# Rewritability in Monadic Disjunctive Datalog, MMSNP, and Expressive Description Logics

## TL;DR

This paper investigates the decidability and complexity of rewriting problems for monadic disjunctive Datalog, MMSNP, and description logic-based queries, providing new constructions and complexity bounds.

## Contribution

It establishes decidability and complexity results for rewritability into FO, Datalog, and MDLog, and introduces a new canonical Datalog construction applicable to formulas with free variables.

## Key findings

- Rewritability into FO and monadic Datalog is decidable.
- Rewritability into Datalog is decidable under certain equality conditions.
- Complexity is 2NExpTime-complete for most cases, with some gaps remaining.

## Abstract

We study rewritability of monadic disjunctive Datalog programs, (the complements of) MMSNP sentences, and ontology-mediated queries (OMQs) based on expressive description logics of the ALC family and on conjunctive queries. We show that rewritability into FO and into monadic Datalog (MDLog) are decidable, and that rewritability into Datalog is decidable when the original query satisfies a certain condition related to equality. We establish 2NExpTime-completeness for all studied problems except rewritability into MDLog for which there remains a gap between 2NExpTime and 3ExpTime. We also analyze the shape of rewritings, which in the MMSNP case correspond to obstructions, and give a new construction of canonical Datalog programs that is more elementary than existing ones and also applies to formulas with free variables.