Showing papers on "Normal modal logic published in 2007"

Handbook of Modal Logic

Abstract: The Handbook of Modal Logic contains 20 articles, which collectively introduce contemporary modal logic, survey current research, and indicate the way in which the field is developing. The articles survey the field from a wide variety of perspectives: the underling theory is explored in depth, modern computational approaches are treated, and six major applications areas of modal logic (in Mathematics, Computer Science, Artificial Intelligence, Linguistics, Game Theory, and Philosophy) are surveyed. The book contains both well-written expository articles, suitable for beginners approaching the subject for the first time, and advanced articles, which will help those already familiar with the field to deepen their expertise. Please visit: http://people.uleth.ca/~woods/RedSeriesPromo_WP/PubSLPR.html

Handbook of Spatial Logics

Abstract: What is Spatial Logic?.- First-Order Mereotopology.- Axioms, Algebras and Topology.- Qualitative Spatial Reasoning Using Constraint Calculi.- Modal Logics of Space.- Topology and Epistemic Logic.- Logical Theories for Fragments of Elementary Geometry.- Locales and Toposes as Spaces.- Spatial Logic + Temporal Logic = ?.- Dynamic Topological Logic.- Logic of Space-Time and Relativity Theory.- Discrete Spatial Models.- Real Algebraic Geometry and Constraint Databases.- Mathematical Morphology.- Spatial Reasoning and Ontology: Parts, Wholes, and Locations.

14 Hybrid logics

Abstract: Publisher Summary This chapter discusses the proof theory, expressivity, and complexity of a number of the well-known hybrid logics and provides a snapshot of the logical territory lying between the basic modal languages and their classical companions. Standard modal logics use modalities for talking about the relations in relational structure. Hybrid logics are extensions of standard modal logics, involving symbols that name individual states in models. Hybrid logic arises when mechanisms for naming and asserting identity of worlds are added; to give an analogy, they are to standard modal systems what first-order languages with equality are to equality-free languages. The chapter provides an overview of the field of hybrid logic and presents a coherent picture of the current state of affairs in the field of hybrid logic. The most important techniques and results in the field, with respect to completeness, expressive power, frame definability, interpolation, and complexity are discussed. The proof systems for hybrid languages such as sequents, natural deduction, or tableaux, and some issues concerning the development of automated provers based on them are also described.

First Order Modal Logic

Abstract: Publisher Summary First-order modal logics are modal logics in which the underlying propositional logic is replaced by a first-order predicate logic. They pose some of the most difficult mathematical challenges. This chapter surveys basic first-order modal logics and examines recent attempts to find a general mathematical setting in which to analyze them. A number of logics that make use of constant domain, increasing domain, and varying domain semantics is discussed, and a first-order intensional logic and a first-order version of hybrid logic is presented. One criterion for selecting these logics is the availability of sound and complete proof procedures for them, typically axiom systems and/or tableau systems. The first-order modal logics are compared to fragments of sorted first-order logic through appropriate versions of the standard translation. Both positive and negative results concerning fragment decidability, Kripke completeness, and axiomatizability are reviewed. Modal hyperdoctrines are introduced as a unifying tool for analyzing the alternative semantics. These alternative semantics range from specific semantics for non-classical logics, to interpretations in well-established mathematical framework. The relationship between topological semantics and D. Lewis's counterpart semantics is investigated and an axiomatization is presented.

Model theory of modal logic.

Abstract: Publisher Summary This chapter presents a theoretical analysis of modal logic that can be applied to many application areas. It presents the central core of contemporary insight into the mathematical structure of modal logic. Model theory is about semantics; it studies the interplay between a logical language (logic) and the models (structures) for that language. Modal logics come as members of a loosely knit family and have various links to other logics––classical first- and second-order logic as well as, temporal and process logics stemming from particular applications. The chapter introduces some key notions; in particular, the different levels of semantics, followed by a discussion of the concept of bisimulation and bisimulation respecting model constructions. The study of modal logic as logic of frames is described. This comprises advanced constructions such as ultrafilter extensions and ultraproducts, basic model theory of general frames, and a survey of classical results on frame definability and relations with second-order logic. The semantics of modal logic has two emblematic features, which have a crucial impact on its model theory––namely, modal logic is local and multi-layered.

