Modular algorithms for heterogeneous modal logics via multi-sorted coalgebra
Mathematical Structures in Computer Science
Connections of coalgebra and semantic modeling
Proceedings of the 13th Conference on Theoretical Aspects of Rationality and Knowledge
On the fusion of coalgebraic logics
CALCO'11 Proceedings of the 4th international conference on Algebra and coalgebra in computer science
Generalized product of coalgebraic hybrid logics
CALCO'11 Proceedings of the 4th international conference on Algebra and coalgebra in computer science
ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part II
The Logic of Exact Covers: Completeness and Uniform Interpolation
LICS '13 Proceedings of the 2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science
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Coalgebras provide a unifying semantic framework for a wide variety of modal logics. It has previously been shown that the class of coalgebras for an endofunctor can always be axiomatized in rank 1. Here we establish the converse, i.e. every rank-1 modal logic has a sound and strongly complete coalgebraic semantics. This is achieved by constructing for a given modal logic a canonical coalgebraic semantics, consisting of a signature functor and interpretations of modal operators, which turns out to be final among all such structures. The canonical semantics may be seen as a coalgebraic reconstruction of neighbourhood semantics, broadly construed. A finitary restriction of the canonical semantics yields a canonical weakly complete semantics which moreover enjoys the Hennessy–Milner property. As a consequence, the machinery of coalgebraic modal logic, in particular generic decision procedures and upper complexity bounds, becomes applicable to arbitrary rank-1 modal logics, without regard to their semantic status; we thus obtain purely syntactic versions of such results. As an extended example, we apply our framework to recently defined deontic logics. In particular, our methods lead to the new result that these logics are strongly complete.