Results 11  20
of
37
ICategories as a framework for solving domain equations
, 1993
"... An abstract notion of category of information systems or Icategory is introduced as a generalisation of Scott's wellknown category of information systems. As in the theory of partial orders, Icategories can be complete or !algebraic, and it is shown that !algebraic Icategories can be obt ..."
Abstract

Cited by 7 (1 self)
 Add to MetaCart
An abstract notion of category of information systems or Icategory is introduced as a generalisation of Scott's wellknown category of information systems. As in the theory of partial orders, Icategories can be complete or !algebraic, and it is shown that !algebraic Icategories can be obtained from a certain completion of countable Icategories. The proposed axioms for a complete Icategory introduce a global partial order on the morphisms of the category, making them a cpo. An initial algebra theorem for a class of functors continuous on the cpo of morphisms is proved, thus giving canonical solution of domain equations; an effective version of these results for !algebraic Icategories is also provided. Some basic examples of Icategories representing the categories of sets, Boolean algebras, Scott domains and continuous Scott domains are constructed. 1 Introduction A distinctive feature of information systems representing Scott domains, as expressed in [Sco82, LW84], is that th...
Induction and recursion on the partial real line with applications to Real PCF
 Theoretical Computer Science
, 1997
"... The partial real line is an extension of the Euclidean real line with partial real numbers, which has been used to model exact real number computation in the programming language Real PCF. We introduce induction principles and recursion schemes for the partial unit interval, which allow us to verify ..."
Abstract

Cited by 6 (1 self)
 Add to MetaCart
(Show Context)
The partial real line is an extension of the Euclidean real line with partial real numbers, which has been used to model exact real number computation in the programming language Real PCF. We introduce induction principles and recursion schemes for the partial unit interval, which allow us to verify that Real PCF programs meet their specification. They resemble the socalled Peano axioms for natural numbers. The theory is based on a domainequationlike presentation of the partial unit interval. The principles are applied to show that Real PCF is universal in the sense that all computable elements of its universe of discourse are definable. These elements include higherorder functions such as integration operators. Keywords: Induction, coinduction, exact real number computation, domain theory, Real PCF, universality. Introduction The partial real line is the domain of compact real intervals ordered by reverse inclusion [28,21]. The idea is that singleton intervals represent total rea...
On a Question of Friedman
 Information and Computation
, 1995
"... In this paper we answer a question of Friedman, providing an !separable model M of the fijcalculus. There therefore exists an ffseparable model for any ff 0. The model M permits no nontrivial enrichment as a partial order; neither does it permit an enrichment as a category with an initial ob ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
(Show Context)
In this paper we answer a question of Friedman, providing an !separable model M of the fijcalculus. There therefore exists an ffseparable model for any ff 0. The model M permits no nontrivial enrichment as a partial order; neither does it permit an enrichment as a category with an initial object. The open term model embeds in M: by way of contrast we provide a model which cannot embed in any nontrivial model separating all pairs of distinct elements. 1 Introduction Separability is a recurring topic in the calculus. It is usually defined syntactically; there is also an interesting modeltheoretic definition. Say that a subset A of an applicative structure (X; \Delta) is separable if any function f : A ! X is realised by some f in X , by which is meant, that for all a in A, f(a) = f \Delta a. This idea first appears in work of Flagg and Myhill [FM]. They termed the concept "discreteness," employing a topological analogy; we prefer to extend the usual calculus terminology....
Compact Metric Spaces as MinimalLimit Sets in Domains of Bottomed Sequences
, 2003
"... It is shown that every compact metric space X is homeomorphically embedded in an !algebraic domain D as the set of minimal limit elements. ..."
Abstract

