Completeness for the coalgebraic cover modality

TL;DR

This paper develops a sound and complete axiomatization for a finitary coalgebraic logic with a cover modality, extending classical propositional logic, and employs algebraic and categorical methods for proof and semantics.

Contribution

It introduces a derivation system for coalgebraic logic, proves its soundness and completeness, and establishes algebraic and categorical foundations for the logic's semantics.

Findings

The derivation system is sound and complete for coalgebraic inequalities.

The logic's Lindenbaum-Tarski algebra is the initial algebra for a specific endo-functor.

The proof employs Pattinson's stratification method and new concepts in relation lifting.

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.