Showing papers on "Normal modal logic published in 2011"

Admissibility of Logical Inference Rules

Abstract: Preface and acknowledgments. Introduction. Syntaxes and Semantics. Syntax of formal logic systems. First-order semantics and universal algebra. Algebraic semantics for propositional logics. Admissible rules in algebraic logics. Logical consequence relations. Algebraizable consequence relations. Admissibility for consequence relations. Lattices of logical consequences. Semantics for Non-Standard Logics. Algebraic semantics for intuitionistic logic. Algebraic semantics, modal and tense logics. Kripke semantics for modal and temporal logic. Kripke semantics for intuitionistic logic. Stone's theory and Kripke semantics. The finite model property. Relation of intuitionistic and modal logics. Advanced tools for the finite model property. Kripke incomplete logics. Advanced tools for Kripke completeness.Criteria for Admissibility. Reduced forms. T--translation of inference rules. Semantic criteria for admissibility. Some technical lemmas. Criteria for determining admissibility. Elementary theories of free algebras. Scheme--logics of first--order theories. Some counterexamples. Admissibility through reduced forms. Bases for Inference Rules. Initial auxiliary results. The absence of finite bases. Bases for some strong logics. Bases for valid rules of tabular logics. Tabular logics without independent bases. Structural Completeness. General properties, descriptions. Quasi--characteristic inference rules. Some preliminary technical results. Hereditary structural completeness. Structurally complete fragments. Related Questions. Rules with meta--variables. The preservation of admissible for S4 rules. The preservation of admissibility for H. Non-compact modal logics. Decidable Kripke non--compact logics. Non--compact superintuitionistic logics. Index. Bibliography.

On the Minimum Many-Valued Modal Logic over a Finite Residuated Lattice

Abstract: This article deals with many-valued modal logics, based only on the necessity operator, over a residuated lattice. We focus on three basic classes, according to the accessibility relation, of Kripke frames: the full class of frames evaluated in the residuated lattice (and so defining the minimum modal logic), the ones evaluated in the idempotent elements and the ones only evaluated in 0 and 1. We show how to expand an axiomatization, with canonical truth-constants in the language, of a finite residuated lattice into one of the modal logic, for each one of the three basic classes of Kripke frames. We also provide axiomatizations for the case of a finite MV chain but this time without canonical truth-constants in the language.

Proof Theory for Modal Logic

Abstract: The axiomatic presentation of modal systems and the standard formulations of natural deduction and sequent calculus for modal logic are reviewed, together with the difficulties that emerge with these approaches. Generalizations of standard proof systems are then presented. These include, among others, display calculi, hypersequents, and labelled systems, with the latter surveyed from a closer perspective.

Unification in modal and description logics

Abstract: Unification was originally introduced in automated deduction and term rewriting, but has recently also found applications in other fields. In this article, we give a survey of the results on unification obtained in two closely related, yet different, application areas of unification: description logics and modal logics.

On the progression of knowledge in the situation calculus

Abstract: In a seminal paper, Lin and Reiter introduced the notion of progression for basic action theories in the situation calculus. Earlier works by Moore, Scherl and Levesque extended the situation calculus to account for knowledge. In this paper, we study progression of knowledge in the situation calculus. We first adapt the concept of bisimulation from modal logic and extend Lin and Reiter's notion of progression to accommodate knowledge. We show that for physical actions, progression of knowledge reduces to forgetting predicates in first-order modal logic. We identify a class of first-order modal formulas for which forgetting an atom is definable in first-order modal logic. This class of formulas goes beyond formulas without quantifyingin. We also identify a simple case where forgetting a predicate reduces to forgetting a finite number of atoms. Thus we are able to show that for local-effect physical actions, when the initial KB is a formula in this class, progression of knowledge is definable in first-order modal logic. Finally, we extend our results to the multi-agent case.

Generalized arrow update logic

Abstract: This paper presents a logic for reasoning about information change in multi-agent settings based on epistemic arrow deletion in Kripke models.

