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STRONG COMPLETENESS OF COALGEBRAIC MODAL LOGICS
 SYMPOSIUM ON THEORETICAL ASPECTS OF COMPUTER SCIENCE
"... Canonical models are of central importance in modal logic, in particular as they witness strong completeness and hence compactness. While the canonical model construction is well understood for Kripke semantics, nonnormal modal logics often present subtle difficulties – up to the point that canoni ..."
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Cited by 4 (4 self)
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Canonical models are of central importance in modal logic, in particular as they witness strong completeness and hence compactness. While the canonical model construction is well understood for Kripke semantics, nonnormal modal logics often present subtle difficulties – up to the point that canonical models may fail to exist, as is the case e.g. in most probabilistic logics. Here, we present a generic canonical model construction in the semantic framework of coalgebraic modal logic, which pinpoints coherence conditions between syntax and semantics of modal logics that guarantee strong completeness. We apply this method to reconstruct canonical model theorems that are either known or folklore, and moreover instantiate our method to obtain new strong completeness results. In particular, we prove strong completeness of graded modal logic with finite multiplicities, and of the modal logic of exact probabilities.
Exemplaric Expressivity of Modal Logics
, 2008
"... This paper investigates expressivity of modal logics for transition systems, multitransition systems, Markov chains, and Markov processes, as coalgebras of the powerset, finitely supported multiset, finitely supported distribution, and measure functor, respectively. Expressivity means that logically ..."
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Cited by 3 (0 self)
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This paper investigates expressivity of modal logics for transition systems, multitransition systems, Markov chains, and Markov processes, as coalgebras of the powerset, finitely supported multiset, finitely supported distribution, and measure functor, respectively. Expressivity means that logically indistinguishable states, satisfying the same formulas, are behaviourally indistinguishable. The investigation is based on the framework of dual adjunctions between spaces and logics and focuses on a crucial injectivity property. The approach is generic both in the choice of systems and modalities, and in the choice of a “base logic”. Most of these expressivity results are already known, but the applicability of the uniform setting of dual adjunctions to these particular examples is what constitutes the contribution of the paper.
Functorial coalgebraic logic: The case of manysorted varieties
 Electron. Notes Theor. Comput. Sci
"... Following earlier work, a modal logic for Tcoalgebras 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 generaliz ..."
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Cited by 3 (2 self)
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Following earlier work, a modal logic for Tcoalgebras 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 onesorted varieties to functors between manysorted varieties. This yields an equational logic for the presheaf semantics of higherorder abstract syntax. As another application, we show how the move to functors between manysorted varieties allows to modularly combine syntax and proof systems of different logics. Second, we show how to associate to any setfunctor T a complete (finitary) logic L consisting of modal operators and Boolean connectives.
Equational Coalgebraic Logic
 MFPS
, 2009
"... Coalgebra develops a general theory of transition systems, parametric in a functor T; the functor T specifies the possible onestep behaviours of the system. A fundamental question in this area is how to obtain, for an arbitrary functor T, a logic for Tcoalgebras. We compare two existing proposals, ..."
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Cited by 1 (1 self)
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Coalgebra develops a general theory of transition systems, parametric in a functor T; the functor T specifies the possible onestep behaviours of the system. A fundamental question in this area is how to obtain, for an arbitrary functor T, a logic for Tcoalgebras. We compare two existing proposals, Moss’s coalgebraic logic and the logic of all predicate liftings, by providing onestep translations between them, extending the results in [21] by making systematic use of Stone duality. Our main contribution then is a novel coalgebraic logic, which can be seen as an equational axiomatization of Moss’s logic. The three logics are equivalent for a natural but restricted class of functors. We give examples showing that the logics fall apart in general. Finally, we argue that the quest for a generic logic for Tcoalgebras is still open in the general case.
Free Heyting algebras: revisited
"... Abstract. We use coalgebraic methods to describe finitely generated free Heyting algebras. Heyting algebras are axiomatized by rank 01 axioms. In the process of constructing free Heyting algebras we first apply existing methods to weak Heyting algebras—the rank 1 reducts of Heyting algebras—and the ..."
