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Geometry of Interaction and Linear Combinatory Algebras
, 2000
"... this paper was quite di#erent, stemming from the axiomatics of categories of tangles (although the authors were aware of possible connections to iteration theories. In fact, similar axiomatics in the symmetric case, motivated by flowcharts and "flownomials" had been developed some years earlier by S ..."
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Cited by 44 (10 self)
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this paper was quite di#erent, stemming from the axiomatics of categories of tangles (although the authors were aware of possible connections to iteration theories. In fact, similar axiomatics in the symmetric case, motivated by flowcharts and "flownomials" had been developed some years earlier by Stefanescu (Stefanescu 2000).) However, the first author realized, following a stimulating discussion with Gordon Plotkin, that traced monoidal categories provided a common denominator for the axiomatics of both the Girardstyle and AbramskyJagadeesanstyle versions of the Geometry of Interaction, at the basic level of the multiplicatives. This insight was presented in (Abramsky 1996), in which Girardstyle GoI was dubbed "particlestyle", since it concerns information particles or tokens flowing around a network, while the AbramskyJagadeesan style GoI was dubbed "wavestyle", since it concerns the evolution of a global information state or "wave". Formally, this distinction is based on whether the tensor product (i.e. the symmetric monoidal structure) in the underlying category is interpreted as a coproduct (particle style) or as a product (wave style). This computational distinction between coproduct and product interpretations of the same underlying network geometry turned out to have been partially anticipated, in a rather di#erent context, in a pioneering paper by E. S. Bainbridge (Bainbridge 1976), as observed by Dusko Pavlovic. These two forms of interpretation, and ways of combining them, have also been studied recently in (Stefanescu 2000). He uses the terminology "additive" for coproductbased (i.e. our "particlestyle") and "multiplicative" for productbased (i.e. our "wavestyle"); this is not suitable for our purposes, because of the clash with Linear Logic term...
Equilogical Spaces
, 1998
"... It is well known that one can build models of full higherorder dependent type theory (also called the calculus of constructions) using partial equivalence relations (PERs) and assemblies over a partial combinatory algebra (PCA). But the idea of categories of PERs and ERs (total equivalence relation ..."
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Cited by 31 (12 self)
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It is well known that one can build models of full higherorder dependent type theory (also called the calculus of constructions) using partial equivalence relations (PERs) and assemblies over a partial combinatory algebra (PCA). But the idea of categories of PERs and ERs (total equivalence relations) can be applied to other structures as well. In particular, we can easily dene the category of ERs and equivalencepreserving continuous mappings over the standard category Top 0 of topological T 0 spaces; we call these spaces (a topological space together with an ER) equilogical spaces and the resulting category Equ. We show that this categoryin contradistinction to Top 0 is a cartesian closed category. The direct proof outlined here uses the equivalence of the category Equ to the category PEqu of PERs over algebraic lattices (a full subcategory of Top 0 that is well known to be cartesian closed from domain theory). In another paper with Carboni and Rosolini (cited herein) a more abstract categorical generalization shows why many such categories are cartesian closed. The category Equ obviously contains Top 0 as a full subcategory, and it naturally contains many other well known subcategories. In particular, we show why, as a consequence of work of Ershov, Berger, and others, the KleeneKreisel hierarchy of countable functionals of nite types can be naturally constructed in Equ from the natural numbers object N by repeated use in Equ of exponentiation and binary products. We also develop for Equ notions of modest sets (a category equivalent to Equ) and assemblies to explain why a model of dependent type theory is obtained. We make some comparisons of this model to other, known models. 1
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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Cited by 20 (6 self)
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We investigate the development of theories of types and computability via realizability.
A Uniform Approach to Domain Theory in Realizability Models
 Mathematical Structures in Computer Science
, 1996
"... this paper we provide a uniform approach to modelling them in categories of modest sets. To do this, we identify appropriate structure for doing "domain theory" in such "realizability models". In Sections 2 and 3 we introduce PCAs and define the associated "realizability" categories of assemblies an ..."