A new four-valued approach to modal logic

Abstract: In this paper several systems of modal logic based on four-valued matrices are presented. We start with pure modal logics, i.e. modal logics with modal operators as the only operators, using the Polish framework of structural consequence relation. We show that with a four-valued matrix we can define modal operators which have the same behavior as in pure S5 (S5 with only modal operators). We then present modal logics with conjunction and disjunction based on four-valued matrices. We show that if we use partial order instead of linear order, we are avoiding Lukasiewicz’s paradox. We then introduce classical negation and we show than defining implication in the usual way using negation and disjunction Kripke law is valid using either linear or partial order. On the other hand we show that the difference between linear and partial order appears at the level of the excluded middle and the replacement theorem.

An Algebraic Approach to Subframe Logics. Modal Case

Abstract: We prove that if a modal formula is refuted on a wK4-algebra (B, 2), then it is refuted on a finite wK4-algebra which is isomorphic to a subalgebra of a relativization of (B, 2). As an immediate consequence, we obtain that each subframe and cofinal subframe logic over wK4 has the finite model property. On the one hand, this provides a purely algebraic proof of the results of Fine [11] and Zakharyaschev [22] for K4. On the other hand, it extends the Fine-Zakharyaschev results to wK4.

Finite-valued Lukasiewicz modal logic Is PSPACE-complete

Abstract: It is well-known that satisfiability (and hence validity) in the minimal classical modal logic is a PSPACE-complete problem. In this paper we consider the satisfiability and validity problems (here they are not dual, although mutually reducible) for the minimal modal logic over a finite Lukasiewicz chain, and show that they also are PSPACE-complete. This result is also true when adding either the Delta operator or truth constants in the language, i.e. in all these cases it is PSPACE-complete.

Cut-free Gentzen calculus for multimodal CK

Abstract: This paper extends previous work on the modal logic CK as a reference system, both proof-theoretically and model-theoretically, for a correspondence theory of constructive modal logics. First, the fundamental nature of CK is discussed and compared with the intuitionistic modal logic IK which is traditionally taken to be the base line. Then, it is shown, that CK admits of a cut-free Gentzen sequent calculus G-CK which has (i) a local interpretation in constructive Kripke models and (ii) does not require explicit world labels. Finally, the paper demonstrates how non-classical modal logics such as IK, CS4, CL, or [email protected]?s deontic system of 2-sequents arise as theories of CK, presented both as special rules and as frame classes.

Generic modal cut elimination applied to conditional logics

Abstract: We develop a general criterion for cut elimination in sequent calculi for propositional modal logics, which rests on absorption of cut, contraction, weakening and inversion by the purely modal part of the rule system. Our criterion applies also to a wide variety of logics outside the realm of normal modal logic. We give extensive example instantiations of our framework to various conditional logics. For these, we obtain fully internalised calculi which are substantially simpler than those known in the literature, along with leaner proofs of cut elimination and complexity. In one case, conditional logic with modus ponens and conditional excluded middle, cut elimination and complexity were explicitly stated as open in the literature.

Schematic validity in dynamic epistemic logic: decidability

Abstract: Unlike standard modal logics, many dynamic epistemic logics are not closed under uniform substitution. The classic example is Public Announcement Logic (PAL), an extension of epistemic logic based on the idea of information acquisition as elimination of possibilities. In this paper, we address the open question of whether the set of schematic validities of PAL, the set of formulas all of whose substitution instances are valid, is decidable. We obtain positive answers for multi-agent PAL, as well as its extension with relativized common knowledge, PAL-RC. The conceptual significance of substitution failure is also discussed.

Logics of Rational Interaction

Abstract: This paper provides an overview of temporal and dynamic epistemic logic, two important logical approaches to rational interaction. There are may approaches to rational interaction and a natural question is how the approaches under consideration are related. We will also discuss relations to other logics and applications.

Probabilistic modal µ-calculus with independent product

Abstract: The probabilistic modal µ-calculus pLµ (often called the quantitative µ-calculus) is a generalization of the standard modal µ-calculus designed for expressing properties of probabilistic labeled transition systems. The syntax of pLµ formulas coincides with that of the standard modal µ-calculus. Two equivalent semantics have been studied for pLµ, both assigning to each process-state p a value in [0, 1] representing the probability that the property expressed by the formula will hold in p: a denotational semantics and a game semantics given by means of two player stochastic games. In this paper we extend the logic pLµ with a second conjunction called product, whose semantics interprets the two conjuncts as probabilistically independent events. This extension allows one to encode useful operators, such as the modalities with probability one and with non-zero probability. We provide two semantics for this extended logic: one denotational and one based on a new class of games which we call tree games. The main result is the equivalence of the two semantics. The proof is carried out in ZFC set theory extended with Martin's Axiom at the first uncountable cardinal.

