Showing papers on "Normal modal logic published in 2013"

Many-Dimensional Modal Logics: Theory and Applications

Abstract: I Introduction 1 Modal logic basics 1.1 Modal axiomatic systems 1.2 Possible world semantics 1.3 Classical first-order logic and the standard translation 1.4 Multimodal logics 1.5 Algebraic semantics 1.6 Decision, complexity and axiomatizability problems 2 Applied modal logic 2.1 Temporal logic 2.2 Interval temporal logic 2.3 Epistemic logic 2.4 Dynamic logic 2.5 Description logic 2.6 Spatial logic 2.7 Intuitionistic logic 2.8 'Model level' reductions between logics 3 Many-dimensional modal logics 3.1 Fusions 3.2 Spatio-temporal logics 3.3 Products 3.4 Temporal epistemic logics 3.5 Classical first-order logic as a propositional multimodal logic 3.6 First-order modal logics 3.7 First-order temporal logics 3.8 Description logics with modal operators 3.9 HS as a two-dimensional logic 3.10 Modal transition logics 3.11 Intuitionistic modal logics II Fusions and products 4 Fusions of modal logics 4.1 Preserving Kripke completeness and the finite model property 4.2 Algebraic preliminaries 4.3 Preserving decidability of global consequence 4.4 Preserving decidability 4.5 Preserving interpolation 4.6 On the computational complexity of fusions 5 Products of modal logics: introduction 5.1 Axiomatizing products 5.2 Proving decidability with quasimodels 5.3 The finite model property 5.4 Proving undecidability 5.5 Proving complexity with tilings 6 Decidable products 6.1 Warming up: Kn x Km 6.2 CPDL x K_m 6.3 Products of epistemic logics with Km 6.4 Products of temporal logics with Km 6.5 Products with S5 6.6 Products with multimodal S5 7 Undecidable products 7.1 Products of linear orders with infinite ascending chains 7.2 Products of linear orders with infinite descending chains 7.3 Products of Dedekind complete linear orders 7.4 Products of finite linear orders 7.5 More undecidable products 8 Higher-dimensional products 8.1 S5 x S5 x ... x S5 8.2 Products between K4 x K4 x ... x K4 and S5 x S5 x ... x S5 8.3 Products with the fmp 8.4 Between K x K x ... x K and S5 x S5 x ... x S5 8.5 Finitely axiomatizable and decidable products 9 Variations on products 9.1 Relativized products 9.2 Valuation restrictions 10 Intuitionistic modal logics 10.1 Intuitionistic modal logics with Box 10.2 Intuitionistic modal logics with Box and Diamond 10.3 The finite model property III First-order modal logics 11 Fragments of first-order temporal logics 11.1 Undecidable fragments 11.2 Monodic formulas, decidable fragments 11.3 Embedding into monadic second-order theories 11.4 Complexity of decidable fragments of QLogSU(N) 11.5 Satisfiability in models over (N,<) with finite domains 11.6 Satisfiability in models over (R,<) with finite domains 11.7 Axiomatizing monodic fragments 11.8 Monodicity and equality 12 Fragments of first-order dynamic and epistemic logics 12.1 Decision problems 12.2 Axiomatizing monodic fragments IV Applications to knowledge representation 13 Temporal epistemic logics 13.1 Synchronous systems 13.2 Agents who know the time and neither forget nor learn 14 Modal description logics 14.1 Concept satisfiability 14.2 General formula satisfiability 14.3 Restricted formula satisfiability 14.4 Satisfiability in models with finite domains 15 Tableaux for modal description logics 15.1 Tableaux for ALC 15.2 Tableaux for K(ALC) with constant domains 15.3 Adding expressive power to K(ALC) 16 Spatio-temporal logics 16.1 Modal formalisms for spatio-temporal reasoning 16.2 Embedding spatio-temporal logics in first-order temporal logic 16.3 Complexity of spatio-temporal logics 16.4 Models based on Euclidean spaces Epilogue. Bibliography. List of tables. List of languages and logics. Symbol index. Subject index.

