Computation of the greatest simulations and bisimulations between fuzzy automata

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 efficient 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 Ignjatović et al. [On the greatest solutions to weakly linear systems of fuzzy relation inequalities and equations, Fuzzy Sets Syst. 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.