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Cited by 19 (6 self)
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this paper we provide a uniform approach to modelling them in categories of modest sets. To do this, we identify appropriate structure for doing "domain theory" in such "realizability models". In Sections 2 and 3 we introduce PCAs and define the associated "realizability" categories of assemblies and modest sets. Next, in Section 4, we prepare for our development of domain theory with an analysis of nontermination. Previous approaches have used (relatively complicated) categorical formulations of partial maps for this purpose. Instead, motivated by the idea that A provides a primitive programming language, we consider a simple notion of "diverging" computation within A itself. This leads to a theory of divergences from which a notion of (computable) partial function is derived together with a lift monad classifying partial functions. The next task is to isolate a subcategory of modest sets with sufficient structure for supporting analogues of the usual domaintheoretic constructions. First, we expect to be able to interpret the standard constructions of total type theory in this category, so it should inherit cartesianclosure, coproducts and the natural numbers from modest sets. Second, it should interact well with the notion of partiality, so it should be closed under application of the lift functor. Third, it should allow the recursive definition of partial functions. This is achieved by obtaining a fixpoint object in the category, as defined in (Crole and Pitts 1992). Finally, although there is in principle no definitive list of requirements on such a category, one would like it to support more complicated constructions such as those required to interpret polymorphic and recursive types. The central part of the paper (Sections 5, 6, 7 and 9) is devoted to establish...
An Extension of Models of Axiomatic Domain Theory to Models of Synthetic Domain Theory
 In Proceedings of CSL 96
, 1997
"... . We relate certain models of Axiomatic Domain Theory (ADT) and Synthetic Domain Theory (SDT). On the one hand, we introduce a class of nonelementary models of SDT and show that the domains in them yield models of ADT. On the other hand, for each model of ADT in a wide class we construct a model of ..."
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Cited by 17 (6 self)
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. We relate certain models of Axiomatic Domain Theory (ADT) and Synthetic Domain Theory (SDT). On the one hand, we introduce a class of nonelementary models of SDT and show that the domains in them yield models of ADT. On the other hand, for each model of ADT in a wide class we construct a model of SDT such that the domains in it provide a model of ADT which conservatively extends the original model. Introduction The aim of Axiomatic Domain Theory (ADT) is to axiomatise the structure needed on a category so that its objects can be considered to be domains (see [11, x Axiomatic Domain Theory]). Models of axiomatic domain theory are given with respect to an enrichment base provided by a model of intuitionistic linear type theory [2, 3]. These enrichment structures consist of a monoidal adjunction C \Gamma! ? /\Gamma D between a cartesian closed category C and a symmetric monoidal closed category with finite products D, as well as with an !inductive fixedpoint object (Definition 1...
A Relationship between Equilogical Spaces and Type Two Effectivity
"... In this paper I compare two well studied approaches to topological semantics the domaintheoretic approach, exemplied by the category of countably based equilogical spaces, Equ, and Type Two Eectivity, exemplied by the category of Baire space representations, Rep(B ). These two categories are both ..."
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Cited by 16 (0 self)
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In this paper I compare two well studied approaches to topological semantics the domaintheoretic approach, exemplied by the category of countably based equilogical spaces, Equ, and Type Two Eectivity, exemplied by the category of Baire space representations, Rep(B ). These two categories are both locally cartesian closed extensions of countably based T 0 spaces. A natural question to ask is how they are related.
Logical Full Abstraction and PCF
 Tbilisi Symposium on Language, Logic and Computation. SiLLI/CSLI
, 1996
"... ion and PCF John Longley Gordon Plotkin March 15, 1996 Abstract We introduce the concept of logical full abstraction, generalising the usual equational notion. We consider the language PCF and two extensions with "parallel" operations. The main result is that, for standard interpretations, lo ..."
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Cited by 13 (5 self)
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ion and PCF John Longley Gordon Plotkin March 15, 1996 Abstract We introduce the concept of logical full abstraction, generalising the usual equational notion. We consider the language PCF and two extensions with "parallel" operations. The main result is that, for standard interpretations, logical full abstraction is equivalent to equational full abstraction together with universality; the proof involves constructing enumeration operators. We also consider restrictions on logical complexity and on the level of types. 1 Introduction The study of denotational semantics seeks to provide mathematical descriptions of programming languages by giving denotations of programs in terms of previously understood mathematical structures. For example, if P is a program that takes an input and produces an output, we might take its denotation to be a function from a set of inputvalues to a set of outputvalues. The most widelyknown approach to denotational semantics is that of traditiona...
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