Showing papers on "Normal modal logic published in 1995"
TL;DR: A general theory of normal forms is presented, based on a categorial result (Dubuc, 1974) for the free monoid construction, mainly for proposictional modal logic, although it seems to have a wider range of applications.
85 citations
TL;DR: This paper proves decidability for a largèbounded fragment' of predicate logic, and points out several applications, including a fresh look at the landscape of possible predicate logics, including candi-685.
Abstract: Modal Logic is traditionally concerned with the intensional operators \\possibly\" and \
ecessary\", whose intuitive correspondence with the standard quantiiers \\there ex-ists\" and \\for all\" comes out clearly in the usual Kripke semantics. This observation underlies the well-known translation from modal logic into the rst-order language over possible worlds models 12, 13]. In this way, modal formalisms correspond to fragments of a full rst-order (or sometimes higher-order) language over these models , which are both expressively perspicuous and deductively tractable. In this paper, we shall enquire which features of`modal fragments' are responsible for these attractions. Throughout, we shall concentrate on the basic language of modal propositional logic, which still serves as thèpure paradigm' in a rapidly expanding eld of more expressive modal formalisms 56, 51]. What precisely arèmodal fragments' of classical rst-order logic? Perhaps the most innuential answer is that of Gabbay 25], which identiies them with so-called``nite-variable fragments', using only some xed nite number of variables (free or bound). This viewpoint has been endorsed by many authors (cf. 16]). Our paper presents a critical review of its supporting evidence, adding some new results about nite-variable fragments, including failures of the Lo s{Tarski preservation theorem. But there is also a second answer to our question, implicit in much of the literature, which emphasizes so-called`bounded quantiication'. As our positive contribution, we shall develop the latter perspective here, showing its utility as a guide towards generalization of modal notions and techniques to larger fragments of classical logics. In particular, we prove decidability for a largèbounded fragment' of predicate logic, and point out several applications. One can also combine the two views on modal logic, as will be illustrated. Finally, we shall make another move. The above analogy works both ways. Modal operators are like quantiiers, but quan-tiiers behave like modal operators. This observation inspires a generalized modal semantics for rst-order predicate logic using accessibility constraints on assignments (cf. 45, 47]) which moves the earlier quantiier restrictions into the semantics. This provides a fresh look at the landscape of possible predicate logics, including candi-685
74 citations
TL;DR: In this article, the modal logic of non-contingency is considered in a general setting, without making special assumptions about the accessibility relation, and some of its extensions are discussed.
Abstract: We consider the modal logic of non-contingency in a general setting, without making special assumptions about the accessibility relation. The basic logic in this setting is axiomatized, and some of its extensions are discussed, with special attention to the expressive weakness of the language whose sole modal primitive is non-contingency (or equivalently, contingency), by comparison with the usual language based on necessity (or equivalently, possibility).
74 citations
TL;DR: In this paper, simple finite axiomatizations are given for versions of the modal logics K and K4 with non-contingency (or contingency) as the sole modal primitive.
Abstract: Simple finite axiomatizations are given for versions of the modal logics K and K4 with non-contingency (or contingency) as the sole modal primitive. This answers two questions of I. L. Humberstone.
64 citations
TL;DR: The formalism of cylindric modal logic can be motivated from two directions as discussed by the authors : it forms an interesting bridge over the gap between propositional formalisms and first-order logic, in that it formalizes firstorder logic as if it were a modal formalism: the assignments of firstorder variables can be seen as states or possible worlds of the modality formalism, and the quantifiers ∃ and ∀ may be studied as special cases of modal operators ♢ and ☐, respectively.
Abstract: The formalism of cylindric modal logic can be motivated from two directions. In its own right, it forms an interesting bridge over the gap between propositional formalisms and first-order logic, in that it formalizes first-order logic as if it were a modal formalism: The assignments of first-order variables can be seen as states or possible worlds of the modal formalism, and the quantifiers ∃ and ∀ may be studied as special cases of the modal operators ♢ and ☐, respectively. Elaborating this idea, we find that from this modal viewpoint, the standard semantics of first-order logic corresponds to just one of many possible classes of Kripke frames, and that other classes might be of interest as well.
60 citations
Book•
11 Sep 1995
TL;DR: This book discusses the Computational Model of Nonmonotonic Logic, a model for Logic Programming and Deductive Data Bases, and the Architecture of IKBS: Components, which describes the design of PAYE and its components.