Modal Logic as Metaphysics

Abstract: Preface 1. Contingentism and Necessitism 2. The Barcan Formula and its Converse: Early Developments 3. Possible Worlds Model Theory 4. Predication and Modality 5. From First-Order to Higher-Order Modal Logic 6. Intensional Comprehension Principles and Metaphysics 7. Mappings between Contingentist and Necessitist Discourse 8. Consequences of necessitism Methodological Afterword Bibliography Index

Complexity Results for Modal Dependence Logic

Abstract: Modal dependence logic was introduced recently by Vaananen. It enhances the basic modal language by an operator = (). For propositional variables p 1, . . . , p n , = (p 1, . . . , p n-1, p n ) intuitively states that the value of p n is determined by those of p 1, . . . , p n-1. Sevenster (J. Logic and Computation, 2009) showed that satisfiability for modal dependence logic is complete for nondeterministic exponential time. In this paper we consider fragments of modal dependence logic obtained by restricting the set of allowed propositional connectives. We show that satisfiability for poor man's dependence logic, the language consisting of formulas built from literals and dependence atoms using $${\wedge, \square, \lozenge}$$ (i. e., disallowing disjunction), remains NEXPTIME-complete. If we only allow monotone formulas (without negation, but with disjunction), the complexity drops to PSPACE-completeness. We also extend Vaananen's language by allowing classical disjunction besides dependence disjunction and show that the satisfiability problem remains NEXPTIME-complete. If we then disallow both negation and dependence disjunction, satisfiability is complete for the second level of the polynomial hierarchy. Additionally we consider the restriction of modal dependence logic where the length of each single dependence atom is bounded by a number that is fixed for the whole logic. We show that the satisfiability problem for this bounded arity dependence logic is PSPACE-complete and that the complexity drops to the third level of the polynomial hierarchy if we then disallow disjunction. In this way we completely classify the computational complexity of the satisfiability problem for all restrictions of propositional and dependence operators considered by Vaananen and Sevenster.

Extending Łukasiewicz Logics with a Modality: Algebraic Approach to Relational Semantics

Abstract: This paper presents an algebraic approach of some many-valued generalizations of modal logic. The starting point is the definition of the [0, 1]-valued Kripke models, where [0, 1] denotes the well known MV-algebra. Two types of structures are used to define validity of formulas: the class of frames and the class of Łn-valued frames. The latter structures are frames in which we specify in each world u the set (a subalgebra of Łn) of the allowed truth values of the formulas in u. We apply and develop algebraic tools (namely, canonical and strong canonical extensions) to generate complete modal n + 1-valued logics and we obtain many-valued counterparts of Shalqvist canonicity result.

From Frame Properties to Hypersequent Rules in Modal Logics

Abstract: We provide a general method for generating cut-free and/or analytic hyper sequent Gent Zen-type calculi for a variety of normal modal logics. The method applies to all modal logics characterized by Kripke frames, transitive Kripke frames, or symmetric Kripke frames satisfying some properties, given by first-order formulas of a certain simple form. This includes the logics KT, KD, S4, S5, K4D, K4.2, K4.3, KBD, KBT, and other modal logics, for some of which no Gentzen calculi was presented before. Cut-admissibility (or analyticity in the case of symmetric Kripke frames) is proved semantically in a uniform way for all constructed calculi. The decidability of each modal logic in this class immediately follows.

First-Order Resolution Methods for Modal Logics

Abstract: In this paper we give an overview of results for modal logic which can be shown using techniques and methods from first-order logic and resolution. Because of the breadth of the area and the many applications we focus on the use of first-order resolution methods for modal logics. In addition to traditional propositional modal logics we consider more expressive PDL-like dynamic modal logics closely related to description logics. Without going into too much detail, we survey different ways of translating modal logics into first-order logic, we explore different ways of using first-order resolution theorem provers to solve a range of reasoning problems for modal logics, and we discuss a variety of results which have been obtained in the setting of first-order resolution.

