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Strongly Complete Logics For Coalgebras, Alexander Kurz, Jiří Rosický
Strongly Complete Logics For Coalgebras, Alexander Kurz, Jiří Rosický
Engineering Faculty Articles and Research
Coalgebras for a functor model different types of transition systems in a uniform way. This paper focuses on a uniform account of finitary logics for set-based coalgebras. In particular, a general construction of a logic from an arbitrary set-functor is given and proven to be strongly complete under additional assumptions. We proceed in three parts.
Part I argues that sifted colimit preserving functors are those functors that preserve universal algebraic structure. Our main theorem here states that a functor preserves sifted colimits if and only if it has a finitary presentation by operations and equations. Moreover, the presentation of the …
Completeness For The Coalgebraic Cover Modality, Clemens Kupke, Alexander Kurz, Yde Venema
Completeness For The Coalgebraic Cover Modality, Clemens Kupke, Alexander Kurz, Yde Venema
Engineering Faculty Articles and Research
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, …