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

arXiv: https://arxiv.org/abs/1511.01807 · HTML: https://mlsanity.com/paper/1511.01807/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Nov 2015.

## Built on by (most cited first, 4 of 4)

- [Scattered Factor Universality -- The Power of the Remainder](https://mlsanity.com/paper/2104.09063/) (2021)
- [m-Nearly k-Universal Words -- Investigating Simon Congruence](https://mlsanity.com/paper/2202.07981/) (2022)
- [α-β-Factorization and the Binary Case of Simon's Congruence](https://mlsanity.com/paper/2306.14192/) (2023)
- [Directed Regular and Context-Free Languages](https://mlsanity.com/paper/2401.07106/) (2024)

## Abstract (arXiv)

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 mathsfFO2(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.