Abstract: Part I: Introduction: Background, Historical Developments, and Bibliography. Declarative Knowledge: Propositional Logic. Predicate Logic. Declarative Semantics. Clausal Form. Deduction and Inference: Meaning and Interpretation. Model--Theoretic and Proof, Theoretic Approaches. Query Evaluation. Proof Tree. Inference Procedures. Automated Reasoning Systems: Theoron Provers. Substitution and Unification. Resolution. Resolution Strategies. Rewrite Rules. Logic Programming and Deductive Data Bases. Frames, Semantic Nets, and Production Systems: Frame Structure. Slots--Assertions. ISA--Hierarchy. Inference with Frames. Production Systems. Computation Methods. Search: General: Search Algorithm. Domain Specific Search. Semantic Information and Strategies to Control the Search. Part II: Modal and Intentional Logic: Propositional Modal Logic. Predicate Modal Logic. Modal Structure, Kripke Structure. Modal Operation. IntentionalLogic. Nonmonotonic Reasoning: Nonmonotonic Logic. The Closed-World Assumption. Negation by Failure. Circumscription. Computational Model of Nonmonotonic Logic. Induction: Basic Properties. Concept Formation. Generalization and Specialization. Matching. Learning. Uncertainty: Probabilities. Bayes Law. Fuzzy Logic. Probabilistic Logic. Computation. Meta Knowledge. Temporal Systems. Planning Actions: States, Actions, and the Frame Problem. Action Ordering. A Basic Plan Interpreter. Conditional Plans. Goals, Restricted Goals. Goal Regression. Domains and Application. Decision Theory. Part III: Architecture of IKBS: Components. Data Flow. Control Flow. Case Study PAYE Tax System: Introduction and Basic Properties. Temporal Aspect. Architecture of PAYE. Design of PAYE. Computational Model. Execution. The Link to Imperative Systems. Knowledge Acquisition. The Future: Parallel Systems. Intelligent Systems.
58 citations
20 Aug 1995
TL;DR: This paper presents a framework for integrating modal operators into terminological knowledge representation languages, and introduces syntax and semantics of the extended language, and shows that satisfiability of finite sets of formulas is decidable.
Abstract: Terminological knowledge representation formalisms can be used to represent objective, time-independent facts about an application domain. Notions like belief, intentions, time - which are essential for the representation of multi-agent environments - can only be expressed in a very limited way. For such notions, modal logics with possible worlds semantics provides a formally well-founded and well-investigated basis. This paper presents a framework for integrating modal operators into terminological knowledge representation languages. These operators can be used both inside of concept expressions and in front of terminological and assertional axioms. The main restrictions are that all modal operators are interpreted in the basic logic K, and that we consider increasing domains instead of constant domains. We introduce syntax and semantics of the extended language, and show that satisfiability of finite sets of formulas is decidable.
52 citations
TL;DR: A formal logical system dealing with both uncertainty (possibility) and vagueness (fuzziness) is investigated and a completeness theorem is exhibited.
51 citations
TL;DR: A novel cut-free tableau formulation is presented, and its completeness is proved, and it is shown that this formulation can be formulations using a cut rule in an essential way.
Abstract: We continue a series of papers on a family of many-valued modal logics, a family whose Kripke semantics involves many-valued accessibility relations. Earlier papers in the series presented a motivation in terms of a multiple-expert semantics. They also proved completeness of sequent calculus formulations for the logics, formulations using a cut rule in an essential way. In this paper a novel cut-free tableau formulation is presented, and its completeness is proved.
48 citations
TL;DR: Versions and extensions of intuitionistic and modal logic involving biHeyting and bimodal operators, the axiom of constant domains and Barcan's formula are formulated as structured categories and representation theorems for the resulting concepts are proved.
41 citations
TL;DR: An extension of the concept description language ALC used in KL-ONE-like terminological reasoning is presented in this article, which includes multi-modal operators that either stand for the usual role quantifications or for modalities such as belief, time, etc.
Abstract: An extension of the concept description language ALC used in KL-ONE-like terminological reasoning is presented. The extension includes multi-modal operators that either stand for the usual role quantifications or for modalities such as belief, time, etc. The modal operators can be used at all levels of the concept terms, and they can be used to modify both concepts and roles. This is an instance of a new kind of combination of modal logics where the modal operators of one logic may operate directly on the operators of the other logic. Different versions of this logic are investigated and various results about decidability and undecidability are presented. The main problem, however, decidability of the basic version of the logic, remains open.
TL;DR: An aset-theoretic translation method for polymodal logics that reduces derivability in a large class of propositional polymodAL logics to derivable in a very weak first-order set theory Ω is presented.
Abstract: The paper presents aset-theoretic translation method for polymodal logics that reduces derivability in a large class of propositional polymodal logics to derivability in a very weak first-order set theory Ω. Unlike most existing translation methods, the one we propose applies to any normal complete finitely axiomatizable polymodal logic, regardless of whether it is first-order complete or an explicit semantics is available. The finite axiomatizability of Ω allows one to implement mechanical proof-search procedures via the deduction theorem. Alternatively, more specialized and efficient techniques can be employed. In the last part of the paper, we briefly discuss the application ofset T-resolution to support automated derivability in (a suitable extension of) Ω.