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Abstract. We use coalgebraic methods to describe finitely generated free Heyting algebras. Heyting algebras are axiomatized by rank 01 axioms. In the process of constructing free Heyting algebras we first apply existing methods to weak Heyting algebras—the rank 1 reducts of Heyting algebras—and then adjust them to the mixed rank 01 axioms. On the negative side, our work shows that one cannot use arbitrary axiomatizations in this approach. Also, the adjustments made for the mixed rank axioms are not just purely equational, but rely on properties of implication as a residual. On the other hand, the duality and coalgebra perspective does allow us, in the case of Heyting algebras, to derive Ghilardi’s (Ghilardi, 1992) powerful representation of finitely generated free Heyting algebras in a simple, transparent, and modular way using Birkhoff duality for finite distributive lattices. 1
Traces for Coalgebraic Components
 MATH. STRUCT. IN COMP. SCIENCE
, 2010
"... This paper contributes a feedback operator, in the form of a monoidal trace, to the theory of coalgebraic, statebased modelling of components. The feedback operator on components is shown to satisfy the trace axioms of Joyal, Street and Verity. We employ McCurdy’s tube diagrams, an extension of sta ..."
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This paper contributes a feedback operator, in the form of a monoidal trace, to the theory of coalgebraic, statebased modelling of components. The feedback operator on components is shown to satisfy the trace axioms of Joyal, Street and Verity. We employ McCurdy’s tube diagrams, an extension of standard string diagrams for monoidal categories, for representing and manipulating component diagrams. The microcosm principle then yields a canonical “inner” traced monoidal structure on the category of resumptions (elements of final coalgebras / components). This generalises an observation by Abramsky, Haghverdi and Scott.
www.stacsconf.org STRONG COMPLETENESS OF COALGEBRAIC MODAL LOGICS
"... ABSTRACT. Canonical models are of central importance in modal logic, in particular as they witness strong completeness and hence compactness. While the canonical model construction is well understood for Kripke semantics, nonnormal modal logics often present subtle difficulties – up to the point th ..."
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ABSTRACT. Canonical models are of central importance in modal logic, in particular as they witness strong completeness and hence compactness. While the canonical model construction is well understood for Kripke semantics, nonnormal modal logics often present subtle difficulties – up to the point that canonical models may fail to exist, as is the case e.g. in most probabilistic logics. Here, we present a generic canonical model construction in the semantic framework of coalgebraic modal logic, which pinpoints coherence conditions between syntax and semantics of modal logics that guarantee strong completeness. We apply this method to reconstruct canonical model theorems that are either known or folklore, and moreover instantiate our method to obtain new strong completeness results. In particular, we prove strong completeness of graded modal logic with finite multiplicities, and of the modal logic of exact probabilities. In modal logic, completeness proofs come in two flavours: weak completeness, i.e. derivability of all universally valid formulas, is often proved using finite model constructions, and strong completeness, which additionally allows for a possibly infinite set of assumptions. The latter entails recursive enumerability of the set of consequences of a recursively enumerable set of assumptions, and is usually established using (infinite) canonical models. The appeal of the first method is that it
Expressiveness of Positive Coalgebraic Logic
"... From the point of view of modal logic, coalgebraic logic over posets is the natural coalgebraic generalisation of positive modal logic. From the point of view of coalgebra, posets arise if one is interested in simulations as opposed to bisimulations. From a categorical point of view, one moves from ..."
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From the point of view of modal logic, coalgebraic logic over posets is the natural coalgebraic generalisation of positive modal logic. From the point of view of coalgebra, posets arise if one is interested in simulations as opposed to bisimulations. From a categorical point of view, one moves from ordinary categories to enriched categories. We show that the basic setup of coalgebraic logic extends to this more general setting and that every finitary functor on posets has a logic that is expressive, that is, has the HennessyMilner property. Keywords: Coalgebra, Modal Logic, Poset