![]() ![]() data LCExpr LCVar Id LCApp LCExpr LCExpr LCLam Id LCExpr deriving ( Show, Eq) Write a function freeVars that collects the set of free variables in a given lambda calculus expression. import qualified as Map data Val Num Integer Fun (Val -> Val) Wrong data Term Cons Integer Var String Lam String Term App Term Term Add Term Term type Env Map. The original interpreter of Poly is slow because it emulates beta reduction on top of Haskell, but our implementation avoids this overhead by utilizing the host system’s support for beta-reduction. The pure lambda calculus has three forms: variables, applications, and lambda abstractions. This may not always be good: lists are less general than trees because monoids are fussier than magmas. ![]() The key idea is to replace Lam and Ap constructors with Haskell’s built-in lambda and application constructs. Haskell is more deserving of the term 'list processor' because its lists are true singly-linked lists, and the language encourages their use thanks to extensive library support and notation such as 1, 2, 3 and (h:t). The next step is to compile expressions into a fixed, finite combinators. (Note that $IF can be a function because we piggy back on the lazy evaluation of the host language, Haskell.) Compilation to SKI combinators $IF, $FIX, $ADD, $SUB, $MUL, $EQL will be provided as primitive functions. In the 1930s Alonso Church developed the lambda-calculus as an alternative to set theory for the foundations of mathematics and Haskell B. If, Fix are Op are desugared into function applications. For students, it is a tool to learn lambda-calculus and intuitionistic logic by coding. N = case op of Add -> "$ADD" Sub -> "$SUB" Mul -> "$MUL" Eql -> "$EQL"ĭesugar function converts let x = e in body into (\x -> body) e. Mikrokosmos is an educational untyped and simply typed lambda-calculus interpreter. normal order reduction and normal form (no further reductions possible) It has to be able to solve at least factorial of 5 or similiarly difficult expression ( Y (\\f n. With memoization this might not be so bad.Desugar ( App fun arg) = App (desugar fun) (desugar arg)ĭesugar ( Lam x body) = Lam x (desugar body)ĭesugar ( Let x e body) = App ( Lam x (desugar body)) (desugar e)ĭesugar ( If cond tr fl) = foldl App ( Var "$IF") argsĭesugar ( Fix e) = App ( Var "$FIX") (desugar e)ĭesugar ( Op op a b) = foldl App ( Var n) args Interpreter has to understand basics of -calcul: encapsulation, abstraction, application. The evalWhile function also seems like a poor way to check if we've reached normal form, though I have seen other code that does this. Perhaps there is a better way to do this. 5.2 REPRESENTATION OF DATA TYPES BY FUNCTIONS In the lambda calculus. The eval function feels particularly awkward to me, as every type in Expr is preceded by it's equivalent function. In section 5.4 we define an interpreter for this language based on straightforward. e) where x could be any legal variable name and e any legal expression. We define the untyped lambda calculus as follows: Syntax There are the following three kinds of expressions: A lambda expression has the form ( x. S = Abst "x" (Abst "y" (Abst "z" (App (App (Var "x") (Var "z")) (App (Var "y") (Var "z")))))įor instance: λ> evalWhile (App (App (App s k) k) k) The challenge is to write an interpreter for the untyped lambda calculus in as few characters as possible. Here are a few applicative combinators for debugging. Sub (Sub (App a b) var env) = (App (Sub a var env) (Sub b var env))Įval a b c) = sub (Sub (eval a) b (eval c)) | otherwise = (Abst var' (Sub body var env)) module Eval whereĪpp (App (Abst var body) env) = (Sub body var env) This code is slower than many other interpreters, and if there is a way to make the code more efficient while still conveying the reduction rules clearly, that would be an improvement. My goal is to represent the transformation rules listed here accurately and explicitly. Syntax: type Id String - The 'builtin' functions. ![]() (Sub (App a b) var env) -> (App (Sub a var env) (Sub b var env)) (Sub (Abst var' body) var env) -> (Abst var' (Sub body var env)) (Sub (Abst var body) var env) -> (Abst var body) ![]() (App (Abst var body) env) -> (Sub body var env) This code is a representation of lambda calculus using an AST instead of text. I made a more basic site a while ago that also reduces lambda calculus step-by-step, but also handles substitution correctly, and handles infinite recursion by displaying reductions steps asynchronously: Basic infinite recursion for your site: (Lx.x x)(Ly. ![]()
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