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Relations in Concurrency
"... The theme of this paper is profunctors, and their centrality and ubiquity in understanding concurrent computation. Profunctors (a.k.a. distributors, or bimodules) are a generalisation of relations to categories. Here they are first presented and motivated via spans of event structures, and the seman ..."
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Cited by 263 (33 self)
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The theme of this paper is profunctors, and their centrality and ubiquity in understanding concurrent computation. Profunctors (a.k.a. distributors, or bimodules) are a generalisation of relations to categories. Here they are first presented and motivated via spans of event structures, and the semantics of nondeterministic dataflow. Profunctors are shown to play a key role in relating models for concurrency and to support an interpretation as higherorder processes (where input and output may be processes). Two recent directions of research are described. One is concerned with a language and computational interpretation for profunctors. This addresses the duality between input and output in profunctors. The other is to investigate general spans of event structures (the spans can be viewed as special profunctors) to give causal semantics to higherorder processes. For this it is useful to generalise event structures to allow events which “persist.”
Syntax and Semantics of Dependent Types
 Semantics and Logics of Computation
, 1997
"... ion is written as [x: oe]M instead of x: oe:M and application is written M(N) instead of App [x:oe] (M; N ). 1 Iterated abstractions and applications are written [x 1 : oe 1 ; : : : ; x n : oe n ]M and M(N 1 ; : : : ; N n ), respectively. The lacking type information can be inferred. The universe ..."
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Cited by 40 (4 self)
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ion is written as [x: oe]M instead of x: oe:M and application is written M(N) instead of App [x:oe] (M; N ). 1 Iterated abstractions and applications are written [x 1 : oe 1 ; : : : ; x n : oe n ]M and M(N 1 ; : : : ; N n ), respectively. The lacking type information can be inferred. The universe is written Set instead of U . The Eloperator is omitted. For example the \Pitype is described by the following constant and equality declarations (understood in every valid context): ` \Pi : (oe: Set; : (oe)Set)Set ` App : (oe: Set; : (oe)Set; m: \Pi(oe; ); n: oe) (m) ` : (oe: Set; : (oe)Set; m: (x: oe) (x))\Pi(oe; ) oe: Set; : (oe)Set; m: (x: oe) (x); n: oe ` App(oe; ; (oe; ; m); n) = m(n) Notice, how terms with free variables are represented as framework abstractions (in the type of ) and how substitution is represented as framework application (in the type of App and in the equation). In this way the burden of dealing correctly with variables, substitution, and binding is s...
Domain Theoretic Models Of Polymorphism
, 1989
"... We give an illustration of a construction useful in producing and describing models of Girard and Reynolds' polymorphic calculus. The key unifying ideas are that of a Grothendieck fibration and the category of continuous sections associated with it, constructions used in indexed category theory; th ..."
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Cited by 34 (2 self)
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We give an illustration of a construction useful in producing and describing models of Girard and Reynolds' polymorphic calculus. The key unifying ideas are that of a Grothendieck fibration and the category of continuous sections associated with it, constructions used in indexed category theory; the universal types of the calculus are interpreted as the category of continuous sections of the fibration. As a major example a new model for the polymorphic calculus is presented. In it a type is interpreted as a Scott domain. In fact, understanding universal types of the polymorphic calculus as categories of continuous sections appears to be useful generally. For example, the technique also applies to the finitary projection model of Bruce and Longo, and a recent model of Girard. (Indeed the work here was inspired by Girard's and arose through trying to extend the construction of his model to Scott domains.) It is hoped that by pinpointing a key construction this paper will help towards...
A CategoryTheoretic Account of Program Modules
 Mathematical Structures in Computer Science
, 1994
"... The typetheoretic explanation of modules proposed to date (for programming languages like ML) is unsatisfactory, because it does not capture that evaluation of typeexpressions is independent from evaluation of programexpressions. We propose a new explanation based on \programming languages as inde ..."
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Cited by 23 (6 self)
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The typetheoretic explanation of modules proposed to date (for programming languages like ML) is unsatisfactory, because it does not capture that evaluation of typeexpressions is independent from evaluation of programexpressions. We propose a new explanation based on \programming languages as indexed categories" and illustrates how ML can be extended to support higher order modules, by developing a categorytheoretic semantics for a calculus of modules with dependent types. The paper outlines also a methodology, which may lead to a modular approach in the study of programming languages. Introduction The addition of module facilities to programming languages is motivated by the need to provide a better environment for the development and maintenance of large programs. Nowadays many programming languages include such facilities. Throughout the paper Standard ML (see [Mac85, HMM86, MTH90]) is taken as representative for these languages. The implementation of module facilities has been ...
First Order Linear Logic in Symmetric Monoidal Closed Categories
, 1991
"... There has recently been considerable interest in the development of `logical frameworks ' which can represent many of the logics arising in computer science in a uniform way. Within the Edinburgh LF project, this concept is split into two components; the first being a general proof theoretic encodin ..."
