Polymorphic Typ e Assignment and CPS Conversion (To app ear: ACM SIGPLAN Workshop on Continuations, San Francisco, June 1992) y Rob ert Harp er Mark Lillibridge Carnegie Mellon University Pittsburgh, PA15213 April 10, 1992 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-valu e CPS transform. This typing prop ertymaybe extended to Scheme-likecontinuation-passi ng 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 olymorphi c let is restricted to values, and for the call-by-name interpretation with no restrictions. We prove that there is no call-by-value CPS transform for the full Damas-Milner language that validates the Meyer-Wand typing prop erty and is equivalentto the standard call-by-value transform up to -conversion. 1 Intro duction In their study of the relationship b etween direct and continuation semantics 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 wayto the typ e of its call-by-value continuation passing style (CPS) transform [8]. This result may b e extended ! to the calculus that results from extending with Scheme-like continuation-passing primitives callcc and ! ! throw ( + cont)[1, 3]. Since under a call-by-value op erational semantics is \typ e safe" in the sense of Milner [9, 2], and since the call-by-value CPS transform faithfully mimics the call-by-value semantics [12], ! + cont under a call-by-value op erational semantics is also typ e safe. it follows that In a subsequent study Duba, Harp er, and MacQueen studied the addition of callcc and throw to Stan- ! dard ML [10]. The extension of the Meyer-Wand transform to + cont establishes the soundness of the monomorphicfragment 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 [7] that the full p olymorphic language is unsound when extended with callcc and throw. The source of this discrepancy may b e 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 [6, 14]. 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 restricted to values, and for the call-by-name transform, under no restrictions. Finally,we 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. y Supp orted by a National Science Foundation Graduate Fellowship. 1 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 -conversion. In particular, the standard call-by-value CPS transform fails to preservetypability. 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 olymorphic typ e assignment. callcc and throw are continuation-passing primitives whose de nitions are derived from analogous constructs in Scheme [1] and Standard ML of New Jersey [3]. We consider two CPS transforms for untyp ed terms, corresp onding to the call-by-value and call-by-name op erational semantics [12]. Each CPS transform consists of a transformation jj for untyp ed terms and a transformation jjjj for untyp ed values. Exactly what is considered a value dep ends on which op erational 1 semantics is b eing used. Under call-by-value, variables, -abstractions, and constants are considered values. Under call-by-name, only -abstractions and constants are considered values. We shall use v as a meta- variable for call-by-value values and w as a meta-variable for call-by-name values. De nition 2.1 (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 Lemma 2.2 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 0 We shall also haveneedofavariant call-by-value CPS transform (cbv ) de ned on untyp ed terms sat- isfying the restriction that all let expressions are of the form let x be v in e. I.e., the let-b ound expression is required to b e a (call-by-value) value. Because of this restriction, a simpler rule can b e given for the let case: 0 0 0 jlet x be v in ej = k :let x be jjv jj in (jej k ) cbv cbv cbv This simpler rule for let expressions is the only di erence b etween the two transforms. 0 Lemma 2.3 Let v and v be values obeying the restriction on let expressions and e be a term obeying the restriction on let expressions. Then 0 0 0 0 0 . =x ] jjv jj =[jjv jj 1. jj [v=x]v jj cbv cbv cbv 0 0 0 2. j [v=x]e j =[jjv jj =x ] jej . cbv cbv cbv 1 Note that callcc and throw are considered to b e constants. 2 De nition 2.4 (Call-by-Name CPS Transform) jw j = k :k jjw jj cbn cbn jxj = x cbn je e j = k :je j (x :x je j k ) 1 2 cbn 1 cbn 1 1 2 cbn jlet x be e in e j = k :let x be je j in (je j k ) 1 2 cbn 1 cbn 2 cbn jjx:ejj = x:jej cbn cbn 0 0 jjcallccjj = f :k :f (f :f (l :l k ) k ) cbn 0 0 0 0 jjthrowjj = c:k :k (x:l :c (c :x (x :c x ))) cbn Lemma 2.5 1. jj [e=x]w jj =[jej =x ] jjw jj . cbn cbn cbn 0 0 2. j [e=x]e j =[jej =x ] je j . cbn cbn cbn The correctness of these transforms may either b e established by relating them to an indep endently- de ned op erational semantics (as in [12, 4]), or else taken as the de nition of call-by-value and call-by-name semantics. 3 Simple Typ e Assignment In this section we review Meyer and Wand's typing theorem for the call-by-value CPS transform for the ! simply-typ ed -calculus ( ), and present an analogous result for the call-by-name CPS transform. ! De nition3.1( Typ es and Contexts) ty pes ::= b j ! 1 2 contexts ::= j;x: Here b ranges overacountable set of base typ es. We assume that among the base typ es there is a distinguished typ e , which will b e used in what follows to represent the \answer" typ e of a CPS transform. ! De nition3.2( Typing Rules) .x:(x) (var) ;x: .e : 1 2 (x 62 dom()) (abs) . x:e : ! 1 2 .e : ! .e : 1 2 2 2 (app) .e e : 1 2 .e : ;x: .e : 1 1 1 2 (mono-let) . let x be e in e : 1 2 ! The typ e system + cont is de ned by adding the typ e expression cont and the following typing rules for the continuation-passing primitives: . callcc :( cont ! ) ! (callcc) 0 . throw : cont ! ! (throw) 3 ! De nition 3.3 (Call-by-Value Typ e Transform for ) j j = (jj jj ! ) ! cbv cbv jjbjj = b cbv jj ! jj = jj jj !j j 1 2 cbv 1 cbv 2 cbv The typ e transform is extended to contexts by de ning jjjj (x)= jj(x)jj for each x 2 dom(). cbv cbv Theorem 3.4 (Meyer-Wand) ! ! 1. If ` .v : , then `jjjj . jjv jj : jj jj . cbv cbv cbv ! ! 2. If ` .e : , then `jjjj . jej : j j . cbv cbv cbv ! The call-by-value typ e transform is extended to + cont by de ning jj cont jj = jj jj ! .Itis cbv cbv ! straightforward to verify that Theorem 3.4 extends to + cont in this way[3].