Kripke Models Research Articles

THE LOGIC OF PARTITIONS: INTRODUCTION TO THE DUAL OF THE LOGIC OF SUBSETS

Modern categorical logic as well as the Kripke and topological models of intuitionistic logic suggest that the interpretation of ordinary “propositional” logic should in general be the logic of subsets of a given universe set. Partitions on a set are dual to subsets of a set in the sense of the category-theoretic duality of epimorphisms and monomorphisms—which is reflected in the duality between quotient objects and subobjects throughout algebra. If “propositional” logic is thus seen as the logic of subsets of a universe set, then the question naturally arises of a dual logic of partitions on a universe set. This paper is an introduction to that logic of partitions dual to classical subset logic. The paper goes from basic concepts up through the correctness and completeness theorems for a tableau system of partition logic.

Standard Gödel Modal Logics

We prove strong completeness of the □-version and the ◊-version of a Godel modal logic based on Kripke models where propositions at each world and the accessibility relation are both infinitely valued in the standard Godel algebra [0,1]. Some asymmetries are revealed: validity in the first logic is reducible to the class of frames having two-valued accessibility relation and this logic does not enjoy the finite model property, while validity in the second logic requires truly fuzzy accessibility relations and this logic has the finite model property. Analogues of the classical modal systems D, T, S4 and S5 are considered also, and the completeness results are extended to languages enriched with a discrete well ordered set of truth constants.

Hierarchical logical consequence

The modern view of logical reasoning as modeled by a consequence operator (instead of simply by a set of theorems) has allowed for huge developments in the study of logic as an abstract discipline. Still, it is unable to explain why it is often the case that the same designation is used, in an ambiguous way, to describe several distinct modes of reasoning over the same logical language. A paradigmatic example of such a situation is ‘modal logic’, a designation which can encompass reasoning over Kripke frames, but also over Kripke models, and in any case either locally (at a fixed world) or globally (at all worlds). Herein, we adopt a novel abstract notion of logic presented as a lattice-structured hierarchy of consequence operators, and explore some common proof-theoretic and model-theoretic ways of presenting such hierarchies through a collection of meaningful examples. In order to illustrate the usefulness of the notion of hierarchical consequence operators we address a few questions in the theory of combined logics, where a suitable abstract presentation of the logics being combined is absolutely essential, and we show how to define and achieve a number of interesting preservation results for fibring, in the context of 2-hierarchies.

Multi-Valued Modal Fixed Point Logics for Model Checking

In this paper, I will show how multi-valued logics are used for model checking. Model checking is an automatic technique to analyze correctness of hardware and software systems. A model checker is based on a temporal logic or a modal fixed point logic. That is to say, a system to be checked is formalized as a Kripke model, a property to be satisfied by the system is formalized as a temporal formula or a modal formula, and the model checker checks that the Kripke model satisfies the formula. Although most existing model checkers are based on 2-valued logics, recently new attempts have been made to extend the underlying logics of model checkers to multi-valued logics. I will summarize these new results.

A propositional p-adic probability logic

We present the p-adic probability logic LpPP based on the paper [5] by A. Khrennikov et al. The logical language contains formulas such as P=s(?) with the intended meaning 'the probability of ? is equal to s', where ? is a propositional formula. We introduce a class of Kripke-like models that combine properties of the usual Kripke models and finitely additive p-adic probabilities. We propose an infinitary axiom system and prove that it is sound and strongly complete with respect to the considered class of models. In the paper the terms finitary and infinitary concern the meta language only, i.e., the logical language is countable, formulas are finite, while only proofs are allowed to be infinite. We analyze decidability of LpPP and provide a procedure which decides satisfiability of a given probability formula.

Kripke models for subtheories of CZF

In this paper a method to construct Kripke models for subtheories of constructive set theory is introduced that uses constructions from classical model theory such as constructible sets and generic extensions. Under the main construction all axioms except the collection axioms can be shown to hold in the constructed Kripke model. It is shown that by carefully choosing the classical models various instances of the collection axioms, such as exponentiation, can be forced to hold as well. The paper does not contain any deep results. It consists of first observations on the subject, and is meant to introduce some notions that could serve as a foundation for further research.

On fuzzy modal logics [formula omitted

The fuzzy variant S 5 ( C ) of the well-known modal logic S5 is studied, C being a recursively axiomatized fuzzy propositional logic extending the basic fuzzy logic BL. Three kinds of Kripke models are introduced and corresponding deductive systems are found.