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Cited by 11 (0 self)
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There has recently been considerable interest in the development of `logical frameworks ' which can represent many of the logics arising in computer science in a uniform way. Within the Edinburgh LF project, this concept is split into two components; the first being a general proof theoretic encoding of logics, and the second a uniform treatment of their model theory. This thesis forms a case study for the work on model theory. The models of many first and higher order logics can be represented as fibred or indexed categories with certain extra structure, and this has been suggested as a general paradigm. The aim of the thesis is to test the strength and flexibility of this paradigm by studying the specific case of Girard's linear logic. It should be noted that the exact form of this logic in the first order case is not entirely certain, and the system treated here is significantly different to that considered by Girard.
Coherence and Consistency in Domains
 IN THIRD ANNUAL SYMPOSIUM ON LOGIC IN COMPUTER SCIENCE
, 1990
"... Almost all of the categories normally used as a mathematical foundation for denotational semantics satisfy a condition known as consistent completeness. The goal of this paper is to explore the possibility of using a different conditionthat of coherencewhich has its origins in topology and log ..."
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Cited by 8 (4 self)
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Almost all of the categories normally used as a mathematical foundation for denotational semantics satisfy a condition known as consistent completeness. The goal of this paper is to explore the possibility of using a different conditionthat of coherencewhich has its origins in topology and logic. In particular, we concentrate on those posets whose principal ideals are algebraic lattices and whose topologies are coherent. These form a cartesian closed category which has fixed points for domain equations. It is shown that a "universal domain" exists. Since the construction of this domain seems to be of general significance, a categorical treatment is provided and applied to other classes of domains. Universal domains constructed in this fashion enjoy an additional property: they are saturated. We show that there is exactly one such domain in each of the classes under consideration.
Categorical Properties of Logical Frameworks
, 1993
"... In this paper we give a new presentation of ELF which is wellsuited for semantic analysis. We introduce the notions of internal codability, internal definability, internal typed calculi and frame languages. These notions are central to our perspective of logical frameworks. We will argue that a ..."
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Cited by 1 (1 self)
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In this paper we give a new presentation of ELF which is wellsuited for semantic analysis. We introduce the notions of internal codability, internal definability, internal typed calculi and frame languages. These notions are central to our perspective of logical frameworks. We will argue that a logical framework is a typed calculus which formalizes the relationship between internal typed languages and frame languages. In the second half of the paper, we demonstrate the advantage of our logical framework by showing some categorical properties of it and of encodings in it. By doing so we hope to indicate a sensible model theory of encodings. Copyright c fl1993. All rights reserved. Reproduction of all or part of this work is permitted for educational or research purposes on condition that (1) this copyright notice is included, (2) proper attribution to the author or authors is made and (3) no commercial gain is involved. Technical Reports issued by the Department of Computer Sc...
The Classification of Continuous Domains (Extended Abstract)
"... Achim Jung y Technische Hochschule Darmstadt and Imperial College of Science and Technology, London Abstract The longstanding problem of finding the maximal cartesian closed categories of continuous domains is solved. The solution requires the definition of a new class of continuous domains, cal ..."
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Cited by 1 (0 self)
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Achim Jung y Technische Hochschule Darmstadt and Imperial College of Science and Technology, London Abstract The longstanding problem of finding the maximal cartesian closed categories of continuous domains is solved. The solution requires the definition of a new class of continuous domains, called FSdomains, which contains all retracts of SFPobjects. The properties of FSdomains are discussed in some detail. Keywords: continuous domains, SFPobjects, Lawsontopology, Smyth's Theorem, FSdomains, Ldomains 1 Introduction The first spaces suitable for the interpretation of programming language constructs were continuous lattices discovered by Dana Scott in the late sixties. Continuous lattices turned out to have numerous connections to other fields of mathematics such as algebra, topology, and convex analysis. An indication of this is the voluminous Bibliography of Continuous Lattices contained in [4]. In Computer Science, however, it was soon recognized that the subclass of al...
Foundations for Computable Topology
, 2009
"... Foundations should be designed for the needs of mathematics and not vice versa. We propose a technique for doing this that exploits the correspondence between category theory and logic and is potentially applicable to several mathematical disciplines. Stone Duality. We express the duality between al ..."
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Foundations should be designed for the needs of mathematics and not vice versa. We propose a technique for doing this that exploits the correspondence between category theory and logic and is potentially applicable to several mathematical disciplines. Stone Duality. We express the duality between algebra and geometry as an abstract monadic adjunction that we turn into a new type theory. To this we add an equation that is satisfied by the Sierpiński space, which plays a key role as the classifier for both open and closed subspaces. In the resulting theory there is a duality between open and closed concepts. This captures many basic properties of compact and closed subspaces, despite the absence of any explicitly infinitary axiom. It offers dual results that link general topology to recursion theory. The extensions and applications of ASD elsewhere that this paper survey include a purely recursive theory of elementary real analysis in which, unlike in previous approaches, the real closed interval [0, 1] in ASD is compact.