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Topological Semantics and Bisimulations for Intuitionistic Modal Logics and Their Classical Companion Logics ⋆
"... Abstract. We take the wellknown intuitionistic modal logic of Fischer Servi with semantics in birelational Kripke frames, and give the natural extension to topological Kripke frames. Fischer Servi’s two interaction conditions relating the intuitionistic preorder (or partialorder) with the modal ..."
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Abstract. We take the wellknown intuitionistic modal logic of Fischer Servi with semantics in birelational Kripke frames, and give the natural extension to topological Kripke frames. Fischer Servi’s two interaction conditions relating the intuitionistic preorder (or partialorder) with the modal accessibility relation generalise to the requirement that the relation and its inverse be lower semicontinuous with respect to the topology. We then investigate the notion of topological bisimulation relations between topological Kripke frames, as introduced by Aiello and van Benthem, and show that their topologypreserving conditions are equivalent to the properties that the inverserelation and the relation are lower semicontinuous with respect to the topologies on the two models. Our first main result is that this notion of topological bisimulation yields semantic preservation w.r.t. topological Kripke models for both intuitionistic tense logics, and for their classical companion multimodal logics in the setting of the Gödel translation. After giving canonical topological Kripke models for the Hilbertstyle axiomatizations of the Fischer Servi logic and its classical multimodal companion logic, we show that the syntactic Gödel translation induces a natural semantic map from the intuitionistic canonical model into the canonical model of the classical companion logic, and this map is itself a topological bisimulation. 1
M.: Preorders for reasoning about stability
 In: HSCC’12
, 2012
"... Preorders between processes, like simulation, have played a central role in the verification and analysis of discretestate systems. Logical characterization of such preorders have allowed one to verify the correctness of a system by analyzing an abstraction of the system. In this paper, we invest ..."
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Preorders between processes, like simulation, have played a central role in the verification and analysis of discretestate systems. Logical characterization of such preorders have allowed one to verify the correctness of a system by analyzing an abstraction of the system. In this paper, we investigate whether this approach can be feasibly applied to reason about stability properties of a system. Stability is an important property of systems that have a continuous component in their state space; it stipulates that when a system is started somewhere close to its ideal starting state, its behavior is close to its ideal, desired behavior. In [6], it was shown that stability with respect to equilibrium states is not preserved by bisimulation and hence additional continuity constraints were imposed on the bisimulation re
Nonfinitely axiomatisable twodimensional modal logics
, 2011
"... We show the first examples of recursively enumerable (even decidable) twodimensional products of finitely axiomatisable modal logics that are not finitely axiomatisable. In particular, we show that any axiomatisation of some bimodal logics that are determined by classes of product frames with linea ..."
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We show the first examples of recursively enumerable (even decidable) twodimensional products of finitely axiomatisable modal logics that are not finitely axiomatisable. In particular, we show that any axiomatisation of some bimodal logics that are determined by classes of product frames with linearly ordered first components must be infinite in two senses: It should contain infinitely many propositional variables, and formulas of arbitrarily large modal nestingdepth. 1
Products Of `transitive' Modal Logics Without The (abstract) Finite Model Property
"... It is well known that many twodimensional products of modal logics with at least one `transitive' (but not `symmetric') component lack the product finite model property. Here we show that products of two `transitive' logics (such as, e.g., K4 K4, S4 S4, GrzGrz and GLGL) do not ..."
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It is well known that many twodimensional products of modal logics with at least one `transitive' (but not `symmetric') component lack the product finite model property. Here we show that products of two `transitive' logics (such as, e.g., K4 K4, S4 S4, GrzGrz and GLGL) do not have the (abstract) finite model property either. These are the first known examples of 2D modal product logics without the finite model property where both components are natural unimodal logics having the finite model property.
topological semantics
, 2008
"... On intuitionistic modal and tense logics and their classical companion logics: ..."
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On intuitionistic modal and tense logics and their classical companion logics:
Chapter 1 TOPOLOGY AND EPISTEMIC LOGIC
"... Abstract We present the main ideas behind a number of logical systems for reasoning about points and sets that incorporate knowledgetheoretic ideas, and also the main results about them. Some of our discussions will be about applications of modal ideas to topology, and some will be on applications ..."
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Abstract We present the main ideas behind a number of logical systems for reasoning about points and sets that incorporate knowledgetheoretic ideas, and also the main results about them. Some of our discussions will be about applications of modal ideas to topology, and some will be on applications of topological ideas in modal logic, especially in epistemic logic. In the former area, we would like to present the basic ideas and results of topologic, the study of twosorted bimodal logical systems interpreted on subset spaces; these are arbitrary sets with collections of subsets called opens. Many of the papers in this field deal with questions of axiomatizing the logics of particular classes of subset spaces determined by conditions on the “opens”, such as being closed under intersection, being topologies, or satisfying various chain conditions. In the area of applications of topological ideas in epistemic logic, we include a section on the following topics: a topological semantics and completeness proof for the logic of belief KD45. 1.
Products of `Transitive' Modal Logics. Part I: `negative' results
, 2004
"... Here we solve a number of major open problems concerning computational properties of products and commutators of two `transitive' (but not `symmetric') standard modal logics (such as, e.g., K4, S4, S4.1, Grz, or GL) by showing that all of them are undecidable and lack the (abstract) fin ..."
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Here we solve a number of major open problems concerning computational properties of products and commutators of two `transitive' (but not `symmetric') standard modal logics (such as, e.g., K4, S4, S4.1, Grz, or GL) by showing that all of them are undecidable and lack the (abstract) finite model property. Some of these products turn out to be even not recursively enumerable and some commutators Kripke incomplete.
12a. DISTRIBUTION/AVAILABILITY STATEMENT
, 2001
"... Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments ..."
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Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this