On intuitionistic modal and tense logics and their classical companion logics: Topological semantics and bisimulations

We take the well-known intuitionistic modal logic of Fischer Servi with semantics in bi-relational Kripke frames, and give the natural extension to topological Kripke frames. Fischer Servi’s two interaction conditions relating the intuitionistic pre-order (or partial-order) with the modal accessibility relation generalize to the requirement that the relation and its inverse be lower semi-continuous 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 topology-preserving conditions are equivalent to the properties that the inverse relation and the relation are lower semi-continuous with respect to the topologies on the two models. The 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 multi-modal logics in the setting of the Gödel translation. After giving canonical topological Kripke models for the Hilbert-style axiomatizations of the Fischer Servi logic and its classical companion logic, we use the canonical model in a second main result to characterize a Hennessy–Milner class of topological models between any pair of which there is a maximal topological bisimulation that preserve the intuitionistic semantics.

Fibred Security Language

We study access control policies based on the says operator by introducing a logical framework called Fibred Security Language (FSL) which is able to deal with features like joint responsibility between sets of principals and to identify them by means of first-order formulas. FSL is based on a multimodal logic methodology. We first discuss the main contributions from the expressiveness point of view, we give semantics for the language both for classical and intuitionistic fragment), we then prove that in order to express well-known properties like ‘speaks-for’ or ‘hand-off’, defined in terms of says, we do not need second-order logic (unlike previous approaches) but a decidable fragment of first-order logic suffices. We propose a model-driven study of the says axiomatization by constraining the Kripke models in order to respect desirable security properties, we study how existing access control logics can be translated into FSL and we give completeness for the logic.

Kripke semantics for provability logic GLP

A well-known polymodal provability logic GLP due to Japaridze is complete w.r.t. the arithmetical semantics where modalities correspond to reflection principles of restricted logical complexity in arithmetic. This system plays an important role in some recent applications of provability algebras in proof theory. However, an obstacle in the study of GLP is that it is incomplete w.r.t. any class of Kripke frames. In this paper we provide a complete Kripke semantics for GLP . First, we isolate a certain subsystem J of GLP that is sound and complete w.r.t. a nice class of finite frames. Second, appropriate models for GLP are defined as the limits of chains of finite expansions of models for J . The techniques involves unions of n -elementary chains and inverse limits of Kripke models. All the results are obtained by purely modal-logical methods formalizable in elementary arithmetic.

On the Logical Formalization of Possibilistic Counterparts of States over n-valued Lukasiewicz Events

Possibility and necessity measures are commonly defined over Boolean algebras. This work considers a generalization of these kinds of measures over MV-algebras as a possibilistic counterpart of the (probabilistic) notion of state on MV-algebras. Two classes of possibilistic states over MV-algebras of functions are characterized in terms of (generalized) Sugeno integrals. For reasoning about these representable classes of possibilistic states, we introduce many-valued modal logics based on the Rational Łukasiewicz Logic, that are shown to be complete with respect to corresponding classes of Kripke models equipped with those states.

Predicate Logical Extensions of some Subintuitionistic Logics

The paper presents predicate logical extensions of some subintuitionistic logics. Subintuitionistic logics result if conditions of the accessibility relation in Kripke models for intuitionistic logic are dropped. The accessibility relation which interprets implication in models for the propositional base subintuitionistic logic considered here is neither persistent on atoms, nor reflexive, nor transitive. Strongly complete predicate logical extensions are modeled with a second accessibility relation, which is a partial order, for the interpretation of the universal quantifier.

Preservation theorems for Kripke models

AbstractThere are several ways for defining the notion submodel for Kripke models of intuitionistic first‐order logic. In our approach a Kripke model A is a submodel of a Kripke model B if they have the same frame and for each two corresponding worlds Aα and Bα of them, Aα is a subset of Bα and forcing of atomic formulas with parameters in the smaller one, in A and B, are the same. In this case, B is called an extension of A. We characterize theories that are preserved under taking submodels and also those that are preserved under taking extensions as universal and existential theories, respectively. We also study the notion elementary submodel defined in the same style and give some results concerning this notion. In particular, we prove that the relation between each two corresponding worlds of finite Kripke models A ≤ B is elementary extension (in the classical sense) (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

On a Modal Epistemic Axiom Emerging from McDermott-Doyle Logics

An important question in modal nonmonotonic logics concerns the limits of propositional definability for logics of the McDermott-Doyle family. Inspired by this technical question we define a variant of autoepistemic logic which provably corresponds to the logic of the McDermott-Doyle family that is based on themodal axiom p5: ◊p⊃ (¬sp⊃s¬sp). This axiomis a naturalweakening of classical negative introspection restricting its scope to possible facts. It closely resembles the axiom w5: p⊃ (¬sp⊃s¬sp) which restricts the effect of negative introspection to true facts. We examine p5 in the context of classical possible-worlds Kripke models, providing results for correspondence, completeness and the finite model property. We also identify the corresponding condition for p5 in the context of neighbourhood semantics. Although rather natural epistemically, this axiom has not been investigated in classical modal epistemic reasoning, probably because its addition to S4 gives the well-known strong modal system S5.

