G. Longo and E. Moggi. Cartesian closed categories of enumerations for effective type-structures. In G. Kahn, D. MacQueen, and G. Plotkin, editors, Symposium on Semantics of Data Types, volume 173 of Lecture Notes in Computer Science. Springer-Verlag, 1984.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....treatment of sums, products, exponentials and recursive types via the usual universal properties. However, it was via a direct semantic analysis of non terminating computations [Plo85] involving categories of partial maps [RR88] and, in particular, via the notion of partial cartesian closure [LM84] that an appropriate categorical setting emerged. With this background it was possible, for the first time, to consider categorical models for a rich type theory with recursive types. In [Fio94a, FP94] a notion of categorical model for the metalanguage FPC a type theory with sums, products, ....

G. Longo and E. Moggi. Cartesian closed categories of enumerations for effective type-structures. In G. Kahn, D. MacQueen, and G. Plotkin, editors, Symposium on Semantics of Data Types, volume 173 of Lecture Notes in Computer Science. Springer-Verlag, 1984.

....an adjunction. We write for the counit of this adjunction and call each A B a partial exponential . Partial exponentials extend to a functor : pK op Theta pK pK sending pairs of objects (A; B) to A B and pairs of partial maps (u; v) to p(v ffi ffi (id Omega u) Definition 4. 2 [LM84] A category of partial maps pK is partial cartesian closed if K is cartesian and pK has partial exponentials. Proposition 4.3 Let (K; D) be a uniform domain structure and let v be a Cpo enrichment for K such that (pK; is a Cpo category. Assume pK is partial cartesian closed. If Theta : K ....

G. Longo and E. Moggi. Cartesian closed categories of enumerations for effective typestructures. In G. Kahn, D. MacQueen, and G. Plotkin, editors, Symposium on Semantics of Data Types, volume 173 of Lecture Notes in Computer Science. SpringerVerlag, 1984.

....and functional programming languages, e.g. 1, 40, 47] in which one evaluates arguments before applying them to a concerned procedure. The semantics of higher order computation based on call by value evaluation has been widely studied by many researchers in the context of domain theory, cf. [46, 47, 31, 42, 19, 53, 16, 17], through which it has become clear that the semantic framework needed to capture the call by value computation has a basic difference from the one for call by name computation (see [23, 55] for basic introduction to the topic) The difference between the semantics of call by value and that of ....

....1 ; oe; 2 ii r where i denote projections. Then A Omega and Omega A both define functors on CBV which CPO enrich. We then define, for oe : A B and : C D: i) oe Omega l = oe Omega C) B Omega ) and (ii) oe Omega r = A Omega ) oe Omega D) iii) partial exponential [31]) The functor Delta Omega A viewed as going from CBV t to CBV has the right adjoint A Delta : CBV CBV t , which CPO enriches. Equivalently, there exists an arrow ev : A B) Omega A B such that, for any oe : C Omega A B, there is a unique total arrow p(oe) C A B satisfying ....

Longo, G. and Moggi, E., Cartesian closed categories of enumerations for effective tyep-structures, LNCS 173, Springer-Varlag, 1984.

....approaches to computation. Various significant constructions of categories of partial maps include the dominical categories of DiPaola and Heller [DH87] the concrete categories of domains of Moggi [Mog86, Mog88b] the partial cartesian closed categories of Asperti, Longo, and Moggi [LM84, AL86], and the partial ni ueness ro erties isa ear. cartesian categories of Curien and Obtulowicz [CO89] These constructions are complex and not easy to express. This section discusses the work of Robinson and Rosolini [Ros86, RR88] that unifies these above mentioned categories (and others) and the ....

G. Longo and E. Moggi, Cartesian Closed Categories of Enumerations for Effective Type Structures. In Intl. Symposium of Data Types, Lecture Notes in Computer Science 173, Springer-Verlag, 1984.

....structure) which is a product in the subcategory, G t , of total innocent strategies [15] This situation, which is typical in models of FPC [9, 10] is the one that we axiomatise. The structure for interpreting the exponential type constructor arises as in partial cartesian closed categories [21, 27, 34]. We have already mentioned that the pretensor constructor Omega on the category of games G is not a product. However, for every pair of arenas A and B, we have endofunctors A Omega ( and ( Omega B on G such that A Omega (B) A) Omega B [15] That is, in the terminology of [31] G ....

G. Longo and E. Moggi. Cartesian closed categories of enumerations for effective type-structures. In Symposium on Semantics of Data Types, volume 173 of LNCS. Springer-Verlag, 1984.

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