New Foundations for Fixpoint Computations Roy L. Crole* Andrew M. Pittst Computer Laboratory, University of Cambridge CB2 3QG, England Abstract computations using monads introduced by Moggi [SI, where there is a distinction between the ele- This paper introduces a new higher-order typed ments of a type a and computations of elements constructive predicate logic for fixpoint compu- of that type-the latter being grouped into a tations, which exploits the categorical semantics new type T(a). Moggi’s computational meta- of computations introduced by Moggi [8] and language XMLT [lo], contains the following for- contains a strong version of Martin-Lof’s ‘iter- mation rules: ation type’ [ll].The type system enforces a sep- a type aration of computations from values. The logic contains a novel form of fixpoint induction and can express partial and total correctness state- MEa ments about evaluation of computations to val- Val(M) E T(a) ues. The constructive nature of the logic is wit- nessed by strong metalogical properties which are proved using a category-theoretic version of the ‘logical relations’ method. Let x+-E.F(x) E T(P) 1 Computation types Note These rules, and the others which appear in this paper, are presented in natural deduc- It is well known that primitive recursion at tion style, with discharged hypotheses enclosed higher types can be given a categorical character- in square brackets. Since there are several unfa- isation in terms of Lawvere’s concept of natural miliar variable binding operations in the syntax, number object [6]. We show that a similar char- we will also adopt Martin-Lof’s theory of expres- acterisation can be given for general recursion via sions and arities-a Pq-lambda calculus over a fixpoint operators of higher types, in terms of a single ground type of expressions with abstrac- new concept-that of a fixpoint object in a suit- tion denoted (x)e,application denoted f(e) and ably structured category. This notion was partly a multiple application such as (f(e))(e’)abbre- inspired by contemplation of Martin-Lof’s non- viated to f(e,e’); see [la], for example. Finally, standard ‘iteration type’ in his domain theoretic it should be noted that our syntax is a slight interpretation of type theory [ll].However, the variant of Moggi’s. key ingredient which allows the formulation of the concept of fixpoint object is the treatment of Intuitively, Val(M) is the value M regarded as a trivial computation which immediately eval- *Research supported by a SERC Research Let Studentship. uates to itself; and xeE.F(x) denotes the ‘Research supported by the CLICS project (EEC ES- computation which firstly tries to evaluate E to PRIT BR nr 3003). some value x E a and then proceeds to evaluate CH2897-7/90/0000/0489$01.OO 0 1990 IEEE 4H9 F(x). These intended meanings are captured by 0 Lift(Snd; 77) = Snd : a x T(P)-i T(P). three equational axioms: 0 If F : a x ,B -+ T(y) and G : a x y --+ T(S), then Lift((Fst,F); Lift(G)) = Let x+Val(M).F(x) = F(M) (Fst, Lift(F)); Lift(G) : a x T(P)-+ T(S). Letx+E.Val(x) = E Let y+(Let x+E.F(x)).G(y) = Proposition 1.1 There are mutually inverse Let x-e E.Let yeF(z).G( y) translations between the above combinators, F : a ---t P, and XMLpterms, G, satisfying G(z) E In addition, XMLT extends the simply typed lambda calculus: there are function types a-+@ P [x E al. with lambda abstractions Xz E a.F(x)and ap- Combining this proposition with the fact that plications (MN) satisfying the usual ,8 and 77 equational XMLT theories correspond to let-ccc’s equalities. The system also contains product [lo], one concludes that the categorical combina- types a x /? with (surjective) pairing (M,N) and tors give a simple presentation of this variety of projections Fst(M), Snd(M); and it contains a category; in particular, the action of T on mor- type unit with unique element () E unit. phisms, the monad multiplication and the monad The categorical counterpart of this basic meta- strength are all definable from the combinators. language is the notion of a ‘Cartesian closed cat- In terms of the XMLT language, the action of T egory equipped with a strong monad T’ [8, sec- on a morphism F : a -+ /3 yields tion 21. We shall refer to such structures as let Cartesian closed categories, or just let-ccc ’s. Such