Extended Modal Dependence Logic.

Abstract: In this paper we extend modal dependence logic $\mathcal{MDL}$ by allowing dependence atoms of the form depi¾ź 1,',i¾ź n where i¾ź i , 1≤i≤n, are modal formulas in $\mathcal{MDL}$ , only propositional variables are allowed in dependence atoms. The reasoning behind this extension is that it introduces a temporal component into modal dependence logic. E.g., it allows us to express that truth of propositions in some world of a Kripke structure depends only on a certain part of its past. We show that $\mathcal{EMDL}$ strictly extends $\mathcal{MDL}$ , i.e., there exist $\mathcal{EMDL}$ -formulas which are not expressible in $\mathcal{MDL}$ . However, from an algorithmic point of view we do not have to pay for this since we prove that the complexity of satisfiability and model checking of $\mathcal{EMDL}$ and $\mathcal{MDL}$ coincide. In addition we show that $\mathcal{EMDL}$ is equivalent to $\mathcal{ML}$ extended by a certain propositional connective.

Four-valued modal logic: Kripke semantics and duality.

Abstract: Combining multi-valued and modal logics into a single system is a long-standing concern in mathematical logic and computer science, see for example [7] and the literature cited there. Recent work in this trend [15, 17, 14] develops modal expansions of many-valued systems that are also inconsistencytolerant, along the tradition initiated by Belnap with his “useful four-valued logic” [3]. Our contribution continues on this line, and the specific problem we address is that of defining and axiomatizing the least modal logic over the four-element Belnap lattice. The problem was inspired by [5], but our solution is quite different from (and in some respects more satisfactory than) that of [5] in that we make an extensive and profitable use of algebraic and topological techniques. In fact, our algebraic and topological analyses of the logic have, in our opinion, an independent interest and contribute to the appeal of our approach. Kripke frames provide a semantics for modal logics that is both flexible with regards to intended applications and interpretations, and highly intuitive. When the non-modal part is multi-valued, though, one may wonder whether the accessibility relation between worlds should remain two-valued or be allowed to assume the same range of truth values as the logic itself. Starting from the point of view of AI applications, [7] argues forcefully that multiple values are an appropriate and useful modeling device. This is the approach taken in [5] and here, too. Our aim is to study the least modal logic over the Belnap lattice, that is, the logic determined by the class of all Kripke frames where the accessibility relation as well as semantic valuations are four-valued.

A Finite Model Property for Gödel Modal Logics

Abstract: A new semantics with the finite model property is provided and used to establish decidability for Godel modal logics based on crisp or fuzzy Kripke frames combined locally with Godel logic. A similar methodology is also used to establish decidability, and indeed co-NP-completeness for a Godel S5 logic that coincides with the one-variable fragment of first-order Godel logic.

Products of modal logics

Abstract: ● In the early days of modal logic (before 1980s) there was interest in studying multiple particular systems. Contemporary modal logic also investigates classes of logics and general constructions combining different systems. ● Products were introduced in the 1970s; their intensive study started in the 1990s. Motivations for studying products of modal propositional logics • A natural type of combined modal logics

Kripke Semantics for Modal Bilattice Logic

Abstract: We employ the well-developed and powerful techniques of algebraic semantics and Priestley duality to set up a Kripke semantics for a modal expansion of Arieli and Avron's bilattice logic, itself based on Belnap's four-valued logic. We obtain soundness and completeness of a Hilbert-style derivation system for this logic with respect to four-valued Kripke frames, the standard notion of model in this setting. The proof is via intermediary relational structures which are analysed through a topological reading of one of the axioms of the logic. Both local and global consequence on the models are covered.

