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On modal logics of linear inequalities
 Proc. AiML 2010
, 2010
"... We consider probabilistic modal logic, graded modal logic and stochastic modal logic, where linear inequalities may be used to express numerical constraints between quantities. For each of the logics, we construct a cutfree sequent calculus and show soundness with respect to a natural class of mode ..."
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We consider probabilistic modal logic, graded modal logic and stochastic modal logic, where linear inequalities may be used to express numerical constraints between quantities. For each of the logics, we construct a cutfree sequent calculus and show soundness with respect to a natural class of models. The completeness of the associated sequent calculi is then established with the help of coalgebraic semantics which gives completeness over a (typically much smaller) class of models. With respect to either semantics, it follows that the satisfiability problem of each of these logics is decidable in polynomial space. Keywords: Probabilistic modal logic, graded modal logic, linear inequalities
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
Global Caching for Coalgebraic Description Logics
"... Coalgebraic description logics offer a common semantic umbrella for extensions of description logics with reasoning principles outside relational semantics, e.g. quantitative uncertainty, nonmonotonic conditionals, or coalitional power. Specifically, we work in coalgebraic logic with global assump ..."
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Coalgebraic description logics offer a common semantic umbrella for extensions of description logics with reasoning principles outside relational semantics, e.g. quantitative uncertainty, nonmonotonic conditionals, or coalitional power. Specifically, we work in coalgebraic logic with global assumptions (i.e. a general TBox), nominals, and satisfaction operators, and prove soundness and completeness of an associated tableau algorithm of optimal complexity EXPTIME. The algorithm uses the (known) tableau rules for the underlying modal logics, and is based on on global caching, which raises hopes of practically feasible implementation. Instantiation of this result to concrete logics yields new algorithms in all cases including standard relational hybrid logic.
Flat Coalgebraic Fixed Point Logics
"... Abstract. Fixed point logics are widely used in computer science, in particular in artificial intelligence and concurrency. The most expressive logics of this type are the µcalculus and its relatives. However, popular fixed point logics tend to trade expressivity for simplicity and readability, and ..."
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Abstract. Fixed point logics are widely used in computer science, in particular in artificial intelligence and concurrency. The most expressive logics of this type are the µcalculus and its relatives. However, popular fixed point logics tend to trade expressivity for simplicity and readability, and in fact often live within the single variable fragment of the µcalculus. The family of such flat fixed point logics includes, e.g., CTL, the ∗nestingfree fragment of PDL, and the logic of common knowledge. Here, we extend this notion to the generic semantic framework of coalgebraic logic, thus covering a wide range of logics beyond the standard µcalculus including, e.g., flat fragments of the graded µcalculus and the alternatingtime µcalculus (such as ATL), as well as probabilistic and monotone fixed point logics. Our main results are completeness of the KozenPark axiomatization and a timedout tableaux method that matches EXPTIME upper bounds inherited from the coalgebraic µcalculus but avoids using automata. 1
EXPTIME TABLEAUX FOR THE COALGEBRAIC µCALCULUS ∗
"... Vol.? (?:?) 2???, ? pages www.lmcsonline.org ..."