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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...
Homology of Higher Dimensional Automata
, 1992
"... . Higher dimensional automata can model concurrent computations. The topological structure of the higher dimensional automata determines certain properties of the concurrent computation. We introduce bicomplexes as an algebraic tool for describing these automata and develop a simple homology theory ..."
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Cited by 43 (11 self)
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. Higher dimensional automata can model concurrent computations. The topological structure of the higher dimensional automata determines certain properties of the concurrent computation. We introduce bicomplexes as an algebraic tool for describing these automata and develop a simple homology theory for higher dimensional automata. We then show how the homology of automata has applications in the study of branchingtime equivalences of processes such as bisimulation. 1 Introduction Geometry has been suggested as a tool for modeling concurrency using higher dimensional objects to describe the concurrent execution of processes. This contrasts with earlier models based on the interleaving of computation steps to capture all possible behaviours of a concurrent system. Interleaving models must necessarily commit themselves to a specific choice of atomic action which makes them unable to distinguish between the execution of two truly concurrent actions and two mutually exclusive actions as t...
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
Constructive Category Theory
 IN PROCEEDINGS OF THE JOINT CLICSTYPES WORKSHOP ON CATEGORIES AND TYPE THEORY, GOTEBORG
, 1998
"... ..."
Subtractive Logic
, 1999
"... This paper is the first part of a work whose purpose is to investigate duality in some related frameworks (cartesian closed categories, lambdacalculi, intuitionistic and classical logics) from syntactic, semantical and computational viewpoints. We start with category theory and we show that any ..."
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Cited by 20 (1 self)
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This paper is the first part of a work whose purpose is to investigate duality in some related frameworks (cartesian closed categories, lambdacalculi, intuitionistic and classical logics) from syntactic, semantical and computational viewpoints. We start with category theory and we show that any bicartesian closed category with coexponents is degenerated (i.e. there is at most one arrow between two objects). The remainder of the paper is devoted to logical issues. We examine the propositional calculus underlying the type system of bicartesian closed categories with coexponents and we show that this calculus corresponds to subtractive logic: a conservative extension of intuitionistic logic with a new connector (subtraction) dual to implication. Eventually, we consider first order subtractive logic and we present an embedding of classical logic into subtractive logic. Introduction This paper is the first part of a work whose purpose is to investigate duality in some related ...
A Structural Approach to Reversible Computation
 Theoretical Computer Science
, 2001
"... Reversibility is a key issue in the interface between computation and physics, and of growing importance as miniaturization progresses towards its physical limits. Most foundational work on reversible computing to date has focussed on simulations of lowlevel machine models. By contrast, we develop ..."
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Cited by 18 (3 self)
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Reversibility is a key issue in the interface between computation and physics, and of growing importance as miniaturization progresses towards its physical limits. Most foundational work on reversible computing to date has focussed on simulations of lowlevel machine models. By contrast, we develop a more structural approach. We show how highlevel functional programs can be mapped compositionally (i.e. in a syntaxdirected fashion) into a simple kind of automata which are immediately seen to be reversible. The size of the automaton is linear in the size of the functional term. In mathematical terms, we are building a concrete model of functional computation. This construction stems directly from ideas arising in Geometry of Interaction and Linear Logic—but can be understood without any knowledge of these topics. In fact, it serves as an excellent introduction to them. At the same time, an interesting logical delineation between reversible and irreversible forms of computation emerges from our analysis. 1
The logic of entanglement
"... Abstract. We expose the information flow capabilities of pure bipartite entanglement as a theorem — which embodies the exact statement on the ‘seemingly acausal flow of information ’ in protocols such as teleportation [13]. We use this theorem to redesign and analyze known protocols (e.g. logic gate ..."
