Some results on the structure of unary unambiguous automata

The paper focuses on deterministic and unambiguous finite automata (DFAʼs and UNFAʼs respectively for short) in the case of a one-letter alphabet. We present a structural characterization of unary UNFAʼs and some considerations relating minimal UNFAʼs with minimum DFAʼs 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 UNFAʼs 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 an UNFA recognizing a unary language with the same parameters in an UNFA recognizing its complement.