LISP AND SYMBOLIC COMPUTATION: An International Journal, ?, ??{ ??, 1993 c 1993 Kluwer Academic Publishers { Manufactured in The Netherlands Polymorphic Typ e Assignment and CPS Conversion y ROBERT HARPER ([email protected] ) z MARK LILLIBRIDGE (md [email protected] ) School of Computer Science Carnegie Mel lon University 5000 Forbes Avenue Pittsburgh, PA15213 Keywords: Polymorphism, continuations Abstract. Meyer and Wand established that the typ e of a term in the simply typ ed - calculus may b e related in a straightforward manner to the typ e of its call-by-value CPS transform. This typing prop ertymay b e extended to Scheme-likecontinuation-pas sin g primitives, from which the soundness of these extensions follows. We study the extension of these results to the Damas-Milner p olymorphic typ e assignment system under b oth the call-by-value and call-by-name interpretations. We obtain CPS transforms for the call-by-value interpretation, provided that the p olymorphic let is restricted to values, and for the call-by-name interpretation with no restrictions. We provethatthereis no call-by-value CPS transform for the full Damas-Milner language that validates the Meyer-Wand typing prop erty and is equivalent to the standard call-by-value transform up to op erational equivalence. 1. Intro duction In their study of the relationship b etween direct and continuation seman- ! tics for the simply typ ed -calculus ( ), Meyer and Wand note that the ! typ e of a term in may b e related in a simple and natural way to the typ e of its call-by-value continuation passing style (CPS) transform [13]. This result may b e extended to the calculus that results from extend- ! ing with Scheme-likecontinuation-passing primitives callcc and throw ! ! ( + cont )[1, 4]. Since under a call-by-value op erational semantics is \typ e safe" in the sense of Milner [14, 2], and since the call-by-value CPS transform faithfully mimics the call-by-value semantics [17], it follows that This is a revised version of a pap er presented at the ACM SIGPLAN Workshop on Continuations, San Francisco, June 1992. y This work was sp onsored by the Defense Advanced Research Pro jects Agency, CSTO, under the title \The Fox Pro ject: Advanced Development of Systems Software", ARPA Order No. 8313, issued by ESD/AVS under Contract No. F19628{91{C{0168. z Supp orted by a National Science Foundation Graduate Fellowship. 2 HARPER AND LILLIBRIDGE ! + cont under a call-by-value op erational semantics is also typ e safe. In a subsequent study Duba, Harp er, and MacQueen studied the addition of callcc and throw to Standard ML [15]. The extension of the Meyer-Wand ! transform to + cont establishes the soundness of the monomorphic frag- ment of the language, but the soundness of the p olymorphic language with continuation-passing primitives was left op en. It was subsequently proved by the authors [9] that the full p olymorphic language is unsound when extended with callcc and throw. The source of this discrepancy maybe traced to the interaction b etween the p olymorphic let construct and the typing rules for callcc.Several ad-hoc metho ds for restricting the language to recover soundness have b een prop osed [8, 23]. In this pap er we undertake a systematic study of the interaction b etween continuations and p olymorphism by considering the typing prop erties of the CPS transform for b oth the call-by-value and call-by-name variants of the Damas-Milner language [2] and its extension with continuation-passing primitives. We obtain suitable extensions of the Meyer-Wand theorem for the call-by-value CPS transform, provided that the p olymorphic let is re- stricted to values, and for the call-by-name transform, under no restrictions. Finally,we prove that there is no call-by-value CPS transform for the full Damas-Milner language that b oth satis es the Meyer-Wand typing prop- erty and is equivalent to the usual transform up to op erational equivalence. In particular, the standard call-by-value CPS transform fails to preserve typability. 