Abstract
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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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