README.md

@@ -1,4 +1,4 @@
-# untyped NbE in racket
+# Kripke-style untyped NbE in racket
 [![status-badge](https://woodpecker.electriclam.com/api/badges/4/status.svg)](https://woodpecker.electriclam.com/repos/4)
 
 An implementation of normalization by evaluation loosely based on [A
@@ -10,3 +10,6 @@ Since the implementation is untyped, the `normalize` function only
 gives you β-normal forms. However, you can get a little bit of
 βη-equivalence by invoking Coquand's algorithm (`η-eq?`) on
 β-normal forms.
+
+A more efficient version of NbE based on de Bruijn levels (inverted de
+bruijn indices) can be found in the `debruijn-levels` branch.

nbe-test.rkt

@@ -1,6 +1,6 @@
-#lang typed/racket
+#lang racket
 
-(require typed/rackunit "nbe.rkt")
+(require rackunit "nbe.rkt")
 
 (define-syntax tm-app
   (syntax-rules ()
@@ -11,85 +11,64 @@
 (define-syntax-rule (tm-var a) `(var ,a))
 (define-syntax-rule (tm-abs a) `(λ ,a))
 
-(: tm-id Term)
 (define tm-id '(λ (var 0)))
 
-(: tm-fst Term)
 (define tm-fst '(λ (λ (var 1))))
 
-(: tm-snd Term)
 (define tm-snd '(λ (λ (var 0))))
 
-(: tm-pair Term)
 (define tm-pair `(λ (λ (λ ,(tm-app '(var 0) '(var 2) '(var 1))))))
 
-(: tm-fix Term)
 (define tm-fix
   (let ([g (tm-abs (tm-app (tm-var 1) (tm-app (tm-var 0) (tm-var 0))))])
     (tm-abs (tm-app g g))))
 
-(: tm-zero Term)
 (define tm-zero tm-snd)
 
-(: tm-suc (-> Term Term))
 (define (tm-suc a) (tm-abs (tm-abs (tm-app (tm-var 1) (tm-app a (tm-var 1) (tm-var 0))))))
 
-(: tm-add (-> Term Term Term))
 (define (tm-add a b)
   (tm-abs (tm-abs
            (tm-app b (tm-var 1) (tm-app a (tm-var 1) (tm-var 0))))))
 
-(: tm-ind (-> Term Term Term Term))
-(define (tm-ind a b c)
-  `(ind ,a ,b ,c))
-
-(: tm-padd (-> Term Term Term))
-(define (tm-padd a b)
-  (tm-app (tm-ind a tm-id (tm-abs (tm-psuc (tm-app (tm-var 1) (tm-var 0))))) b))
-
-(: tm-compose (-> Term Term Term))
 (define (tm-compose a b)
   (tm-abs (tm-app a (tm-app b (tm-var 0)))))
 
-(: tm-mult (-> Term Term Term))
 (define (tm-mult a b) (tm-compose a b))
 
-(: tm-nat (-> V Term))
 (define (tm-nat n)
   (if (positive? n)
       (tm-suc (tm-nat (- n 1)))
       tm-zero))
 
-(: tm-const Term)
-(define tm-const tm-fst)
-
-(: tm-loop Term)
-(define tm-loop
-  (let ([g (tm-abs (tm-app (tm-var 0) (tm-var 0)))])
-    (tm-app g g)))
-
-(: tm-nat-to-pnat Term)
-(define tm-nat-to-pnat
-  (tm-abs (tm-app (tm-var 0) (tm-abs '(succ (var 0))) 'zero)))
-
-(: tm-pnat (-> V Term))
 (define (tm-pnat n)
   (if (positive? n)
       `(succ ,(tm-pnat (- n 1)))
       'zero))
 
-(: tm-psuc (-> Term Term))
+(define (tm-ifz a b c)
+  `(if-zero ,a ,b ,c))
+
 (define (tm-psuc a)
   `(succ ,a))
 