TL;DR: The authors present λ→□, a proof term calculus for intuitionistic modal logic S4 that is well-suited for practical applications and is equivalent to other formulations in the literature, with respect to provability.
22 Sep 1995
TL;DR: In this article, a description language for finite trees is presented, and a sound and complete proof system is provided using standard axioms from modal provability logic and modal logics of programs, and prove completeness by extending techniques due to Van Benthem and Meyer-Violent.
Abstract: In this paper we introduce a description language for finite trees. Although we briefly note some of its intended applications, the main goal of the paper is to provide it with a sound and complete proof system.We do so using standard axioms from modal provability logic and modal logics of programs, and prove completeness by extending techniques due to Van Benthem and Meyer-Viol [2] and Blackburn and Meyer-Viol [5]. We conclude with a proof of the EXPTIME-completeness of the satisfiability problem, and a discussion of issues related to complexity and theorem proving.
TL;DR: The similarities and diierences between quantiiers and modal operators are investigated and proof theoretical abduction methods for the modal systems K, D, T and S4 are deened, that are sound and complete.
Abstract: In this work, the problem of performing abduction in modal logics is addressed, along the lines of 3], where a proof theoretical abduction method for full rst order classical logic is deened, based on tableaux and Gentzen-type systems. This work applies the same methodology to face modal abduction. The non-classical context enforces the value of analytical proof systems as tools to face the meta-logical and proof-theoretical questions involved in abductive reasoning. The similarities and diierences between quantiiers and modal operators are investigated and proof theoretical abduction methods for the modal systems K, D, T and S4 are deened, that are sound and complete. The construction of the abductive explanations is in strict relation with the expansion rules for the modal logics, in a modular manner that makes local modiications possible. The method given in this paper is general, in the sense that it can be adapted to any propositional modal logic for which analytic tableaux are provided. Moreover, the way towards an extension to rst order modal logic is straightforward.
TL;DR: There are exactly twenty maximal structurally incomplete modal logics above K4 and they are all tabular: a modal logic λ is hereditarily structurally complete iffλ is not included in any logic from the list of twenty special tabular logics.
Abstract: We consider structural completeness in modal logics. The main result is the necessary and sufficient condition for modal logics over K4 to be hereditarily structurally complete: a modal logic λ is hereditarily structurally complete iff λ is not included in any logic from the list of twenty special tabular logics. Hence there are exactly twenty maximal structurally incomplete modal logics above K4 and they are all tabular.
TL;DR: Ray Jennings and Peter Schotch have developed a generalized relational frame theory which articulates an infinite hierarchy of sublogics of K which expressing a species of “weakly aggregative necessity”.
Abstract: We are accustomed to regarding K as the weakest modal logic admitting of a relational semantics in the style made popular by Kripke. However, in a series of papers which demonstrates a startling connection between modal logic and the theory of paraconsistent inference, Ray Jennings and Peter Schotch have developed a generalized relational frame theory which articulates an infinite hierarchy of sublogics of K, each expressing a species of “weakly aggregative necessity”. Recall that K is axiomatized, in the presence of N and RM, by the schema of “binary aggregation”For each n ≥ 1, the weakly aggregative modal logic Kn is axiomatized by replacing K with the schema of “n-ary aggregation”which is an n-ary relaxation, or weakening, of K. Note that K1 = K.In [3], the authors claim without proof that Kn is determined by the class of frames F = (W, R), where W is a nonempty set and R is an (n + 1)-ary relation on W, under the generalization of Kriple's truth condition according to which □α is true at a point w in W if and only if α is true at one of x1,…,xn for all x1,…, xn in W such that Rw, x1,…, xn.
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28 Feb 1995
TL;DR: In this article, a uniform presentation of modal analogues of well-known definability and preservation results from first-order logic is given, including algebraic characterizations of modality equivalence and modally definable classes of models.
Abstract: This paper contributes to the model theory of modal logic using bisimulations as the fundamental tool. A uniform presentation is given of modal analogues of well-know definability and preservation results from first-order logic. These results include algebraic characterizations of modal equivalence, and of modally definable classes of models; the preservation results concern preservation of modal formulas under submodels, unions of chains and homomorphisms.
TL;DR: An overview of decidability results for modal logics having a binary modality is given, and the demonstration of proof-techniques is put on to help in finding the borderlines between decidable and undecidable fragments of usual first-order logic.
Abstract: We give an overview of decidability results for modal logics having a binary modality. We put an emphasis on the demonstration of proof-techniques, and hope that this will also help in finding the borderlines between decidable and undecidable fragments of usual first-order logic.