Modal logic: a semantic perspective

Abstract: Publisher Summary This chapter discusses the semantic ideas underlying modern modal logic, and in particular, Kripke semantics—or relational semantics. It introduces the basic model theoretic constructions, explores links between modal logic and classical logic, both on models and on frames, and examines the extent to which the key semantic ideas transfer to richer modal logics and languages while maintaining a relatively low computational complexity. The basic modal languages and the graphs over which they are interpreted are discussed. The chapter also introduces the notion of bisimulation, based on which, modal logic as a fragment of first-order logic is characterized. The computability and computational complexity of modal logic is examined. The level of frames and the link between modal and classical logic are explored. Three alternatives to relational semantics––namely, algebraic semantics, neighborhood semantics, and topological semantics are also discussed.

Modal Logics of Space

Abstract: 1 The interest of Space When thinking about the physical world, modal logicians have taken Time as their main theme, because it fits so well with an interest in the flow of information and computation. Spatial logics have been footnotes to the tradition – even though the axiomatic method was largely geometrical. An exception is Tarski's early work on deviant geometrical primitives, and his decidable first-order axiomatization of elementary geometry. Today Space remains intriguing – both for mathematical reasons, and given the amount of work in CS and AI on visual reasoning and image processing. These two concerns are by no means the same, but both involve logic of spatial structures.

Algebras and Coalgebras

Abstract: Publisher Summary This chapter outlines the development of the algebraic semantics of modal logic and introduces an alternative coalgebraic approach Algebraic semantics is important because it allows general techniques from universal algebra to be applied to the study of modal logic It rises to some of the most penetrating analyses of the mathematics of modality Coalgebras are simple but fundamental mathematical structures that capture the essence of dynamic or evolving systems The theory of universal coalgebra provides a general framework for the study of notions related to behavior such as invariance and observational indistinguishability The more recent coalgebraic approach, which also links up with category theory, is valuable because it offers a uniform mathematical setting to analyze dynamic systems in terms of modal logic The concepts and ideas are discussed and important techniques and landmark results; proofs, or proof sketches, are presented in the chapter

Combining modal logics

Abstract: Publisher Summary This chapter surveys two key combination methods––namely, fusions and products. It also examines some other combination methods. The properties of combined logics depend on those of their components plus the particular method of combination. The idea of combining modal logics is natural for many applications. The chapter explores the way in which modal logics can be combined and highlights the consequences of combining them. The formation of fusions is the simplest and the most natural way of combining modal logics. The formation of Cartesian products of various structures—vector and topological spaces, algebras—is a standard mathematical way of capturing the multidimensional character of the world. In modal logic, products of Kripke frames are natural constructions allowing it to reflect interactions between modal operators representing time, space, knowledge, or actions. The product construction as a combination method on modal logics is introduced and is used in applications in computer science and artificial intelligence.

12 Modal mu-calculi

Abstract: Publisher Summary Modal mu-calculus is a logic used extensively in certain areas of computer science and is of considerable intrinsic mathematical and logical interest. Its defining feature is the addition of inductive definitions to modal logic; thereby it achieves a great increase in expressive power and an equally great increase in difficulty of understanding. It includes many of the logics used in systems verification, and is quite straightforward to evaluate. It also provides one of the strongest examples of the connections between modal and temporal logics, automata theory, and the theory of games. It provides second-order expressive power sufficient to generalize the most common temporal logics, but is still decidable and has the finite model property. It raises many intriguing issues about the interface between modal logic, complexity theory, and automata theory. This chapter surveys a range of the questions and results about the modal mu-calculus and related logics. The logic is defined, some approaches to gaining an intuitive understanding of formulae are described, and the main theorem about the semantics is established. The modal mu-calculus has the tree model property and relates to some other temporal logics, to automata and to games.

Ten Philosophical Problems in Deontic Logic

Abstract: The paper discusses ten philosophical problems in deontic logic: how to formally represent norms, when a set of norms may be termed 'coherent', how to deal with normative conflicts, how contrary- to-duty obligations can be appropriately modeled, how dyadic deontic operators may be redefined to relate to sets of norms instead of pref- erence relations between possible worlds, how various concepts of per- mission can be accommodated, how meaning postulates and counts-as conditionals can be taken into account, and how sets of norms may be revised and merged. The problems are discussed from the viewpoint of input/output logic as developed by van der Torre & Makinson. We ar- gue that norms, not ideality, should take the central position in deontic semantics, and that a semantics that represents norms, as input/output logic does, provides helpful tools for analyzing, clarifying and solving the problems of deontic logic.