+(define (tm-double m)
+  (tm-app tm-fix
+          (tm-abs (tm-abs (tm-ifz (tm-var 0) 'zero 'zero))) m ))
+
+(define (tm-padd m n)
+  (tm-app tm-fix
+          (tm-abs (tm-abs (tm-abs (tm-ifz (tm-var 1) (tm-var 0) (tm-psuc (tm-app (tm-var 3) (tm-var 0) (tm-var 1))))))) m n))
+
 (check-equal? (normalize `(app ,tm-id ,tm-id)) tm-id)
 (check-equal? (normalize `(app (app (app ,tm-pair ,tm-id) ,tm-fst) ,tm-snd)) tm-fst)
 (check-equal? (normalize `(app (app (app ,tm-pair ,tm-id) ,tm-fst) ,tm-fst)) tm-id)
 (check-equal? (normalize (tm-app tm-snd (tm-app tm-pair tm-id tm-fst) tm-fst)) tm-fst)
 (check-equal? (normalize (tm-add (tm-nat 499) (tm-nat 777))) (normalize (tm-add (tm-nat 777) (tm-nat 499))))
-(check-equal? (normalize (tm-app tm-const (tm-nat 0) tm-loop)) (normalize (tm-nat 0)))
+(check-equal? (normalize (tm-mult (tm-nat 3) (tm-nat 2))) (normalize (tm-nat 6)))
-(check-equal? (normalize (tm-app tm-nat-to-pnat (tm-nat 10))) (tm-pnat 10))
+(check-equal? (normalize (tm-mult (tm-nat 11) (tm-nat 116))) (normalize (tm-nat 1276)))
-(check-equal? (normalize `(ind ,(tm-pnat 3) ,(tm-pnat 0) (var 1))) (tm-pnat 2))
+(check η-eq? (normalize (tm-add (tm-nat 499) (tm-nat 777))) (normalize (tm-add (tm-nat 777) (tm-nat 499))))
-(check-equal? (normalize `(ind ,(tm-pnat 3) ,tm-loop (var 1))) (tm-pnat 2))
+(check βη-eq? (tm-mult (tm-nat 6) (tm-nat 7)) (tm-nat 42))
-(check-equal? (normalize (tm-padd (tm-pnat 10000) (tm-pnat 2000)))
+(check βη-eq? '(if-zero (succ (succ zero)) zero (succ (succ (var 0)))) (tm-pnat 3))
-              (tm-pnat 12000))
+(check βη-eq? (tm-padd (tm-pnat 8) (tm-pnat 11)) (tm-pnat 19))
+(check βη-eq? (tm-padd (tm-pnat 2) (tm-psuc (tm-var 0))) '(succ (succ (succ (var 0)))))

nbe.rkt

@@ -1,159 +1,138 @@
-#lang typed/racket
+#lang racket
 ;; Grammar (Λ)
 ;; t := λ t | app t t | i
 
-(define-type denv (-> V (Promise D)))
+;; Domain
-(define-type V Nonnegative-Integer)
+;; D := neu D_ne | fun [(var -> var) -> D → D]
-(define-type Term (∪ 'zero
+;; D_ne := var i | app D_ne D
-                     (List 'succ Term)
-                     (List 'var V)
-                     (List 'λ Term)
-                     (List 'app Term Term)
-                     (List 'ind Term Term Term)))
 
-(define-type D (∪ 'zero
+(define (tm? a)
-                  (List 'succ (Promise D))
+  (match a
-                  (List 'fun (-> (Promise D) D))
+    ['zero true]
-                  (List 'neu D-ne)))
+    [`(succ ,a) (tm? a)]
+    [`(if-zero ,a ,b ,c) (and (tm? a) (tm? b) (tm? c))]
+    [`(λ ,a) (tm? a)]
+    [`(app ,a ,b) (and (tm? a) (tm? b))]
+    [`(var ,i) (exact-nonnegative-integer? i)]
+    [_ false]))
 
