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A modal proof theory for final polynomial coalgebras. Theoret
 Comput. Sci
"... An infinitary proof theory is developed for modal logics whose models are coalgebras of polynomial functors on the category of sets. The canonical model method from modal logic is adapted to construct a final coalgebra for any polynomial functor. The states of this final coalgebra are certain “maxim ..."
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An infinitary proof theory is developed for modal logics whose models are coalgebras of polynomial functors on the category of sets. The canonical model method from modal logic is adapted to construct a final coalgebra for any polynomial functor. The states of this final coalgebra are certain “maximal ” sets of formulas that have natural syntactic closure properties. The syntax of these logics extends that of previously developed modal languages for polynomial coalgebras by adding formulas that express the “termination ” of certain functions induced by transition paths. A completeness theorem is proven for the logic of functors which have the Lindenbaum property that every consistent set of formulas has a maximal extension. This property is shown to hold if if the deducibility relation is generated by countably many inference rules. A counterexample to completeness is also given. This is a polynomial functor that is not Lindenbaum: it has an uncountable set of formulas that is deductively consistent but has no maximal extension and is unsatisfiable, even though all of its countable subsets are satisfiable. 1
Coalgebraic Lindström Theorems
"... We study modal Lindström theorems from a coalgebraic perspective. We provide three different Lindström theorems for coalgebraic logic, one of which is a direct generalisation of de Rijke’s result for Kripke models. Both the other two results are based on the properties of bisimulation invariance, ..."
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We study modal Lindström theorems from a coalgebraic perspective. We provide three different Lindström theorems for coalgebraic logic, one of which is a direct generalisation of de Rijke’s result for Kripke models. Both the other two results are based on the properties of bisimulation invariance, compactness, and a third property: ωbisimilarity, and expressive closure at level ω, respectively. These also provide new results in the case of Kripke models. Discussing the relation between our work and a recent result by van Benthem, we give an example showing that only requiring bisimulation invariance together with compactness does not suffice to characterise basic modal logic.
A HennessyMilner Property for ManyValued Modal Logics
"... A HennessyMilner property, relating modal equivalence and bisimulations, is defined for manyvalued modal logics that combine a local semantics based on a complete MTLchain (a linearly ordered commutative integral residuated lattice) with crisp Kripke frames. A necessary and sufficient algebraic c ..."
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A HennessyMilner property, relating modal equivalence and bisimulations, is defined for manyvalued modal logics that combine a local semantics based on a complete MTLchain (a linearly ordered commutative integral residuated lattice) with crisp Kripke frames. A necessary and sufficient algebraic condition is then provided for the class of imagefinite models of these logics to admit the HennessyMilner property. Complete characterizations are obtained in the case of manyvalued modal logics based on BLchains (divisible MTLchains) that are finite or have universe [0,1], including crisp Lukasiewicz, Gödel, and product modal logics.