Bisimulations for fuzzy automata

TL;DR

This paper explores the use of bisimulations and uniform fuzzy relations to determine equivalence between fuzzy automata, reducing the problem to fuzzy automata isomorphism testing, which relates to the graph isomorphism problem.

Contribution

It establishes a novel characterization of UFB-equivalence of fuzzy automata via isomorphism of their factor automata, linking bisimulation testing to automata isomorphism.

Findings

UFB-equivalence reduces to fuzzy automata isomorphism testing.

Provides conditions for forward bisimulation via kernels and co-kernels.

Includes a comprehensive overview of bisimulation concepts across automata types.

Abstract

Bisimulations have been widely used in many areas of computer science to model equivalence between various systems, and to reduce the number of states of these systems, whereas uniform fuzzy relations have recently been introduced as a means to model the fuzzy equivalence between elements of two possible different sets. Here we use the conjunction of these two concepts as a powerful tool in the study of equivalence between fuzzy automata. We prove that a uniform fuzzy relation between fuzzy automata $\cal A$ and $\cal B$ is a forward bisimulation if and only if its kernel and co-kernel are forward bisimulation fuzzy equivalences on $\cal A$ and $\cal B$ and there is a special isomorphism between factor fuzzy automata with respect to these fuzzy equivalences. As a consequence we get that fuzzy automata $\cal A$ and $\cal B$ are UFB-equivalent, i.e., there is a uniform forward bisimulation between them, if and only if there is a special isomorphism between the factor fuzzy automata of $\cal A$ and $\cal B$ with respect to their greatest forward bisimulation fuzzy equivalences. This result reduces the problem of testing UFB-equivalence to the problem of testing isomorphism of fuzzy automata, which is closely related to the well-known graph isomorphism problem. We prove some similar results for backward-forward bisimulations, and we point to fundamental differences. Because of the duality with the studied concepts, backward and forward-backward bisimulations are not considered separately. Finally, we give a comprehensive overview of various concepts on deterministic, nondeterministic, fuzzy, and weighted automata, which are related to bisimulations.