MK-fuzzy Automata and MSO Logics
Manfred Droste (University of Leipzig), Temur Kutsia (RISC), George, Rahonis (Aristotle University of Thessaloniki), Wolfgang Schreiner (RISC)

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.
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