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Non-deterministic and fuzzy automata in toposes. (English) Zbl 0920.18001
We consider here some topos-theoretic aspects of categorical automata theory. First, we observe that the equivalence of non-deterministic dynamics with categories enriched in a suitable monoidal category (built from the input monoid) [R. Betti, Boll. Unione Mat. Ital., V. Ser., B 17, 44-58 (1980; Zbl 0456.18003)] can be extended to automata internal to an elementary topos. Next, after introducing one way to view fuzzy sets as sheaves, and hence as objects in a Grothendieck topos, following M. Barr [Can. Math. Bull. 29, 501-508 (1986; Zbl 0563.03040)], we show that the universal minimal realization theory of J. A. Goguen [Int. J. Man-Mach. Stud. 6, 513-561 (1974; Zbl 0321.68055); Math. Syst. Theory 6(1972), 359-374 (1973; Zbl 0248.18015); Bull. Am. Math. Soc. 78, 777-783 (1972; Zbl 0277.18003)] can be internalized, thus providing a minimal realization for fuzzy automata.
18B20 Categories of machines, automata
68Q45 Formal languages and automata
68Q70 Algebraic theory of languages and automata
18B25 Topoi
03G30 Categorical logic, topoi
03E72 Theory of fuzzy sets, etc.