The modal logic of forcing

Abstract: A set theoretical assertion psi is forceable or possible, written lozenge psi, if psi holds in some forcing extension, and necessary, written square psi, if psi holds in all forcing extensions. In this forcing interpretation of modal logic, we establish that if ZFC is consistent, then the ZFC-provable principles of forcing are exactly those in the modal theory S4.2.

Topology and Epistemic Logic

Abstract: We present the main ideas behind a number of logical systems for reasoning about points and sets that incorporate knowledge-theoretic 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

A normal simulation of coalition logic and an epistemic extension

Abstract: In this paper we show how coalition logic can be reduced to the fusion of a normal modal STIT logic for agency and a standard normal temporal logic for discrete time, and how this multi-modal system can be suitably extended with an epistemic modality. Both systems are complete, and we provide a new axiomatization for the STIT-fragment. The epistemic extension enables us to express that agents see to something under uncertainty about the present state or uncertainty about which action is being taken. In accordance with established terminology in the planning community, we call this version of STIT the 'conformant STIT'. The conformant STIT enables us to express that agents are able to perform a uniform strategy. As a final word of recommendation for this paper we want to point out that its subject is at the junction of four academic fields, viz. modal logic, philosophy, game-theory and AI-planning.

Modal Logics for Region-based Theories of Space

Abstract: We dedicate this paper to Professor Andrzej Grzegorczyk. His paper "Axiomatization of geometry without points" [20] is one of the first contributions to the region-based theory of space.

15 Combining modal logics

Abstract: Publisher Summary This chapter surveys two key combination methods––namely, fusions and products. It also examines some other combination methods. The properties of combined logics depend on those of their components plus the particular method of combination. The idea of combining modal logics is natural for many applications. The chapter explores the way in which modal logics can be combined and highlights the consequences of combining them. The formation of fusions is the simplest and the most natural way of combining modal logics. The formation of Cartesian products of various structures—vector and topological spaces, algebras—is a standard mathematical way of capturing the multidimensional character of the world. In modal logic, products of Kripke frames are natural constructions allowing it to reflect interactions between modal operators representing time, space, knowledge, or actions. The product construction as a combination method on modal logics is introduced and is used in applications in computer science and artificial intelligence.

A Deep Inference System for the Modal Logic S5

Abstract: We present a cut-admissible system for the modal logic S5 in a formalism that makes explicit and intensive use of deep inference. Deep inference is induced by the methods applied so far in conceptually pure systems for this logic. The system enjoys systematicity and modularity, two important properties that should be satisfied by modal systems. Furthermore, it enjoys a simple and direct design: the rules are few and the modal rules are in exact correspondence to the modal axioms.

Using tableau to decide expressive description logics with role negation

Abstract: This paper presents a tableau approach for deciding description logics outside the scope of OWL DL/1.1 and current state-of-the-art tableau-based description logic systems. In particular, we define a sound and complete tableau calculus for the description logic ALBO and show that it provides a basis for decision procedures for this logic and numerous other description logics with full role negation. ALBO is the extension of ALC with the Boolean role operators, inverse of roles, domain and range restriction operators and it includes full support for nominals (individuals). ALBO is a very expressive description logic which subsumes Boolean modal logic and the two-variable fragment of first-order logic and reasoning in it is NExpTime-complete. An important novelty is the use of a generic, unrestricted blocking rule as a replacement for standard loop checking mechanisms implemented in description logic systems. An implementation of our approach exists in the METTEL system.

Multimo dal Logics of Products of Topologies

Abstract: We introduce the horizontal and vertical topologies on the product of topological spaces, and study their relationship with the standard product topology. We show that the modal logic of products of topological spaces with horizontal and vertical topologies is the fusion S4 ⊕ S4. We axiomatize the modal logic of products of spaces with horizontal, vertical, and standard product topologies.We prove that both of these logics are complete for the product of rational numbers ℚ × ℚ with the appropriate topologies.