Epistemic Updates on Algebras

Abstract: We develop the mathematical theory of epistemic updates with the tools of duality theory. We focus on the Logic of Epistemic Actions and Knowledge (EAK), introduced by Baltag-Moss- Solecki, without the common knowledge operator. We dually characterize the product update construction of EAK as a certain construction transforming the complex algebras associated with the given model into the complex algebra associated with the updated model. This dual characterization naturally generalizes to much wider classes of algebras, which include, but are not limited to, arbitrary BAOs and arbitrary modal expansions of Heyting algebras (HAOs). As an application of this dual characterization, we axiomatize the intuitionistic analogue of the logic of epistemic knowledge and actions, which we refer to as IEAK, prove soundness and completeness of IEAK w.r.t. both algebraic and relational models, and illustrate how IEAK encodes the reasoning of agents in a concrete epistemic scenario.

Correspondence between Modal Hilbert Axioms and Sequent Rules with an Application to S5

Abstract: Which modal logics can be ‘naturally’ captured by a sequent system? Clearly, this question hinges on what one believes to be natural, i.e. which format of sequent rules one is willing to accept. This paper studies the relationship between the format of sequent rules and the corresponding syntactical shape of axioms in an equivalent Hilbert-system. We identify three different such formats, the most general of which captures most logics in the S5-cube. The format is based on restricting the context in rule premises and the correspondence is established by translating axioms into rules of our format and vice versa. As an application we show that there is no set of sequent rules of this format which is sound and cut-free complete for S5 and for which cut elimination can be shown by the standard permutation-of-rules argument.

The Logic of Exact Covers: Completeness and Uniform Interpolation

Abstract: We show that all (not necessarily normal or monotone) modal logics that can be axiomatised in rank-1 have the interpolation property, and that in fact interpolation is uniform if the logics just have finitely many modal operators. As immediate applications, we obtain previously unknown interpolation theorems for a range of modal logics, containing probabilistic and graded modal logic, alternating temporal logic and some variants of conditional logic. Technically, this is achieved by translating to and from a new (coalgebraic) logic introduced in this paper, the logic of exact covers. It is interpreted over coalgebras for an endofunctor on the category of sets that also directly determines the syntax. Apart from closure under bisimulation quantifiers (and hence interpolation), we also provide a complete tableaux calculus and establish both the Hennessy-Milner and the small model property for this logic.

Modal Logic and Distributed Message Passing Automata

Abstract: In a recent article, Lauri Hella and co-authors identify a canonical connection between modal logic and deterministic distributed constant-time algorithms. The paper reports a variety of highly natural logical characterizations of classes of distributed message passing automata that run in constant time. The article leaves open the question of identifying related logical characterizations when the constant running time limitation is lifted. We obtain such a characterization for a class of finite message passing automata in terms of a recursive bisimulation invariant logic which we call modal substitution calculus (MSC). We also give a logical characterization of the related class A of infinite message passing automata by showing that classes of labelled directed graphs recognizable by automata in A are exactly the classes co-definable by a modal theory. A class C is co-definable by a modal theory if the complement of C is definable by a possibly infinite set of modal formulae. We also briefly discuss expressivity and decidability issues concerning MSC. We establish that MSC contains the Sigma^\mu_1 fragment of the modal \mu-calculus in the finite. We also observe that the single variable fragment MSC^1 of MSC is not contained in MSO, and that the SAT and FINSAT problems of MSC^1 are complete for PSPACE.

Constructing Cut Free Sequent Systems with Context Restrictions Based on Classical or Intuitionistic Logic

Abstract: We consider a general format for sequent rules for not necessarily normal modal logics based on classical or intuitionistic propositional logic and provide relatively simple local conditions ensuring cut elimination for such rule sets. The rule format encompasses e.g. rules for the boolean connectives and transitive modal logics such as S4 or its constructive version. We also adapt the method of constructing suitable rule sets by saturation to the intuitionistic setting and provide a criterium for translating axioms for intuitionistic modal logics into sequent rules. Examples include constructive modal logics and conditional logic \(\mathbb{VA}\).