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Cited by 11 (1 self)
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Abstract. We expose the information flow capabilities of pure bipartite entanglement as a theorem — which embodies the exact statement on the ‘seemingly acausal flow of information ’ in protocols such as teleportation [13]. We use this theorem to redesign and analyze known protocols (e.g. logic gate teleportation [9] and entanglement swapping [14]) and show how to produce some new ones (e.g. parallel composition of logic gates). We also show how our results extend to the multipartite case and how they indicate that entanglement can be measured in terms of ‘information flow capabilities’. Ultimately, we propose a scheme for automated design of protocols involving measurements, local unitary transformations and classical communication. 1
Topology, Domain Theory and Theoretical Computer Science
, 1997
"... In this paper, we survey the use of ordertheoretic topology in theoretical computer science, with an emphasis on applications of domain theory. Our focus is on the uses of ordertheoretic topology in programming language semantics, and on problems of potential interest to topologists that stem from ..."
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Cited by 10 (2 self)
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In this paper, we survey the use of ordertheoretic topology in theoretical computer science, with an emphasis on applications of domain theory. Our focus is on the uses of ordertheoretic topology in programming language semantics, and on problems of potential interest to topologists that stem from concerns that semantics generates. Keywords: Domain theory, Scott topology, power domains, untyped lambda calculus Subject Classification: 06B35,06F30,18B30,68N15,68Q55 1 Introduction Topology has proved to be an essential tool for certain aspects of theoretical computer science. Conversely, the problems that arise in the computational setting have provided new and interesting stimuli for topology. These problems also have increased the interaction between topology and related areas of mathematics such as order theory and topological algebra. In this paper, we outline some of these interactions between topology and theoretical computer science, focusing on those aspects that have been mo...
Process Realizability
 In Foundations of Secure Computation
, 2000
"... This paper aims to give a readable and reasonably accessible account of some ideas linking the currently still largely separate worlds of concurrency theory and process algebra, on the one hand, and type theory, categorical models and realizability on the other. Background in process algebra may be ..."
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Cited by 9 (1 self)
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This paper aims to give a readable and reasonably accessible account of some ideas linking the currently still largely separate worlds of concurrency theory and process algebra, on the one hand, and type theory, categorical models and realizability on the other. Background in process algebra may be found in standard texts such as [Hoa85, Hen88, Mil89, Ros97]; while background in realizability, categorical models etc. is provided by texts such as [GLT89, AL91, Cro93, AC98, BW99]. A modest background in either or both of these fields should suffice to understand the main ideas. Most of the detailed verification of properties of the formal definitions we will present is left as a series of exercises. The diligent reader who attempts a number of these should get some feeling for the interplay between concrete processtheoretic notions, and more abstract logical and categorical ideas, which is characteristic of this topic. It is this interplay which makes the topic a fascinating one for the author; I hope this brief introduction, to a field which is still wide open for further development, succeeds in conveying something of this fascination to the reader.
An Algebraic Framework For The Definition Of Compositional Semantics Of Normal Logic Programs
, 1994
"... ion) Given two normal programs P1 and P2, the following three facts are equivalent: (i) Sem(P 1) = Sem(P 2) (ii) For every program P , Sem(P [ P 1) = Sem(P [ P 2) (iii) For every program P , MP[P1 = MP[P2 . Proof. It is enough to prove that (iii) implies (i), because the other implications are d ..."
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Cited by 8 (6 self)
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ion) Given two normal programs P1 and P2, the following three facts are equivalent: (i) Sem(P 1) = Sem(P 2) (ii) For every program P , Sem(P [ P 1) = Sem(P [ P 2) (iii) For every program P , MP[P1 = MP[P2 . Proof. It is enough to prove that (iii) implies (i), because the other implications are direct consequences of lemma 3.1 and theorem 5.1. Let us suppose that there exists a model A in Mod(; ;) such that F 1(A) 6= F 2(A), where F1 = Sem(P 1) and F2 = Sem(P 2). Then, we will show that there exists a program P such that MP[P1 6= MP[P2 . Let j 2 IN be the least layer such that F 1(A) + j 6= F 2(A) + j or F 1(A) j 6= F 2(A) j . Then we can consider two cases. First, if there exists the given level k 2 IN , and F 1(A) + j 6= F 2(A) + j , for some j < k, then F 1(B) 6= F 2(B) for all models B 2 Mod(; ;) such that A + j = B + i and A j 1 = B i 1 for some layer i. This is the case for the model B such that, for all i 2 IN : B + i = A + j B i = A j 1 In any other cas...