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15
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 45 (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
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 23 (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.
Higher Dimensional Automata Revisited
 MATHEMATICAL STRUCTURES IN COMPUTER SCIENCE
, 2000
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Dipaths and dihomotopies in a cubical complex
 Advances in Applied Mathematics
, 2003
"... In the geometric realization of a cubical complex without degeneracies, a ✷set, dipaths and dihomotopies may not be combinatorial, i.e., not geometric realizations of combinatorial dipaths and equivalences. When we want to use geometric/topological tools to classify dipaths on the 1skeleton, combi ..."
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Cited by 12 (1 self)
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In the geometric realization of a cubical complex without degeneracies, a ✷set, dipaths and dihomotopies may not be combinatorial, i.e., not geometric realizations of combinatorial dipaths and equivalences. When we want to use geometric/topological tools to classify dipaths on the 1skeleton, combinatorial dipaths, up to dihomotopy, and in particular up to combinatorial dihomotopy, we need that all dipaths are in fact dihomotopic to a combinatorial dipath. And moreover that two combinatorial dipaths which are dihomotopic are then combinatorially dihomotopic. We prove that any dipath from a vertex to a vertex is dihomotopic to a combinatorial dipath, in a non selfintersecting ✷set. And that two combinatorial dipaths which are dihomotopic through a non combinatorial dihomotopy are in fact combinatorially dihomotopic, in a geometric ✷set. Moreover, we prove that in a geometric ✷set, the dhomotopy introduced in [6] coincides with the dihomotopy in [5]
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 11 (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.
Investigation of Concurrent Processes By Means of Homotopy Functors
, 1999
"... this paper . . . . . . . . . . . . . . . . . . . . . . . . . 4 ..."
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Cited by 9 (2 self)
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this paper . . . . . . . . . . . . . . . . . . . . . . . . . 4
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 8 (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 = ×.
Homotopy in Concurrent Processes
, 1998
"... Homotopy in concurrent processes In theories of job scheduling and of distributed computing, there have been many attempts to introduce tools originating from algebraic and combinatorial topology, such as homotopy groups. Informally, the fundamental (or first homotopy) group gives an account of the ..."
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Cited by 6 (2 self)
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Homotopy in concurrent processes In theories of job scheduling and of distributed computing, there have been many attempts to introduce tools originating from algebraic and combinatorial topology, such as homotopy groups. Informally, the fundamental (or first homotopy) group gives an account of the nature of "holes" in a topological space. In the realm of processes, such holes may correspond to forbidden configurations; e.g. where more than one process is within the same critical region. However, many topological properties, technically necessary for the construction of the fundamental group, have no counterparts, or only artificial ones, in concurrent processes. The paths corresponding to process executions are not cyclic, because time only flows forwards, and they cannot be as naturally composed as looping paths in topological spaces. The fundamental "group" lacks therefore its group operations. This paper puts forward a simple remedy for the shortcomings: if you cannot find a usefu...
Classifying Holes of Arbitrary Dimensions in Partially Ordered Cubes
, 2000
"... this report belongs to topology. Its motivations are, however, derived from investigations of concurrent processes. For a general discussion of concurrent processes giving rise to topological spaces, the reader is referred for instance to Pratt [7] or [8]; or to Fajstrup, Goubault, Raussen [1]; or t ..."
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Cited by 6 (0 self)
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this report belongs to topology. Its motivations are, however, derived from investigations of concurrent processes. For a general discussion of concurrent processes giving rise to topological spaces, the reader is referred for instance to Pratt [7] or [8]; or to Fajstrup, Goubault, Raussen [1]; or to Goubault [6]. In the most general terms, the mathematical structures modeling cooperation between processes are endowed with a partial order and with a topology. The partial order relates events to other events they may be causes of, which captures causality and time succession. Every component of a system of processes may have its own time and these private times have only to be in tune when interprocess communication is taking place. Because of this, the order does not have to be total. The topology in a model provides a notion of closeness between particular execution trajectories. For instance, when a process is slightly slowed down without changing the order in which it communicates with other processes (cooperates or competes for shared resources), the resulting execution is close to the original one. On the other hand, if two executions differ in the way the communication takes place, they are far from each other. In this sense, communications create "holes" in the spaces modeling concurrency. Topology makes a convenient language to discuss this. A standard topological toolbox containing homotopies and homologies is not readily applicable for a classification of the holes, since it disregards the time or causalityinduced ordering which is paramount in computer science applications. Researchers have, therefore, attempted to adapt these concepts. Some discussion and comparison to earlier work may be found in Sec. 5. This report is a step towards finding useful invar...
Comparing Transition Systems with Independence and Asynchronous Transition Systems
 Department of Computer Science, University of Aarhus
, 1996
"... . Transition systems with independence and asynchronous transition systems are noninterleaving models for concurrency arising from the same simple idea of decorating transitions with events. They differ for the choice of a derived versus a primitive notion of event which induces considerable differe ..."
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Cited by 6 (3 self)
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. Transition systems with independence and asynchronous transition systems are noninterleaving models for concurrency arising from the same simple idea of decorating transitions with events. They differ for the choice of a derived versus a primitive notion of event which induces considerable differences and makes the two models suitable for different purposes. This opens the problem of investigating their mutual relationships, to which this paper gives a fully comprehensive answer. In details, we characterise the category of extensional asynchronous transitions systems as the largest full subcategory of the category of (labelled) asynchronous transition systems which admits TSI, the category of transition systems with independence, as a coreflective subcategory. In addition, we introduce eventmaximal asynchronous transitions systems and we show that their category is equivalent to TSI, so providing an exhaustive characterisation of transition systems with independence in terms of asyn...