Algorithms for computing the greatest simulations and bisimulations between fuzzy automata

TL;DR

This paper introduces effective algorithms for determining and computing the greatest simulations and bisimulations between fuzzy automata, extending the theoretical framework with practical computational methods.

Contribution

It provides the first algorithms for deciding and computing the greatest simulations and bisimulations of various types between fuzzy automata.

Findings

Algorithms for all types of simulations and bisimulations are developed.

The algorithms are based on computing the greatest post-fixed point in fuzzy relation lattices.

The methods are effective and applicable to fuzzy automata analysis.

Abstract

Recently, two types of simulations (forward and backward simulations) and four types of bisimulations (forward, backward, forward-backward, and backward-forward bisimulations) between fuzzy automata have been introduced. If there is at least one simulation/bisimulation of some of these types between the given fuzzy automata, it has been proved that there is the greatest simulation/bisimulation of this kind. In the present paper, for any of the above-mentioned types of simulations/bisimulations we provide an effective algorithm for deciding whether there is a simulation/bisimulation of this type between the given fuzzy automata, and for computing the greatest one, whenever it exists. The algorithms are based on the method developed in [J. Ignjatovi\'c, M. \'Ciri\'c, S. Bogdanovi\'c, On the greatest solutions to certain systems of fuzzy relation inequalities and equations, Fuzzy Sets and Systems 161 (2010) 3081-3113], which comes down to the computing of the greatest post-fixed point, contained in a given fuzzy relation, of an isotone function on the lattice of fuzzy relations.