The height of piecewise-testable languages and the complexity of the logic of subwords

TL;DR

This paper introduces new methods to measure the complexity of piecewise-testable languages through their height and explores the expressive power and complexity of a fragment of first-order logic over subword structures.

Contribution

It develops novel techniques for bounding the height of languages and applies these to analyze the logical expressiveness and complexity of the two-variable fragment of first-order logic with subword ordering.

Findings

Bounding techniques for language height established

FO^2 logic over subwords only expresses piecewise-testable properties

The logic has elementary complexity

Abstract

The height of a piecewise-testable language $L$ is the maximum length of the words needed to define $L$ by excluding and requiring given subwords. The height of $L$ is an important descriptive complexity measure that has not yet been investigated in a systematic way. This article develops a series of new techniques for bounding the height of finite languages and of languages obtained by taking closures by subwords, superwords and related operations. As an application of these results, we show that $\mathsf{FO}^2(A^*,\sqsubseteq)$, the two-variable fragment of the first-order logic of sequences with the subword ordering, can only express piecewise-testable properties and has elementary complexity.