Functorial Coalgebraic Logic: The Case of Many-sorted Varieties

Authors:
Alexander Kurz;Daniela Petrişan
Affiliations:
Department of Computer Science, University of Leicester, UK;Department of Computer Science, University of Leicester, UK
Venue:
Electronic Notes in Theoretical Computer Science (ENTCS)
Year:
2008

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Abstract

Following earlier work, a modal logic for T-coalgebras is a functor L on a suitable variety. Syntax and proof system of the logic are given by presentations of the functor. This paper makes two contributions. First, a previous result characterizing those functors that have presentations is generalized from endofunctors on one-sorted varieties to functors between many-sorted varieties. This yields an equational logic for the presheaf semantics of higher-order abstract syntax. As another application, we show how the move to functors between many-sorted varieties allows to modularly combine syntax and proof systems of different logics. Second, we show how to associate to any set-functor T a complete (finitary) logic L consisting of modal operators and Boolean connectives.