Showing papers on "Modal operator published in 2001"

A judgmental reconstruction of modal logic

Abstract: We reconsider the foundations of modal logic, following Martin-Lof's methodology of distinguishing judgments from propositions. We give constructive meaning explanations for necessity and possibility, which yields a simple and uniform system of natural deduction for intuitionistic modal logic that does not exhibit anomalies found in other proposals. We also give a new presentation of lax logic and find that the lax modality is already expressible using possibility and necessity. Through a computational interpretation of proofs in modal logic we further obtain a new formulation of Moggi's monadic metalanguage.

Modal Logics and mu-Calculi: An Introduction

Abstract: We briefly survey the background and history of modal and temporal logics. We then concentrate on the modal mu-calculus, a modal logic which subsumes most other commonly used logics. We provide an informal introduction, followed by a summary of the main theoretical issues. We then look at model-checking, and finally at the relationship of modal logics to other formalisms.

Decidable fragments of first-order modal logics

Abstract: The paper considers the set of first-order polymodal formulas the modal operators in which can be applied to subformulas of at most one free variable. Using a mosaic technique, we prove a general satisfiability criterion for formulas in , which reduces the modal satisfiability to the classical one. The criterion is then used to single out a number of new, in a sense optimal, decidable fragments of various modal predicate logics.

Multi‐agent Only Knowing

Abstract: Levesque introduced a notion of ``only knowing'', with the goal of capturing certain types of nonmonotonic reasoning. Levesque's logic dealt with only the case of a single agent. Recently, both Halpern and Lakemeyer independently attempted to extend Levesque's logic to the multi-agent case. Although there are a number of similarities in their approaches, there are some significant differences. In this paper, we reexamine the notion of only knowing, going back to first principles. In the process, we simplify Levesque's completeness proof, and point out some problems with the earlier definitions. This leads us to reconsider what the properties of only knowing ought to be. We provide an axiom system that captures our desiderata, and show that it has a semantics that corresponds to it. The axiom system has an added feature of interest: it includes a modal operator for satisfiability, and thus provides a complete axiomatization for satisfiability in the logic K45.

Modal Logic and the Two-Variable Fragment

Abstract: We introduce a modal language L which is obtained from standard modal logic by adding the difference operator and modal operators interpreted by boolean combinations and the converse of accessibility relations. It is proved that L has the same expressive power as the two-variable fragment FO^2 of first-order logic but speaks less succinctly about relational structures: if the number of relations is bounded, then L-satisfiability is ExpTime-complete but FO^2 satisfiability is NExpTime-complete. We indicate that the relation between L and FO^2 provides a general framework for comparing modal and temporal languages with first-order languages.

Term-Modal Logics

Abstract: Many powerful logics exist today for reasoning about multi-agent systems, but in most of these it is hard to reason about an infinite or indeterminate number of agents. Also the naming schemes used in the logics often lack expressiveness to name agents in an intuitive way. To obtain a more expressive language for multi-agent reasoning and a better naming scheme for agents, we introduce a family of logics called term-modal logics. A main feature of our logics is the use of modal operators indexed by the terms of the logics. Thus, one can quantify over variables occurring in modal operators. In term-modal logics agents can be represented by terms, and knowledge of agents is expressed with formulas within the scope of modal operators. This gives us a flexible and uniform language for reasoning about the agents themselves and their knowledge. This article gives examples of the expressiveness of the languages and provides sequent-style and tableau-based proof systems for the logics. Furthermore we give proofs of soundness and completeness with respect to the possible world semantics.

Beyond modalities: sufficiency and mixed algebras

Abstract: In [24] a generalisation of relation algebras to Boolean algebras with normal and additive operators is introduced. These operators are the counterparts to the modal operators of possibility. In this paper we introduce a class of Boolean algebras with co-normal and co-additive operators referred to as sufficiency operators. They are the algebraic counterpart to the logical sufficiency operators introduced in [17] for an extension of modal logics. Next, we define a class of mixed algebras i.e., Boolean algebras with an additional modal operator and a sufficiency operator. We study representation and duality theory for these new classes of algebras. The motivation for those algebras comes from the problems of reasoning with incomplete information and spatial reasoning.

