# Model Comparison Games for Horn Description Logics

## TL;DR

This paper introduces model comparison games for Horn description logics, providing new definability and expressive completeness results, and analyzing model indistinguishability complexity, with extensions to more expressive Horn fragments.

## Contribution

It develops model comparison games for hornALC, establishing definability, expressive completeness, and complexity results, and extends the approach to more expressive Horn fragments.

## Key findings

- Checking hornALC indistinguishability is ExpTime-complete.
- HornALC is less efficiently checkable than ALC.
- Extensions to more expressive Horn fragments are proposed.

## Abstract

Horn description logics are syntactically defined fragments of standard description logics that fall within the Horn fragment of first-order logic and for which ontology-mediated query answering is in PTime for data complexity. They were independently introduced in modal logic to capture the intersection of Horn first-order logic with modal logic. In this paper, we introduce model comparison games for the basic Horn description logic hornALC (corresponding to the basic Horn modal logic) and use them to obtain an Ehrenfeucht-Fra\"iss\'e type definability result and a van Benthem style expressive completeness result for hornALC. We also establish a finite model theory version of the latter. The Ehrenfeucht-Fra\"iss\'e type definability result is used to show that checking hornALC indistinguishability of models is ExpTime-complete, which is in sharp contrast to ALC indistinguishability (i.e., bisimulation equivalence) checkable in PTime. In addition, we explore the behavior of Horn fragments of more expressive description and modal logics by defining a Horn guarded fragment of first-order logic and introducing model comparison games for it.