Fuzzy alternating $\mathrm{B\ddot{u}chi}$ automata over distributive lattices

TL;DR

This paper introduces a new model of fuzzy alternating Büchi automata over distributive lattices with leaf-based weights, enabling easier complementation and showing equivalence with nondeterministic versions, along with closure properties and decision problem analysis.

Contribution

It proposes a novel leaf-weighted fuzzy alternating Büchi automata model, proves their expressive equivalence with nondeterministic automata, and explores their closure properties and decision problems.

Findings

Equivalent expressive power between fuzzy nondeterministic and alternating Büchi automata.

Languages recognized by fuzzy alternating co-Büchi automata are fuzzy ω-regular.

Closure properties and decision problems are established for these automata.

Abstract

We give a new version of fuzzy alternating $\mathrm{B\ddot{u}chi}$ automata over distributive lattices: weights are putting in every leaf node of run trees rather than along with edges from every node to its children. Such settings are great benefit to obtain complement just by taking dual operation and replacing each final weight with its complement. We prove that $L$-fuzzy nondeterministic $\mathrm{B\ddot{u}chi}$ automata have the same expressive power as $L$-fuzzy alternating $\mathrm{B\ddot{u}chi}$ ones. A direct construction (without related knowledge about $L$-fuzzy nondeterministic $\mathrm{B\ddot{u}chi}$ ones such as: above equivalence relation and their closure properties) is given to show that the languages recognized by $L$-fuzzy alternating co-$\mathrm{B\ddot{u}chi}$ automata are also $L$-fuzzy $\omega$-regular. Furthermore, the closure properties and the discussion about decision problems for fuzzy alternating $\mathrm{B\ddot{u}chi}$ automata are illustrated in our paper.