 | 5.6 | 17.3 | 20.6 | 20.1 | 23.8 | 21.9 | 20.1 | |
| ID Hard | 0 | 0.4 | 1.0 | 0 | 0 | 0.1 | 0 | 0 | 0 | 0 | | |
| OOD Hard | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | | |
| 4M | Mult | ID Easy | 97.7 | 95.5 | 98.6 | 93.8 | 96.6 | 95.7 | 94.9 | 97.2 | 96.9 | 94.6 |
| ID Hard | 63.0 | 52.8 | 68.6 | 54.7 | 54.0 | 54.6 | 45.5 | 58.7 | 61.7 | 56.8 | | |
| OOD Hard | 2.2 | 3.1 | 2.9 | 1.6 | 5.3 | 3.9 | 2.2 | 4.4 | 3.3 | 3.7 | | |
| Sudoku | ID Easy | 56.4 | 88.4 | 97.3 | 97.6 | 49.3 | 82.1 | 80.6 | 76.2 | 94.1 | 97.3 | |
| ID Hard | 0 | 28.6 | 47.4 | 47.7 | 0 | 15.1 | 35.9 | 15.2 | 35.3 | 55.6 | | |
| OOD Hard | 0 | 0.2 | 1.6 | 3.3 | 3.1 | 0.4 | 0.9 | 0 | 1.1 | 2.7 | | |
| 16M | Mult | ID Easy | 99.3 | 99.0 | 99.0 | 98.2 | 98.5 | 98.7 | 97.8 | 99.0 | 99.5 | 99.2 |
| ID Hard | 64.8 | 62.9 | 75.7 | 71.9 | 63.2 | 68.6 | 65.6 | 65.1 | 77.1 | 74.6 | | |
| OOD Hard | 1.9 | 1.0 | 1.2 | 1.1 | 9.1 | 8.1 | 7.5 | 5.4 | 5.6 | 5.4 | | |
| Sudoku | ID Easy | 96.5 | 91.8 | 97.3 | 96.7 | 87.6 | 98.1 | 98.9 | 94.5 | 96.7 | 97.1 | |
| ID Hard | 49.0 | 41.0 | 51.4 | 52.7 | 34.7 | 55.7 | 66.3 | 47.8 | 53.8 | 53.0 | | |
| OOD Hard | 0.6 | 0 | 2.4 | 4.0 | 0 | 1.1 | 2.0 | 0 | 3.8 | 2.9 | | |
Table 12: The difference of accuracy (%) of the 1M, 4M, and 16M transformers after PPO. Positive values mean that PPO raises the accuracy of the models above SFT.
| 1M | Mult | ID Easy | $+16.0$ | $+0.7$ | $-0.4$ | $-2.8$ | $+6.3$ | $-3.3$ | $+3.0$ |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| ID Hard | $+5.8$ | $-3.1$ | $-0.9$ | $-3.3$ | $+0.2$ | $-1.7$ | $-0.4$ | | |
| OOD Hard | $+0.1$ | $-1.1$ | $-0.4$ | $+0.0$ | $-0.3$ | $+0.0$ | $+0.3$ | | |
| Sudoku | ID Easy | $+0.3$ | $+3.1$ | $+1.3$ | $+3.2$ | $-0.1$ | $+1.9$ | $+0.7$ | |
| ID Hard | $+0.0$ | $+0.1$ | $+0.9$ | $+0.0$ | $-0.1$ | $+0.1$ | $+0.0$ | | |
| OOD Hard | $+0.0$ | $+0.0$ | $+0.0$ | $+0.0$ | $+0.0$ | $+0.0$ | $+0.0$ | | |
| 4M | Mult | ID Easy | $+5.7$ | $-2.2$ | $+1.0$ | $-3.5$ | $+2.1$ | $+1.9$ | $+1.6$ |
| ID Hard | $+25.7$ | $-4.1$ | $+6.4$ | $+1.7$ | $+10.6$ | $+7.0$ | $+3.1$ | | |
| OOD Hard | $+0.0$ | $+0.2$ | $+1.1$ | $+0.5$ | $+1.6$ | $+0.6$ | $-0.5$ | | |
| Sudoku | ID Easy | $+4.2$ | $-3.7$ | $+0.5$ | $+1.6$ | $-5.1$ | $+0.2$ | $-7.9$ | |
| ID Hard | $-3.3$ | $-12.3$ | $+1.1$ | $-5.6$ | $-5.2$ | $-1.8$ | $-9.8$ | | |
| OOD Hard | $+0.0$ | $+0.2$ | $+1.6$ | $+3.3$ | $+2.7$ | $-3.6$ | $-5.8$ | | |
| 16M | Mult | ID Easy | $+0.1$ | $+0.2$ | $+0.1$ | $-0.6$ | $-0.7$ | $-0.8$ | $-0.7$ |
| ID Hard | $-1.1$ | $-2.3$ | $-1.0$ | $-3.0$ | $-2.7$ | $-7.8$ | $-7.9$ | | |
| OOD Hard | $-0.6$ | $-0.1$ | $-0.1$ | $-0.2$ | $-0.1$ | $-1.3$ | $+0.3$ | | |
| Sudoku | ID Easy | $+0.8$ | $-5.3$ | $-0.6$ | $+4.2$ | $-5.4$ | $-0.9$ | $-0.8$ | |
| ID Hard | $+0.2$ | $-9.1$ | $-1.7$ | $-2.1$ | $-12.2$ | $-2.2$ | $-4.4$ | | |
| OOD Hard | $+0.2$ | $-0.9$ | $-2.0$ | $-2.0$ | $-0.7$ | $-7.1$ | $-12.4$ | | |
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1. Declaration of LLM usage
1. Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.
1. Answer: [N/A]
1. Justification: Although this research is related to LLM reasoning, we focus on tiny transformers. The appendix includes the evaluation of LLMs, yet these results do not impact our core methodology and originality.
1. Guidelines:
- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components.
- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described.