## Sunday, 1 February 2009

### CPS transformation using delimited continuations

`;; Idea: Use reified continuations to implement a CPS converter;;;; in the expression: (+ (* x x) (* y y));; the continuation of (* x x) is (lambda (x-squared) (+ x-squared (* y y)));;;; similiarly,;;;; in the expression: (list '+ (list '* 'x 'x) (list '* 'y 'y));; the continuation of (list '* 'x 'x) is (lambda (x-squared) (list '+ x-squared (list '* 'y 'y)))(require scheme/control) ;; shift/reset delimited continuations;; > (reset (list '+ (shift k (list '* 'x 'x))     (list '* 'y 'y)));; (* x x);; > (reset (list '+ (shift k (k '?))              (list '* 'y 'y)));; (+ ? (* y y));; > (reset (list '+ (shift k (k (list '* 'x 'x))) (list '* 'y 'y)));; (+ (* x x) (* y y));; > (reset (list '+ (shift k `(let ((x-squared ,(list '* 'x 'x))) ,(k 'x-squared))) (list '* 'y 'y)));; (let ((x-squared (* x x))) (+ x-squared (* y y)));; CPS applications should throw the result value into a continuation;; so (f x y z) turns into (f x y z (lambda (result) (continuation result)))(define (apply# f . args) (shift k (let ((g (gensym "g"))) (reset `(,f ,@args (lambda (,g) ,(k g)))))));; Some examples of apply# in action:;;;; > `(k ,(apply# 'f--> 'x 'y 'z));; (f--> x y z (lambda (g349) (k g349)));; > (apply# 'f--> (apply# '+--> 2 3) 'y 'z);; (+--> 2 3 (lambda (g345) (f--> g345 y z (lambda (g346) g346))));; > (apply# 'f--> (apply# '+--> 2 3) (apply# '*--> 'x 'y) 'z);; (+--> 2 3 (lambda (g415) (*--> x y (lambda (g416) (f--> g415 g416 z (lambda (g417) g417))))));; What about syntax like IF? clearly apply# would be wrong (due to evaluation order) so define new syntax!(define-syntax if#  (syntax-rules ()    ((if# <cond> <then> <else>) (shift k `(if ,<cond> ,(reset (k <then>)) ,(reset (k <else>)))))));; Examples:;;;; > (if# (apply# 'zero?--> 'n) ''yes ''no);; (zero?--> n (lambda (g374) (if g374 'yes 'no)));; > `(display ,(if# (apply# 'zero?--> 'n) ''yes ''no));; (zero?--> n (lambda (g850) (if g850 (display 'yes) (display 'no))))(define-syntax define#  (syntax-rules ()    ((define# (name/args ...) body) `(define (name/args ... k-->) ,(reset `(k--> ,body))))));; That's enough now to CPS convert entire procedures:;; > (define# (fact-iter--> n acc);;     (if# (apply# 'zero?--> 'n);;          'acc;;          (apply# 'fact-iter--> (apply# '---> 'n 1) (apply# '*--> 'acc 'n))))(define (fact-iter--> n acc k-->)  (zero?--> n (lambda (g875)   (if g875       (k--> acc)       (---> n 1 (lambda (g876)        (*--> acc n (lambda (g877)         (fact-iter--> g876 g877 (lambda (g878) (k--> g878)))))))))));; Test it! (this CPS format is a subset of Scheme)(define (zero?--> n k) (k (zero? n)))(define (*--> x y k) (k (* x y)))(define (---> x y k) (k (- x y)));; > (fact-iter--> 7 1 display);; 5040;; Now a function to CPS convert based on all this is trivial, it's just a fold;; that replaces if with if#, define with define# and applications with apply#!(define (if#-thunked cond then-thunk else-thunk) (if# cond (then-thunk) (else-thunk)))(define (define#-thunked name/args body-thunk)  (let ((k--> (gensym "k-->"))) `(define (,@name/args ,k-->) ,(reset `(,k--> ,(body-thunk))))))(define (cps term)  (if (pair? term)      (case (car term)        ((quote) `',term)        ((if) (if#-thunked (cps (cadr term)) (lambda () (cps (caddr term))) (lambda () (cps (cadddr term)))))        ((define) (define#-thunked (cadr term) (lambda () (cps (caddr term)))))        (else (apply apply# (map cps term))))      term));; > (cps '(define (fact-iter-2 n acc) (if (zero? n) acc (fact-iter-2 (- n 1) (* n acc)))))(define (fact-iter-2--> n acc k-->)  (zero?--> n (lambda (g443)   (if g443       (k--> acc)       (---> n 1 (lambda (g444)        (*--> n acc (lambda (g445)         (fact-iter-2--> g444 g445 (lambda (g446)          (k--> g446)))))))))));; > (fact-iter-2--> 8 1 display);; 40320`