Uniform Representation of Content and Structure for structured document retrieval

Abstract: Documents often display a hierarchical structure. For example, a SGML document contains a title, several sections, which themselves contain paragraphs. In this paper, we develop a formal model to represent in a uniform manner structured documents by their content and structure. As a result, querying structured documents can be done with respect to their content, their structure, or both. The model is based on a possible worlds approach, modal operators and uncertainty distributions.

The Convergence of Scientific Knowledge: A view from the limit

Abstract: List of Figures. List of Tables. Preface. Acknowledgments. Formal Prerequisites. Interdependence Scheme for Topics. 1. Introduction. Part I: The Philosophy of Convergence. 2. Knowledge, Method and Reliability. 3. Knowledge and Skepticism. 4. The Epistemology of Convergence. Part II: Modal Operator Theory. 5. The Ontology of Convergence. 6. Science and Setup. 7. Two Relations of Correctness. 8. Methods and Methodology. 9. Forcing. 10. Definitions of Knowledge. 11. Modal Formalization. 12. Systems for Convergent Knowledge. 13. Knowledge in Time. 14. Forcing, Convergence - and Method. 15. Transmissibility. Part III: Convergence In Sum. 16. Knowledge in the End. Appendices. Index. Nomenclature. References.

A Fixed-point Characterization of a Deontic Logic of Regular Action

Abstract: We define a deontic logic of regular action as a characterization within a modal μ-calculus of action. First a semantics of deontic notions for regular action is given in terms of conditions on modal action structures. Then modal μ-calculus formulas characterizing these conditions are constructed by closely following the structure of deterministic finite automatons for regular action.

Logical Foundations for Reasoning about Trust in Secure Digital Communication

Abstract: This paper investigates foundations for the description of, and reasoning about, trust in secure digital communication We propose a logic, called the Typed Modal Logic (TML), which extends first-order logic with typed variables and modal operators to express agent beliefs Based on the logic, the theory of trust for a specific security system can be established Such trust theories provide a foundation for reasoning about trust in digital communication

First-order expressivity for S5-models: Modal vs. two-sorted languages

Abstract: Standard models for model predicate logic consist of a Kripke frame whose worlds come equipped with relational structures Both modal and two-sorted predicate logic are natural languages for speaking about such models In this paper we compare their expressivity We determine a fragment of the two-sorted language for which the modal language is expressively complete on S5-models Decidable criteria for modal definability are presented

Possibly alternative modality

Abstract: Despite the recognition of secondary ?modal? resources such as modal adjectives and adverbs, there has been relatively little discussion of the full extent of their contribution to the expression of modal meaning in general. In this corpus-based study, I focus exclusively on the adverb possibly and describe the range of environments that are revealed by a data set of 2000 randomly selected citations. On the basis of the observed data, I argue for a single core sense of possibly that distinguishes it both from modal operators, such as may/ might and from closely related adverbs such as perhaps and maybe. I also argue that, beyond the stereotypical function of possibly as a modal adjunct, there is massive evidence to suggest that it functions additionally as a modalising element in units at the lower rank of group. I therefore propose a revision to the structure of these units to incorporate the expression of modal meaning.

An Application of Deontic Logic to Information System Constraints

Abstract: In the field of information systems the term "constraint" is applied to statements of various kinds. Here we start from the analysis of a simple example to characterise the different kinds of constraints. It is shown that constraints may be necessary truths or deontic constraints. Moreover, deontic constraints are classified into three different types: deontic constraints about the world, deontic constraints about the representation of the world (self-completeness), and deontic constraints about the links between the world and its representation (validity and completeness). We describe a modal logical framework to define the different types of constraints, to characterise their violations, and to show how to repair their violations. Two different general forms of deontic constraints are considered, namely O(p→ψ) and p→Op, and it is shown that, except for deontic constraints about the world, the latter is more appropriate. Special issues related to the definition of quantifiers in the context of modal operators are also considered.

Model Existence in Non-Compact Modal Logic

Abstract: Predicate modal logics based on Kwith non-compact extra axioms are discussed and a sufficient condition for the model existence theorem is presented. We deal with various axioms in a general way by an algebraic method, instead of discussing concrete non-compact axioms one by one.

