A generalized fuzzy automaton based on regular measures over tribes of fuzzy sets

This paper introduces a generalized framework for fuzzy automata whose state and transition structures are defined in terms of tribes of fuzzy subsets evaluated under regular measures. Based on the foundational results of Navara and Pták on regular measures over tribes, we develop a formal semantics for fuzzy computation by integrating measure theory with automata theory. In this model, language recognition is redefined through μ -recognizability, which quantifies the acceptance degree of input strings by applying a regular measure to fuzzy state distributions.

We investigate the main theoretical properties of the proposed automata, including closure under union and intersection, equivalence under varying regular measures, and the effect of homomorphisms on language recognition. We also introduce a minimization technique based on behavioral equivalence with respect to a given regular measure. Furthermore, under the stated closure assumptions on the tribe and the underlying t-norm, a Kleene-type representation theorem is proved, connecting μ -recognizable languages with regular expressions defined over tribes.

The results establish a robust and extensible framework that unifies fuzzy logic, automata theory, and abstract measure theory, and provides a basis for future developments in uncertainty modeling and soft computing.