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A Note on Ordinal DFAs

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 Added by Stephen Bloom
 Publication date 2010
and research's language is English




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We prove the following theorem. Suppose that $M$ is a trim DFA on the Boolean alphabet $0,1$. The language $L(M)$ is well-ordered by the lexicographic order $slex$ 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 $slex$-descending sequence of words. This property is used to obtain a polynomial time algorithm to determine, given a DFA $M$, whether $L(M)$ is well-ordered by the lexicographic order. Last, we apply an argument in cite{BE,BEa} to give a proof that the least nonregular ordinal is $omega^omega $.



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66 - Michiel de Bondt 2018
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The purposes of this note are the following two; we first generalize Okada-Takeutis well quasi ordinal diagram theory, utilizing the recent result of Dershowitz-Tzamerets version of tree embedding theorem with gap conditions. Second, we discuss possible use of such strong ordinal notation systems for the purpose of a typical traditional termination proof method for term rewriting systems, especially for second-order (pattern-matching-based) rewriting systems including a rewrite-theoretic version of Buchholzs hydra game.
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