Kleen’s Theorem for 𝐵𝐿-general L-fuzzy automata

The contribution of generl fuzzy automata to neural networks has been considerable, and dynamical fuzzy systems are becoming more and more popular and useful. Basic logic, or 𝐵𝐿 for short, has been introduced by Hájek [5] in order to provide a general framework for formalizing statements of fuzzy nature. In this note, some of the closure properties of the 𝐵𝐿-general fuzzy automaton based on lattice valued such as union, intersection, connection and a serial connection are considered, after that, the behavior of them are discussed. Moreover, for a given 𝐵𝐿-general fuzzy automaton on the basis of lattice valued, a complete 𝐵𝐿-general fuzzy automaton on the basis of lattice valued is presented. Afterward, we may test the Pumping Lemma for the 𝐵𝐿-general fuzzy automaton based on lattice valued. In particular, a connection between the behavior of 𝐵𝐿-general fuzzy automaton based on lattice valued and its language is presented. Also, it is proven that L is a recognizable set if and only if L is rational. Also, it is driven that Kleen’s Theorem is valid for the 𝐵𝐿-general fuzzy automaton on the basis of lattice valued. Finally, we give some examples to clarify these notions.