# Languages ordered by the subword order

## TL;DR

This paper investigates the decidability of logical theories over languages with subword and cover relations, extending previous results by including regular predicates and counting quantifiers.

## Contribution

It introduces four new decidable logical theories for languages with subword, cover, and regular predicates, expanding the scope of earlier decidability results.

## Key findings

- Four new decidable logical theories identified
- Decidability depends on language, predicates, and logic fragment
- Extends previous work by including cover relation and counting quantifiers

## Abstract

We consider a language together with the subword relation, the cover relation, and regular predicates. For such structures, we consider the extension of first-order logic by threshold- and modulo-counting quantifiers. Depending on the language, the used predicates, and the fragment of the logic, we determine four new combinations that yield decidable theories. These results extend earlier ones where only the language of all words without the cover relation and fragments of first-order logic were considered.