2. Untyp ed Terms The language of untyp ed terms is given by the following grammar: e ::= x j x:e j e e j let x be e in e j callcc j throw 1 2 1 2 Here x ranges over a countably in nite set of variables. We include the let construct as a primitive b ecause it is needed in the discussion of p olymor- phic typ e assignment. The constants callcc and throw stand for continuation- passing primitives whose de nitions are derived from analogous constructs in Scheme [1] and Standard ML of New Jersey [4]. We consider two CPS transforms for untyp ed terms, corresp onding to the call-by-value and call-by-name op erational semantics [17]. EachCPS transform consists of a transformation jj for untyp ed terms and a trans- formation jjjj for untyp ed values. Exactly what is considered a value de- p ends on which op erational semantics is b eing used. Under a call-by-value interpretation, the set of values is given by the following grammar: v ::= x j x:e j callcc j throw POLYMORPHIC TYPE ASSIGNMENT AND CPS CONVERSION 3 Since throw is a two-argument function, it is p ossible to regard throw v as avalue. Wecho ose to not regard it as a value b ecause to do so compli- cates somewhat b oth the transform and the asso ciated theorems due to the resulting overlap b etween the application and value cases. Under a call-by- name interpretation, the set of values is given by the following grammar: n ::= x:e j callcc j throw 1 De nition 1 (Call-by-Value CPS Transform) jv j = k :k jjv jj cbv cbv je e j = k :je j (x :je j (x :x x k )) 1 2 cbv 1 cbv 1 2 cbv 2 1 2 jlet x be e in e j = k :je j (x:je j k ) 1 2 cbv 1 cbv 2 cbv jjxjj = x cbv jjx:ejj = x:jej cbv cbv jjcallccjj = f :k :f k k cbv jjthrowjj = c:k :k (x:l :c x) cbv The transformation of let expressions is chosen to re ect the usual \sequen- tial" evaluation strategy for let's in a call-by-value language in which the let-b ound expression, e , is fully evaluated prior to evaluation of the b o dy, 1 e . (Compare this with the call-by-name transform in De nition 2 b elow.) 2 The transformation of callcc di ers from that in Scheme since continuations are not here represented as functions which are applied to their arguments, but rather are represented directly as continuations which are invoked us- ing the throw primitive. (See Harp er, Duba, and MacQueen [8] for further discussion of this p oint.) The call-by-value CPS transform is comp ositional in the sense that it commutes with substitution. Lemma 1 0 0 1. jj [v=x]v jj =[jjv jj =x ] jjv jj . cbv cbv cbv 2. j [v=x]e j =[jjv jj =x ] jej . cbv cbv cbv Pro of: The pro of pro ceeds by simultaneous induction on the struc- 0 ture of v and e.Wegive here a few illustrative cases; the remainder followby a similar argument. We drop the \cbv " subscript for the sake of readability. 1 In each equation any b ound variable o ccurring on the right that do es not also o ccur on the left is assumed to b e chosen so as to avoid capture. 4 HARPER AND LILLIBRIDGE 0 v = x : 0 0 Wehaveby de nition jjxjj = x, and hence [jjv jj=x]jjxjj = jjv jj = 0 jj[v =x]xjj. 0 0 v = x 6= x : 0 0 0 0 0 Wehaveby de nition jjx jj = x , and hence [jjv jj=x]jjx jj = x = 0 0 jj[v =x]x jj. 0 v = y :e : Wemay assume without loss of generality that y 6= x and that y do es not o ccur freely in jjv jj. jj[v=x]y :ejj = jjy :[v=x]ejj = y :j[v=x]ej = y :[jjv jj=x]jej (by induction) = [jjv jj=x]y :jej = [jjv jj=x] jjy :ejj 0 e = v : Wemay assume without loss of generalitythat k do es not o ccur 0 freely in jjv jj. Note that values are stable under substitution of avalue for a variable. 0 0 0 j[v=x]v j = k :k jj[v=x]v jj ( [v=x]v is a value) 0 = k :k [jjv jj=x]jjv jj (by induction) 0 = [jjv jj=x]k :k jjv jj 0 = [jjv jj=x] jjv jj e = e e : 1 2 Wemay assume without loss of generalitythatthevariables x , 1 x , and k do not o ccur freely in jjv jj. 2 j[v=x](e e )j = j[v=x]e [v=x]e j 1 2 1 2 = k :j[v=x]e j (x :j[v=x]e j (x :x x k )) 1 1 2 2 1 2 = k :[jjv jj=x]je j (x :[jjv jj=x]je j (x :x x k )) 1 1 2 2 1 2 = [jjv jj=x]k :je j (x :je j (x :x x k )) 1 1 2 2 1 2 = [jjv jj=x]je e j 1 2 0 We shall also haveneedofavariant call-by-value CPS transform (cbv ) de ned on untyp ed terms satisfying the restriction that all let expressions are of the form let x be v in e | i.e.,thelet-b ound expression is required to b e a (call-by-value) value.