## Diagram: Graph Representations of Lambda Calculus Expressions ### Overview The image presents two directed graph representations of lambda calculus expressions. Each graph consists of nodes labeled with symbols (λ, ⋎) and directed edges indicating the flow of computation. The left graph represents a more complex expression with branching, while the right graph represents a simpler expression with self-reference. ### Components/Axes * **Nodes:** Represented by circles containing symbols. The symbols are: * λ (lambda): Represents lambda abstraction. * ⋎: Represents application. * y: Represents a free variable. * **Edges:** Represented by arrows, indicating the direction of computation or data flow. ### Detailed Analysis **Left Graph:** * **Top Node:** A single ⋎ node at the top, with an outgoing arrow. * **Second Level:** Two λ nodes, each connected to the top ⋎ node by an arrow. * **Third Level:** Each λ node connects to two ⋎ nodes. * **Bottom Level:** The two ⋎ nodes connected to each λ node are interconnected with arrows forming a loop. **Right Graph:** * **Top Node:** A single ⋎ node at the top, with an outgoing arrow. * **Second Level:** Two λ nodes, each connected to the top ⋎ node by an arrow. * **Third Level:** Each λ node connects to a ⋎ node. * **Bottom Level:** Each ⋎ node has an outgoing arrow labeled "y" and a self-loop arrow. ### Key Observations * The ⋎ symbol seems to represent function application, while λ represents lambda abstraction. * The arrows indicate the direction of data flow or computation. * The left graph represents a more complex expression with branching and interconnected nodes. * The right graph represents a simpler expression with self-reference (loops) and free variables (y). ### Interpretation The diagrams illustrate how lambda calculus expressions can be represented as directed graphs. The nodes represent operations (abstraction and application), and the edges represent the flow of data or computation. The left graph likely represents a more complex expression involving multiple applications and abstractions, while the right graph represents a simpler expression with self-application, potentially related to recursion or fixed-point combinators. The "y" labels in the right graph indicate free variables, which are variables not bound by any lambda abstraction within the expression. The self-loops in the right graph suggest a recursive process where the output of an application is fed back into itself.