scispace - formally typeset
Search or ask a question

Showing papers on "Normal modal logic published in 2001"


Book
28 Jun 2001

2,698 citations


Book
01 Jan 2001
TL;DR: In this paper, a historical perspective of classical logic and the material conditional is presented, along with a set of relevant logics, including Intuitionist logic, Basic relevant logic, Mainstream relevant logic and Fuzzy logic.
Abstract: Introduction 1. Classical logic and the material conditional 2. Basic modal logic 3. Normal modal logics 4. Non-normal worlds strict conditionals 5. Conditional logics 6. Intuitionist logic 7. Many-valued logics 8. First degree entailment 9. Basic relevant logic 10. Mainstream relevant logics 11. Fuzzy logic 12. Conclusion: a historical perspective.

368 citations


Journal ArticleDOI
TL;DR: This work reconsiders the foundations of modal logic, following Martin-Löf's methodology of distinguishing judgments from propositions, and gives a new presentation of lax logic, finding that the lax modality is already expressible using possibility and necessity.
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.

348 citations


Book ChapterDOI
02 Apr 2001
TL;DR: Two major applications, model checking partial state spaces and three-valued program shape analysis, are presented as evidence of the suitability of Kripke MTSs as a foundation for three- valued analyses.
Abstract: We present Kripke modal transition systems (Kripke MTSs), a generalization of modal transition systems [27, 26], as a foundation for three-valued program analysis. The semantics of Kripke MTSs are presented by means of a mixed power domain of states; soundness and consistency are proved. Two major applications, model checking partial state spaces and three-valued program shape analysis, are presented as evidence of the suitability of Kripke MTSs as a foundation for three-valued analyses.

240 citations


Book ChapterDOI
01 Jan 2001
TL;DR: The modal mu-calculus is concentrated on, a modal logic which subsumes most other commonly used logics and which looks at model-checking, and the relationship of modal logics to other formalisms.
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.

187 citations


Proceedings ArticleDOI
16 Jun 2001
TL;DR: A uniform type theory that integrates intensionality, extensionality and proof irrelevance as judgmental concepts is developed that contrasts with previous approaches that, a priori, distinguished propositions from specifications.
Abstract: We develop a uniform type theory that integrates intensionality, extensionality and proof irrelevance as judgmental concepts. Any object may be treated intensionally (subject only to /spl alpha/-conversion), extensionally (subject also to /spl beta//spl eta/-conversion), or as irrelevant (equal to any other object at the same type), depending on where it occurs. Modal restrictions developed by R. Harper et al. (2000) for single types are generalized and employed to guarantee consistency between these views of objects. Potential applications are in logical frameworks, functional programming and the foundations of first-order modal logics. Our type theory contrasts with previous approaches that, a priori, distinguished propositions (whose proofs are all identified - only their existence is important) from specifications (whose implementations are subject to some definitional equalities).

119 citations


Book ChapterDOI
10 Sep 2001
TL;DR: Duality results are proved which show how to relate Kripke models to algebraic models and these in turn to the appropriate categorical models for these logics.
Abstract: We consider two systems of constructive modal logic which are computationally motivated. Their modalities admit several computational interpretations and are used to capture intensional features such as notions of computation, constraints, concurrency, etc. Both systems have so far been studied mainly from type-theoretic and category-theoretic perspectives, but Kripke models for similar systems were studied independently. Here we bring these threads together and prove duality results which show how to relate Kripke models to algebraic models and these in turn to the appropriate categorical models for these logics.

114 citations


Journal ArticleDOI
TL;DR: This paper gives a semantical underpinning for a many- sorted modal logic associated with certain dynamical systems, like tran- sition systems, automata or classes in object-oriented languages, as coalgebras of so-called polynomial func- tors, built up from constants and identities, using products, coproducts and powersets.
Abstract: This paper gives a semantical underpinning for a many- sorted modal logic associated with certain dynamical systems, like tran- sition systems, automata or classes in object-oriented languages. These systems will be described as coalgebras of so-called polynomial func- tors, built up from constants and identities, using products, coproducts and powersets. The semantical account involves Boolean algebras with operators indexed by polynomial functors, called MBAOs, for Many- sorted Boolean Algebras with Operators, combining standard (categor- ical) models of modal logic and of many-sorted predicate logic. In this setting we will see Lindenbaum MBAO models as initial objects, and canonical coalgebraic models of maximally consistent sets of formulas as nal objects. They will be used to (re)prove completeness results, and Hennessey{Milner style characterisation results for the modal logic, rst established by Roiger. Mathematics Subject Classication. 03G05, 03G30, 06E25.

