Showing papers on "Normal modal logic published in 2002"

Deontic Logic and Contrary-to-Duties

Abstract: Deontic logic is concerned with the logical analysis of such normative notions as obligation, permission, right and prohibition. Although its origins lie in systematic legal and moral philosophy, deontic logic has begun to attract the interest of researchers in other areas, particularly computer science, management science and organisation theory. Among the application areas which have already received some attention in the literature are: issues of knowledge representation in the design of legal expert systems; the formal specification of aspects of computer systems, for instance in regard to security and access control policies, fault tolerance, and database integrity constraints; the formal characterisation of aspects of organisational structure, pertaining for example to the responsibilities and powers which agents are required or authorised to exercise. The “AEON” workshop proceedings provide some illustrations of work in these areas (see [ΔEON91; ΔEON94; ΔEON96]).

Description logics of minimal knowledge and negation as failure

Abstract: We present description logics of minimal knowledge and negation as failure (MKNF-DLs), which augment description logics with modal operators interpreted according to Lifschitz's nonmonotonic logic MKNF. We show the usefulness of MKNF-DLs for a formal characterization of a wide variety of nonmonotonic features that are both commonly available inframe-based systems, and needed in the development of practical knowledge-based applications: defaults, integrity constraints, role, and concept closure. In addition, we provide a correct and terminating calculus for query answering in a very expressive MKNF-DL.

Multi-Dimensional Modal Logic as a Framework for Spatio-Temporal Reasoning

Abstract: In this paper we advocate the use of multi-dimensional modal logics as a framework for knowledge representation and, in particular, for representing spatio-temporal information. We construct a two-dimensional logic capable of describing topological relationships that change over time. This logic, called PSTL (Propositional Spatio-Temporal Logic) is the Cartesian product of the well-known temporal logic PTL and the modal logic S4u, which is the Lewis system S4 augmented with the universal modality. Although it is an open problem whether the full PSTL is decidable, we show that it contains decidable fragments into which various temporal extensions (both point-based and interval based) of the spatial logic RCC-8 can be embedded. We consider known decidability and complexity results that are relevant to computation with multi-dimensional formalisms and discuss possible directions for further research.

Sequent Systems for Modal Logics

Abstract: This chapter surveys the application of various kinds of sequent systems to modal and temporal logic, also called tense logic. The starting point are ordinary Gentzen sequents and their limitations both technically and philosophically. The rest of the chapter is devoted to generalizations of the ordinary notion of sequent. These considerations are restricted to formalisms that do not make explicit use of semantic parameters like possible worlds or truth values, thereby excluding, for instance, Gabbay’s labelled deductive systems, indexed tableau calculi, and Kanger-style proof systems from being dealt with. Readers interested in these types of proof systems are referred to [Gabbay, 1996], [Gore, 1999] and [Pliuskeviene, 1998]. Also Orlowska’s [1988; 1996] Rasiowa-Sikorski-style relational proof systems for normal modal logics will not be considered in the present chapter. In relational proof systems the logical object language is associated with a language of relational terms. These terms may contain subterms representing the accessibility relation in possible-worlds models, so that semantic information is available at the same level as syntactic information. The derivation rules in relational proof systems manipulate finite sequences of relational formulas constructed from relational terms and relational operations. An overview of ordinary sequent systems for non-classical logics is given in [Ono, 1998], and for a general background on proof theory the reader may consult [Troelstra and Schwichtenberg, 2000].

