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Magic numbers in the state hierarchy of finite automata. (English) Zbl 1130.68069
Summary: A number \(d\) is magic for \(n\), if there is no regular language for which an optimal nondeterministic finite state automaton (nfa) uses exactly \(n\) states and, at the same time, the optimal deterministic finite state automaton (dfa) uses exactly \(d\) states. We show that, in the case of unary regular languages, the state hierarchy of dfa’s, for the family of languages accepted by \(n\)-state nfa’s, is not contiguous. There are some “holes” in the hierarchy, i.e., magic numbers in between values that are not magic. This solves, for automata with a single letter input alphabet, an open problem of existence of magic numbers. Actually, most of the numbers is magic in the unary case. As an additional bonus, we also get a new universal lower bound for the conversion of unary \(d\)-state dfa’s into equivalent nfa’s: nondeterminism does not reduce the number of states below \(\log ^{2} d\), not even in the best case.

68Q45 Formal languages and automata
Full Text: DOI
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