Results 11  20
of
248
Generic trace semantics via coinduction
 Logical Methods in Comp. Sci
, 2007
"... Abstract. Trace semantics has been defined for various kinds of statebased systems, notably with different forms of branching such as nondeterminism vs. probability. In this paper we claim to identify one underlying mathematical structure behind these “trace ..."
Abstract

Cited by 35 (10 self)
 Add to MetaCart
Abstract. Trace semantics has been defined for various kinds of statebased systems, notably with different forms of branching such as nondeterminism vs. probability. In this paper we claim to identify one underlying mathematical structure behind these “trace
The Discrete Objects in the Effective Topos
 Proc. London Math. Soc
, 1990
"... The original aim of this paper was to give a rather quick and undemanding proof that the effective topos contains two nontrivial small (i.e. internal) full subcategories which are closed under all small limits in the topos (and hence in particular are internally complete). The interest in such subc ..."
Abstract

Cited by 34 (7 self)
 Add to MetaCart
(Show Context)
The original aim of this paper was to give a rather quick and undemanding proof that the effective topos contains two nontrivial small (i.e. internal) full subcategories which are closed under all small limits in the topos (and hence in particular are internally complete). The interest in such subcategories arises from
Local Realizability Toposes and a Modal Logic for Computability (Extended Abstracts)
 Presented at Tutorial Workshop on Realizability Semantics, FLoC'99
, 1999
"... ) Steven Awodey 1 Lars Birkedal 2y Dana S. Scott 2z 1 Department of Philosophy, Carnegie Mellon University 2 School of Computer Science, Carnegie Mellon University April 15, 1999 Abstract This work is a step toward developing a logic for types and computation that includes both the usual ..."
Abstract

Cited by 27 (8 self)
 Add to MetaCart
) Steven Awodey 1 Lars Birkedal 2y Dana S. Scott 2z 1 Department of Philosophy, Carnegie Mellon University 2 School of Computer Science, Carnegie Mellon University April 15, 1999 Abstract This work is a step toward developing a logic for types and computation that includes both the usual spaces of mathematics and constructions and spaces from logic and domain theory. Using realizability, we investigate a configuration of three toposes, which we regard as describing a notion of relative computability. Attention is focussed on a certain local map of toposes, which we study first axiomatically, and then by deriving a modal calculus as its internal logic. The resulting framework is intended as a setting for the logical and categorical study of relative computability. 1 Introduction We report here on the current status of research on the Logic of Types and Computation at Carnegie Mellon University [SAB + ]. The general goal of this research program is to develop a logical fra...
Categorical and combinatorial aspects of descent theory, [arXiv:math/0303175
"... There is a construction which lies at the heart of descent theory. The combinatorial aspects of this paper concern the description of the construction in all dimensions. The description is achieved precisely for strict ncategories and outlined for weak ncategories. The categorical aspects concern ..."
Abstract

Cited by 27 (2 self)
 Add to MetaCart
(Show Context)
There is a construction which lies at the heart of descent theory. The combinatorial aspects of this paper concern the description of the construction in all dimensions. The description is achieved precisely for strict ncategories and outlined for weak ncategories. The categorical aspects concern the development of descent theory in low dimensions in order to provide a template for a theory in all dimensions. The theory involves nonabelian cohomology, stacks, torsors, homotopy, and higherdimensional categories. Many of the ideas are scattered through the literature or are folklore; a few are new. Section Headings
Solving Recursive Domain Equations with Enriched Categories
, 1994
"... Both preorders and metric spaces have been used at various times as a foundation for the solution of recursive domain equations in the area of denotational semantics. In both cases the central theorem states that a `converging' sequence of `complete' domains/spaces with `continuous' ..."
Abstract

Cited by 27 (0 self)
 Add to MetaCart
Both preorders and metric spaces have been used at various times as a foundation for the solution of recursive domain equations in the area of denotational semantics. In both cases the central theorem states that a `converging' sequence of `complete' domains/spaces with `continuous' retraction pairs between them has a limit in the category of complete domains/spaces with retraction pairs as morphisms. The preorder version was discovered first by Scott in 1969, and is referred to as Scott's inverse limit theorem. The metric version was mainly developed by de Bakker and Zucker and refined and generalized by America and Rutten. The theorem in both its versions provides the main tool for solving recursive domain equations. The proofs of the two versions of the theorem look astonishingly similar, but until now the preconditions for the preorder and the metric versions have seemed to be fundamentally different. In this thesis we establish a more general theory of domains based on the noti...
Deliverables: A Categorical Approach to Program Development in Type Theory
, 1992
"... This thesis considers the problem of program correctness within a rich theory of dependent types, the Extended Calculus of Constructions (ECC). This system contains a powerful programming language of higherorder primitive recursion and higherorder intuitionistic logic. It is supported by Pollack&a ..."
Abstract

Cited by 25 (1 self)
 Add to MetaCart
This thesis considers the problem of program correctness within a rich theory of dependent types, the Extended Calculus of Constructions (ECC). This system contains a powerful programming language of higherorder primitive recursion and higherorder intuitionistic logic. It is supported by Pollack's versatile LEGO implementation, which I use extensively to develop the mathematical constructions studied here. I systematically investigate Burstall's notion of deliverable, that is, a program paired with a proof of correctness. This approach separates the concerns of programming and logic, since I want a simple program extraction mechanism. The \Sigmatypes of the calculus enable us to achieve this. There are many similarities with the subset interpretation of MartinLof type theory. I show that deliverables have a rich categorical structure, so that correctness proofs may be decomposed in a principled way. The categorical combinators which I define in the system package up much logical bo...
Remarks on piecewiselinear algebra
 Paci J. Math
, 1982
"... This note studies some of the basic properties of the category whose objects are finite unions of (open and closed) polyhedra and whose morphisms are (not necessarily continuous) piecewiselinear maps. Introduction * A function f: V —>W between real vector spaces is piecewiselinear (PL) if ther ..."
Abstract

Cited by 25 (5 self)
 Add to MetaCart
(Show Context)
This note studies some of the basic properties of the category whose objects are finite unions of (open and closed) polyhedra and whose morphisms are (not necessarily continuous) piecewiselinear maps. Introduction * A function f: V —>W between real vector spaces is piecewiselinear (PL) if there exists a partition of V into "open polyhedra " X (i.e., relative interiors of polyhedra) such that / is
Developing Theories of Types and Computability via Realizability
, 2000
"... We investigate the development of theories of types and computability via realizability. ..."
Abstract

Cited by 23 (6 self)
 Add to MetaCart
(Show Context)
We investigate the development of theories of types and computability via realizability.
Geometric Theories and Databases
 APPLICATIONS OF CATEGORIES IN COMPUTER SCIENCE, VOLUME 177 OF LONDON MATHEMATICAL SOCIETY LECTURE NOTES
, 1992
"... Domain theoretic understanding of databases as elements of powerdomains is modified to allow multisets of records instead of sets. This is related to geometric theories and classifying toposes, and it is shown that algebraic base domains lead to algebraic categories of models in two cases analogous ..."
Abstract

Cited by 22 (5 self)
 Add to MetaCart
Domain theoretic understanding of databases as elements of powerdomains is modified to allow multisets of records instead of sets. This is related to geometric theories and classifying toposes, and it is shown that algebraic base domains lead to algebraic categories of models in two cases analogous to the lower (Hoare) powerdomain and Gunter's mixed powerdomain.