A Modal BI Logic for Dynamic Resource Properties

Abstract: The logic of Bunched implications (BI) and its variants or extensions provide a powerful framework to deal with resources having static properties. In this paper, we propose a modal extension of BI logic, called DBI, which allows us to deal with dynamic resource properties. After defining a Kripke semantics for DBI, we illustrate the interest of DBI for expressing some dynamic properties and then we propose a labelled tableaux calculus for this logic. This calculus is proved sound and complete w.r.t. the Kripke semantics. Moreover, we also give a method for countermodel generation in this logic.

Three-valued Logics in Modal Logic

Abstract: Every truth-functional three-valued propositional logic can be conservatively translated into the modal logic S5. We prove this claim constructively in two steps. First, we define a Translation Manual that converts any propositional formula of any three-valued logic into a modal formula. Second, we show that for every S5-model there is an equivalent three-valued valuation and vice versa. In general, our Translation Manual gives rise to translations that are exponentially longer than their originals. This fact raises the question whether there are three-valued logics for which there is a shorter translation into S5. The answer is affirmative: we present an elegant linear translation of the Logic of Paradox and of Strong Three-valued Logic into S5.

Quantificational Logic and Empty Names

Abstract: © 2013, Philosophers’ Imprint This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License The classical theory of quantification is subject to a number of difficulties relating to contingent existence and to the treatment of both empty and non-empty names. In this paper I propose and defend a modal metaphysics, couched in a weakening of classical logic, that avoids these objections. It is well known that classical quantification theory does not provide a straightforward treatment of empty names. According to the simplest way of translating between English and first-order logic there are false sentences of English which translate to theorems of first-order logic. For a given first-order language, L, classical quantification theory proves every instance of the schema

A logic for epistemic two-dimensional semantics

Abstract: Epistemic two-dimensional semantics is a theory in the philosophy of language that provides an account of meaning which is sensitive to the distinction between necessity and apriority. While this theory is usually presented in an informal manner, I take some steps in formalizing it in this paper. To do so, I define a semantics for a propositional modal logic with operators for the modalities of necessity, actuality, and apriority that captures the relevant ideas of epistemic two-dimensional semantics. I also describe some properties of the logic that are interesting from a philosophical perspective, and apply it to the so-called nesting problem.

Hennessy-Milner logic with greatest fixed points as a complete behavioural specification theory

Abstract: There are two fundamentally different approaches to specifying and verifying properties of systems. The logical approach makes use of specifications given as formulae of temporal or modal logics and relies on efficient model checking algorithms; the behavioural approach exploits various equivalence or refinement checking methods, provided the specifications are given in the same formalism as implementations. In this paper we provide translations between the logical formalism of Hennessy-Milner logic with greatest fixed points and the behavioural formalism of disjunctive modal transition systems. We also introduce a new operation of quotient for the above equivalent formalisms, which is adjoint to structural composition and allows synthesis of missing specifications from partial implementations. This is a substantial generalisation of the quotient for deterministic modal transition systems defined in earlier papers.

Moving Up and Down in the Generic Multiverse

Abstract: We investigate the modal logic of the generic multiverse which is a bimodal logic with operators corresponding to the relations “is a forcing extension of” and “is a ground model of”. The fragment of the first relation is the modal logic of forcing and was studied by the authors in earlier work. The fragment of the second relation is the modal logic of grounds and will be studied here for the first time. In addition, we discuss which combinations of modal logics are possible for the two fragments.

A General Lindström Theorem for Some Normal Modal Logics

Abstract: There are several known Lindstrom-style characterization results for basic modal logic. This paper proves a generic Lindstrom theorem that covers any normal modal logic corresponding to a class of Kripke frames definable by a set of formulas called strict universal Horn formulas. The result is a generalization of a recent characterization of modal logic with the global modality. A negative result is also proved in an appendix showing that the result cannot be strengthened to cover every first-order elementary class of frames. This is shown by constructing an explicit counterexample.