-(define-type D-ne (∪ (List 'app D-ne D)
+(define-syntax-rule (ap a b)
-                     (List 'idx V)
+  (match (force a)
-                     (List 'ind D-ne D (-> (Promise D) (Promise D) D))))
+    [`(fun ,f) (force (f identity b))]
+    [`(neu ,u) `(neu (app ,u ,b))]
+    [_ (error "ap: type error")]))
 
-(: ext (-> denv (Promise D) denv))
+(define-syntax-rule (ifz a b c)
-(define (ext ρ a)
+  (match (force a)
+    ['zero (force b)]
+    [`(succ ,u) (ap c u)]
+    [`(neu ,u) `(neu (if-zero ,u ,b ,c))]))
+
+(define compose-ren compose)
+(define (compose-ren-sub ξ ρ)
+  (compose (curry ren-dom ξ) ρ))
+(define-syntax-rule (ext ρ a)
   (lambda (i)
     (if (zero? i)
         a
         (ρ (- i 1)))))
 
-
+(define (ren-ne-dom ξ a)
-
-(: interp-fun (-> Term denv D))
-(define (interp-fun a ρ)
-  (list 'fun (λ (x) (interp a (ext ρ x)))))
-
-(: interp-fun2 (-> Term denv (-> (Promise D) (Promise D) D)))
-(define (interp-fun2 a ρ)
-  (λ (x y) (interp a (ext (ext ρ x) y))))
-
-(: interp-ind (-> D (Promise D) (-> (Promise D) (Promise D) D) D))
-(define (interp-ind a b c)
   (match a
-    [`(neu ,u) `(neu (ind ,u ,(force b) ,c))]
+    [`(var ,i) `(var ,(ξ i))]
-    ['zero (force b)]
+    [`(app ,a ,b) `(app ,(ren-ne-dom ξ a) ,(ren-dom ξ b))]
-    [`(succ ,a) (c a (delay (interp-ind (force a) b c)))]))
+    [`(if-zero ,a ,b ,c) `(if-zero ,(ren-ne-dom ξ a) ,(ren-dom ξ b) ,(ren-dom ξ c))]))
 
-(: ap (-> D Term denv D))
+(define (ren-dom ξ a)
-(define (ap a b ρ)
+  (delay (match (force a)
-  (match a
+    ['zero 'zero]
-    ['zero (error "type-error: ap zero")]
+    [`(succ ,a) `(succ ,(ren-dom ξ a))]
-    [`(succ ,_) (error "type-error: ap succ")]
+    [`(neu ,a) `(neu ,(ren-ne-dom ξ a))]
-    [`(fun ,f) (f (delay (interp b ρ)))]
+    [`(fun ,f) `(fun ,(λ (ξ0 α) (f (compose-ren ξ0 ξ) α)))])))
-    [`(neu ,u) `(neu (app ,u ,(interp b ρ)))]))
+
+(define-syntax-rule (interp-fun a ρ)
+  (list 'fun (λ (ξ x) (interp a (ext (compose-ren-sub ξ ρ) x)))))
 
-(: interp (-> Term denv D))
 (define (interp a ρ)
-  (match a
+  (delay (match a
     [`(var ,i) (force (ρ i))]
     ['zero 'zero]
-    [`(succ ,a) `(succ ,(delay (interp a ρ)))]
+    [`(succ ,a) `(succ ,(interp a ρ))]
-    [`(ind ,a ,b ,c) (interp-ind
+    [`(if-zero ,a ,b ,c) (ifz (interp a ρ) (interp b ρ) (interp-fun c ρ))]
-                      (interp a ρ)
-                      (delay (interp b ρ))
-                      (interp-fun2 c ρ))]
     [`(λ ,a) (interp-fun a ρ)]
-    [`(app ,a ,b) (ap (interp a ρ) b ρ)]))
+    [`(app ,a ,b) (ap (interp a ρ) (interp b ρ))])))
 
