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22
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.
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 14 (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 = ×.
EventState Duality: The Enriched Case
"... Enriched categories have been applied in the past to both eventoriented true concurrency models and stateoriented information systems, with no evident relationship between the two. Ordinary Chu spaces expose a natural duality between partially ordered temporal spaces (pomsets, event structures), a ..."
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Cited by 5 (0 self)
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Enriched categories have been applied in the past to both eventoriented true concurrency models and stateoriented information systems, with no evident relationship between the two. Ordinary Chu spaces expose a natural duality between partially ordered temporal spaces (pomsets, event structures), and partially ordered information systems.
ORDINARY AND DIRECTED COMBINATORIAL HOMOTOPY, APPLIED TO IMAGE ANALYSIS AND CONCURRENCY
 HOMOLOGY, HOMOTOPY AND APPLICATIONS, VOL.5(2), 2003, PP.211–231
, 2003
"... Combinatorial homotopical tools developed in previous works, and consisting essentially of intrinsic homotopy theories for simplicial complexes and directed simplicial complexes, can be applied to explore mathematical models representing images, or directed images, or concurrent processes. An image, ..."
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Cited by 5 (4 self)
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Combinatorial homotopical tools developed in previous works, and consisting essentially of intrinsic homotopy theories for simplicial complexes and directed simplicial complexes, can be applied to explore mathematical models representing images, or directed images, or concurrent processes. An image, represented by a metric space X, can be explored at a variable resolution ɛ> 0, by equipping it with a structure tɛX of simplicial complex depending on ɛ; this complex can be further analysed by homotopy groups π ɛ n(X) = πn(tɛX) and homology groups H ɛ n(X) = Hn(tɛX). Loosely speaking, these objects detect singularities which can be captured by an ndimensional grid, with edges bound by ɛ; this works equally well for continuous or discrete regions of euclidean spaces. Similarly, a directed image, represented by an “asymmetric metric space”, produces a family of directed simplicial complexes sɛX and can be explored by the fundamental ncategory ↑Π ɛ n(X) of the latter. The same directed tools can be applied to combinatorial models of concurrent automata, like Chuspaces.
Bifinite Chu spaces
"... ABSTRACT. This paper studies colimits of sequences of finite Chu spaces and their ramifications. We consider three base categories of Chu spaces: the generic Chu spaces (C), the extensional Chu spaces (E), and the biextensional Chu spaces (B). The main results are: (1) a characterization of monics i ..."
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ABSTRACT. This paper studies colimits of sequences of finite Chu spaces and their ramifications. We consider three base categories of Chu spaces: the generic Chu spaces (C), the extensional Chu spaces (E), and the biextensional Chu spaces (B). The main results are: (1) a characterization of monics in each of the three categories; (2) existence (or the lack thereof) of colimits and a characterization of finite objects in each of the corresponding categories using monomorphisms/injections (denoted as iC, iE, and iB, respectively); (3) a formulation of bifinite Chu spaces with respect to iC; (4) the existence
The Glory of the Past and Geometrical Concurrency
"... This paper contributes to the general understanding of the geometrical model of concurrency that was named higher dimensional automata (HDAs) by Pratt and van Glabbeek. In particular we provide some understanding of the modal logics for such models and their expressive power in terms of the bisimula ..."
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This paper contributes to the general understanding of the geometrical model of concurrency that was named higher dimensional automata (HDAs) by Pratt and van Glabbeek. In particular we provide some understanding of the modal logics for such models and their expressive power in terms of the bisimulation that can be captured. The geometric model of concurrency is interesting from two main reasons: its generality and expressiveness, and the natural way in which autoconcurrency and action refinement are captured. Logics for this model, though, are not well investigated, where a simple, yet adequate, modal logic over HDAs was only recently introduced. As this modal logic, with two existential modalities, during and after, captures only split bisimulation, which is rather low in the spectrum of van Glabbeek and Vaandrager, the immediate question was what small extension of this logic could capture the more finegrained hereditary history preserving bisimulation (hh)? In response, the work in this paper provides several insights. One is the fact that the geometrical aspect of HDAs makes it possible to use for capturing the hhbisimulation, a standard modal logic that does not employ event variables, opposed to the two logics (over
Sheaves and geometric logic and applications to modular verification of complex systems
 ENTCS
, 2009
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Chu Spaces, Concept Lattices and Information Systems in n Dimensions
"... Abstract Three areas of computer science that were developed independently but have strong interconnections were brought together in recent work of G.Q. Zhang and his collaborators: Chu spaces and concept lattices, on the one hand, and domains and information systems on the other. In a different d ..."
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Abstract Three areas of computer science that were developed independently but have strong interconnections were brought together in recent work of G.Q. Zhang and his collaborators: Chu spaces and concept lattices, on the one hand, and domains and information systems on the other. In a different direction, inspired by the work of Wille on formal concept analysis, the author developed a theory of nadic concept analysis that led to the definitions of nclosure systems, nclosure operators and of ninformation systems. The infinitary versions of the ordinary concepts, discussed in Zhang's work, are special cases of the ndimensional ones when restricted to the 2 dimensions. In this work, some of the interconnections revealed in Zhang's work between the 2dimensional concepts are lifted to the ndimensional framework. The hope is that the present work may help further clarify these relationships and also provide some impetus for considering applications of some of the recent work in this multidimensional framework in fields, such as datamining, knowledge discovery, ontology and ontological engineering.