Cited by 5 (3 self)
 Add to MetaCart
It is shown that every compact metric space X is homeomorphically embedded in an !algebraic domain D as the set of minimal limit elements.
Induction and recursion on the partial real line via biquotients of bifree algebras (Extended Abstract)
 IN PROCEEDINGS OF THE TWELVETH ANNUAL IEEE SYMPOSIUM ON LOGIC IN COMPUTER SCIENCE
, 1997
"... ..."
Information Categories
 Applied Categorical Structures
, 1991
"... \Information systems" have been introduced by Dana Scott as a convenient means of presenting a certain class of domains of computation, usually known as Scott domains. Essentially the same idea has been developed, if less systematically, by various authors in connection with other classes o ..."
Abstract

Cited by 3 (3 self)
 Add to MetaCart
\Information systems" have been introduced by Dana Scott as a convenient means of presenting a certain class of domains of computation, usually known as Scott domains. Essentially the same idea has been developed, if less systematically, by various authors in connection with other classes of domains. In previous work, the present authors introduced the notion of an Icategory as an abstraction and enhancement of this idea, with emphasis on the solution of domain equations of the form D = F (D), with F a functor. An important feature of the work is that we are not conned to domains of computation as usually understood; other classes of spaces, more familiar to mathematicians in general, become also accessible. Here we present the idea in terms of what we call information categories, which are concrete Icategories in which the objects are structured sets of \tokens" and morphisms are relations between tokens. This is more in the spirit of information system work, and...
Computability on the Interval Space: A Domain Approach
"... The effectively given continuous domains aims characterize the computable functions (as opposed to the merely continuous ones) and the computable elements of types represented by continuous domains. In this paper we show that computability on arbitrary effectively given continuous domain depends str ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
The effectively given continuous domains aims characterize the computable functions (as opposed to the merely continuous ones) and the computable elements of types represented by continuous domains. In this paper we show that computability on arbitrary effectively given continuous domain depends strongly upon the ChurchTuring computability (classical computability on countable sets) on a countable base. In order to introduce the notion of computability on the interval space we need the concepts of effectively given continuous domain. In this approach, several desired properties of a computable interval analysis are obtained.
Multivalued Logics, Effectiveness and Domains
"... Abstract. Effective domain theory is applied to fuzzy logic to give suitable notions of semidecidable and decidable Lsubset. The connection with the notions of fuzzy Turing machines and fuzzy grammar given in literature is also investigated. This shows the inadequateness of these definitions and t ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
(Show Context)
Abstract. Effective domain theory is applied to fuzzy logic to give suitable notions of semidecidable and decidable Lsubset. The connection with the notions of fuzzy Turing machines and fuzzy grammar given in literature is also investigated. This shows the inadequateness of these definitions and the difficulties in formulating an analogue of Church Thesis for fuzzy logic. 1
Interactive Computation: Stepping Stone in the Pathway From Classical to Developmental Computation ∗
"... This paper reviews and extends previous work on the domaintheoretic notion of Machine Development. It summarizes the concept of Developmental Computation and shows how Interactive Computation can be understood as a stepping stone in the pathway from Classical to Developmental Computation. A critica ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
(Show Context)
This paper reviews and extends previous work on the domaintheoretic notion of Machine Development. It summarizes the concept of Developmental Computation and shows how Interactive Computation can be understood as a stepping stone in the pathway from Classical to Developmental Computation. A critical appraisal is given of Classical Computation, showing in which ways its shortcomings tend to restrict the possible evolution of real computers, and how Interactive and Developmental Computation overcome such shortcomings. A formal conceptual framework is sketched, in order to frame the future development of the formal theory of Developmental Computation. Finally, the current frontier of the work on Developmental Computation is briefly exposed. 1
Noetherian spaces in verification
 In ICALP’10
, 2010
"... Abstract. Noetherian spaces are a topological concept that generalizes well quasiorderings. We explore applications to infinitestate verification problems, and show how this stimulated the search for infinite procedures à la KarpMiller. 1 ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
(Show Context)
Abstract. Noetherian spaces are a topological concept that generalizes well quasiorderings. We explore applications to infinitestate verification problems, and show how this stimulated the search for infinite procedures à la KarpMiller. 1