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Developing Theories of Types and Computability via Realizability
, 2000
"... We investigate the development of theories of types and computability via realizability. ..."
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We investigate the development of theories of types and computability via realizability.
A Dependent Type Theory with Names and Binding
 In Proceedings of the 2004 Computer Science Logic Conference, number 3210 in Lecture notes in Computer Science
, 2004
"... We consider the problem of providing formal support for working with abstract syntax involving variable binders. Gabbay and Pitts have shown in their work on FraenkelMostowski (FM) set theory how to address this through firstclass names: in this paper we present a dependent type theory for prog ..."
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Cited by 15 (1 self)
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We consider the problem of providing formal support for working with abstract syntax involving variable binders. Gabbay and Pitts have shown in their work on FraenkelMostowski (FM) set theory how to address this through firstclass names: in this paper we present a dependent type theory for programming and reasoning with such names. Our development is based on a categorical axiomatisation of names, with freshness as its central notion. An associated adjunction captures constructions known from FM theory: the freshness quantifier N , namebinding, and unique choice of fresh names. The Schanuel topos  the category underlying FM set theory  is an instance of this axiomatisation.
Exact Completions and Toposes
 University of Edinburgh
, 2000
"... Toposes and quasitoposes have been shown to be useful in mathematics, logic and computer science. Because of this, it is important to understand the di#erent ways in which they can be constructed. Realizability toposes and presheaf toposes are two important classes of toposes. All of the former and ..."
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Cited by 13 (4 self)
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Toposes and quasitoposes have been shown to be useful in mathematics, logic and computer science. Because of this, it is important to understand the di#erent ways in which they can be constructed. Realizability toposes and presheaf toposes are two important classes of toposes. All of the former and many of the latter arise by adding "good " quotients of equivalence relations to a simple category with finite limits. This construction is called the exact completion of the original category. Exact completions are not always toposes and it was not known, not even in the realizability and presheaf cases, when or why toposes arise in this way. Exact completions can be obtained as the composition of two related constructions. The first one assigns to a category with finite limits, the "best " regular category (called its regular completion) that embeds it. The second assigns to
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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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.
On the Role of Category Theory in the Area of Algebraic Specifications
 In LNCS , Proc. WADT11
, 1996
"... . The paper summarizes the main concepts and paradigms of category theory and explores some of their applications to the area of algebraic specifications. In detail we discuss different approaches to an abstract theory of specification logics. Further we present a uniform framework for developing pa ..."
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. The paper summarizes the main concepts and paradigms of category theory and explores some of their applications to the area of algebraic specifications. In detail we discuss different approaches to an abstract theory of specification logics. Further we present a uniform framework for developing particular specification logics. We make use of `classifying categories', to present categories of algebras as functor categories and to obtain necessary basic results for particular specification logics in a uniform manner. The specification logics considered are: equational logic for total algebras, conditional equational logic for partial algebras, and rewrite logic for concurrent systems. 1 Category Theory and Applications in Computer Science Category theory has been developed as a mathematical theory over 50 years and has influenced not only almost all branches of structural mathematics but also the development of several areas of computer science. It is the aim of this paper to review t...
Constructive set theories and their categorytheoretic models
 IN: FROM SETS AND TYPES TO TOPOLOGY AND ANALYSIS
, 2005
"... We advocate a pragmatic approach to constructive set theory, using axioms based solely on settheoretic principles that are directly relevant to (constructive) mathematical practice. Following this approach, we present theories ranging in power from weaker predicative theories to stronger impredicat ..."
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We advocate a pragmatic approach to constructive set theory, using axioms based solely on settheoretic principles that are directly relevant to (constructive) mathematical practice. Following this approach, we present theories ranging in power from weaker predicative theories to stronger impredicative ones. The theories we consider all have sound and complete classes of categorytheoretic models, obtained by axiomatizing the structure of an ambient category of classes together with its subcategory of sets. In certain special cases, the categories of sets have independent characterizations in familiar categorytheoretic terms, and one thereby obtains a rich source of naturally occurring mathematical models for (both predicative and impredicative) constructive set theories.
Complexity Doctrines
, 1995
"... vii Introduction ix 1 Tensor and Linear Time 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1 1.1 Almost Equational Specification : : : : : : : : : : : : : : : : : 3 1.1.1 Sketches : : : : : : : : : : : : : : : : : : : : : : : : : : 3 1.1.2 Orthogonality : : : : : : : : : : ..."
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vii Introduction ix 1 Tensor and Linear Time 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1 1.1 Almost Equational Specification : : : : : : : : : : : : : : : : : 3 1.1.1 Sketches : : : : : : : : : : : : : : : : : : : : : : : : : : 3 1.1.2 Orthogonality : : : : : : : : : : : : : : : : : : : : : : : 5 1.1.3 Essentially Algebraic Specification : : : : : : : : : : : : 8 1.2 Tensor and System T : : : : : : : : : : : : : : : : : : : : : : : 10 1.2.1 Serial Composition : : : : : : : : : : : : : : : : : : : : 10 1.2.2 Parallel Composition : : : : : : : : : : : : : : : : : : : 11 1.2.3 Unary Numbers : : : : : : : : : : : : : : : : : : : : : : 13 1.2.4 System T : : : : : : : : : : : : : : : : : : : : : : : : : 13 1.3 Comprehensions and Tiers : : : : : : : : : : : : : : : : : : : : 15 1.3.1 Comprehensions : : : : : : : : : : : : : : : : : : : : : : 15 1.3.2 Extents : : : : : : : : : : : : : : : : : : : : : : : : : : 18 1.3.3 Dyadic Numbers : : : : : : : : : : : : : : : ...
Classifying Categories for Partial Equational Logic
, 2002
"... Along the lines of classical categorical type theory for total functions, we establish equivalence results between certain classes of partial equational theories on the one hand and corresponding classes of categories on the other hand, staying close to standard categorical notions. ..."
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Cited by 5 (3 self)
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Along the lines of classical categorical type theory for total functions, we establish equivalence results between certain classes of partial equational theories on the one hand and corresponding classes of categories on the other hand, staying close to standard categorical notions.