Weak bisimulations for fuzzy automata

Forward and backward bisimulations have been introduced recently by Ćirić et al. (2012) [9] as a means for modeling the equivalence between states of fuzzy automata and approximating the language-equivalence, as well as for use in the state reduction of fuzzy automata. The main aim of the present paper is to introduce two new kinds of bisimulations, weak forward and backward bisimulations, which provide better approximations of the language-equivalence than forward and backward bisimulations, and when employed in the state reduction, they provide better reductions. We give procedures for deciding whether there exist weak forward and backward simulations and bisimulations, and for computing the greatest ones, whenever they exist. Using weak bisimulations in conjunction with the concept of a uniform fuzzy relation, we determine necessary and sufficient conditions under which two fuzzy automata are weak bisimulation equivalent. We also characterize uniform weak backward and forward bisimulations between two fuzzy automata in terms of isomorphisms between their Nerode and reverse Nerode automata.