Back and forth between guarded and modal logics

Abstract: Guarded fixed-point logic μGF extends the guarded fragment by means of least and greatest fixed points, and thus plays the same role within the domain of guarded logics as the modal μ-calculus plays within the modal domain. We provide a semantic characterization of μGF within an appropriate fragment of second-order logic, in terms of invariance under guarded bisimulation. The corresponding characterization of the modal μ-calculus, due to Janin and Walukiewicz, is lifted from the modal to the guarded domain by means of model theoretic translations. Guarded second-order logic, the fragment of second-order logic which is introduced in the context of our characterization theorem, captures a natural and robust level of expressiveness with several equivalent characterizations. For a wide range of issues in guarded logics it may take up a role similar to that of monadic second-order in relation to modal logics. At the more general methodological level, the translations between the guarded and modal domains make the intuitive analogy between guarded and modal logics available as a tool in the further analysis of the model theory of guarded logics.

Types, Tableaus, and Gödel's God

Abstract: Preface. Part I: Classical Logic. 1. Classical Logic - Syntax. 2. Classical Logic - Semantics. 3. Classical Logic - Basic Tableaus. 4. Soundness and Completeness. 5. Equality. 6. Extensionality. Part II: Modal Logic. 7. Modal Logic, Syntax and Semantics. 8. Modal Tableaus. 9. Miscellaneous Matters. Part III: Ontological Arguments. 10. Godel's Argument, Background. 11. Godel's Argument, Formally. References. Index.

A modal walk through space

Abstract: We investigate the major mathematical theories of space from a modal standpoint: topology, affine geometry, metric geometry, and vector algebra. This allows us to see new fine-structure in spatial patterns which suggests analogies across these mathematical theories in terms of modal, temporal, and conditional logics. Throughout the modal walk through space, expressive power is analyzed in terms of language design, bisimulations, and correspondence phenomena. The result is both unification across the areas visited, and the uncovering of interesting new questions.

Fusions of description logics and abstract description systems

Abstract: Fusions are a simple way of combining logics. For normal modal logics, fusions have been investigated in detail. In particular, it is known that, under certain conditions, decidability transfers from the component logics to their fusion. Though description logics are closely related to modal logics, they are not necessarily normal. In addition, ABox reasoning in description logics is not covered by the results from modal logics. In this paper, we extend the decidability transfer results from normal modal logics to a large class of description logics. To cover different description logics in a uniform way, we introduce abstract description systems, which can be seen as a common generalization of description and modal logics, and show the transfer results in this general setting.

Combinations of Modal Logics

Abstract: There is increasing use of combinations of modal logics in both foundational and applied research areas. This article provides an introduction to both the principles of such combinations and to the variety of techniques that have been developed for them. In addition, the article outlines many key research problems yet to be tackled within this callenging area of work.

On modal logics between K × K × K and S5 × S5 × S5

Abstract: We prove that every n-modal logic between Kn and S5n is undecidable, whenever n ≥ 3 We also show that each of these logics is non-finitely axiomatizable, lacks the product finite model property, and there is no algorithm deciding whether a finite frame validates the logic These results answer several questions of Gabbay and Shehtman The proofs combine the modal logic technique of Yankov–Fine frame formulas with algebraic logic results of Halmos, Johnson and Monk, and give a reduction of the (undecidable) representation problem of finite relation algebras

Martin's Axiom, Omitting Types, and Complete Representations in Algebraic Logic

Abstract: We give a new characterization of the class of completely representable cylindric algebras of dimension 2 #lt; n ≤ w via special neat embeddings. We prove an independence result connecting cylindric algebra to Martin's axiom. Finally we apply our results to finite-variable first order logic showing that Henkin and Orey's omitting types theorem fails for L n, the first order logic restricted to the first n variables when 2 #lt; n#lt;w. L n has been recently (and quite extensively) studied as a many-dimensional modal logic.

On Lukasiewicz's Four-Valued Modal Logic.

Abstract: Łukasiewicz's four-valued modal logic is surveyed and analyzed, together with Łukasiewicz's motivations to develop it. A faithful interpretation of it in classical (non-modal) two-valued logic is presented, and some consequences are drawn concerning its classification and its algebraic behaviour. Some counter-intuitive aspects of this logic are discussed in the light of the presented results, Łukasiewicz's own texts, and related literature.

