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 $I_{\epsilon }$, for any $\epsilon \in (0,1)$. 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 $A_{\epsilon }$ over $I_{\epsilon }$, 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 $A_{\epsilon }$ and $B_{\epsilon }$. Those problems concerning the automata $A_{\epsilon }$ and $B_{\epsilon }$ are easier to solve, due to the local finiteness of the semiring reduct of $I_{\epsilon }$, 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 $A_{\epsilon }$ and $B_{\epsilon }$ 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.