MK-fuzzy Automata and MSO Logics

TL;DR

This paper introduces MK-fuzzy automata based on a bimonoid structure, explores their closure properties, and establishes an expressive equivalence with a fragment of MK-fuzzy MSO logic, advancing the theoretical foundation for fuzzy automata and logic.

Contribution

It presents the first formal definition of MK-fuzzy automata, analyzes their closure properties, and proves an expressive equivalence with a fragment of MK-fuzzy MSO logic.

Findings

Closure properties of MK-fuzzy automata are established.

A Nivat-like decomposition theorem is proved for MK-fuzzy languages.

Expressive equivalence between MK-fuzzy automata and a fragment of MK-fuzzy MSO logic is shown.

Abstract

We introduce MK-fuzzy automata over a bimonoid K which is related to the fuzzification of the McCarthy-Kleene logic. Our automata are inspired by, and intend to contribute to, practical applications being in development in a project on runtime network monitoring based on predicate logic. We investigate closure properties of the class of recognizable MK-fuzzy languages accepted by MK-fuzzy automata as well as of deterministically recognizable MK-fuzzy languages accepted by their deterministic counterparts. Moreover, we establish a Nivat-like result for recognizable MK-fuzzy languages. We introduce an MK-fuzzy MSO logic and show the expressive equivalence of a fragment of this logic with MK-fuzzy automata, i.e., a B\"uchi type theorem.