-(: reify (-> V D Term))
+(define (reify a)
-(define (reify n a)
+  (match (force a)
-  (match a
     ['zero 'zero]
-    [`(succ ,a) `(succ ,(reify n (force a)))]
+    [`(succ ,a) `(succ ,(reify a))]
-    [`(fun ,f) (list 'λ (reify (+ n 1) (f (delay `(neu (idx ,n))))))]
+    [`(fun ,f) (list 'λ (reify (f  (curry + 1) '(neu (var 0)))))]
-    [`(neu ,a) (reify-neu n a)]))
+    [`(neu ,a) (reify-neu a)]))
 
-(: reify-neu (-> V D-ne Term))
+(define (extract-body a)
-(define (reify-neu n a)
   (match a
-    [`(ind ,a ,b ,c) (list 'ind
+    [`(λ ,a) a]
-                           (reify-neu n a)
+    [_ (error "reify-neu: not reifiable")]))
-                           (reify n b)
-                           (reify (+ n 2)
-                                  (c (delay `(neu (idx ,n)))
-                                     (delay `(neu (idx ,(+ 1 n)))))))]
-    [`(app ,u ,v) (list 'app (reify-neu n u) (reify n v))]
-    [`(idx ,i) (list 'var (max 0 (- n (+ i 1))))]))
 
-(: idsub (-> V V D))
+(define (reify-neu a)
-(define (idsub s i) `(neu (idx ,(max 0 (- s (+ i 1))))))
-
-(: scope (-> Term V))
-(define (scope a)
   (match a
-    ['zero 0]
+    [`(if-zero ,a ,b ,c) (list 'if (reify-neu a) (reify b) (extract-body (reify c)))]
-    [`(succ ,a) (scope a)]
+    [`(app ,u ,v) (list 'app (reify-neu u) (reify v))]
-    (`(ind ,a ,b ,c) (max (scope a) (scope b) (- (scope c) 2)))
+    [`(var ,i) a]))
-    [`(if-zero ,a ,b ,c) (max (scope a) (scope b) (scope c))]
+
-    [`(λ ,a) (max 0 (- (scope a) 1))]
+(define (idsub i) `(neu (var ,i)))
-    [`(app ,a ,b) (max (scope a) (scope b))]
-    [`(var ,i) (+ i 1)]))
 
-(: normalize (-> Term Term))
 (define (normalize a)
-  (let ([sa (scope a)])
+  (reify (interp a idsub)))
-    (reify sa (interp a (λ (x) (delay (idsub sa x)))))))
 
-;; (define (subst ρ a)
+(define (subst ρ a)
-;;   (match a
+  (match a
-;;     [`(var ,i) (ρ i)]
+    [`(var ,i) (ρ i)]
-;;     [`(app ,a ,b) `(app ,(subst ρ a) ,(subst ρ b))]
+    [`(app ,a ,b) `(app ,(subst ρ a) ,(subst ρ b))]
-;;     [`(λ ,a) `(λ ,(subst (ext (compose (curry subst (λ (i) `(var ,(+ i 1)))) ρ)
+    [`(λ ,a) `(λ ,(subst (ext (compose (curry subst (λ (i) `(var ,(+ i 1)))) ρ)
-;;                               '(var 0)) a))]))
+                              '(var 0)) a))]))
 
-;; (define (idsub-tm i) `(var ,i))
+(define (idsub-tm i) `(var ,i))
-;; (define (subst1 b a)
+(define (subst1 b a)
-;;   (subst (ext idsub-tm b) a))
+  (subst (ext idsub-tm b) a))
 
