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SOME GEOMETRIC PERSPECTIVES IN CONCURRENCY THEORY
 HOMOLOGY, HOMOTOPY AND APPLICATIONS, VOL.5(2), 2003, PP.95–136
, 2003
"... Concurrency, i.e., the domain in computer science which deals with parallel (asynchronous) computations, has very strong links with algebraic topology; this is what we are developing in this paper, giving a survey of “geometric” models for concurrency. We show that the properties we want to prove on ..."
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Cited by 57 (4 self)
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Concurrency, i.e., the domain in computer science which deals with parallel (asynchronous) computations, has very strong links with algebraic topology; this is what we are developing in this paper, giving a survey of “geometric” models for concurrency. We show that the properties we want to prove on concurrent systems are stable under some form of deformation, which is almost homotopy. In fact, as the “direction ” of time matters, we have to allow deformation only as long as we do not reverse the direction of time. This calls for a new homotopy theory: “directed ” or dihomotopy. We develop some of the geometric intuition behind this theory and give some hints about the algebraic objects one can associate with it (in particular homology groups). For some historic as well as for some deeper reasons, the theory is at a stage where there is a nice blend between cubical, ωcategorical and topological techniques.
Detecting Deadlocks In Concurrent Systems
 IN CONCUR’98: CONCURRENCY THEORY (NICE
, 1998
"... We study deadlocks using geometric methods based on generalized process graphs [11], i.e., cubical complexes or HigherDimensional Automata (HDA) [23, 24, 30, 35], describing the semantics of the concurrent system of interest. A new algorithm is described and fully assessed, both theoretically a ..."
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Cited by 53 (12 self)
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We study deadlocks using geometric methods based on generalized process graphs [11], i.e., cubical complexes or HigherDimensional Automata (HDA) [23, 24, 30, 35], describing the semantics of the concurrent system of interest. A new algorithm is described and fully assessed, both theoretically and practically and compared with more wellknown traversing techniques. An implementation is
Presheaf Models for Concurrency
, 1999
"... In this dissertation we investigate presheaf models for concurrent computation. Our aim is to provide a systematic treatment of bisimulation for a wide range of concurrent process calculi. Bisimilarity is defined abstractly in terms of open maps as in the work of Joyal, Nielsen and Winskel. Their wo ..."
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Cited by 49 (19 self)
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In this dissertation we investigate presheaf models for concurrent computation. Our aim is to provide a systematic treatment of bisimulation for a wide range of concurrent process calculi. Bisimilarity is defined abstractly in terms of open maps as in the work of Joyal, Nielsen and Winskel. Their work inspired this thesis by suggesting that presheaf categories could provide abstract models for concurrency with a builtin notion of bisimulation. We show how
Hundreds of Impossibility Results for Distributed Computing
 Distributed Computing
, 2003
"... We survey results from distributed computing that show tasks to be impossible, either outright or within given resource bounds, in various models. The parameters of the models considered include synchrony, faulttolerance, different communication media, and randomization. The resource bounds refe ..."
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Cited by 49 (5 self)
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We survey results from distributed computing that show tasks to be impossible, either outright or within given resource bounds, in various models. The parameters of the models considered include synchrony, faulttolerance, different communication media, and randomization. The resource bounds refer to time, space and message complexity. These results are useful in understanding the inherent difficulty of individual problems and in studying the power of different models of distributed computing.
Algebraic Topology And Concurrency
 Theoretical Computer Science
, 1998
"... This article is intended to provide some new insights about concurrency theory using ideas from geometry, and more specifically from algebraic topology. The aim of the paper is twofold: we justify applications of geometrical methods in concurrency through some chosen examples and we give the mathem ..."
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Cited by 48 (12 self)
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This article is intended to provide some new insights about concurrency theory using ideas from geometry, and more specifically from algebraic topology. The aim of the paper is twofold: we justify applications of geometrical methods in concurrency through some chosen examples and we give the mathematical foundations needed to understand the geometric phenomenon that we identify. In particular we show that the usual notion of homotopy has to be refined to take into account some partial ordering describing the way time goes. This gives rise to some new interesting mathematical problems as well as give some common grounds to computerscientific problems that have not been precisely related otherwise in the past. The organization of the paper is as follows. In Section 2 we explain to which extent we can use some geometrical ideas in computer science: we list a few of the potential or well known areas of application and try to exemplify some of the properties of concurrent (and distributed) systems we are interested in. We first explain the interest of using some geometric ideas for semantical reasons. Then we take the example of concurrent databases with the problem of finding deadlocks and with some aspects of serializability theory. More general questions about schedules can be asked as well and related to some geometric considerations, even for scheduling microinstructions (and not only coarsegrained transactions as for databases). The final example is the one of faulttolerant protocols for distributed systems, where subtle scheduling properties go into play. In Section 3 we give the first few definitions needed for modeling the topological spaces arising from Section 2. Basically, we need to define a topological space containing all traces of executions of the concu...
