Abstract
Iwama et al [1] showed that there exists an n-state binary nondeterministic finite automaton such that its equivalent minimal deterministic finite automaton has exactly 2n– α states, for all n ≥ 7 and 5 ≤ α ≤ 2n – 2, subject to certain coprimality conditions. We investigate the same question for both unary and binary symmetric difference nondeterministic finite automata [2]. In the binary case, we show that for any n ≥ 4, there is an n-state ⊕-NFA which needs 2n − − 1 + 2k − − 1 –1 states, for 2< k ≤ n – 1. In the unary case, we prove the following result for a large practical subclass of unary symmetric difference nondeterministic finite automata: For all n ≥ 2, we show that there are many values of α such that there is no n-state unary symmetric difference nondeterministic finite automaton with an equivalent deterministic finite automaton with 2n – α states, where 0 < α< 2n − 1. For each n ≥ 2, we quantify such values of α precisely.
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References
Iwama, K., Matsuura, A., Paterson, M.: A Family of NFAs which need 2n − α Deterministic States. Theoretical Computer Science 301, 451–462 (2003)
Van Zijl, L.: Random Number Generation with Symmetric Difference NFAs. In: Watson, B.W., Wood, D. (eds.) CIAA 2001. LNCS, vol. 2494, pp. 263–273. Springer, Heidelberg (2003)
Van Zijl, L.: Nondeterminism and Succinctly Representable Regular Languages. In: Proceedings of SAICSIT 2002, ACM International Conference Proceedings Series, pp. 212–223 (2002)
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van Zijl, L. (2005). Magic Numbers for Symmetric Difference NFAs. In: Domaratzki, M., Okhotin, A., Salomaa, K., Yu, S. (eds) Implementation and Application of Automata. CIAA 2004. Lecture Notes in Computer Science, vol 3317. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-30500-2_41
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DOI: https://doi.org/10.1007/978-3-540-30500-2_41
Publisher Name: Springer, Berlin, Heidelberg
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