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Some results on the structure of unary unambiguous automata. (English) Zbl 1223.68062
Summary: The paper focuses on deterministic and unambiguous finite automata (DFAs and UNFAs, respectively, for short) in the case of a one-letter alphabet. We present a structural characterization of unary UNFAs and some considerations relating minimal UNFAs with minimum DFAs recognizing a given unary language. We also present an algorithm for the construction of a minimal UNFA for a unary regular language. Then we establish a correspondence between pairs of UNFAs recognizing a unary language and its complement respectively, and the disjoint covering systems of number theory. It allows us to provide some conditions relating the number of successful simple paths and the lengths of cycles in a UNFA recognizing a unary language with the same parameters in an UNFA recognizing its complement.

MSC:
68Q45 Formal languages and automata
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