Algorithms for computing the greatest simulations and bisimulations between fuzzy automata

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.