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Modular Markovian Logic
"... Abstract. We introduce Modular Markovian Logic (MML) for compositional continuoustime and continuousspace Markov processes. MML combines operators specific to stochastic logics with operators that reflect the modular structure of the semantics, similar to those used by spatial and separation logic ..."
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Abstract. We introduce Modular Markovian Logic (MML) for compositional continuoustime and continuousspace Markov processes. MML combines operators specific to stochastic logics with operators that reflect the modular structure of the semantics, similar to those used by spatial and separation logics. We present a complete Hilbertstyle axiomatization for MML, prove the small model property and analyze the relation between the stochastic bisimulation and the logical equivalence relation induced by MML on models. 1
Completeness of the finitary Moss logic
 In Areces and Goldblatt [3
"... abstract. We give a sound and complete derivation system for the valid formulas in the finitary version of Moss ’ coalgebraic logic, for coalgebras of arbitrary type. ..."
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abstract. We give a sound and complete derivation system for the valid formulas in the finitary version of Moss ’ coalgebraic logic, for coalgebras of arbitrary type.
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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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.
Optimal Tableau Algorithms for Coalgebraic Logics
"... Abstract. Deciding whether a modal formula is satisfiable with respect to a given set of (global) assumptions is a question of fundamental importance in applications of logic in computer science. Tableau methods have proved extremely versatile for solving this problem for many different individual l ..."
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Abstract. Deciding whether a modal formula is satisfiable with respect to a given set of (global) assumptions is a question of fundamental importance in applications of logic in computer science. Tableau methods have proved extremely versatile for solving this problem for many different individual logics but they typically do not meet the known complexity bounds for the logics in question. Recently, it has been shown that optimality can be obtained for some logics while retaining practicality by using a technique called “global caching”. Here, we show that global caching is applicable to all logics that can be equipped with coalgebraic semantics, for example, classical modal logic, graded modal logic, probabilistic modal logic and coalition logic. In particular, the coalgebraic approach also covers logics that combine these various features. We thus show that global caching is a widely applicable technique and also provide foundations for optimal tableau algorithms that uniformly apply to a large class of modal logics. 1
On the Fusion of coalgebraic logics
"... Abstract. Fusion is arguably the simplest way to combine modal logics. For normal modal logics with Kripke semantics, many properties such as completeness and decidability are known to transfer from the component logics to their fusion. In this paper we investigate to what extent these results can b ..."
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Abstract. Fusion is arguably the simplest way to combine modal logics. For normal modal logics with Kripke semantics, many properties such as completeness and decidability are known to transfer from the component logics to their fusion. In this paper we investigate to what extent these results can be generalised to the case of arbitrary coalgebraic logics. Our main result generalises a construction of Kracht and Wolter and confirms that completeness transfers to fusion for a large class of logics over coalgebraic semantics. This result is independent of the rank of the logics and relies on generalising the notions of distance and box operator to coalgebraic models. 1
EXPTIME TABLEAUX FOR THE COALGEBRAIC µCALCULUS ∗
"... Vol.? (?:?) 2???, ? pages www.lmcsonline.org ..."
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"... This article 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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This article 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 article.
Hybrid Logic with the Difference Modality for Generalisations of Graphs
"... We discuss recent work generalising the basic hybrid logic with the difference modality to any reasonable notion of transition. This applies equally to both subrelational transitions such as monotone neighbourhood frames or selection function models as well as those with more structure such as Marko ..."
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We discuss recent work generalising the basic hybrid logic with the difference modality to any reasonable notion of transition. This applies equally to both subrelational transitions such as monotone neighbourhood frames or selection function models as well as those with more structure such as Markov chains and alternating temporal frames. We provide a generic canonical cutfree sequent system and a terminating proofsearch strategy for the fragment without the difference modality but including the global modality. Keywords: Global Modality, Difference Modality, Coalgebraic Semantics, Cutfree Sequent System
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