Directable Fuzzy and Nondeterministic Automata

TL;DR

This paper explores three types of directability in fuzzy automata, establishing their decidability, and analyzing their closure properties, by extending concepts from nondeterministic automata theory.

Contribution

It introduces and studies three notions of directability in fuzzy automata, proving their decidability and analyzing their closure properties, extending nondeterministic automata theory.

Findings

Directability properties are decidable for fuzzy automata.

Closure properties under subautomata, homomorphic images, and products are established.

Results are derived from the theory of nondeterministic automata.

Abstract

We study three notions of directability of fuzzy automata akin to the D1-, D2- and D3-directability of nondeterministic automata. Thus an input word $w$ of a fuzzy automaton is D1-directing if a fixed single state is reachable by $w$ from all states, D2-directing if exactly the same states are reachable by $w$ from every state, and D3-directing if there is a state reachable by $w$ from every state. We study the various sets of directing words of fuzzy automata, prove that the directability properties are decidable, and show how such results can be deduced from the theory of directable nondeterministic automata. Moreover, we establish the closure properties of the different classes of directable fuzzy automata under the class operations of forming subautomata, homomorphic images and finite direct products.