Towards a Proof Theory of G\"odel Modal Logics

Abstract: Analytic proof calculi are introduced for box and diamond fragments of basic modal fuzzy logics that combine the Kripke semantics of modal logic K with the many-valued semantics of G\"odel logic. The calculi are used to establish completeness and complexity results for these fragments.

Tangled modal logic for spatial reasoning

Abstract: We consider an extension of the propositional modal logic S4 which allows ⋄ to act not only on isolated formulas, but also on sets of formulas. The interpretation of ⋄Γ is then given by the tangled closure of the valuations of formulas in Γ, which over finite transitive, reflexive models indicates the existence of a cluster satisfying Γ. This extension has been shown to be more expressive than the basic modal language: for example, it is equivalent to the bisimulation-invariant fragment of FOL over finite S4 models, whereas the basic modal language is weaker. However, previous analyses of this logic have been entirely semantic, and no proof system was available. In this paper we present a sound proof system for the polyadic S4 and prove that it is complete. The axiomatization is fairly standard, adding only the fixpoint axioms of the tangled closure to the usual S4 axioms. The proof proceeds by explicitly constructing a finite model from a consistent set of formulas.

Logics for information systems and their dynamic extensions

Abstract: The article proposes logics for information systems, which provide information about a set of objects regarding a set of attributes. Both “complete” and “incomplete” information systems are dealt with. The language of these logics contains modal operators, and constants corresponding to attributes and attribute values. Sound and complete deductive systems for these logics are presented, and the problem of decidability is addressed. Furthermore, notions of information and information update are defined, and dynamic extensions of the above logics are presented to accommodate these notions. A set of reduction axioms enables us to obtain a complete axiomatization of the dynamic logics.

Dualities for Algebras of Fitting's Many-Valued Modal Logics

Abstract: Stone-type duality connects logic, algebra, and topology in both conceptual and technical senses. This paper is intended to be a demonstration of this slogan. In this paper we focus on some versions of Fitting's L-valued logic and L-valued modal logic for a finite distributive lattice L. Building upon the theory of natural dualities, which is a universal algebraic theory of categorical dualities, we establish a Jonsson-Tarski-style duality for algebras of L-valued modal logic, which encompasses Jonsson-Tarski duality for modal algebras as the case L = 2. We also discuss how the dualities change when the algebras are enriched by truth constants. Topological perspectives following from the dualities provide compactness theorems for the logics and the effective classification of categories of algebras involved, which tells us that Stone-type duality makes it possible to use topology for logic and algebra in significant ways. The author is grateful to Professor Susumu Hayashi for his encouragement, to Shohei Izawa for his comments and discussions, and to Kentaro Sato for his suggesting a similar result to Theorem 2.5 for the category of algebras of Lukasiewicz n-valued logic.

On bisimulation and model-checking for concurrent systems with partial order semantics