T(F) %f Xe E T(a).Letz-ee.Val(Fz) categories can be presented very simply using a : T(a)-+ T(P). language of categorical combinators for XMLT, extending Curien’s ccc combinators for the sim- Similarly, the monad multiplication is defined by ply typed lambda calculus [2]. dzf pff - Xe’ E T(Ta).Let e+e’.e Definition 1.1 The let-categorical combinators are defined by the following grammar: : Tya)+ T(a) and the monad strength is defined by dzf tff$ - A typing statement for these combinators takes Xz E a x TP.Let ycSnd(z).Val((Fst(z),y)) the form F : a + P; an equality statement takes : cr x T(P) -+ T(a x P). the form F = F‘ : a + P. The rules for deduc- ing these statements are those for ccc’s (see [6]) A concept which is intimately bound up with augmented by the following rules (where F x G the correspondence between the metalanguage abbreviates (Fst; F, Snd; G)): XMLT and let-ccc’s is functional completeness, as used by Lambek and Scott [6] in the con- 0 q:a+T(a). text of ordinary Cartesian closed categories. Here one must consider polynomial categories over let- 0 If F : a x P -+ T(y), then ccc’s and show that they are functionally com- Lift(F) : a x T(P)+ T(y) (and plete in a suitable sense. Lift(F) = Lift($’’) if F = F’). Definition 1.2 Let C be a let-ccc and let a be 0 If F : a i P and G : /3 x y -+ T(S),then (F x Id); Lift(G) = Lift((F x Id); G) : an object of C. The polynomial category C[X]is specified by the following data: a x T(7)-i T(S). 0 If F : a x p 3 T(y), then 0 The objects of C[X]are just the objects of (Id x 7); Lift(F) = F : a x /3 + T(y). C. 490 0 The morphisms of C[X] are equivalence The usual category-theoretic considerations classes of the let-categorical combinators imply that the structure fix,U, w is determined generated by the morphisms of C, together uniquely up to isomorphism, within the given with an indeterminate global element X : let-ccc, by the above properties. One should unit + a. The equivalence relation is that also note that U, being the structure morphism generated by the equality rules in defini- for the intial algebra of an endofunctor, is it- tion 1.1 and the existing identities in C- self an isomorphism. (This fact will be used so that there is a canonical inclusion of let- later in Proposition 3.1.) Using the relation be- categories i : C - C[X]. tween syntax and category theory discussed in section 1, one can translate the definition of a Theorem 1.1 (Functional Completeness) fixpoint object into a corresponding extension of Let C[X] be as above. For every morphism the metalanguage XMLT. This entails adding a U : ,8 + y in C[X] there is a unique mor- new type fix, together with the term-forming and phism A(U) : a x ,B --+ y in C such that equality rules shown in Figure 1. The last rule ((();X),ld);A(U) = U. (In particular, each in this figure, which expresses the uniqueness of global element of y in C[X] is of the form X;F It,(F), will be redundant in the full FIX logic of for some unique morphism F : a ---f y in C.) section 3: it is derivable from the induction rule for fix introduced in that section. It is possible to prove the functional complete- Fixpoint objects are so called because they en- ness result simply by calculation with the cat- able one to define fixpoint combinators at all egorical combinators. Then one may derive types of the form a+T(P).' the correspondence between the metalanguage and the category theory using the recipe de- Proposition 2.1 (Definability of fixpoint scribed for Cartesian closed categories by Lam- combinators) In the presence of a fixpoint ob- bek and Scott [6]. We make the comment that ject, one may define a combinator when specifying the metalanguage terms gener- ated by a given category, terms Letx+E.U in the metalanguage are instances of morphisms which satisfies (Ya,p F) = (F (Y,,p F))for all (Id, E);Lift((Snd, Fst); A(U)). When construct- F E (a+TP)+a+TP. Indeed, defining ing the category from the metalanguage, all mor- phisms in the category are formed in the way one ya,p = (F)lta+Tp(F, u(w)) would expect. where F is (e)(XxE cr.Let f-+e.F(s,f)), then 2 The fixpoint type [f E a+TP,x E a] F(f,z) E TP Definition 2.1 In a let-ccc, a fixpoint object is specified by the following data: and An initial algebra 0 : T(fix) + fix for the functor T(-).