Geffert V. (2006) Magic Numbers in the State Hierarchy of Finite Automata. In: Královič R., Urzyczyn P. (eds) Mathematical Foundations of Computer Science 2006. MFCS 2006. Lecture Notes in Computer Science, vol 4162. Springer, Berlin, Heidelberg
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, but for which 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 .
As an additional bonus, we get a 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 log2d, not even in the best case.