### Abstract

We study the ﬁnitary version of the coalgebraic logic introduced by L. Moss.

The syntax of this logic, which is introduced uniformly with respect to a coalgebraic type functor, required to preserve weak pullbacks, extends that of classical propositional logic with a so-called coalgebraic cover modality depending on the type functor. Its semantics is deﬁned in terms of a categorically deﬁned relation lifting operation. As the main contributions of our paper we introduce a derivation system, and prove that it provides a sound and complete axiomatization for the collection of coalgebraically valid inequalities. Our soundness and completeness proof is algebraic, and we employ

Pattinson’s stratiﬁcation method, showing that our derivation system can be stratiﬁed in countably many layers, corresponding to the modal depth of the formulas involved.

In the proof of our main result we identify some new concepts and obtain some auxiliary results of independent interest. We survey properties of the notion of relation lifting, induced by an arbitrary but ﬁxed set functor. We introduce a category of Boolean algebra presentations, and establish an adjunction between it and the category of Boolean algebras.

Given the fact that our derivation system involves only formulas of depth one, it can be encoded as a endo-functor on Boolean algebras. We show that this functor is ﬁnitary and preserves embeddings, and we prove that the Lindenbaum-Tarski algebra of our logic can be identiﬁed with the initial algebra for this functor.

The syntax of this logic, which is introduced uniformly with respect to a coalgebraic type functor, required to preserve weak pullbacks, extends that of classical propositional logic with a so-called coalgebraic cover modality depending on the type functor. Its semantics is deﬁned in terms of a categorically deﬁned relation lifting operation. As the main contributions of our paper we introduce a derivation system, and prove that it provides a sound and complete axiomatization for the collection of coalgebraically valid inequalities. Our soundness and completeness proof is algebraic, and we employ

Pattinson’s stratiﬁcation method, showing that our derivation system can be stratiﬁed in countably many layers, corresponding to the modal depth of the formulas involved.

In the proof of our main result we identify some new concepts and obtain some auxiliary results of independent interest. We survey properties of the notion of relation lifting, induced by an arbitrary but ﬁxed set functor. We introduce a category of Boolean algebra presentations, and establish an adjunction between it and the category of Boolean algebras.

Given the fact that our derivation system involves only formulas of depth one, it can be encoded as a endo-functor on Boolean algebras. We show that this functor is ﬁnitary and preserves embeddings, and we prove that the Lindenbaum-Tarski algebra of our logic can be identiﬁed with the initial algebra for this functor.

Original language | English |
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Article number | 2 |

Journal | Logical Methods in Computer Science |

Volume | 8 |

Issue number | 3 |

DOIs | |

Publication status | Published - 31 Jul 2012 |

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### Keywords

- Coalgebra
- modal logic
- relation lifting
- completeness
- cover modality
- presentations by generators and relations

### Cite this

Kupke, C., Kurz, A., & Venema, Y. (2012). Completeness for the coalgebraic cover modality.

*Logical Methods in Computer Science*,*8*(3), [2]. https://doi.org/10.2168/LMCS-8(3:2)2012