Logic Programming in a Fragment of Intuitionistic Temporal Linear Logic

Abstract: Recent development of logic programming languages based on linear logic suggests a successful direction to extend logic programming to be more expressive and more efficient. The treatment of formulasas-resources gives us not only powerful expressiveness, but also efficient access to a large set of data. However, in linear logic, whole resources are kept in one context, and there is no straight way to represent complex data structures as resources. For example, in order to represent an ordered list and time-dependent data, we need to put additional indices for each resource formula. This paper describes a logic programming language, called TLLP, based on intuitionistic temporal linear logic. This logic, an extension of linear logic with some features from temporal logics, allows the use of the modal operators '?'(next-time) and '?'(always) in addition to the operators used in intuitionistic linear logic. The intuitive meaning of modal operators is as follows: ? B means that B can be used exactly once at the next moment in time; ? B means that B can be used exactly once any time; !B means that B can be used arbitrarily many times (including 0 times) at any time. We first give a proof theoretic formulation of the logic of the TLLP language. We then present a series of resource management systems designed to implement not only interpreters but also compilers based on an extension of the standard WAM model.

Especificação normativa de agentes institucionais e da interacção entre agentes

Abstract: The main objective of this work was to contribute to the formal study of concepts and models suited to the normative specification of organized collective entities, that also support a rigorous analysis of them. Based on the legal concept of artificial person and on the legal relationships of mandate and representation, the concepts of role, action in a role, representation, contract and institutional agent are introduced. Such concepts are the basis for the characterization of a model for organized collective entities. They also allow the characterization of normative relationships that agents in a society may establish between each other. The formal characterization of those concepts in done through the definition of deontic and action modal logics, following the tradition initiated by S. Kanger, I. Porn and L. Lindahl. It is proposed a first-order, many sorted and multimodal logic, including a new action operator that captures the notion of action of an agent in a role. The properties of this logic are analyzed, and the soundness of the given axiomatization (with respect to the semantic defined which is based in the minimal models) is proved. This logic is used in the formal specification of institutional agents, societies of agents and of normative interactions between agents. With this logical model it is possible to analyze in a rigorous way the effects of agents actions in a society. Namely, it is possible to analyze the effects of an action of an agent, when he is acting in a role, in the actions of other agents, in the attribution of new obligations or permissions to the same agent or to other agents, or in the detection of non-ideal behaviour (unfulfillment of obligations). Finally, some extensions to the proposed logic are discussed. Some new concepts are introduced as well as some modal operators that express them. The extended logic is then explored in the representation of problems related with the recognition of an action in a role and with the detection of frauds that may occur when an agent tries to act in a role.

Deciding modal logics using tableaux and set theory.

Abstract: We propose a tableau-like decision procedure for deciding the satisfiability of set-theoretical formulae with restricted universal quantifiers and the powerset operator. Our result apply to a rather large class of set theories. The procedure we define can be used as a subroutine to decide the same class of formulae both in Set Theory and in non well-founded set theories, since we assume neither Regularity nor any form of anti-foundation axiom. Moreover, the decidability result presented allow to characterize a class of decidable modal logics. Thanks to the 2-as-P (box-as-powerset) translation our procedure can be used to uniformly study a large class of modal logics which includes K, T , S4, S5, S4.3.

Decidable Navigation Logics for Object Structures

Abstract: In this paper, we introduce decidable multimodal logics to describe and reason about navigation across object structures. The starting point of these navigation logics is the modelling of object structures as Kripke models that contain a family of deterministic accessibility relations; one for each pointer attribute. These pointer attributes are used in the logics both as first-order terms in equalities and as modal operators. To handle the ambiguities of pointer attributes the logics also cover a mechanism to bind logical variables to objects that are reachable by a pointer. The main result of this paper is a tableau construction for deciding the validity of formulas in the navigation logics.

A New Approach to Reasoning About Accountability in Cryptographic Protocols for E-Commerce

Abstract: This chapter presents a generic belief logic and demonstrates how it can be used to reason about accountability in cryptographic protocols for electronic commerce. First, we explain why the analysis of accountability properties can be treated in terms of belief. Different from other logics that have been proposed earlier to deal with accountability, our logic uses more general logical terms to deal with accountability, instead of the specific predicate “canprove”. We argue that the essence of accountability is actually the ability to “make” someone “believe” something, and the notion of “make” is just another modal operator in a generic belief logic. We then describe our belief logic and present an axiomatization system for analyzing cryptographic protocols for e-commerce. Finally, we illustrate with two examples how our logic can be used for our intended purpose.