, Volume 30, Issue 1, pp 151–164 | Cite as

A Note on Ordinal DFAs

  • Stephen L. Bloom
  • Yi Di Zhang


We prove the following theorem. Suppose that M is a trim DFA. Then \(\mathcal{L}(M)\) is well-ordered by the lexicographic order <  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 < -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 .


Regular ordinal Lexicographic order Deterministic finite automaton 


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Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Department of Computer ScienceStevens Institute of TechnologyHobokenUSA
  2. 2.Department of MathematicsStevens Institute of TechnologyHobokenUSA

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