-;; (define (eval-tm a)
+(define (eval-tm a)
-;;   (match a
+  (match a
-;;     [(list 'var _) a]
+    [(list 'var _) a]
-;;     [(list 'λ a) `(λ ,(eval-tm a))]
+    [(list 'λ a) `(λ ,(eval-tm a))]
-;;     [(list 'app a b)
+    [(list 'app a b)
-;;      (match (eval-tm a)
+     (match (eval-tm a)
-;;        [(list 'λ a) (eval-tm (subst1 b a))]
+       [(list 'λ a) (eval-tm (subst1 b a))]
-;;        [v `(app ,v ,(eval-tm b))])]))
+       [v `(app ,v ,(eval-tm b))])]))
 
-;; (define (eval-tm-strict a)
+(define (eval-tm-strict a)
-;;   (match a
+  (match a
-;;     [(list 'var _) a]
+    [(list 'var _) a]
-;;     [(list 'λ a) `(λ ,(eval-tm-strict a))]
+    [(list 'λ a) `(λ ,(eval-tm-strict a))]
-;;     [(list 'app a b)
+    [(list 'app a b)
-;;      (match (eval-tm-strict a)
+     (match (eval-tm-strict a)
-;;        [(list 'λ a) (eval-tm-strict (subst1 (eval-tm-strict b) a))]
+       [(list 'λ a) (eval-tm-strict (subst1 (eval-tm-strict b) a))]
-;;        [v `(app ,v ,(eval-tm-strict b))])]))
+       [v `(app ,v ,(eval-tm-strict b))])]))
 
-;; ;; Coquand's algorithm but for β-normal forms
+;; Coquand's algorithm but for β-normal forms
-;; (: η-eq? (-> Term Term Boolean))
+(define (η-eq? a b)
-;; (define (η-eq? a b)
+  (match (list a b)
-;;   (match (list a b)
+    ['(zero zero) true]
-;;     ['(zero zero) true]
+    [`((succ ,a) (succ ,b)) (η-eq? a b)]
-;;     [`((succ ,a) (succ ,b)) (η-eq? a b)]
+    [`((if-zero ,a ,b ,c) (if-zero ,a0 ,b0 ,c0))
-;;     [`((if-zero ,a ,b ,c) (if-zero ,a0 ,b0 ,c0))
+     (and (η-eq? a a0) (η-eq? b b0) (η-eq? c c0))]
-;;      (and (η-eq? a a0) (η-eq? b b0) (η-eq? c c0))]
+    [`((λ ,a) (λ ,b)) (η-eq? a b)]
-;;     [`((λ ,a) (λ ,b)) (η-eq? a b)]
+    [`((λ ,a) ,u) (η-eq? a `(app ,(subst (λ (i) `(var ,(+ i 1))) u) (var 0)))]
-;;     [`((λ ,a) ,u) (η-eq? a `(app ,(subst (λ (i) `(var ,(+ i 1))) u) (var 0)))]
+    [`(,u (λ ,a)) (η-eq? `(app ,(subst (λ (i) `(var ,(+ i 1))) u) (var 0)) a)]
-;;     [`(,u (λ ,a)) (η-eq? `(app ,(subst (λ (i) `(var ,(+ i 1))) u) (var 0)) a)]
+    [`((app ,u0 ,v0) (app ,u1 ,v1)) (and (η-eq? u0 u1) (η-eq? v0 v1))]
-;;     [`((app ,u0 ,v0) (app ,u1 ,v1)) (and (η-eq? u0 u1) (η-eq? v0 v1))]
+    [`((var ,i) (var ,j)) (eqv? i j)]
-;;     [`((var ,i) (var ,j)) (eqv? i j)]
+    [_ false]))
-;;     [_ false]))
 
 
-;; (define (βη-eq? a b)
+(define (βη-eq? a b)
-;;   (η-eq? (normalize a) (normalize b)))
+  (η-eq? (normalize a) (normalize b)))
 
-(: β-eq? (-> Term Term Boolean))
 (define (β-eq? a b)
   (equal? (normalize a) (normalize b)))
 
-(provide reify interp normalize β-eq? Term D V)
+(provide eval-tm eval-tm-strict reify interp normalize tm? η-eq? βη-eq? β-eq?)