## Due Date: Fri 20 AprPair Project

In this project, you will use F# to implement an interpreter for a simple call-by-name lambda calculus (the computational model underlying functional programming languages such as Haskell). This amounts to an almost verbatim implementation of the calculus described here (except that we do not yet have a list type).

## Abstract Syntax

Specifically, you will define an algebraic data type (discriminated union type) that includes the variants for arithmetic expressions along with the following functional language constructs:
• lambda x . e: anonymous function, that is, a designated formal argument x along with a function body in which x occurs zero or more times
• x: variable (presumably occurring in an expression with ancestor that is a lambda expression introducing x as a formal argument)
• e1 e2: function application, that is, an expression (presumably representing a function) followed by another expression (presumably representing an actual argument)
• if c then e1 else e2: conditional
We will worry about concrete syntax (strings that would be parsed into an equivalent abstract syntax tree) in a later phase of this project.

## Semantics

The semantic domain (values) is a subdomain of the abstract syntactic domain consisting only of
• integer constants
• function closures
Next, you will define a function for evaluating expressions represented using the abstract syntax defined above using call-by-name (normal-order evaluation)This function takes an expression that follows the abstract syntax defined above and reduces it to a value from the semantic domain.
• To evaluate an arithmetic expression, simply port the Scala example.
• An anonymous function definition evaluates to itself.
• A variable evaluates to itself.
• A function application is evaluated as follows (this is where the evaluation order is decided):
1. The first expression is evaluated. (If it the result is not a function closure, lambda x . b, then there is an error.)
2. Beta substitution is performed to substitute the second expression (without evaluating it) for every free occurrence of x in b.
3. The resulting function body (after the substitution) is evaluated.
• A conditional is evaluated as follows. (Note that it is non-strict in e1 and e2, otherwise it would defeat the purpose.)
1. The condition is evaluated. (If the result is not an integer constant, then there is an error.)
2. If the condition is non-zero, the result is the result of evaluating e1.
3. Otherwise the result is the result of evaluating e2.
If an (unbound) variable occurs in the result of an evaluation, then there is an error.

In particular, you should implement a separate auxiliary function for beta substitution. Then use this function in the evaluation of function application. There will be no environment anymore for evaluation or function closures.

## Examples

eval (Var "x") -> error (* your eval function should no longer include a branch for Var  *)
eval (Fun("x",Plus(Const 7,Var("x")))) -> Fun("x",Plus(Const 7,Var("x")))
eval (App(Fun("x",Plus(Const 7,Var("x"))),Const 3)) -> Const 10
eval (App(Var("x"),Const 3) -> error
eval (If(Const 7,Const 3,Const 4)) -> Const 3
eval (If(Const 0,Const 3,Const 4)) -> Const 4
eval (If(Fun("x",Var "x"),Const 3,Const 4)) -> Const 3
eval (If(Fun("x",Var "y"),Const 3,Const 4)) -> Const 3
let Y = (* you get to define it yourself, as in the handout -- this is cool! *)
eval (App(App(Y, Fun("f",Fun("n",If(Var "n",Times(Var "n",App(Var("f"),Minus(Var "n",Const 1))),Const 1)))), Const 5) -> Const 120