# Decidability of Quasi-Dense Modal Logics

From International Center for Computational Logic

# Decidability of Quasi-Dense Modal Logics

##### Piotr Ostropolski-NalewajaPiotr Ostropolski-Nalewaja, Tim LyonTim Lyon

Piotr Ostropolski-Nalewaja, Tim Lyon

**Decidability of Quasi-Dense Modal Logics***Proceedings of the 39th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS 2024)*, 2024. ACM**Kurzfassung****Abstract**

The decidability of axiomatic extensions of the modal logic K with modal reduction principles, i.e. axioms of the form ♢ⁱp → ♢ⁿp, has remained a long-standing open problem. In this paper, we make significant progress toward solving this problem and show that decidability holds for a large subclass of these logics, namely, for 'quasi-dense logics.' Such logics are extensions of K with with modal reduction axioms such that 0 < i < n (dubbed 'quasi-density axioms'). To prove decidability, we define novel proof systems for quasi-dense logics consisting of disjunctive existential rules, which are first-order formulae typically used to specify ontologies in the context of database theory. We show that such proof systems can be used to generate proofs and models of modal formulae, and provide an intricate model-theoretic argument showing that such generated models can be encoded as finite objects called 'templates.' By enumerating templates of bound size, we obtain an EXPSPACE decision procedure as a consequence.**Weitere Informationen unter:****Further Information:**Link**Projekt:****Project:**DeciGUT**Forschungsgruppe:****Research Group:**Computational LogicComputational Logic

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@inproceedings{OL2024,
author = {Piotr Ostropolski-Nalewaja and Tim Lyon},
title = {Decidability of Quasi-Dense Modal Logics},
booktitle = {Proceedings of the 39th Annual {ACM/IEEE} Symposium on Logic in
Computer Science (LICS 2024)},
publisher = {ACM},
year = {2024}
}
```