Tight bounds on the number of states of DFAs that are equivalent to n-state NFAs
Theoretical Computer Science
A family of NFAs which need 2n - α deterministic states
Theoretical Computer Science
On the state complexity of reversals of regular languages
Theoretical Computer Science
Introduction to Automata Theory, Languages, and Computation (3rd Edition)
Introduction to Automata Theory, Languages, and Computation (3rd Edition)
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Quite recently, it has been shown that, for each n, and each d between n and 2n, there exists a regular language for which each optimal nondeterministic one-way finite state automaton (nfa) uses exactly n states, but its optimal deterministic counterpart (dfa) exactly d states. This gives the complete state hierarchy for the relation between nfa's and dfa's. However, in literature, either the size of the input alphabet for these automata is very large, namely, 2n-1+1, or the argument is "non-constructive," proving the mere existence without an explicit exhibition of the witness language. We shall give a simpler "constructive" proof for this state hierarchy, displaying explicitly the witness automata and, at the same time, reduce the input alphabet size. That is, we shall present a construction of an optimal nfa with n states, and with the input alphabet size bounded by n+2, for which the equivalent optimal dfa uses exactly d states, for each given n and d satisfying n≤d≤2n.