Nondeterministic fuzzy automata with membership values in complete residuated lattices

Automata theory based on complete residuated lattice-valued logic has been initiated by Qiu, and further systematically established by Qiu et al. In this paper, we propose the notion of nondeterministic fuzzy automata with membership values in a complete residuated lattice L, called lattice-valued nondeterministic fuzzy automata (L-NFAs). In our setting, a state of an L-NFA may have more than one transition labeled by the same input symbol, which reflects nondeterminism. To compare the behaviors of L-NFAs, we introduce two language equivalence relations which have different discriminating power. Furthermore, we extend the two relations to the complete residuated lattice-valued setting and investigate their properties such as robustness and compositionality. The theory developed here is applicable to the quantitative modeling and verification of fuzzy systems.