A Characterization Theorem for the Alternation-Free Fragment of the Modal µ-Calculus

Abstract: We provide a characterization theorem, in the style of van Ben them and Janin-Walukiewicz, for the alternation-free fragment of the modal mu-calculus. For this purpose we introduce a variant of standard monadic second-order logic (MSO), which we call well-founded monadic second-order logic (WFMSO). When interpreted in a tree model, the second-order quantifiers of WFMSO range over subsets of conversely well-founded sub trees. The first main result of the paper states that the expressive power of WFMSO over trees exactly corresponds to that of weak MSO-automata. Using this automata-theoretic characterization, we then show that, over the class of all transition structures, the bisimulation-invariant fragment of WFMSO is the alternation-free fragment of the modal mu-calculus. As a corollary, we find that the logics WFMSO and WMSO (weak monadic second-order logic, where second-order quantification concerns finite subsets), are incomparable in expressive power.

A modal theorem-preserving translation of a class of three-valued logics of incomplete information

Abstract: There are several three-valued logical systems that form a scattered landscape, even if all reasonable connectives in three-valued logics can be derived from a few of them. Most papers on this subject neglect the issue of the relevance of such logics in relation with the intended meaning of the third truth-value. Here, we focus on the case where the third truth-value means unknown, as suggested by Kleene. Under such an understanding, we show that any truth-qualified formula in a large range of three-valued logics can be translated into KD as a modal formula of depth 1, with modalities in front of literals only, while preserving all tautologies and inference rules of the original three-valued logic. This simple information logic is a two-tiered classical propositional logic with simple semantics in terms of epistemic states understood as subsets of classical interpretations. We study in particular the translations of Kleene, Godel, ᴌukasiewicz and Nelson logics. We show that Priest’s logic of paradox, closely connected to Kleene’s, can also be translated into our modal setting, simply by exchanging the modalities possible and necessary. Our work enables the precise expressive power of three-valued logics to be laid bare for the purpose of uncertainty management.

Actuality in Propositional Modal Logic

Abstract: We show that the actuality operator A is redundant in any propositional modal logic characterized by a class of Kripke models (respectively, neighborhood mod- els). Specifically, we prove that for every formula φ in the propositional modal language with A, there is a formula ψ not containing A such that φ and ψ are materially equivalent at the actual world in every Kripke model (respectively, neighborhood model). Inspection of the proofs leads to corresponding proof-theoretic results concerning the eliminability of the actuality operator in the actuality extension of any normal propositional modal logic and of any "classical" modal logic. As an application, we provide an alternative proof of a result of Williamson's to the effect that the compound operator A behaves, in any normal logic between T and S5, like the simple necessity operator in S5.

RESEARCH ARTICLE A modal theorem-preserving translation of a class of three-valued logics of incomplete information

Abstract: There are several three-valued logical systems that form a scattered landscape, even if all reasonable connectives in three-valued logics can be derived from a few of them. Most papers on this subject neglect the issue of the relevance of such logics in relation with the intended meaning of the third truth value. Here, we focus on the case where the third truth-value means unknown, as suggested by Kleene. Under such an understanding, we show that any truth-qualied formula in a large range of three-valued logics can be translated into KD as a modal formula of depth 1, with modalities in front of literals only, while preserving all tautologies and inference rules of the original three-valued logic. This simple information logic is a two-tiered classical propositional logic with simple semantics in terms of epistemic states understood as subsets of classical interpretations. We study in particular the translations of Kleene, Godel, Lukasiewicz and Nelson logics. We show that Priest logic of paradox, closely connected to Kleene's, can also be translated into our modal setting, just exchanging the modalities possible and necessary. Our work enables the precise expressive power of three-valued logics to be laid bare for the purpose of uncertainty management.