109 citations


28 Feb 2001
TL;DR: The main results are that adding negation of modal parameters to K makes reasoning ExpTime-complete, which is shown by using an automata-theoretic approach, and that adding atomic negation and conjunction to K even yields a NExpTime- complete logic, which was shown by a reduction of a variant of the domino problem.
Abstract: Boolean Modal Logics extend multi-modal K by allowing the use of boolean operators to define complex relation terms. In this paper, we investigate the complexity of reasoning with various such logics. The main results are that (1) adding negation of modal parameters to K makes reasoning ExpTime-complete, which is shown by using an automata-theoretic approach, and that (2) adding atomic negation and conjunction to K even yields a NExpTime- complete logic, which is shown by a reduction of a variant of the domino problem. The last result is relativized by the fact that it depends on an infinite number of modal parameters to be available. If the number of modal parameters is bounded, full Boolean Modal Logic becomes ExpTime-complete. This is shown by a reduction to K enriched with the universal modality.

93 citations


Journal ArticleDOI
TL;DR: Modal logic is proposed to be used as a logic for coalgebras and the relationship of this approach with the coalgebraic logic of Moss is discussed.

91 citations


Journal ArticleDOI
TL;DR: A PSPACE algorithm that decides satisfiability of the graded modal lo gic Gr(KR), a natural extension of propositional modal logic KR by counting expressions, is presented, which is the first known algorithm which meets the lower bound for the complexity of the problem.
Abstract: We present a PSPACE algorithm that decides satisfiability of the graded modal lo gic Gr(KR)—a natural extension of propositional modal logic KR by counting expressions—which plays an important role in the area of knowledge representation The algorithm employs a tableaux approach and is the first known algorithm which meets the lower bound for the complexity of the problem Thus, we exactly fix the complexity of the problem and refute a EXPTIME-hardness conjecture We extend the results to the logic Gr(K R 1 \ ), which augments Gr(KR) with inverse relations and intersection of accessibility relations This establishes a kind of “theoretical benchma rk” that all algorithmic approaches can be measured against

Journal ArticleDOI
TL;DR: A general satisfiability criterion is proved for formulas in , which reduces the modal satisfiability to the classical one, and is used to single out a number of new, in a sense optimal, decidable fragments of various modal predicate 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.

Journal ArticleDOI
TL;DR: A general theorem is proved showing that in many cases two‐dimensional Cartesian products of modal logics determined by infinite or arbitrarily long finite linear orders are undecidable, and a sufficient condition for such products to be not recursively enumerable is proved.
Abstract: We study two‐dimensional Cartesian products of modal logics determined by infinite or arbitrarily long finite linear orders and prove a general theorem showing that in many cases these products are undecidable, in particular, such are the squares of standard linear logics like K4.3, S4.3, GL.3, Grz.3, or the logic determined by the Cartesian square of any infinite linear order. This theorem solves a number of open problems posed by Gabbay and Shehtman. We also prove a sufficient condition for such products to be not recursively enumerable and give a simple axiomatization for the square K4.3 × K4.3 of the minimal liner logic using non‐structural Gabbay‐type inference rules.


Journal ArticleDOI
TL;DR: Several consequence relations on the usual language of propositional intuitionistic logic that can be defined semantically by using Kripke frames and the same defining truth conditions for the connectives but without imposing some of the conditions on the Kripkel frames that are required in the intuitionistic case are studied.
Abstract: In the present paper we study systematically several consequence relations on the usual language of propositional intuitionistic logic that can be defined semantically by using Kripke frames and the same defining truth conditions for the connectives as in intuitionistic logic but without imposing some of the conditions on the Kripke frames that are required in the intuitionistic case. The logics so obtained are called subintuitionistic logics in the literature. We depart from the perspective of considering a logic just as a set of theorems and also depart from the perspective taken by Restall in that we consider standard Kripke models instead of models with a base point. We study the relations between subintuitionistic logics and modal logics given by the translation considered by Dosen. Moreover, we classify the logics obtained according to the hierarchy considered in Abstract Algebraic Logic.

