Minimal determinization algorithm for fuzzy automata

The determinization of fuzzy automata is a well-studied problem in theoretical computer science celebrated for its practical applications. Indeed, in the fields of fuzzy discrete event systems, fault diagnosis, clinical monitoring, decision-making systems, and model checking, when a suitable model of a fuzzy automaton is employed, it is desirable to find its language-equivalent deterministic version because of its computational efficiency. Although many methods have been developed to convert a fuzzy automaton to its language equivalent fuzzy deterministic finite automaton (FDfA), they can be applied only for fuzzy automata defined over specific underlying sets of truth values. For example, recently developed determinization methods employ the concept of maximal factorization, which can be defined only on non-locally finite lattices or the Boolean lattice. In addition, not all such determinization methods result in a minimal FDfA. On the other hand, even though such determinization methods have been developed for fuzzy automata over specific underlying structures, these methods cannot be generalized for fuzzy automata over locally finite lattices. This article focuses on filling this gap and develops a novel method for computing a minimal FDfA for a fuzzy automaton defined over a locally finite and divisible residuated lattice. Our method uses the new concept of a reduction graph that emerges from the strict order relation on the resulting fuzzy states, according to which we can construct all minimal FDfAs equivalent to a given fuzzy automaton.