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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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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...
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 32 (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
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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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.
Higher Dimensional Automata Revisited
 MATHEMATICAL STRUCTURES IN COMPUTER SCIENCE
, 2000
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Complete Cuboidal Sets in Axiomatic Domain Theory (Extended Abstract)
 In Proceedings of 12th Annual Symposium on Logic in Computer Science
, 1997
"... ) Marcelo Fiore !mf@dcs.ed.ac.uk? Gordon Plotkin y !gdp@dcs.ed.ac.uk? John Power !ajp@dcs.ed.ac.uk? Department of Computer Science Laboratory for Foundations of Computer Science University of Edinburgh, The King's Buildings Edinburgh EH9 3JZ, Scotland Abstract We study the enrichme ..."
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) Marcelo Fiore !mf@dcs.ed.ac.uk? Gordon Plotkin y !gdp@dcs.ed.ac.uk? John Power !ajp@dcs.ed.ac.uk? Department of Computer Science Laboratory for Foundations of Computer Science University of Edinburgh, The King's Buildings Edinburgh EH9 3JZ, Scotland Abstract We study the enrichment of models of axiomatic domain theory. To this end, we introduce a new and broader notion of domain, viz. that of complete cuboidal set, that complies with the axiomatic requirements. We show that the category of complete cuboidal sets provides a general notion of enrichment for a wide class of axiomatic domaintheoretic structures. Introduction The aim of Axiomatic Domain Theory (ADT) is to provide a conceptual understanding of why domains are adequate as mathematical models of computation. (For a discussion see [12, x Axiomatic Domain Theory ].) The approach taken is to axiomatise the structure needed on a category so that its objects can be considered as domains, and its maps as continuous...
A comparison of Petri net semantics under the collective token philosophy
 SpringerVerlag LNCS
, 1998
"... Abstract. In recent years, several semantics for place/transition Petri nets have been proposed that adopt the collective token philosophy. We investigate distinctions and similarities between three such models, namely configuration structures, concurrent transition systems, and (strictly) symmetric ..."
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Abstract. In recent years, several semantics for place/transition Petri nets have been proposed that adopt the collective token philosophy. We investigate distinctions and similarities between three such models, namely configuration structures, concurrent transition systems, and (strictly) symmetric (strict) monoidal categories. Weusethenotionof adjunction to express each connection. We also present a purely logical description of the collective token interpretation of net behaviours in terms of theories and theory morphisms in partial membership equational logic.
Transition and cancellation in concurrency and branching time
 Mathematical Structures in Computer Science 13(4) (2003
, 2002
"... We review the conceptual development of (true) concurrency and branching time starting from Petri nets and proceeding via Mazurkiewicz traces, pomsets, bisimulation, and event structures up to higher dimensional automata (HDAs), whose acyclic case may be identified with triadic event structures and ..."
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Cited by 15 (1 self)
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We review the conceptual development of (true) concurrency and branching time starting from Petri nets and proceeding via Mazurkiewicz traces, pomsets, bisimulation, and event structures up to higher dimensional automata (HDAs), whose acyclic case may be identified with triadic event structures and triadic Chu spaces. Acyclic HDAs may be understood as the extension of Boolean logic with a third truth value expressing transition. We prove the necessity of such a third value under mild assumptions about the nature of observable events, and show that the expansion of any complete Boolean basis L to L with a third literal �a expressing a = forms an expressively complete basis for the representation of acyclic HDAs. The main contribution is a new event state × of cancellation, sibling to, serving to distinguish a(b + c) from ab + ac while simplifying the extensional definitions of termination �A and sequence AB. We show that every HDAX (acyclic HDA with ×) is representable in the expansion of L to L × with a fourth literal �a expressing a = ×.
Combinatorics of labelling in higher dimensional automata
"... Abstract. The main idea for interpreting concurrent processes as labelled precubical sets is that a given set of n actions running concurrently must be assembled to a labelled ncube, in exactly one way. The main ingredient is the nonfunctorial construction called labelled directed coskeleton. It i ..."
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Abstract. The main idea for interpreting concurrent processes as labelled precubical sets is that a given set of n actions running concurrently must be assembled to a labelled ncube, in exactly one way. The main ingredient is the nonfunctorial construction called labelled directed coskeleton. It is defined as a subobject of the labelled coskeleton, the latter coinciding in the unlabelled case with the right adjoint to the truncation functor. This nonfunctorial construction is necessary since the labelled coskeleton functor of the category of labelled precubical sets does not fulfil the above requirement. We prove in this paper that it is possible to force the labelled coskeleton functor to be wellbehaved by working with labelled transverse symmetric precubical sets. Moreover, we prove that this solution is the only one. A transverse symmetric precubical set is a precubical set equipped with symmetry maps and with a new kind of degeneracy map called transverse degeneracy. Finally, we also prove that the two settings are equivalent from a directed algebraic topological viewpoint. To illustrate, a new semantics of CCS, equivalent to the old one, is given. Contents
TOWARDS A HOMOTOPY THEORY OF HIGHER DIMENSIONAL TRANSITION SYSTEMS
"... Abstract. We proved in a previous work that CattaniSassone’s higher dimensional transition systems can be interpreted as a smallorthogonality class of a topological locally finitely presentable category of weak higher dimensional transition systems. In this paper, we turn our attention to the full ..."
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Abstract. We proved in a previous work that CattaniSassone’s higher dimensional transition systems can be interpreted as a smallorthogonality class of a topological locally finitely presentable category of weak higher dimensional transition systems. In this paper, we turn our attention to the full subcategory of weak higher dimensional transition systems which are unions of cubes. It is proved that there exists a left proper combinatorial model structure such that two objects are weakly equivalent if and only if they have the same cubes after simplification of the labelling. This model structure is obtained by Bousfield localizing a model structure which is left determined with respect to a class of maps which is not the class of monomorphisms. We prove that the higher dimensional transition systems corresponding to two process algebras are weakly equivalent if and only if they are isomorphic. We also construct a second Bousfield localization in which two bisimilar cubical transition systems are weakly equivalent. The appendix contains a technical lemma about smallness of weak factorization systems in coreflective subcategories which can be of independent interest. This paper is a first step towards a homotopical interpretation of bisimulation for higher dimensional transition
Directed Algebraic Topology and Higher Dimensional . . .
, 2009
"... CattaniSassone’s notion of higher dimensional transition system is interpreted as a smallorthogonality class of a locally finitely presentable topological category of weak higher dimensional transition systems. In particular, the higher dimensional transition system associated with the labelled ..."
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Cited by 6 (4 self)
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CattaniSassone’s notion of higher dimensional transition system is interpreted as a smallorthogonality class of a locally finitely presentable topological category of weak higher dimensional transition systems. In particular, the higher dimensional transition system associated with the labelled ncube turns out to be the free higher dimensional transition system generated by one ndimensional transition. As a first application of this construction, it is proved that the category of optimal higher dimensional transition systems (those such that each action is in a 1transition) is equivalent to a locally finitely presentable reflective full subcategory of the category of labelled symmetric precubical sets. As a second application, it is given a factorization of the mapping taking a CCS process name to a flow through higher dimensional transition systems. The second application of this paper can be easily adapted to other process algebras and to other topological models of concurrency than the one of flows.