Undecidability of the unification and admissibility problems for modal and description logics

TL;DR

This paper proves that the unification and admissibility problems are undecidable for certain basic modal and description logics, including K, K4, and hybrid logics, highlighting fundamental limits of automated reasoning in these systems.

Contribution

It demonstrates the undecidability of unification and admissibility in standard modal logics extended with universal modalities and nominals, a novel result in the field.

Findings

Undecidability of unification in K and K4 with universal modality.

Undecidability of unification in hybrid logics with nominals.

Implications for description logics like ALCO, SHIQO, SHI, and SHIQ.

Abstract

We show that the unification problem `is there a substitution instance of a given formula that is provable in a given logic?' is undecidable for basic modal logics K and K4 extended with the universal modality. It follows that the admissibility problem for inference rules is undecidable for these logics as well. These are the first examples of standard decidable modal logics for which the unification and admissibility problems are undecidable. We also prove undecidability of the unification and admissibility problems for K and K4 with at least two modal operators and nominals (instead of the universal modality), thereby showing that these problems are undecidable for basic hybrid logics. Recently, unification has been introduced as an important reasoning service for description logics. The undecidability proof for K with nominals can be used to show the undecidability of unification for boolean description logics with nominals (such as ALCO and SHIQO). The undecidability proof for K with the universal modality can be used to show that the unification problem relative to role boxes is undecidable for Boolean description logic with transitive roles, inverse roles, and role hierarchies (such as SHI and SHIQ).