22 Sep 1995
TL;DR: In this article, contraction free sequent calculi for the three modal logics K, T, and S4 were proposed and proved to provide more efficient decision procedures than those hitherto known.
Abstract: We propose so called contraction free sequent calculi for the three prominent modal logics K, T, and S4. Deduction search in these calculi is shown to provide more efficient decision procedures than those hitherto known. In particular space requirements for our logics are lowered from the previously established bounds of the form n2, n3 and n4 to n log n, n log n, and n2 log n respectively.
TL;DR: Tense logics in the bimodal propositional language are investigated with respect to the Finite Model Property in this paper, and techniques from investigations of modal logics above K4 are extended to tense logic.
Abstract: Tense logics in the bimodal propositional language are investigated with respect to the Finite Model Property. In order to prove positive results techniques from investigations of modal logics above K4 are extended to tense logic. General negative results show the limits of the transfer.
TL;DR: The definition of complete strategies in tableaux calculi for propositional modal logics using a non exhaustive backtracking mechanism, a selective periodicity test and a uniform or a non uniform priority on the order of application of the tableaux rules is defined.
Abstract: The major emphasis of this paper is on the definition of complete strategies in tableaux calculi for propositional modal logics. The strategies use a non exhaustive backtracking mechanism, a selective periodicity test and a uniform or a non uniform priority on the order of application of the tableaux rules. The propositional modal logics treated herein are those having a tableaux calculus with finite sets of formulas possibly occurring in the tableaux. Experimental results with the ATINF modal prover are presented.
TL;DR: It is proved that the Hallden-completeness in any normal modal logic is equivalent to the so-called super-embedding property of a suitable class of modal algebras, and the joint embeddingproperty of a class of alge bras is equivalent of the Pseudo-Relevance Property.
Abstract: In this paper we find an algebraic equivalent of the Hallden property in modal logics, namely, we prove that the Hallden-completeness in any normal modal logic is equivalent to the so-called super-embedding property of a suitable class of modal algebras. The joint embedding property of a class of algebras is equivalent to the Pseudo-Relevance Property. We consider connections of the above-mentioned properties with interpolation and amalgamation. Also an algebraic equivalent of of the principle of variable separation in superintuitionistic logics will be found.
TL;DR: The modal μ-calculus due to Kozen is considered, which is a finitary modal logic with least and greatest fixed points of monotone operators, and the existing duality between the categories of modal algebras with homomorphisms and the category of descriptive modal frames with contractions is extended to the case of having fixed point operators.
Proceedings Article•
01 Jan 1995
TL;DR: This paper considers two variants of Kowalski and Sergot's Event Calculus: the Skeptical EC and the Credulous EC, and proves SKEC and CREC to be the operational counterparts of the modal operators of necessity and possibility in an appropriate modal logic.
Abstract: This paper proposes a modal logic reconstruction of temporal reasoning about partially ordered events in a logic programming framework. It considers two variants of Kowalski and Sergot's Event Calculus (EC): the Skeptical EC (SKEC) and the Credulous EC (CREC). In the presence of partially ordered sequences of events, SKEC and CREC derive the maximal validity intervals over which the relevant properties are necessarily and possibly true, respectively. SKEC and CREC are proved to be the operational counterparts of the modal operators of necessity and possibility in an appropriate modal logic and their properties in relation to EC are studied.
03 Jul 1995
TL;DR: This paper analyzes the difference in the multi preference semantics of the defeasible deontic logic DefDiode, which contains one preference relation for ideality, which can be used to formalizeDeontic paradoxes like the Chisholm and Forrester paradoxes, and another preference relations for normality,Which can be use to formalized exceptions.
Abstract: There is a fundamental difference between a conditional obligation being violated by a fact, and a conditional obligation being overridden by another conditional obligation. In this paper we analyze this difference in the multi preference semantics of our defeasible deontic logic DefDiode. The semantics contains one preference relation for ideality, which can be used to formalize deontic paradoxes like the Chisholm and Forrester paradoxes, and another preference relation for normality, which can be used to formalize exceptions. The interference of the two preference orderings generates new questions about preferential semantics.
20 Mar 1995
TL;DR: To give rigid semantics to graded modal operators, an extended fuzzy-measure-based model is defined as a family of minimal models for modal logic, each of which corresponds to an intermediate value of a fuzzy measure.
Abstract: To give rigid semantics to graded modal operators, an extended fuzzy-measure-based model is defined as a family of minimal models for modal logic, each of which corresponds to an intermediate value of a fuzzy measure. Soundness and completeness results of several systems of modal logic are proved with respect to classes of newly introduced models based on intermediate values of fuzzy, possibility, necessity, and Dirac measures, respectively. It is emphasized that a fuzzy measure inherently contains a multimodal logical structure. >