Homological properties of non-deterministic branchings of mergings in higher dimensional automata

Gaucher, Philippe

doi:10.4310/hha.2005.v7.n1.a4

Cited by 17 publications

(19 citation statements)

References 14 publications

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“…The category of precubical sets is also not convenient for the study of these homology theories because of the absence of degenerate cubes (i.e. of "thin" cubes, that is without volume) and of composition of cubes: see the introduction and especially Figure 3 of [12] for further explanations. However, the construction of this paper does use as an intermediate category the category of precubical sets.…”

Section: Presentation Of the Resultsmentioning

confidence: 99%

Towards a homotopy theory of process algebra

Gaucher

2008

Homology, Homotopy and Applications

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This paper proves that labelled flows are expressive enough to contain all process algebras which are a standard model for concurrency. More precisely, we construct the space of execution paths and of higher dimensional homotopies between them for every process name of every process algebra with any synchronization algebra using a notion of labelled flow. This interpretation of process algebra satisfies the paradigm of higher dimensional automata (HDA): one non-degenerate full n-dimensional cube (no more no less) in the underlying space of the time flow corresponding to the concurrent execution of n actions. This result will enable us in future papers to develop a homotopical approach of process algebras. Indeed, several homological constructions related to the causal structure of time flow are possible only in the framework of flows.

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Section: Presentation Of the Resultsmentioning

confidence: 99%

Towards a homotopy theory of process algebra

Gaucher

2008

Homology, Homotopy and Applications

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“…Gaucher [42][43][44][45][46][47], like Pratt [24], takes a globular approach to HDA, and relates it to the cubical approach. Variations and generalisations of this approach are studied by Gaucher and Goubault [52,[48][49][50][51].…”

Section: Definitionmentioning

confidence: 99%

“…Homology theories for HDA are proposed by Goubault and Gaucher [62,55,[42][43][44][45][46][47][48][49][50][51][52]. Gaucher [42][43][44][45][46][47], like Pratt [24], takes a globular approach to HDA, and relates it to the cubical approach.…”

Section: Definitionmentioning

confidence: 99%

On the expressiveness of higher dimensional automata

Glabbeek

2006

Theoretical Computer Science

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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 (HDA), 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 HDA. These include the history preserving bisimulation, which is the coarsest equivalence that fully respects branching time, causality and their interplay, as well as the ST-bisimulation, 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 HDA, it is now well-defined whether members of different models of concurrency are equivalent. Fig. 1 lists the main models of concurrency proposed in the literature, ordered by expressive power. Of all models, the labelled variant is understood. Here, I only treat models of processes that perform actions a, b, c, . . . whose internal structure is not further examined, and real-time and stochastic aspects of processes are completely ignored. Furthermore, I only study the representation of processes, not the representation of operators on processes. I restrict myself to models that take branching time fully into account, and hence skip models that represent processes by the sets of their executions. Thus, the expressive power of the models differs only to the extent that certain forms of causal dependence are expressible. I limit myself to models that take a fully asynchronous view on parallelism: whenever a number of actions can happen simultaneously, this must be because they are causally independent, and hence they can also happen in any order. Because of this, I do not include Petri nets with inhibitor arcs, or Chu spaces [25]. A hierarchy of concurrency modelsThere is an arrow from model A to model B in Fig. 1 iff there exists a translation from processes representable in model A to processes representable in model B that fully respects branching time, causality, and their interplay. Thus, a process and its translation ought to be history preserving bisimulation equivalent [26,11].

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“…As you might imagine, there are problems in finding a formula in still higher dimensions. In the groupoid case, this is handled by a homotopy addition lemma and thin elements [22], but in the category case a formula for just a commutative 4-cube is complicated [55].…”

Section: The Search For Higher Homotopy Groupoidsmentioning

confidence: 99%