Reduction graph for minimal determinization of fuzzy automata

We introduce a minimal determinization procedure for fuzzy finite automata (FfAs) with membership values in a complete residuated lattice (CRL). The method is based on the well-known determinization method via factorization of fuzzy states. However, different to other determinization methods, we do not assume that the CRL is zero divisors free. This fact requires modifying the functions that define the factorization to avoid the zero divisor values when creating the fuzzy states in the determinization procedure. After generating a right-irreducible fuzzy deterministic finite automaton (FDfA) equivalent to the original FfA by determinization via factorization, we construct the so-called reduction graph of this fuzzy automaton, where each arc represents the notion that a fuzzy state is left-reducible by another fuzzy state. By making these left-reductions, we obtain the equivalent minimal FDfA. It is worth mentioning that an empty fuzzy state is always reducible by a nonempty fuzzy state. This behavior, specific for a CRL with zero divisors, has also to be taken into account when the state reduction is carried out.