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A Note on Ordinal DFAs

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We prove the following theorem. Suppose that M is a trim DFA. Then \(\mathcal{L}(M)\) is well-ordered by the lexicographic order < ℓ iff whenever the non sink states q, q.0 are in the same strong component, then q.1 is a sink. It is easy to see that this property is sufficient. In order to show the necessity, we analyze the behavior of a < ℓ-descending sequence of words. This property is used to obtain a polynomial time algorithm to determine, given a DFA M, whether \(\mathcal{L}(M)\) is well-ordered by the lexicographic order. Last, we apply an argument in Bloom and Ésik (Fundam Inform 99:383–407, 2010, Int J Found Comput Sci, 2011) to give a proof that the least nonregular ordinal is ω ω.

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Authors and Affiliations

  1. Department of Computer Science, Stevens Institute of Technology, Hoboken, NJ, USA

    Stephen L. Bloom

  2. Department of Mathematics, Stevens Institute of Technology, Hoboken, NJ, USA

    Yi Di Zhang

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  1. Stephen L. Bloom
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  2. Yi Di Zhang
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Correspondence to Yi Di Zhang.

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Bloom, S.L., Zhang, Y.D. A Note on Ordinal DFAs. Order 30, 151–164 (2013). https://doi.org/10.1007/s11083-011-9233-1

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  • DOI: https://doi.org/10.1007/s11083-011-9233-1

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