Approximate weak simulations and bisimulations for fuzzy automata over the product structure.

In this article, we modify the product structure and turn it into the so-called ε -truncated product structure I ε , for any ε ∈ (0 , 1). The new structure keeps the property of being a complete residuated lattice, and additionally, its semiring reduct is locally finite. We convert each fuzzy automaton A over the product structure into a fuzzy automaton A ε over I ε , and accordingly, we turn the problems of testing the existence and computing weak simulations and bisimulations between fuzzy automata A and B over the product structure into the corresponding problems for the automata A ε and B ε. Those problems concerning the automata A ε and B ε are easier to solve, due to the local finiteness of the semiring reduct of I ε , and can be solved even in cases where the corresponding problems for the automata A and B cannot be solved. We show that weak simulations and bisimulations between the automata A ε and B ε determine certain kinds of approximate weak simulations and bisimulations between the original automata A and B , which we call ε -weak simulations and bisimulations. We also prove that the existence of an ε -weak simulation or bisimulation between automata A and B witnesses the existence of a certain kind of approximate inclusion or equivalence, where the deviation measure of the fuzzy languages of those automata from language inclusion or equality does not exceed ε. [ABSTRACT FROM AUTHOR]