M. Smith and G. Plotkin. The categorytheoretic solution of recursive domain equations. SIAM Journal of Computing, Volume 11, 1982.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....h : A; f) B; g) are given by h : A B s.t. g h = T (h) f . The terminal T coalgebra is written as ( T ; out T : T T ( T ) and given a coalgebra (A; f) the unique morphism (often called anamorphism) is written unfold T (f) A T . For completeness we review some material from [PS78]: Given an endofunctor T : C C and i 2 we write T i : C C for the ith iteration of T . We de ne Chain T : C and Chain T : op C : Chain T (i) T i (1) Chain T (i j) T i ( ChainT (j i) Chain T (i) T i (0) Chain T (i j) T j ( Chain T (i j) ....

G. D. Plotkin and M. B. Smyth. The category-theoretic solution of recursive domain equations. SIAM Journal on Computing, 11, 1978.

....to single out the general principles among the properties satis ed by the model. Moreover, the theory at the heart of Denotational Semantics, i.e. Domain Theory (see [GS89, Mos89] has focused on the mathematical structures for giving semantics to recursive de nitions of types and functions (see [SP82]) while other structures, that might be relevant to a better understanding of programming languages, have been overlooked. This paper identify one of such structures, i.e. monads , but probably there are others just waiting to be discovered. The categorical semantic of computations presented in ....

M. Smith and G. Plotkin. The category-theoretic solution of recursive domain equations. SIAM Journal of Computing, 11, 1982.

....P y as introduced in [19] These, among others, can be constructed by taking the fixpoint of a suitable process functor. In general, it will be useful to have criteria by which one may know that these fixpoints exist. We start by stating a slight generalization of a result of Plotkin and Smythe [17] Theorem 46 Given two categories C, D, the latter having colimits and an initial object 0, along with a functor F : C Theta D D preserving colimits, there exists an initial fixpoint F : C D. That is, we have ffl a functor F : C D and ffl a canonical natural isomorphism : F ....

.... category of pre fixpoints of F , a pre fixpoint being a pair (G; fl) in which G : C D is a functor and fl : F hidC; Gi Delta G) Furthermore, if C has colimits, then F preserves them Outline of Proof : We can fix an object A of C, in which case the proof follows the same line as in [17], that is, we take the colimit of 0 F (A; 0) F (A; F (A; 0) F (A; F (A; F (A; 0) Delta Delta Delta to obtain F A and a canonical isomorphism : FF A F A. For any morphism, f : A B, we have a natural transformation between the respective diagrams on A and B which ....

G.D. Plotkin and M.B. Smyth. The category-theoretic solution of recursive domain equations. Technical Report Research Report No. 60, D.A.I., 1978.

.... are guaranteed because H (u) is a limit (note that each of the rectangles commutes, so H (t) is a cone for the diagram for which H (u) is the limit) It is a standard domain theoretic result that H is continuous if H is, and if H preserves reflecting functions then so does H (see [32] for details) Having defined various means whereby new parameterised types (functors) may be constructed from old, we now focus our attention on those types which may be entirely constructed using only the operations above. This includes all ground types definable in ML like languages. z We use ....

....(t) is the limit of the diagram there is a unique morphism from C to H (t) which makes the diagram commute. However, both h and k make the diagram commute, and so we deduce that h = k . 10 To prove the same result for epics we appeal to the well known limit colimit coincidence result (again see [32] for the details, or see [29] for a simpler account) and deduce that the limit of a diagram 1 H (1) oe H 2 (1) oe H ( Delta Delta Delta oe is isomorphic to the colimit of the corresponding diagram 1 H (1) H 2 (1) H ( Delta Delta Delta where t : 1 t is given by ....

M. Smith and G. Plotkin, The Category-Theoretic Solution of Recursive Domain Equations, SIAM J. COMPUT, Vol 11, No 4, pp 761-783, Nov 1982.

....cannot handle higher order types, since (unlike position functors) lists are not closed under exponentials. Data categories are defined to provide a setting for the study of data functors. Examples of data categories include Sets, Pos (the category of bottomless, complete partial orders) [10] and Eff , the effective topos [4] To the locos structure (lextensive categories with list objects) used to model shapely type constructors we must add cartesian closure for the higher order structure. The bulk of the paper is devoted to defining terms, establishing the above result, and showing ....

M. Smith and G. Plotkin. The categorytheoretic solution of recursive domain equations. SIAM Journal of Computing, Volume 11, 1982.

....the main result of this paper, as well as too corollaries, which establish the modularity and functoriality of our framework (Section 4.3) Finally, Sections 5 and 6 mention conclusions and directions for future work. 2 The Model Our model consists of ordered categories (similar to O categories [SP82]) with behaviours corresponding to morphisms between them. It can be sketched by means of the following diagram: L B O R Gamma Gamma Gamma Gamma Psi D B 0 D L is a category, which we identify with a programming language: its objects are types and its morphisms are programs. ....

M. Smith and G. Plotkin. The category-theoretic solution of recursive domain equations. SIAM Journal of Computing, 11, 1982.

