Automata theory based on complete residuated lattice-valued logic: Reduction and minimization

Automata theory based on complete residuated lattice-valued logic, called L-valued finite automata (L-VFAs), has been established by Qiu recently. In this paper, we define a kind of Mealy type of L-VFAs (MLFAs), a generalization of L-VFAs. Two kinds of statewise equivalence relations are introduced, and a minimal form is defined. We study the existence of the minimal form of an MLFA. Then, we show that any two states can be distinguished by some word with finite length. Also, a minimization algorithm of the MLFAs is presented. In addition, we obtain a minimization algorithm for L-VFAs as well. Finally, we define L-valued languages (L-VLs) and L-valued regular languages (L-VRLs) recognized by L-VFAs, and provide some properties of L-VRLs.