#lang racket #| ;; Syntax of the language: ;; ;; Plain λ-calculus: ;; x,y,z ::= variable name Variable ;; e ::= (λ x e) Abstraction (lambda) ;; | (e₁ e₂) Application ;; | x variable reference ;; ;; Plain λ-calculus + laziness: ;; e ::= … ;; | (#%app e₁ e₂) Sugar application ;; ;; Translation to λ-calculus ;; (#%app e₁ e₂) => ((e₁ env) (λ _ e₂)) ;; ;; Plain λ-calculus + continuations: ;; e ::= (λ k x e) Abstraction (lambda) ;; | (call/prompt stack-frame-name e₁ continuation e₂) Primitive application ;; | x variable reference ;; | (#%app e₁ e₂) Sugar application ;; | (#%lam x e) Sugar lambda ;; ;; (#%app e₁ e₂) => (call/cc (λ (k) (call/prompt "stack frame" e₁ k e₂)) (f e) => (λ kont . (eval-f k=(λ res-f (eval-e k=(λ res-e (res-f res-e k=kont)))))) ;; translation rules x => (λ k . k x) (λ x e) => (λ k . k (λ (k' x) . [[e]] k' )) (f arg) => (λ k . k ( [[f]] (λ fval . [[arg]] (λ argval . fval k argval) ))) eval k x => k x eval k (λ x e) => can't reduce further eval k (f arg) => (eval f) then (eval arg) then (eval k (fval argval)) ;; Plain λ-calculus + continuations: ;; e ::= (λ k x e) Abstraction (lambda) ;; | (e₁ k e₂) Primitive application ;; | x variable reference ;; | (#%app e₁ e₂) Sugar application is call/cc eval ((λ k x e) kont param) => e[x := param, k := kont] eval (#%app f param) => (call/cc f param) => (f current-continuation param) location of expr current-continuation (λ k x _) k (_ k e₂) (λ outer-continuation evaled-f (f k e₂)) (e₁ _ e₂) ?? (e₁ k _) (λ outer-continuation result (e₁ k result)) (#%app _ e₂) Sugar application is call/cc (#%app e₁ _) Sugar application is call/cc ;; Plain λ-calculus + continuations: ;; e ::= (λ k=x₁ x₂ e) Abstraction (lambda), takes a continuation ;; | (e₁ k=e₂ e₃) Raw aplication ;; | x variable reference ;; | (#%app e₁ e₂) Sugar application ;; ;; Evaluation rules: ;; eval env ((λ k=x₂ x₃ e₁) k=e₂ e₃) => eval env[x₂↦e₂][x₃↦e₃] e₁ ;; x => env[x] ;; ((#%app e₁ e₂) k=e' e'') => ;; (e' k=(#%app e₁ e₂) e'') => ;; (e' k=e'' (#%app e₁ e₂)) => (e₁ k=(λ arg k=? (e' k=e'' arg)) e₂) ;; ;; (#%app f (#%app g x)) => (g k=f x) ;; (f (g (h x))) => ((g f) (h x)) => (h (g f) x) ;; λk.x => k x ;; λk.λx.e => k (λk λk' (#%app e k)) ;; ;; Plain lambda-calculus + first-class environments: ;; "x" ::= "x","y","z"… String ;; e ::= (λ env arg e) Abstraction (lambda) which ;; * an environment (map from strings to values) ;; * takes an argument always named arg which is not added to the env ;; | (e env e) Application ;; | env the env of the innermost lambda containing this expression ;; | arg the arg of the innermost lambda containing this expression ;; prim ::= ;; | get Get variable from environment, type is (→ Environment → String Any) ;; | add Extend environment with new binding, type is (→ Environment String (→ _Environment Any Environment))) ;; ;; Translation to plain lambda-calculus: ;; (λ env arg e) => (λ arg (λ env e)) ;; (e₁ env e₂) => ((e₁ env) e₂) ;; env => env ;; arg => arg ;; get => primitive "get" from an immutable name↦val mapping (could be implemented in plain lambda-calculus) ;; add => primitive "add" to an immutable name↦val mapping (could be implemented in plain lambda-calculus) ;; ;; With laziness: ;; (e₁ env e₂) => ((e₁ env) (λ env (λ _ e₂))) ;; ;; With continuations ;; (e₁ env e₂) => ((e₁ env) (λ env (λ _ e₂))) ;; (f (g x)) => (g k=f x) ;; ;; With #%app ;; |# ;; "x" ::= "x","y","z"… String ;; e ::= (-λ -env -arg -k e) Abstraction (lambda) which takes ;; * an environment always named -env (not in the -env) ;; * a promise for an argument always named -arg (not in the -env) ;; * a continuation always named -k (not in the -env) ;; | (v e-env e-arg e-k) Tail call ;; | (v e-env () e-k) Forcing a promise ;; | (v () e-ret ()) Calling a continuation ;; | -env the -env ;; | -arg the -arg of the innermost lambda ;; | -k the continuation of the innermost lambda ;; | (-get e-env e-str) Get variable from environment ;; | (-add e-env e-str e-val) Extend environment with new binding #| (λ -env -arg -k ((get -env "1+") (-add -env "foo" 42) -arg -k)) (λ -env -arg -k (let (["env2" (-add -env "foo" 42)]) ((get -env "1+") (get -env "env2") -arg -k))) (define -lambda '…) |# #;( ;; lambda calculus: v ::= (λ x e) || "str" || 0 e ::= v || x || (e e) ;; reduction: redex continuation frames (((λ x (λ y x)) 1) (inc 1)) _ => ((λ x (λ y x)) 1) _ (_ (inc 1)) => (λ y 1) _ (_ (inc 1)) => ( (λ y 1) (inc 1)) _ => (inc 1) _ ((λ y 1) _ ) => 2 _ ((λ y 1) _ ) => ( (λ y 1) 2 ) _ => 1 _ ;; state of evaluation: redex = (v1 v2) continuation = (λ result e) ) #;( ;; Using explicit closures: v ::= (λ […] x e) || "str" || 0 e ::= v || (λ ?? x e) || x || (e e) ;; Rules: rule name environment redex continuation frames => environment′ redex′ continuation frames′ APP [E] ((λ [E′] x e) v) … => [E′,x=v] e … CAPTURE [E] (λ ?? x e) … => [E] (λ [E] x e) … APP-F [E] (e-f e-arg) … => [E] e-f … E,(_ e-arg) APP-ARG [E] (v-f e-arg) … => [E] e-arg … E,(v-f _) CONTINUE-F [E] v-f … E′,(_ e-arg) => [E′] (v-f e-arg) … CONTINUE-ARG [E] v-arg … E′,(v-f _) Optimization: [],(v-f _) => [E′] (v-f v-arg) … ;; Reduction example: env redex continuation frames rule to use [inc=…] (((λ ?? x (λ ?? y x)) 1) (inc 1)) … […],_ APP-F => [inc=…] ((λ ?? x (λ ?? y x)) 1) … […],_ [inc=…],(_ (inc 1)) APP-F => [inc=…] (λ ?? x (λ ?? y x)) … […],_ [inc=…],(_ (inc 1)) [inc=…],(_ 1) CAPTURE => [inc=…] (λ [] x (λ ?? y x)) … […],_ [inc=…],(_ (inc 1)) [inc=…],(_ 1) CONTINUE-F => [inc=…] ((λ [] x (λ ?? y x)) 1) … […],_ [inc=…],(_ (inc 1)) APP-ARG => [inc=…] 1 … […],_ [inc=…],(_ (inc 1)) [inc=…],((λ [] x (λ ?? y x)) _) CONTINUE-ARG => [inc=…] ((λ [] x (λ ?? y x)) 1) … […],_ [inc=…],(_ (inc 1)) APP => [inc=…,x=1] (λ ?? y x) … […],_ [inc=…],(_ (inc 1)) CAPTURE => [inc=…,x=1] (λ [x=1] y x) … […],_ [inc=…],(_ (inc 1)) CONTINUE-F => [inc=…] ( (λ [x=1] y x) (inc 1)) … […],_ APP-ARG => [inc=…] (inc 1) … […],_ [inc=…],((λ [x=1] y x) _) APP-F => [inc=…] inc … […],_ [inc=…],((λ [x=1] y x) _) [inc=…],(_ 1) GETVAR => [inc=…] … … […],_ [inc=…],((λ [x=1] y x) _) [inc=…],(_ 1) CONTINUE-F => [inc=…] (… 1) … […],_ [inc=…],((λ [x=1] y x) _) APP-ARG => [inc=…] 1 … […],_ [inc=…],((λ [x=1] y