هم‌ارزی رفتاری بین اتوماتاهای فازی عمومی: رویکردی مبتنی بر تابعگون‌ها و جبرهای BL

In this paper, a categorical framework is proposed for modeling and analyzing the behavior of general fuzzy automata based on BL-algebras. First, a category of BL general fuzzy automata is defined, in which morphisms act as fuzzy mappings between fuzzy state sets and reconstruct the initial, transition, and output structures in a compositional manner. Then, functors from the category of nondeterministic automata to this category are introduced, which model the fuzzy behavior of systems via fuzzy transition and output mappings. To compare the behavior of these functors, the notion of behavioral equivalence is defined based on a residuated operator in the BL lattice, and a set of fuzzy equivalent mappings between functors is introduced. It is proven that this set, if nonempty, contains a maximal element that reconstructs behavioral structures in a fuzzy manner. Furthermore, based on the concept of bisimulation between nondeterministic automata, it is shown that this relation is transferred under the behavioral functor to a fuzzy bisimulation between BL general fuzzy automata. Based on this transfer, it is proven that the existence of a bisimulation between nondeterministic automata guarantees behavioral equivalence between the resulting BL general fuzzy automata.