Existential Definability over the Subword Ordering
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We study first-order logic (FO) over the structure consisting of finite words over some alphabet A, together with the (non-contiguous) subword ordering. In terms of decidability of quantifier alternation fragments, this logic is well-understood: If every word is available as a constant, then even the Σ_1 (i.e., existential) fragment is undecidable, already for binary alphabets A. However, up to now, little is known about the expressiveness of the quantifier alternation fragments: For example, the undecidability proof for the existential fragment relies on Diophantine equations and only shows that recursively enumerable languages over a singleton alphabet (and some auxiliary predicates) are definable. We show that if |A|≥ 3, then a relation is definable in the existential fragment over A with constants if and only if it is recursively enumerable. This implies characterizations for all fragments Σ_i: If |A|≥ 3, then a relation is definable in Σ_i if and only if it belongs to the i-th level of the arithmetical hierarchy. In addition, our result yields an analogous complete description of the Σ_i-fragments for i≥ 2 of the pure logic, where the words of A^* are not available as constants.
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