x) _) [inc=…],(… _) CONTINUE-ARG => [inc=…] (… 1) … […],_ [inc=…],((λ [x=1] y x) _) APP … => [inc=…] 2 … […],_ [inc=…],((λ [x=1] y x) _) CONTINUE-ARG => [inc=…] ( (λ [x=1] y x) 2 ) … […],_ APP => [inc=…,x=1,y=2] x … […],_ GETVAR => [inc=…,x=1,y=2] 2 … […],_ CONTINUE-? => […] 2 … … ) #;( ;; Using first-class environments and lazy evaluations: ;; λ, env, χ, get, push, drop are keywords ;; v-env v ::= (\ env χ e) ;; open term, expects an env to close the term || […] ;; mapping from names to values || "str" || 0 || get || push || pop e ::= v || (@ e e e) TODO: instead of ad-hoc var-to-string conversion, use a functional env ;; Rules: rule name environment redex continuation frames => environment′ redex′ continuation frames′ ;; Primitive application APP env=E, χ=X (@ (\ env χ e) v-env (\ env () e-arg)) … => env=v-env,χ=(\ env () e-arg) e … ;;--------------------------------------------------------------------------------------------------------------------------- ;; Evaluation of sub-parts of an application APP-F env=E, χ=X (@ e-f e-env e-arg) … => env=E, χ=X e-f … [env=E,χ=X],(@ _ e-env e-arg) APP-ENV env=E, χ=X (@ e-f e-env e-arg) … => env=E, χ=X e-env … [env=E,χ=X],(@ v-f _ e-arg) APP-ARG env=E, χ=X (@ e-f e-env e-arg) … => env=E, χ=X e-arg … [env=E,χ=X],(@ v-f v-env _ ) ;;--------------------------------------------------------------------------------------------------------------------------- ;; Syntactic sugar (insertion of #%app) SUGAR-APP env=E, χ=X (#%app e-f e-arg ) … => env=E′, χ=X (@ (@ (get env "#%app") env (\ env () e-f)) env (\ env () e-arg)) … ;; defaults to: => env=E′, χ=X (@ e-f env (\ env () e-arg)) … SUGAR-LAM env=E, χ=X (λ var-name e) … => env=E′, χ=X (#%app (#%app λ var-name) e) … ;; defaults to: => env=E′, χ=X (@ capture env (λ env χ (@ (λ env χ e) (add env "var-name" χ) χ))) ;;--------------------------------------------------------------------------------------------------------------------------- CAPTURE env=E, χ=X (@ capture v-env (λ env χ e)) … => env=E, χ=X (λ env χ (@ (λ env χ e) v-env χ)) … FORCE env=E, χ=(λ env () e-arg) (@ force v-env (λ env χ e)) … => env=E, χ=() TODO … [env=E,χ=(λ env () e-arg)],??? CONTINUE-F [E] v-f … E′,(_ e-arg) => [E′] (v-f e-arg) … CONTINUE-ARG [E] v-arg … E′,(v-f _) Optimization: [],(v-f _) => [E′] (v-f v-arg) … ) ;; "x" ::= "x","y","z"… String ;; ;; v ::= (pλ -env e) promise: (unit) -> env -> α ;; | (kλ -arg e) continuation: α -> void ;; | (cλ -arg e) closure: (α -> β) ;; ;; e ::= (-λ -env -arg -k e) Abstraction (lambda) which takes ;; * an environment always named -env (not in the -env) ;; * a promise for an argument always named -arg (not in the -env) ;; * a continuation always named -k (not in the -env) ;; | (v e-env e-arg e-k) Tail call ;; | (v e-env () e-k) Forcing a promise ;; | (v () e-ret ()) Calling a continuation ;; | -env the -env ;; | -arg the -arg of the innermost lambda ;; | -k the continuation of the innermost lambda ;; | (-get e-env e-str) Get variable from environment ;; | (-add e-env e-str e-val) Extend environment with new binding