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Hybrid Logics
"... This chapter provides a modern overview of the field of hybrid logic. Hybrid logics are extensions of standard modal logics, involving symbols that name individual states in models. The first results that are nowadays considered as part of the field date back to the early work of Arthur ..."
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This chapter provides a modern overview of the field of hybrid logic. Hybrid logics are extensions of standard modal logics, involving symbols that name individual states in models. The first results that are nowadays considered as part of the field date back to the early work of Arthur
Parametric completeness for separation theories
, 2013
"... In this paper, we close the logical gap between provability in the logic BBI, which is the propositional basis for separation logic, and validity in an intended class of separation models, as employed in applications of separation logic such as program verification. An intended class of separation m ..."
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In this paper, we close the logical gap between provability in the logic BBI, which is the propositional basis for separation logic, and validity in an intended class of separation models, as employed in applications of separation logic such as program verification. An intended class of separation models is usually specified by a collection of axioms describing the specific model properties that are expected to hold, which we call a separation theory. Our main contributions are as follows. First, we show that several typical properties of separation theories are not definable in BBI. Second, we show that these properties become definable in a suitable hybrid extension of BBI, obtained by adding a theory of naming to BBI in the same way that hybrid logic extends normal modal logic. The binderfree extension HyBBI captures most of the properties we consider, and the full extension HyBBI(↓) with the usual ↓ binder of hybrid logic covers all these properties. Third, we present an axiomatic proof system for our hybrid logic whose extension with any set of “pure ” axioms is sound and complete with respect to the models satisfying those axioms. As a corollary of this general result, we obtain, in a parametric manner, a sound and complete axiomatic proof system for any separation theory from our considered class. To the best of our knowledge, this class includes all separation theories appearing in the published literature. Categories and Subject Descriptors F.3.1 [Logics and Mean
Model checking strategic equilibria
 Lecture Notes in Artificial Intelligence
"... Abstract. Solution concepts are a fundamental tool for the analysis of gamelike systems, and as a consequence, much effort has been devoted to the problem of characterising solution concepts using logic. However, one problem is that, to characterise solution concepts such as Nash equilibrium, it se ..."
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Abstract. Solution concepts are a fundamental tool for the analysis of gamelike systems, and as a consequence, much effort has been devoted to the problem of characterising solution concepts using logic. However, one problem is that, to characterise solution concepts such as Nash equilibrium, it seems necessary to refer to strategies in the object language, which tends to complicate the object language. We propose a logic in which we can formulate important properties of games (and in particular purestrategy solution concepts) without recourse to naming strategies in the object language. The idea is that instead of using predicates which state that a particular collection of strategies forms a solution, we define formulae of the logic that are true at a state if and only if this state constitutes a particular equilibrium outcome. We demonstrate the usefulness of the logic by model checking equilibria of strategic games. 1
NAMED MODELS IN COALGEBRAIC HYBRID LOGIC
 SYMPOSIUM ON THEORETICAL ASPECTS OF COMPUTER SCIENCE
"... Hybrid logic extends modal logic with support for reasoning about individual states, designated by socalled nominals. We study hybrid logic in the broad context of coalgebraic semantics, where Kripke frames are replaced with coalgebras for a given functor, thus covering a wide range of reasoning pr ..."
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Hybrid logic extends modal logic with support for reasoning about individual states, designated by socalled nominals. We study hybrid logic in the broad context of coalgebraic semantics, where Kripke frames are replaced with coalgebras for a given functor, thus covering a wide range of reasoning principles including, e.g., probabilistic, graded, default, or coalitional operators. Specifically, we establish generic criteria for a given coalgebraic hybrid logic to admit named canonical models, with ensuing completeness proofs for pure extensions on the one hand, and for an extended hybrid language with local binding on the other. We instantiate our framework with a number of examples. Notably, we prove completeness of graded hybrid logic with local binding.
Algebraization of Hybrid Logic with Binders
"... Abstract. This paper introduces an algebraic semantics for hybrid logic with binders H(↓, @). It is known that this formalism is a modal counterpart of the bounded fragment of the firstorder logic, studied by Feferman in the 1960’s. The algebraization process leads to an interesting class of boolea ..."
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Abstract. This paper introduces an algebraic semantics for hybrid logic with binders H(↓, @). It is known that this formalism is a modal counterpart of the bounded fragment of the firstorder logic, studied by Feferman in the 1960’s. The algebraization process leads to an interesting class of boolean algebras with operators, called substitutionsatisfaction algebras. We provide a representation theorem for these algebras and thus provide an algebraic proof of completeness of hybrid logic.
Axiomatizing hybrid logic using modal logic
"... We study hybrid logics with nominals and ‘actuality ’ operators @i. We recall the method of ten Cate, Marx, and Viana to simulate hybrid logic using modalities and ‘nice’ frames, and we show that the hybrid logic of a class of frames is the modal logic of the class of its corresponding nice frames. ..."
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We study hybrid logics with nominals and ‘actuality ’ operators @i. We recall the method of ten Cate, Marx, and Viana to simulate hybrid logic using modalities and ‘nice’ frames, and we show that the hybrid logic of a class of frames is the modal logic of the class of its corresponding nice frames. Using these results, we show how to axiomatize the hybrid logic of any elementary class of frames. Then we study quasimodal logics, which are hybrid logics axiomatized by modal axioms together with basic hybrid axioms common to any hybrid logic, using only orthodox inference rules. We show that the hybrid logic of any elementary modally definable class of frames, or of any elementary class of frames closed under disjoint unions, bounded morphic images, ultraproducts and generated subframes, is quasimodal. We also show that the hybrid analogues of modal logics studied by McKinsey–Lemmon and Hughes are quasimodal.
Reasoning about strategic games with hybrid logic of choice and preferences
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NAMED MODELS IN COALGEBRAIC HYBRID LOGIC
, 2010
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Sahlqvist theory for Hybrid Logic Opgedragen aan Dick de Jongh voor zijn 65ste verjaardag
"... In his work and life, Dick de Jongh often emphasized the importance of bringing together different disciplines, of viewing problems from different perspectives, and of gaining insight, understanding and joy by employing Iran’s official government policy called dialogue among civilisations. All these ..."
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In his work and life, Dick de Jongh often emphasized the importance of bringing together different disciplines, of viewing problems from different perspectives, and of gaining insight, understanding and joy by employing Iran’s official government policy called dialogue among civilisations. All these aspects are combined in the work he did in the last phase of his career: the master of logic program at the ILLC. The dialogue between the world —embodied in the highly mixed group of foreign students — and the ILLC has been enormously beneficial for both sides. We do not exaggerate if we say that the lives of all three authors of this little work have been enormously enriched through this part of Dick’s work. Thanks Dick. 1
Global View On Reactivity: Switch Graphs and Their Logics
"... The notion of reactive graph generalises the one of graph by allowing the base accessibility relation to change when its edges are traversed. Can we represent these more general structures using points and arrows? We prove this can be done by introducing higher order arrows: the switches. The possib ..."
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The notion of reactive graph generalises the one of graph by allowing the base accessibility relation to change when its edges are traversed. Can we represent these more general structures using points and arrows? We prove this can be done by introducing higher order arrows: the switches. The possibility of expressing the dependency of the future states of the accessibility relation on individual transitions by the use of higherorder relations, that is, coding metarelational concepts by means of relations, strongly suggests the use of modal languages to reason directly about these structures. We introduce a hybrid modal logic for this purpose and prove its completeness. 1