Proofnets for S5: sequents and circuits for modal logic

Abstract: . In this paper I introduce a sequent system for the propositional modal logic S5. Derivations of valid sequents in the system are shown to correspond to proofs in a novel natural deduction system of circuit proofs (reminiscient of proofnets in linear logic [9, 15], or multipleconclusion calculi for classical logic [22, 23, 24]). The sequent derivations and proofnets are both simple extensions of sequents and proofnets for classical propositional logic, in which the new machinery—to take account of the modal vocabulary—is directly motivated in terms of the simple, universal Kripke semantics for S5. The sequent system is cut-free (the proof of cut-elimination is a simple generalisation of the systematic cut-elimination proof in Belnap's Display Logic [5, 21, 26]) and the circuit proofs are normalising.

A New Modal Lindström Theorem

Abstract: We prove new Lindstrom theorems for the basic modal propositional language, and for some related fragments of first-order logic. We find difficulties with such results for modal languages without a finite-depth property, high-lighting the difference between abstract model theory for fragments and for extensions of first-order logic. In addition we discuss new connections with interpolation properties, and the modal invariance theorem.

10 Higher order modal logic

Abstract: Publisher Summary This chapter focuses at some of the motivations for combining modality with quantification and abstraction over objects of higher order. The basic ideas of modal logic are extended to higher-order settings and extended in a number of different ways. The chapter discusses Richard Montague's system of “Intensional Logic,” which is by far the most influential of higher order modal logics to date. The logic is not fully intensional, as it validates the axiom of extensionality. This leads to a series of well-known problems centering on “logical omniscience” and the logic is not Church-Rosser. An exposition of a modal type theory is intensional in two ways: in the sense of being a modal logic and in the sense that extensionality does not hold. The chapter explains the basic syntax and semantics of this logic, discusses a tableau calculus, and outlines the elementary model theory in the form of a model existence theorem and its usual corollaries, such as generalized completeness.

A complete and compact propositional deontic logic

Abstract: In this paper we present a propositional deontic logic, with the goal of using it to specify fault-tolerant systems, and an axiomatization of it. We prove several results about this logic: completeness, soundness, compactness and decidability. The main technique used during the completeness proof is based on standard techniques for modal logics, but it has some new characteristics introduced for dealing with this logic. In addition, the logic provides several operators which appear useful for use in practice, in particular to model fault-tolerant systems and to reason about their fault tolerance properties.

Reasoning about Communication Graphs

Abstract: Let us assume that some agents are connected by a communication graph. In the communication graph, an edge from agent i to agent j means that agent i can directly receive information from agent j. Agent i can then refine its own information by learning information that j has, including information acquired by j from another agent, k. We introduce a multi-agent modal logic with knowledge modalities and a modality representing communication among agents. Among other properties, we show that the logic is decidable, that it completely characterizes the communication graph, and that it satisfies the basic properties of the space logic of [18].

Logically Possible Worlds and Counterpart Semantics for Modal Logic

Abstract: Publisher Summary This chapter focuses on logically possible worlds and counterpart semantics for modal logic. Originally conceived as the logic of necessity and possibility, philosophical roots of modal logic go back at least as far as Aristotle and the Stoic Diodorus Cronus. The most common semantics of modal logics introduce the notions of a world, a possibility, or a situation, and impose further structure by means of more or less complex relations, for example, by the notion of accessibility. Modal logic does not embody the commitment to possible worlds of any sort—rather the doctrines of modal realism and anti-realism are subject to considerable philosophical debate. The languages of modal predicate logic that is considered differs from the language of modal propositional logic as follows. First, there are neither propositional variables nor propositional constants. Second, first-order and second-order languages of modal predicate logic differ syntactically as well as with respect to the substitution principles assumed. This chapter discusses the notion of modal predicate logic, objects in counterpart frames, and the semantical impact of Haecceitism. Details of metaframes and status of the modal language are also presented in this chapter.

Probabilistic modal logic

Abstract: A modal logic is any logic for handling modalities: concepts like possibility, necessity, and knowledge. Artificial intelligence uses modal logics most heavily to represent and reason about knowledge of agents about others' knowledge. This type of reasoning occurs in dialog, collaboration, and competition. In many applications it is also important to be able to reason about the probability of beliefs and events. In this paper we provide a formal system that represents probabilistic knowledge about probabilistic knowledge. We also present exact and approximate algorithms for reasoning about the truth value of queries that are encoded as probabilistic modal logic formulas. We provide an exact algorithm which takes a probabilistic Kripke stntcture and answers probabilistic modal queries in polynomial-time in the size of the model. Then, we introduce an approximate method for applications in which we have very many or infinitely many states. Exact methods are impractical in these applications and we show that our method returns a close estimate efficiently.