Gates accept concurrent behavior
 In Proc. 34th Ann. IEEE Symp. on Foundations of Comp. Sci
, 1993
"... We represent concurrent processes as Boolean propositions or gates, cast in the role of acceptors of concurrent behavior. This properly extends other mainstream representations of concurrent behavior such as event structures, yet is defined more simply. It admits an intrinsic notion of duality that ..."
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Cited by 37 (16 self)
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We represent concurrent processes as Boolean propositions or gates, cast in the role of acceptors of concurrent behavior. This properly extends other mainstream representations of concurrent behavior such as event structures, yet is defined more simply. It admits an intrinsic notion of duality that permits processes to be viewed as either schedules or automata. Its algebraic structure is essentially that of linear logic, with its morphisms being consequencepreserving renamings of propositions, and with its operations forming the core of a natural concurrent programming language. 1
Geometry and Concurrency: A User's Guide
, 2000
"... Introduction "Geometry and Concurrency" is not yet a wellestablished domain of research, but is rather made of a collection of seemingly related techniques, algorithms and formalizations, coming from different application areas, accumulated over a long period of time. There is currently ..."
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Cited by 33 (7 self)
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Introduction "Geometry and Concurrency" is not yet a wellestablished domain of research, but is rather made of a collection of seemingly related techniques, algorithms and formalizations, coming from different application areas, accumulated over a long period of time. There is currently a certain amount of effort made for unifying these (in particular see the article (Gunawardena, 1994)), following the workshop "New Connections between Computer Science and Mathematics" held at the Newton Institute in Cambridge, England in November 1995 (and sponsored by HP/BRIMS). More recently, the first workshop on the very same subject has been held in Aalborg, Denmark (see http://www.math.auc.dk/~raussen/admin/workshop/workshop.html where the articles of this issue, among others, have been first sketched. But what is "Geometry and Concurrency" composed of then? It is an area of research made of techniques which use geometrical reasoning for describing and solving problems
Homotopy and Concurrency
 Bulletin of the EATCS
, 1994
"... In this paper we give a homotopy theoretic proof of a wellknown result in database engineering: that 2phase locking is safe. The proof gives an immediate intuitive reason for why the 2phase locking condition implies safety. We point out a number of interesting open questions regarding the interpl ..."
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Cited by 33 (2 self)
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In this paper we give a homotopy theoretic proof of a wellknown result in database engineering: that 2phase locking is safe. The proof gives an immediate intuitive reason for why the 2phase locking condition implies safety. We point out a number of interesting open questions regarding the interplay between homotopy and concurrency. Keywords: Homotopy theory, serializability, 2phase locking, concurrency theory 1 Introduction What has homotopy got to do with concurrency? At first sight it seems unlikely that there should be any relationship between these two subjects. After all, homotopy theory is about continuous objects while concurrency typically deals with discrete structures. In this paper I will try to show that, on the contrary, there may be a very natural relationship between the two. Instead of making a lot of abstract statements about homotopy and concurrency I would like to work through the proof of a theorem which all database engineers learn at their Mother's knee: th...
Higher dimensional transition systems
, 1996
"... We introduce the notion of higher dimensional transition systems as a model of concurrency providing an elementary, settheoretic formalisation of the idea of higher dimensional transition. We show an embedding of the category of higher dimensional transition systems into that of higher dimension ..."
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Cited by 30 (3 self)
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We introduce the notion of higher dimensional transition systems as a model of concurrency providing an elementary, settheoretic formalisation of the idea of higher dimensional transition. We show an embedding of the category of higher dimensional transition systems into that of higher dimensional automata which cuts down to an equivalence when we restrict to nondegenerate automata. Moreovel; we prove that the natural notion of bisimulation for such structures is a generalisation of the strong history preserving bisimulation, and provide an abstract categorical account of it via open maps. Finally, we dejine a notion of unfolding for higher dimensional transition systems and characterise the structures so obtained as a generalisation of event structures.
On the Expressiveness of higher dimensional automata
 EXPRESS 2004, ENTCS
, 2005
"... Abstract In this paper I compare the expressive power of several models of concurrency based on their ability to represent causal dependence. To this end, I translate these models, in behaviour preserving ways, into the model of higher dimensional automata, which is the most expressive model under i ..."
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Cited by 27 (0 self)
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Abstract In this paper I compare the expressive power of several models of concurrency based on their ability to represent causal dependence. To this end, I translate these models, in behaviour preserving ways, into the model of higher dimensional automata, which is the most expressive model under investigation. In particular, I propose four different translations of Petri nets, corresponding to the four different computational interpretations of nets found in the literature. I also extend various equivalence relations for concurrent systems to higher dimensional automata. These include the history preserving bisimulation, which is the coarsest equivalence that fully respects branching time, causality and their interplay, as well as the STbisimulation, a branching time respecting equivalence that takes causality into account to the extent that it is expressible by actions overlapping in time. Through their embeddings in higher dimensional automata, it is now welldefined whether members of different models of concurrency are equivalent.