Abstract
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.
Highlights
1.1 Higher-dimensional automataA higher-dimensional automaton (HDA) is an automaton with a supplementary structure consisting of two- and higher-dimensional cubes linking its states and transitions
As in the case of the trace language and the fundamental monoid, we show that the homology language is compatible with cubical dimaps and establish formulas to compute it for tensor products and coproducts
This paper introduced weak equivalence, a coarse notion of equivalence for higher-dimensional automata
Summary
We present basic material on precubical sets, concurrent alphabets, and higher-dimensional automata. The definition of higher-dimensional automata is essentially the one of van Glabbeek [Gla06], with the difference that we consider HDAs over concurrent alphabets and allow labels to be words. It is not our intention to provide a comprehensive introduction to the material of this section. Explanations, and examples, the reader is referred to, e.g., [Die, Maz, FGH+16, Gla06]
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