Book ChapterDOI
06 Apr 2001
TL;DR: 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 Exp time-complete but FO^1 satisfiability is NExpTime-complete.
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.

Journal ArticleDOI
TL;DR: The proof of the exponential-time upper bound is extended to PDL-like extensions of K m and to global logical consequence and global satisfiability problems and the last part of the paper presents non-trivial classes of exponential time complete regular grammar logics.

Proceedings Article
04 Aug 2001
TL;DR: In this article, a new logical approach to reason explicitly about Dempster-Shafer belief functions is introduced, where one just starts with Boolean formulas φ and a belief function on them; the belief of φ is taken to be the truth degree of the (fuzzy) proposition Bφ standing for "φ is believed".
Abstract: In this paper we introduce a new logical approach to reason explicitly about Dempster-Shafer belief functions. We adopt the following view: one just starts with Boolean formulas φ and a belief function on them; the belief of φ is taken to be the truth degree of the (fuzzy) proposition Bφ standing for "φ is believed" For our complete axiomatization (Hylbert-style) we use one of the possible definitions of belief, namely as probability of (modal) necessity. This enables us to define a logical system combining the modal logic S5 with an already proposed fuzzy logic approach to reason about probabilities. In particular, our fuzzy logic is the logic ŁΠ1/2 which puts Lukasiewicz and Product logics together.

Journal ArticleDOI
TL;DR: Examples of the expressiveness of the languages are given and proofs of soundness and completeness with respect to the possible world semantics are given.
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.

Journal ArticleDOI
TL;DR: Sahlqvist formulas in hybrid polyadic modal languages containing nominals and universal modality or satisfaction operators are defined to ensure first-order definability and canonicity of these formulas immediately transfers to arbitrary polyadic languages.
Abstract: Building on a new approach to polyadic modal languages and Sahlqvist formulas introduced in [10] we define Sahlqvist formulas in hybrid polyadic modal languages containing nominals and universal modality or satisfaction operators. Particularly interesting is the case of reversive polyadic languages, closed under all ‘inverses’ of polyadic modalities because the minimal valuations arising in the computation of the first-order equivalents of polyadic Sahlqvist formulae are definable in such languages and that makes the proof of first-order definability and canonicity of these formulas a simple syntactic exercise. Furthermore, the first-order definability of Sahlqvist formulas immediately transfers to arbitrary polyadic languages, while the direct transfer of canonicity requires a more involved proof-theoretic analysis.

Journal Article
TL;DR: This paper presents a multimodal language which is bisimulation invariant and (under a natural completeness condition) expressive enough to characterise elements of the underlying state space up to bisimulations.
Abstract: Coalgebras for a functor on the category of sets subsume many formulations of the notion of transition system, including labelled transition systems, Kripke models, Kripke frames and many types of automata. This paper presents a multimodal language which is bisimulation invariant and (under a natural completeness condition) expressive enough to characterise elements of the underlying state space up to bisimulation. Like Moss' coalgebraic logic, the theory can be applied to an arbitrary signature functor on the category of sets. Also, an upper bound for the size of conjunctions and disjunctions needed to obtain characteristic formulas is given.

Journal ArticleDOI
TL;DR: In this article, an explicit basis for all admissible rules of the modal logic S4 is given, consisting of an infinite sequence of rules which have compact and simple, readable form and depend on increasing set of variables.
Abstract: We find an explicit basis for all admissible rules of the modal logic S4. Our basis consists of an infinite sequence of rules which have compact and simple, readable form and depend on increasing set of variables. This gives a basis for all quasi-identities valid in the free modal algebra ℱS4(ω) of countable rank.