....The construction of F 0 in the previous section shows how to build particular types, but in order to obtain type constructors we must construct initial algebras in a parametrised fashion. A functor F : C m ThetaC n C n can be used to represent a system of (parametrised) domain equations [SP82], whose solution is can be found by constructing, for each object A in C m , an initial algebra ff A : F (A; F y A) F y A for the functor F (A; Gamma) For example, if F (A; X) A X ThetaX then F y A = TA is the binary trees on A; the leaf and node constructors are given by the ....

M. Smith and G. Plotkin. The category-theoretic solution of recursive domain equations. SIAM Journal of Computing, 11, 1982. This article was processed using the L a T E X macro package and P. Taylor's diagrams package.

....D ) D s D D] has initial solution in bqD equal to lim (Dn ; i n ) where (D 0 ; 0 ) is a pointed coherence space with only one atom, Dn 1 ; n 1 ) Dn s n Dn ] i 0 is the morphism which maps 0 to 1 and i n 1 = i n i n ) Proof. Follows from the general Theorem in [13] since the category bqD is complete and the functor G is continuous (see [13] for further details) ut We have achieved our goal, finally One can easily see, in fact, that the initial solution of the domain equation (D; D ) D s D D] in bqD , can be used to model adequately ....

....(D 0 ; 0 ) is a pointed coherence space with only one atom, Dn 1 ; n 1 ) Dn s n Dn ] i 0 is the morphism which maps 0 to 1 and i n 1 = i n i n ) Proof. Follows from the general Theorem in [13] since the category bqD is complete and the functor G is continuous (see [13] for further details) ut We have achieved our goal, finally One can easily see, in fact, that the initial solution of the domain equation (D; D ) D s D D] in bqD , can be used to model adequately the V calculus. More formally one can show that the structure defined below is a ....

Smith, M. B., Plotkin, G. D.: The category-theoretic solution of recursive domain equations. SIAM J. of Computing 11(5) (1982) 761--783 This article was processed using the L A T E X macro package with LLNCS style

.... 32 Applications mathematical developments (see Gierz et al. 1980) as well as methods which influenced the actual design of programming languages (e.g. Gordon et al. 1979) Specific areas are concerned with original applications of Category Theory to the semantics of programming languages (see Plotkin Smyth (1982) for more results and references) and computability in abstract structures (see Barendregt Longo (1982) or Longo Martini (1984) for simple approaches and references) just to mention two broadly construed topics. What is the relation between the equation X = X X and continuity properties ....

Plotkin, G. & Smyth, M.B. (1982). The category-theoretic solution of recursive domain equations. Dept. of Comp. Sci. Edinburgh.

....a monomorphism. F 0 OE L Omega F 0 h0 hnil; ji L Omega ThetaLF 0 0 C h hnil; ji L Omega ThetaLC id ThetaLh Fig. 1. Factorisation of hC 7. 2 The General Case A functor F : C m ThetaC n C n can be used to represent a system of (parametrised) domain equations [SP82], whose solution is can be found by constructing, for each object A in C m , an initial algebra ff A : F (A; F y A) F y A for the functor F (A; Gamma) If such always exist, then F y extends to a functor whose action on f : A B is the F (A; Gamma) algebra homomorphism induced by the ....

M. Smith and G. Plotkin. The category-theoretic solution of recursive domain equations. SIAM Journal of Computing, 11, 1982. This article was processed using the L a T E X macro package with LLNCS style and P. Taylor's diagrams package.

....There is a cone ae : Delta W A where the ae are inclusion morphisms. This cone is universal in Sys (and also in Chu(Set; 2) Evidently, F preserves such colimiting cones of inclusion morphisms, as it preserves inclusion morphisms and is continuous. It follows by the Basic Lemma in [SP82] that j F : F (F1 ) F1 is the initial F algebra where j F is the identity. Next, suppose that F extends to a functor F 0 on Chu(Set; 2) in the sense that is F 0 o I is naturally equivalent to I o F . Then j F : F (F1 ) F1 is also the initial F 0 algebra, again by applying the ....

....should be. 6 General Recursive Type Equations Here we would like to solve type equations in Chu(K; x) where F is built up out of constants, product, tensor and ( Delta) This implies that Chu(K; x) has fixed points and so a zero object. It is appropriate to use the CPO enriched theory of [SP82]. This says that to solve equations of the above form in a category L it suffices that: ffl L is CPO enriched ffl L has limits ffl F is built up out of locally continuous functors, which may be of mixed variance. A functor is locally continuous if it preserves lubs of increasing ....

Gordon Plotkin and Mike Smyth. The Category-Theoretic Solution of Recursive Domain Equations. SIAM Journal on Computing, Vol. 11, No. 4, pp. 761-783, 1982.

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M. Smith and G. Plotkin. The categorytheoretic solution of recursive domain equations. SIAM Journal of Computing, Volume 11, 1982.

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M. Smith and G. Plotkin. The category-theoretic solution of recursive domain equations. SIAM Journal of Computing, 11, 1982. 18

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M. Smith and G. Plotkin. The category-theoretic solution of recursive domain equations. SIAM Journal of Computing, 11, 1982. This article was processed using the L a T E X macro package with LLNCS style and P. Taylor's diagrams package.

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G.D. Plotkin, M. Smyth, "The category theoretic solution of recursive domain equations", SIAM J. Computing 11, 1982.

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G.D. Plotkin, M. Smyth, "The category theoretic solution of recursive domain equations", SIAM J. of Computing 11, 1982.

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