Abstract: In concurrency theory—the branch of (theoretical) computer science that studies the logical and mathematical foundations of parallel computation—there are two main formal ways of modelling the behaviour of systems where multiple actions or events can happen independently and at the same time: either with interleaving or with partial order semantics. On the one hand, the interleaving semantics approach proposes to reduce concurrency to the nondeterministic, sequential computation of the events the system can perform independently. On the other hand, partial order semantics represent concurrency explicitly by means of an independence relation on the set of events that the system can execute in parallel; following this approach, the so-called ‘true concurrency’ approach, independence or concurrency is a primitive notion rather than a derived concept as in the interleaving framework. Using interleaving or partial order semantics is, however, more than a matter of taste. In fact, choosing one kind of semantics over the other can have important implications—both from theoretical and practical viewpoints—as making such a choice can raise different issues, some of which we investigate here. More specifically, this thesis studies concurrent systems with partial order semantics and focuses on their bisimulation and model-checking problems; the theories and techniques herein apply, in a uniform way, to different classes of Petri nets, event structures, and transition system with independence (TSI) models. Some results of this work are: a number of mu-calculi (in this case, fixpoint extensions of modal logic) that, in certain classes of systems, induce exactly the same identifications as some of the standard bisimulation equivalences used in concurrency. Secondly, the introduction of (infinite) higher-order logic games for bisimulation and for model-checking, where the players of the games are given (local) monadic second-order power on the sets of elements they are allowed to play. And, finally, the formalization of a new order-theoretic concurrent game model that provides a uniform approach to bisimulation and model-checking and bridges some mathematical concepts in order theory with the more operational world of games. In particular, we show that in all cases the logic games for bisimulation and model-checking developed in this thesis are sound and complete, and therefore, also determined—even when considering models of infinite state systems; moreover, these logic games are decidable in the finite case and underpin novel decision procedures for systems verification. Since the mu-calculi and (infinite) logic games studied here generalisewell-known fixpoint modal logics as well as game-theoretic decision procedures for analysing concurrent systems with interleaving semantics, this thesis provides some of the groundwork for the design of a logic-based, game-theoretic framework for studying, in a uniform manner, several concurrent systems regardless of whether they have an interleaving or a partial order semantics.

Model checking for modal intuitionistic dependence logic

Abstract: Modal intuitionistic dependence logic ($\mathcal MIDL $) incorporates the notion of "dependence" between propositions into the usual modal logic and has connectives which correspond to intuitionistic connectives in a certain sense. It is the modal version of a variant of first-order dependence logic (Vaananen 2007) considered by Abramsky and Vaananen (2009) basing on Hodges' team semantics (1997). In this paper, we study the computational complexity of the model checking problem for $\mathcal MIDL$ and its fragments built by restricting the operators allowed in the logics. In particular, we show that the model checking problem for $\mathcal MIDL$ in general is PSPACE-complete and that for propositional intuitionistic dependence logic is coNP-complete.

Combinations and completeness transfer for quantified modal logics

Abstract: This paper focuses on three research questions which are connected with combinations of modal logics: (i) Under which conditions can (frame-)completeness (and related properties) be transferred from a propositional modal logic (PML) to its quantificational counterpart (QML)? (ii) Does (frame-) completeness generally transfer from monomodal QMLs to their multimodal combination? (iii) Can completeness be transferred from QMLs with rigid designators to those with non-rigid designators? The paper reports some recent results on these questions and provides some new results.

Conceptual modeling in full computation-tree logic with sequence modal operator

Abstract: In this paper, we propose a method for modeling concepts in full computation-tree logic with sequence modal operators. An extended full computation-tree logic, CTLS*, is introduced as a Kripke semantics with a sequence modal operator. This logic can appropriately represent hierarchical tree structures in cases where sequence modal operators in CTLS* are applied to tree structures. We prove a theorem for embedding CTLS* into CTL*. The validity, satisfiability, and model-checking problems of CTLS* are shown to be decidable. An illustrative example of biological taxonomy is presented using CTLS* formulas. © 2011 Wiley Periodicals, Inc. (This paper is an extended version of Kamide and Kaneiwa.)

Graphical representation of covariant-contravariant modal formulae

Abstract: Covariant-contravariant simulation is a combination of standard (covariant) simulation, its contravariant counterpart and bisimulation. We have previously studied its logical characterization by means of the covariant-contravariant modal logic. Moreover, we have investigated the relationships between this model and that of modal transition systems, where two kinds of transitions (the so-called may and must transitions) were combined in order to obtain a simple framework to express a notion of refinement over state-transition models. In a classic paper, Boudol and Larsen established a precise connection between the graphical approach, by means of modal transition systems, and the logical approach, based on Hennessy-Milner logic without negation, to system specification. They obtained a (graphical) representation theorem proving that a formula can be represented by a term if, and only if, it is consistent and prime. We show in this paper that the formulae from the covariant-contravariant modal logic that admit a "graphical" representation by means of processes, modulo the covariant-contravariant simulation preorder, are also the consistent and prime ones. In order to obtain the desired graphical representation result, we first restrict ourselves to the case of covariant-contravariant systems without bivariant actions. Bivariant actions can be incorporated later by means of an encoding that splits each bivariant action into its covariant and its contravariant parts.