First Order Extensions of Classical Systems of Modal Logic; The role of the Barcan schemas

Abstract: The paper studies first order extensions of classical systems of modal logic (see (Chellas, 1980, part III)). We focus on the role of the Barcan formulas. It is shown that these formulas correspond to fundamental properties of neighborhood frames. The results have interesting applications in epistemic logic. In particular we suggest that the proposed models can be used in order to study monadic operators of probability (Kyburg, 1990) and likelihood (Halpern-Rabin, 1987).

Independence-friendly modal logic and true concurrency

Abstract: We consider modal analogues of Hintikka et al.'s 'independence-friendly first-order logic', and discuss their relationship to equivalences previously studied in concurrency theory.

A Tableau Decision Algorithm for Modalized ALC with Constant Domains

Abstract: The aim of this paper is to construct a tableau decision algorithm for the modal description logic K ALC with constant domains. More precisely, we present a tableau procedure that is capable of deciding, given an ALC-formula ϕ with extra modal operators (which are applied only to concepts and TBox axioms, but not to roles), whether ϕ is satisfiable in a model with constant domains and arbitrary accessibility relations. Tableau-based algorithms have been shown to be 'practical' even for logics of rather high complexity. This gives us grounds to believe that, although the satisfiability problem for K ALC is known to be NEXPTIME-complete, by providing a tableau decision algorithm we demonstrate that highly expressive description logics with modal operators have a chance to be implementable. The paper gives a solution to an open problem of Baader and Laux [5].

Completeness of certain bimodal logics for subset spaces

Abstract: Subset Spaces were introduced by L. Moss and R. Parikh in [8]. These spaces model the reasoning about knowledge of changing states. In [2] a kind of subset space called intersection space was considered and the question about the existence of a set of axioms that is complete for the logic of intersection spaces was addressed. In [9] the first author introduced the class of directed spaces and proved that any set of axioms for directed frames also characterizes intersection spaces. We give here a complete axiomatization for directed spaces. We also show that it is not possible to reduce this set of axioms to a finite set.

A Modal Logic for Full LOTOS based on Symbolic Transition Systems

Abstract: Symbolic transition systems separate data from process behaviour by allowing the data to be uninstantiated Designing a HML like modal logic for these transition systems is interesting because of the subtle interplay between the quanti ers for the data and the modal operators quanti ers on transitions This paper presents the syntax and semantics of such a logic and discusses the design issues involved in its construction The logic has been shown to be adequate with respect to strong early bisimulation over symbolic transition systems derived from Full LOTOS We de ne what is meant by adequacy and discuss how we can reason about it with the aid of a mechanised theorem prover

The Modal Logic of Agreement and Noncontingency

Abstract: The formula $\triangle$A (it is noncontingent whether A) is true at a point in a Kripke model just in case all points accessible to that point agree on the truth-value of A. We can think of $\triangle$-based modal logic as a special case of what we call the general modal logic of agreement, interpreted with the aid of models supporting a ternary relation, S, say, with OA (which we write instead of $\triangle$A to emphasize the generalization involved) true at a point w just in case for all points x, y, with Swxy, x and y agree on the truth-value of A. The noncontingency interpretation is the special case in which Swxy if and only if Rwx and Rwy, where R is a traditional binary accessibility relation. Another application, related to work of Lewis and von Kutschera, allows us to think of OA as saying that A is entirely about a certain subject matter.

Fibring Modal First-Order Logics: Completeness Preservation

Abstract: Fibring is defined as a mechanism for combining logics with a firstorder base, at both the semantic and deductive levels. A completeness theorem is established for a wide class of such logics, using a variation of the Henkin method that takes advantage of the presence of equality and inequality in the logic. As a corollary, completeness is shown to be preserved when fibring logics in that class. A modal first-order logic is obtained as a fibring where neither the Barcan formula nor its converse hold.

