On characterizations of general type-2 fuzzy grammar and its languages by general type-2 fuzzy finite automata

The motivation of this work is to introduce and study the concept of general type-2 fuzzy grammars and their correspondence with general type-2 fuzzy automata. In particular, we establish the formal equivalence between general type-2 fuzzy weak regular grammars and general type-2 fuzzy regular grammars, showing that both generate the same class of general type-2 fuzzy languages. Furthermore, it is proved that every general type-2 fuzzy regular grammar admits an equivalent general type-2 fuzzy automaton, and conversely, every general type-2 fuzzy automaton can be represented by an equivalent general type-2 fuzzy regular grammar. We also investigate the closure properties of the class of general type-2 fuzzy languages recognized by general type-2 fuzzy automata and show that it is closed under union, intersection, concatenation, and Kleene closure, whereas it is not closed under complementation. Finally, through a suitable example, we demonstrate that a context-free general type-2 fuzzy grammar may generate context-sensitive general type-2 fuzzy languages. These results provide a formal algebraic and automata-theoretic foundation for the further development of general type-2 fuzzy language theory.