Existential Definability over the Subword Ordering

TL;DR

This paper characterizes the expressiveness of first-order logic over finite words with subword ordering, showing that for alphabets of size at least three, definable relations correspond exactly to those in the arithmetical hierarchy.

Contribution

It provides a complete characterization of the quantifier alternation fragments of FO logic over subword ordering for alphabets of size three or more, linking definability to the arithmetical hierarchy.

Findings

For |A| ≥ 3, existential fragment defines exactly recursively enumerable relations.

All Σ_i fragments correspond to the i-th level of the arithmetical hierarchy.

Complete description of Σ_i fragments for i ≥ 2 in the pure logic without constants.

Abstract

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 $\Sigma_1$ (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 $|A|\ge 3$, 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 $\Sigma_i$: If $|A|\ge 3$, then a relation is definable in $\Sigma_i$ 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 $\Sigma_i$-fragments for $i\ge 2$ of the pure logic, where the words of $A^*$ are not available as constants.