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arXiv · 1701.07470

Decidability, Complexity, and Expressiveness of First-Order Logic Over the Subword Ordering

Simon Halfon·Philippe Schnoebelen·Georg Zetzsche
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Abstract

We consider first-order logic over the subword ordering on finite words, where each word is available as a constant. Our first result is that the $Σ_1$ theory is undecidable (already over two letters). We investigate the decidability border by considering fragments where all but a certain number of variables are alternation bounded, meaning that the variable must always be quantified over languages with a bounded number of letter alternations. We prove that when at most two variables are not alternation bounded, the $Σ_1$ fragment is decidable, and that it becomes undecidable when three variables are not alternation bounded. Regarding higher quantifier alternation depths, we prove that the $Σ_2$ fragment is undecidable already for one variable without alternation bound and that when all variables are alternation bounded, the entire first-order theory is decidable.

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Simon Halfon, Philippe Schnoebelen, Georg Zetzsche. 2021-09-24. Decidability, Complexity, and Expressiveness of First-Order Logic Over the Subword Ordering. https://arxiv.org/abs/1701.07470

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