Model Checking Modal Transition Systems Using Kripke Structures

Abstract: We reduce the modal mu-calculus model-checking problem for Kripke modal transition systems to the modal mu-calculus modelchecking problem for Kripke structures This reduction is sound, preserves the alternation-depth fragments of the modal mu-calculus, is linear in the size of formulas and models, and extends the reach of modal mu-calculus model checkers to sound abstraction for the full logic These results specialize to CTL* model-checking and CTL model checking

Local realizability toposes and a modal logic for computability

Abstract: This work is a step toward the development of a logic for types and computation that includes not only the usual spaces of mathematics and constructions, but also spaces from logic and domain theory. Using realizability, we investigate a configuration of three toposes that we regard as describing a notion of relative computability. Attention is focussed on a certain local map of toposes, which we first study axiomatically, and then by deriving a modal calculus as its internal logic. The resulting framework is intended as a setting for the logical and categorical study of relative computability.

Polytime, combinatory logic and positive safe induction

Abstract: We characterize the polynomial time operations as those which are provably total in a first order system, which comprises (untyped) combinatory logic with extensionality, together with positive “safe induction” on the set of binary strings. The formalization of safe induction is inspired by Leivants idea of ramification. We also show how to replace ramification by means of modal logic.

The monadic theory of tree-like structures

Abstract: Initiated by the work of Buchi, Lauchli, Rabin, and Shelah in the late 60s, the investigation of monadic second-order logic (MSO) has received continuous attention. The attractiveness of MSO is due to the fact that, on the one hand, it is quite expressive subsuming - besides first-order logic - most modal logics, in particular the μ-calculus. On the other hand, MSO is simple enough such that model-checking is still decidable for many structures. Hence, one can obtain decidability results for several logics by just considering MSO.

HyLoRes 1.0: Direct Resolution for Hybrid Logics

Abstract: Hybrid languages are modal languages that allow direct reference to the elements of a model Even the basic hybrid language \( {\text{(}}\mathcal{H}{\text{(@))}} \), which only extends the basic modal language with the addition of nominals (i,j, k,) and satisfiability operators (@ i , @ j , @ k ,), increases the expressive power: it can explicitly check whether the point of evaluation is a specific, named point in the model (w ⊩ i), and whether a named point satisfies a given formula (w ⊩ @ iϕ ) The extended expressivity allows one to define elegant decision algorithms, where nominals and @ play the role of labels, or prefixes, which are usually needed during the construction of proofs in the modal setup [5,3] Note that they do so inside the object language All these features we get with no increase in complexity: the complexity of the satisfiability problem for \( \mathcal{H}{\text{(@)}} \) is the same as for the basic modal language, PSPACE [2] When we move to very expressive hybrid languages containing binders, we obtain an impressive boost in expressivity, but usually we also move beyond the boundaries of decidability Classical binders like ∀ and ∃ (together with @) make the logic as expressive as first-order logic (FOL) while adding the more “modal” binder ↓ gives a logic weaker than FOL [1] We refer to the Hybrid Logic site at http://wwwhylonet for a broad on-line bibliography

Kripke Semantics for Modal Substructural Logics

Abstract: We introduce Kripke semantics for modal substructural logics, and prove the completeness theorems with respect to the semantics. The completeness theorems are proved using an extended Ishihara's method of canonical model construction (Ishihara, 2000). The framework presented can deal with a broad range of modal substructural logics, including a fragment of modal intuitionistic linear logic, and modal versions of Corsi's logics, Visser's logic, Mendez's logics and relevant logics.

Products of modal logics, part 3: Products of modal and temporal logics .

Abstract: In this paper we improve the results of [2] by proving the product fmp for the product of minimal n-modal and minimal n-temporal logic For this case we modify the finite depth method introduced in [1] The main result is applied to identify new fragments of classical first-order logic and of the equational theory of relation algebras, that are decidable and have the finite model property