The Logic of Partitions: Introduction to the Dual of the Logic of Subsets

Partitions on a set are dual to subsets of a set in the sense of the category-theoretic duality of epimorphisms and monomorphisms. Modern categorical logic as well as the Kripke models of intuitionistic logic suggest that the interpretation of classical logic might be the logic of subsets of a given universe set. The propositional interpretation is isomorphic to the special case where the truth and falsity of propositions behave like the subsets of a one-element set. If classical logic is thus seen as the logic of subsets of a universe set, then the question naturally arises of a dual logic of partitions on a universe set. This paper is an introduction to that logic of partitions dual to classical logic. The paper goes from basic concepts up through the correctness and completeness theorems for a tableau system of partition logic.

What groups do, can do, and know they can do: an analysis in normal modal logics

We investigate a series of logics that allow to reason about agents' actions, abilities, and their knowledge about actions and abilities. These logics include Pauly's Coalition Logic CL, Alternating-time Temporal Logic ATL, the logic of ‘seeing-to-it-that' (STIT), and epistemic extensions thereof. While complete axiomatizations of CL and ATL exist, only the fragment of the STIT language without temporal operators and without groups has been axiomatized by Xu (called Ldm). We start by recalling a simplification of the Ldm that has been proposed in previous work, together with an alternative semantics in terms of standard Kripke models. We extend that semantics to groups via a principle of superadditivity, and give a sound and complete axiomatization that we call Ldm G. We then add a temporal ‘next' operator to Ldm G, and again give a sound and complete axiomatization. We show that Ldm G subsumes coalition logic CL. Finally, we extend these logics with standard S5 knowledge operators. This enables us to express that agents see to something under uncertainty about the present state or uncertainty about which action is being taken. We focus on the epistemic extension of X-Ldm G, noted E-X-Ldm G. In accordance with established terminology in the planning community, we call this extension of X-Ldm G the conformant X-Ldm G. The conformant X-Ldm G enables us to express that agents are able to perform a uniform strategy. We conclude that in that respect, our epistemic extension of X-Ldm G is better suited than epistemic extensions of ATL.

Agent Communication for Dynamic Belief Update

Thus far, various formalizations of rational / logical agent model have been proposed. In this paper, we include the notion of communication channel and belief modality into update logic, and introduce Belief Update Logic (BUL). First, we discuss that how we can reformalize the inform action of FIPA-ACL into communication channel, which represents a connection between agents. Thus, our agents can send a message only when they believe, and also there actually is, a channel between him / her and a receiver. Then, we present a static belief logic (BL) and show its soundness and completeness. Next, we develop the logic to BUL, which can update Kripke model by the inform action; in which we show that in the updated model the belief operator also satisfies K45. Thereafter, we show that every sentence in BUL can be translated into BL; thus, we can contend that BUL is also sound and complete. Furthermore, we discuss the features of CUL, including the case of inconsistent information, as well as channel transmission. Finally, we summarize our contribution and discuss some future issues.

Vietoris Bisimulations

Building on the fact that descriptive frames are coalgebras for the Vietoris functor on the category of Stone spaces, we introduce and study the concept of a Vietoris bisimulation between two descriptive modal models, together with the associated notion of bisimilarity. We prove that our notion of bisimilarity, which is defined in terms of relation lifting, coincides with Kripke bisimilarity (with respect to the underlying Kripke models), with behavioural equivalence, and with modal equivalence, but not with Aczel–Mendler bisimilarity. As a corollary, we obtain that the Vietoris functor does not preserve weak pullbacks. Comparing Vietoris bisimulations between descriptive models to Kripke bisimulations on the underlying Kripke models, we prove that the closure of such a Kripke bisimulation is a Vietoris bisimulation. As a corollary, we show that the collection of Vietoris bisimulations between two descriptive models forms a complete lattice. Finally, we provide a game-theoretic characterization of Vietoris bisimilarity.

Coalgebraic logic for stochastic right coalgebras

We generalize stochastic Kripke models and Markov transition systems to stochastic right coalgebras. These are coalgebras for a functor F ⋅ S with F as an endofunctor on the category of analytic spaces, and S is the subprobability functor. The modal operators are generalized through predicate liftings which are set-valued natural transformations involving the functor. Two states are equivalent iff they cannot be separated by a formula. This equivalence relation is used to construct a cospan for logical equivalent coalgebras under a separation condition for the set of predicate liftings. Consequently, behavioral and logical equivalence are really the same. From the cospan we construct a span. The central argument is a selection argument giving us the dynamics of a mediating coalgebra from the domains of the cospan. This construction is used to establish that behavioral equivalent coalgebras are bisimilar, yielding the equivalence of all three characterizations of a coalgebra’s behavior as in the case of Kripke models or Markov transition systems.

Which Modal Models are the Right Ones (for Logical Necessity)?

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