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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.

##### MSC:
 68Q45 Formal languages and automata
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##### References:
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