On categories associated with crisp deterministic automata with fuzzy rough outputs and fuzzy rough languages

The categorical concepts, tools, and techniques advanced not only the theory of automata and languages but also developed the theory of fuzzy automata and languages and played a significant role in many other branches of theoretical computer science. Unlike theory of classical/fuzzy automata with output and their languages, this paper aims to incorporate the concept of fuzziness and roughness together in the theory of automata with output and their languages, to introduce the concepts of crisp deterministic automata with fuzzy rough output, and fuzzy rough language and study their categorical aspects. We have shown that the crisp deterministic automata with fuzzy rough outputs and fuzzy rough languages, along with their morphisms, respectively, form categories CDAFRO and FRL. We have also shown the existence of a functor between the categories CDAFRO and FRL. The categorical concepts of product and coproduct of a subfamily of object-class, equalizer, and coequalizer of a pair of parallel morphisms in category CDAFRO are studied. Finally, for a given fuzzy rough language, a minimal crisp deterministic automaton having fuzzy rough output recognizing it is constructed using the fuzzy generalization of foremost Myhill–Nerode’s theory of automata and languages.