Fuzzy finite automata and fuzzy regular expressions with membership values in lattice-ordered monoids

Abstract

We study fuzzy finite automata in which all fuzzy sets are defined by membership functions whose codomain forms a lattice-ordered monoid L . For these L -fuzzy finite automata ( L -FFA, for short), we provide necessary and sufficient conditions for the extendability of the state-transition function. It is shown that nondeterministic L -FFA (N L -FFA, for short) are more powerful than deterministic L -FFA (D L -FFA, for short). Then, we give necessary and sufficient conditions for the simulation of an N L -FFA by an equivalent D L -FFA. Next, we turn to the closure properties of languages defined by L -FFAs: we establish closure under the regular operations and provide conditions for closure under intersection and reversal, in particular we show that the family of fuzzy languages accepted by D L -FFAs is not closed under Kleene closure operation, and the family of fuzzy languages accepted by N L -FFAs is not closed under complement operation. Furthermore, we introduce the notions of L -fuzzy regular expressions and give the Kleene theorem for N L -FFAs. The description of D L -FFAs by L -fuzzy regular expressions is also given. Finally, we investigate the level structures of L -FFAs. Our results provide some insight as to what extend properties of L -FFAs and their languages depend on the algebraic properties of L .