On a maximal NFA without mergible states

Authors:
Igor Grunsky;Oleksiy Kurganskyy;Igor Potapov
Affiliations:
Institute of Applied Mathematics and Mechanics, Ukrainian National Academy of Sciences, Donetsk, Ukraine;Institute of Applied Mathematics and Mechanics, Ukrainian National Academy of Sciences, Donetsk, Ukraine;Department of Computer Science, University of Liverpool, Liverpool, UK
Venue:
CSR'06 Proceedings of the First international computer science conference on Theory and Applications
Year:
2006

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Abstract

In this paper we answer an open question about the exact bound on the maximal number of non-mergible states in nondeterministic finite automaton (NFA). It is shown that the maximal possible number of non-mergible states in a NFA that accepts a given regular language L is not greater than 2n – 1, where n is the number of states in the minimal deterministic finite automaton that accepts L. Next we show that the bound is reachable by constructing a NFA that have exactly 2n – 1 non-mergible states. As a generalization of this result we show that the number of states in a NFA that does not contain a subset of k mergible states, where k 1, is bounded by (k – 1)(2n – 1) and the bound is reachable.