M. Wand. Fixed point constructions in order enriched categories. Theoretical Computer Science, 8:13--30, 1979. 47  Home/Search   Document Not in Database   Summary   Related Articles   Check

....scattered; it concentrated on fixed points, mainly of endofunctors but also of endomorphisms. Concerning fixed points for endofunctors, already in [Sco72] Scott mentions a suggestion by Lawvere aiming at providing a categorical framework for performing the D1 construction. But it was not until [Wan79] that the solution of recursive type equations in categories of domains was first treated abstractly , in the sense that no commitment to a particular category of domains was required. Subsequently this approach was developed in [SP82] The approach was very much appreciated as a unification of ....

M. Wand. Fixed point constructions in order-enriched categories. Theoretical Computer Science, 8:13--30, 1979.

....using recursive equations. We are interested in tools for solving such equations. The first was given by Scott [120] with his inverse limit construction in the category of countably based continuous lattices and continuous functions. Smyth and Plotkin [125] building on previous work of Wand [134] provided a categorical framework based on order enriched categories and a general version of Scott s result applicable to a wider class of categories was given. This is our starting point. We want to generalise Smyth and Plotkin s results so as to cover 105 106 CHAPTER 6. A THEORY OF RECURSIVE ....

Mitchell Wand. Fixed-point constructions in order-enriched categories. Theoretical

....that the results of this paper bear no relevance to the important subject of determining neat sufficient conditions for the existence of recursive types with properly mixed variant bases and methods of constructing them. 3 On that matter, there exists an extensive body of literature [see, e.g. Wand 1979; Lehmann and Smyth 1981; Smyth and Plotkin 1982; Freyd 1990; Freyd 1991] The paper is organized as follows. In Section 2, we recall the standard category theoretic account of the conventional concept of inductive type. In Section 3, we present our account of the Mendler style concept. In Section ....

Wand, M. 1979. Fixed-Point Constructions in Order-Enriched Categories. Theoretical Computer Science 8, 1, 13--30.

....i.e. all monofunctors defined using the above basic functors or any of the map functors introduced in x5. Then there exists a type L and two strict functions in F 2 LF L and out F 2 L LF (omitting subscripts whenever possible) which are each others inverse and even id = in F out) [6, 23, 16, 24, 30, 12]. We let F denote the pair (L; in) and say that it is the least fixed point of F . Since in and out are each others inverses we have that LF is isomorphic to L, and indeed L is upto isomorphism a fixed point of F. For example taking XL = 1 j AkX, we have that (A; in) L defines the ....

M. Wand. Fixed point constructions in order enriched categories. Theoretical Computer Science, 8, 1979.

.... nonnegative reals under addition, for which R Cat becomes the category of (generalized) metric spaces, with the composition law as the triangle inequality and functors as contracting maps [Law73] Enriched categories first appeared in computer science with D = Poset = hPoset ; Theta; 1i [Wan79] yielding order enriched categories, a natural notion for domain theory. Poset itself is definable as (the antisymmetric subcategory of) hhf0; 1g; i; 1i Cat, categories enriched in truthvalues. We may now make the connection with semirings. The enriching monoidal category hD; Omega ; I ; ff; ....

M. Wand. Fixed-point constructions in order-enriched categories. TCS, 8(1):13--30, 1979.

....efficient program, since the result will have the running time of the bounding function 2. 1 Contravariance in recursive types A standard way to construct fixpoints of both covariant and contravariant functors is to consider categories whose homsets have been enriched by an ordering structure [Wan79, SP82]. Under certain additional conditions, these categories allow contravariant functors to be converted into ordinary functors on a related category of embedding projection pairs; the fixpoints of these functors will then transfer back to the original category. This permits solutions to equations ....

M. Wand. Fixed-point constructions in order-enriched categories. Theoretical Computer Science, 8:13--30, 1979.

....as classes of labeled measured sets. The class of sets is the class of measured sets when there is only one possible value for a measurement. A category is a measured class in which the measurements are sets, that is d(a; b) is the set of arrows from a to b. An order enriched category (see Wand [48]) is a measured class where the measurements are partially ordered sets rather than unstructured sets. As noted above a metric space is a class where measurements are positive real numbers. Labeling adds some more possibilities. One example is multisets, which can be seen as labeled sets, although ....

M. Wand. Fixed-point constructions in order-enriched categories. Theoretical Computer Science, 8(1):13--30, 1979.

....scattered; it concentrated on fixed points, mainly of endofunctors but also of endomorphisms. Concerning fixed points for endofunctors, already in [Sco72] Scott mentions a suggestion by Lawvere aiming at providing a categorical framework for performing the D1 construction. But it was not until [Wan79] that the solution of recursive type equations in categories of domains was first treated abstractly , in the sense that no commitment to a particular category of domains was required. Subsequently this approach was developed in [SP82] The approach was very much appreciated as a unification of ....

M. Wand. Fixed point constructions in order-enriched categories. Theoretical Computer Science, 8:13--30, 1979.

....O each of whose homsets O(A;B) is equipped with a partial order such that composition preserves the order. Examples include categories of partial maps, categories generated from typed rewrite systems with reduction as the order [19, 15, 21] or from program refinement [9, 10, 16] O categories [20, 23] and categories of relations [1, 4] In subsequent sections (A; M) and (B; N ) denote categories equipped with realms and O;O 0 denote ordered categories. 3 Total Morphisms An endomorphism ff : A A in O is a deflation if ff 2 = ff id A An embedding projection pair [20] in O consists of a ....

M. Wand, Fixed-Point Constructions in Order-Enriched Categories, Theoretical Computer Science 8 (1979) 13--30.

....a model of the untyped lambda calculus. There are a variety of approaches to solving domain equations. The earliest was the inverse limit construction, which was originally devised by Scott [67] generalized by Reynolds in 1972 [55] and eventually abstracted much further by Wand in 1974 and 1979 [79, 80] and by Smyth and Plotkin in 1982[74] Finally, in 1990 91 Freyd [19, 18, 20] abstracted this work further to give an axiomatic categorical account of recursive domain equations. Also, recently in 1997, Cattani, Fiore, and Winskel gave a very general account of the limit colimit coincidence for ....

M. Wand. Fixed-point constructions in order-enriched categories. Theoretical Computer Science, 8:13--30, 1979.

....approximation ordering among terms. The categorical interpretation of this ordering is that we have a partial order structure on the homsets, reflecting the notion of increasing information content among the functions. A category whose homsets are enriched in this way is known as an O category ([57]; see also [51] For example, among functions in a given homset whose codomain is pointed, the constant bottom function will be less defined than all other functions; it is the least element of the homset. We will CHAPTER 4. RETRACTIVE TYPES AND NON TERMINATION 50 write Gamma . M N : oe for ....

M. Wand. Fixed-point constructions in order-enriched categories. Theoretical Computer Science, 8:13-- 30, 1979.

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M. Wand. Fixed point constructions in order enriched categories. Theoretical Computer Science, 8:13--30, 1979. 47

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