Book
13 Dec 2001
TL;DR: In this paper, the authors present a survey of possible worlds semantics and anti-realism in mathematics, focusing on the Semantics of Modal Sentential Logic and Quantificational Logic.
Abstract: Introduction 1. Possible Worlds Semantics 2. Transworld Identity 3. Modal Realism 4. Forbes's Anti-Modal Realism 5. The Semantics of Classical Predicate Logic 6. Modality without Worlds: The Semantics of Modal Sentential Logic 7. Quantificational Logic 8. Modality without Worlds: Explorations, Developments, and Defences 9. Anti-Realism in Mathematics Bibliography Index

Journal ArticleDOI
TL;DR: In this paper, a resolution-based proof procedure for modal, description and hybrid logic is presented, which avoids translations into large undecidable logics, and works directly on modal or hybrid logic formulas instead.
Abstract: We provide a resolution-based proof procedure for modal, description and hybrid logic that improves on previous proposals in important ways. It avoids translations into large undecidable logics, and works directly on modal, description or hybrid logic formulas instead. In addition, by using the hybrid machinery it avoids the complexities of earlier propositional resolution-based methods for modal logic. It combines ideas from the method of prefixes used in tableaux, and resolution ideas in such a way that some of the heuristics and optimizations devised in either field are applicable.

01 Jan 2001
TL;DR: Results on global definability in basic modal logic are presented and model-theoretic results and proof techniques are contrasted with known results about local definability.
Abstract: We present results on global definability in basic modal logic, and contrast our model-theoretic results and proof techniques with known results about local definability.

Journal ArticleDOI
TL;DR: Using the systems, the Craig interpolation lemma for the modal logics KB, KDB, K5, and KD5 is proved and the systems have the analytic superformula property and can thus give a decision procedure.
Abstract: We give complete sequent-like tableau systems for the modal logics KB, KDB, K5, and KD5. Analytic cut rules are used to obtain the completeness. Our systems have the analytic superformula property and can thus give a decision procedure. Using the systems, we prove the Craig interpolation lemma for the mentioned logics.

Journal ArticleDOI
TL;DR: This paper proves completeness and decidability results for a range of normal and nonnormal but quasi-normal propositional modal logics containing “actually” operators, the weakest of which are conservative extensions of K, using a novel generalisation of the standard semantics.
Abstract: The addition of “actually” operators to modal languages allows us to capture important inferential behaviours which cannot be adequately captured in logics formulated in simpler languages. Previous work on modal logics containing “actually” operators has concentrated entirely upon extensions of KT5 and has employed a particular model-theoretic treatment of them. This paper proves completeness and decidability results for a range of normal and nonnormal but quasi-normal propositional modal logics containing “actually” operators, the weakest of which are conservative extensions of K, using a novel generalisation of the standard semantics.

Journal ArticleDOI
01 Apr 2001-Synthese
TL;DR: It is argued that this shift to hybrid logic has consequences for both modal and dialogical logic, and it is shown how to lift the dialogical conception of modal logic to modal proof theory.
Abstract: The title reflects my conviction that, viewed semantically,modal logic is fundamentally dialogical; this conviction is based on the key role played by the notion of bisimulation in modal model theory. But this dialogical conception of modal logic does not seem to apply to modal proof theory, which is notoriously messy. Nonetheless, by making use of ideas which trace back to Arthur Prior (notably the use of nominals, special proposition symbols which ‘name’ worlds) I will show how to lift the dialogical conception to modal proof theory. I argue that this shift to hybrid logic has consequences for both modal and dialogical logic, and I discuss these in detail.

Proceedings Article
01 Apr 2001
TL;DR: A deontic logic of regular action is defined as a characterization within a modal μ-calculus of action by closely following the structure of deterministic finite automatons for 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.

Journal ArticleDOI
TL;DR: The theoretical foundations to extend free-variable semantic tableaux to propositional modal logics, including non-trivial rigorous proofs of soundness and completeness, are presented.
Abstract: Free-variable semantic tableaux are a well-established technique for first-order theorem proving where free variables act as a meta-linguistic device for tracking the eigenvariables used during proof search. We present the theoretical foundations to extend this technique to propositional modal logics, including non-trivial rigorous proofs of soundness and completeness, and also present various techniques that improve the efficiency of the basic naive method for such tableaux.