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