arXiv subjects
Publications and source records attributed to Thomas Kahl.
Higher-dimensional automata, i.e., pointed labeled precubical sets, are a powerful combinatorial-topological model for concurrent systems. In this paper, we show that for every (nonempty) connected polyhedron there exists a shared-variable system such that the higher-dimensional automaton modeling the state space of the system has the homotopy type of the polyhedron.
This paper studies a notion of directed homology for preordered spaces, called the homology digraph. We show that the homology digraph is a directed homotopy invariant and establish variants of the main results of ordinary singular homology theory for the homology digraph. In particular, we prove a Künneth formula, which enables one to compute the homology digraph of a product of preordered spaces from the homology digraphs of the components.
Given a transition system with an independence relation on the alphabet of labels, one can associate with it a usually very large symmetric higher-dimensional automaton. The purpose of this paper is to show that by choosing an acyclic relation whose symmetric closure is the given independence relation, it is possible to construct a much smaller nonsymmetric HDA with the same homology language.
This paper introduces a notion of equivalence for higher-dimensional automata, called weak equivalence. Weak equivalence focuses mainly on a traditional trace language and a new homology language, which captures the overall independence structure of an HDA. It is shown that weak equivalence is compatible with both the tensor product and the coproduct of HDAs and that, under certain conditions, HDAs may be reduced to weakly equivalent smaller ones by merging and collapsing cubes.
It is shown that a higher-dimensional automaton is hhp-bisimilar to the free symmetric HDA generated by it. Consequently, up to hereditary history-preserving bisimilarity, ordinary HDAs and symmetric HDAs are models of concurrency with the same expressive power.
The purpose of this paper is to provide a construction to model shared-variable systems using higher-dimensional automata which is compositional in the sense that the parallel composition of completely independent systems is modeled by the standard tensor product of HDAs and nondeterministic choice is represented by the coproduct.
We construct labeling homomorphisms on the cubical homology of higher-dimensional automata and show that they are natural with respect to cubical dimaps and compatible with the tensor product of HDAs. We also indicate two possible applications of labeled homology in concurrency theory.
Higher-dimensional automata constitute a very expressive model for concurrent systems. In this paper, we discuss "topological abstraction" of higher-dimensional automata, i.e., the replacement of HDAs by smaller ones that can be considered equivalent from both a computer scientific and a topological point of view. By definition, topological abstraction preserves the homotopy type, the trace category, and the homology graph of an HDA. We establish conditions under which cube collapses yield topological abstractions of HDAs.
Félix, Halperin, and Lemaire have shown that the rational module category Mcat and the rational Toomer invariant $e_0$ coincide for simply connected Poincaré duality complexes. We establish an analogue of this result for the sectional category of a fibration.
Higher dimensional automata, i.e. labelled precubical sets, model concurrent systems. We introduce the homology graph of an HDA, which is a directed graph whose nodes are the homology classes of the HDA. We show that the homology graph is invariant under homeomorphic abstraction, i.e. under weak morphisms that are homeomorphisms.
We introduce weak morphisms of higher dimensional automata and use them to define preorder relations for HDAs, among which homeomorphic abstraction and trace equivalent abstraction. It is shown that homeomorphic abstraction is essentially always stronger than trace equivalent abstraction. We also define the trace language of an HDA and show that, for a large class of HDAs, it is invariant under trace equivalent abstraction.
In this paper, we consider 2-dimensional precubical sets, which can be used to model systems of two concurrently executing processes. From the point of view of concurrency theory, two precubical sets can be considered equivalent if their geometric realizations have the same directed homotopy type relative to the extremal elements in the sense of P. Bubenik. We give easily verifiable conditions under which it is possible to reduce a 2-dimensional precubical set to an equivalent smaller one by collapsing an edge or eliminating a square and one or two free faces. We also look at some simple standard examples in order to illustrate how our results can be used to construct small models of 2-dimensional precubical sets.
We construct an explicit semifree model for the fiber join of two fibrations p: E --> B and p': E' --> B from semifree models of p and p'. Using this model, we introduce a lower bound of the sectional category of a fibration p which can be calculated from any Sullivan model of p and which is closer to the sectional category of p than the classical cohomological lower bound given by the nilpotency of the kernel of p^*: H^*(B;Q) --> H^*(E;Q). In the special case of the evaluation fibration X^I --> X x X we obtain a computable lower bound of Farber's topological complexity TC(X). We show that the difference between this lower bound and the classical cohomological lower bound can be arbitrarily large.
In this work, we compare the two approximations of a path-connected space $X$, by the Ganea spaces $G_n(X)$ and by the realizations $\|Λ_\bullet X\|_{n}$ of the truncated simplicial resolutions emerging from the loop-suspension cotriple $ΣΩ$. For a simply connected space $X$, we construct maps $\|Λ_\bullet X\|_{n-1}\to G_n(X)\to \|Λ_\bullet X\|_{n}$ over $X$, up to homotopy. In the case $n=2$, we prove the existence of a map $G_2(X)\to\|Λ_\bullet X\|_{1}$ over $X$ (up to homotopy) and conjecture that this map exists for any $n$.
Algebraic topological methods have been used successfully in concurrency theory, the domain of theoretical computer science that deals with parallel computing. L. Fajstrup, E. Goubault, and M. Raussen have introduced partially ordered spaces (pospaces) as a model for concurrent systems. In this paper it is shown that the category of pospaces under a fixed pospace is both a fibration and a cofibration category in the sense of H. Baues. The homotopy notion in this fibration and cofibration category is relative directed homotopy. It is also shown that the category of pospaces is a closed model category such that the homotopy notion is directed homotopy.
Algebraic topological methods have been used successfully in concurrency theory, the domain of theoretical computer science that deals with distributed computing. L. Fajstrup, E. Goubault, and M. Raussen have introduced partially ordered spaces (pospaces) as a model for concurrent systems. In this paper it is shown that the category of pospaces under a fixed pospace is both a fibration and a cofibration category in the sense of H. Baues. The homotopy notion in this fibration and cofibration category is relative directed homotopy. It is also shown that the category of pospaces is a closed model category such that the homotopy notion is directed homotopy.