An optimal lower bound for nonregular languages
Information Processing Letters
Tight bounds on the number of states of DFAs that are equivalent to n-state NFAs
Theoretical Computer Science
Introduction to Automata Theory, Languages and Computability
Introduction to Automata Theory, Languages and Computability
Optimal Simulations between Unary Automata
SIAM Journal on Computing
Note on Minimal Finite Automata
MFCS '01 Proceedings of the 26th International Symposium on Mathematical Foundations of Computer Science
Converting two-way nondeterministic unary automata into simpler automata
Theoretical Computer Science - Mathematical foundations of computer science
Nondeterminism and the size of two way finite automata
STOC '78 Proceedings of the tenth annual ACM symposium on Theory of computing
A family of NFAs which need 2n - α deterministic states
Theoretical Computer Science
Errata to: "finite automata and unary languages"
Theoretical Computer Science
Finite automata and their decision problems
IBM Journal of Research and Development
On the State Complexity of Complements, Stars, and Reversals of Regular Languages
DLT '08 Proceedings of the 12th international conference on Developments in Language Theory
Deterministic blow-ups of minimal nondeterministic finite automata over a fixed alphabet
DLT'07 Proceedings of the 11th international conference on Developments in language theory
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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 [7]. 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.