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Showing papers on "Modal operator published in 1991"


Journal ArticleDOI
TL;DR: In this discussion transfer theorems for the most simple case when there are just two modal operators are shown and it will be clear that the proof works in the general case as well.
Abstract: In mono-modal logic there is a fair number of high-powered results on completeness covering large classes of modal systems, witness for example Fine [74,85] and Sahlqvist [75]. Mono-modal logic is therefore a well-understood subject in contrast to poly-modal logic where even the most elementary questions concerning completeness, decidability etc. have been left unanswered. Given that so many applications of modal logic one modality is not sufficient, the lack of general results is acutely felt by the “users” of modal logics, contrary to logicians who might entertain the view that a deep understanding of modality alone provides enough insight to be able to generalize the results to logics with several modalities. Although this view has its justification, the main results we are going to prove are certainly not of this type, for they require a fundamentally new technique. The results obtained are called transfer theorems in Fine and Schurz [91] and are of the following type. Let L 63 ⊥ be an independently axiomatizable bimodal logic and L2 as well as L its mono-modal fragments. Then L has a property P iff L2 and L have P . Properties which will be discussed are completeness, finite model property, compactness, persistence, interpolation and Hallden-completeness. In our discussion we will show transfer theorems for the most simple case when there are just two modal operators but it will be clear that the proof works in the general case as well.

176 citations


Proceedings Article
24 Aug 1991
TL;DR: A new version of the Lin/Shoham logic, similar in spirit to the Levesque/Reiter theory of epistemic queries, is described, which can give meaning to Epistemic queries in the context of nonmonotonic databases, including logic programs with negation as failure.
Abstract: The approach to database query evaluation developed by Levesque and Reiter treats databases as first order theories, and queries as formulas of the language which includes, in addition to the language of the database, an epistemic modal operator. In this epistemic query language, one can express questions not only about the external world described by the database, but also about the database itself-- about what the database knows. On the other hand, epistemic formulas are used in knowledge representation for the purpose of expressing defaults. Autoepistemic logic is the best known epistemic nonmonotonic formalism; the logic of grounded knowledge, proposed recently by Lin and Shoham, is another such system. This paper brings these two directions of research together. We describe a new version of the Lin/Shoham logic, similar in spirit to the Levesque/Reiter theory of epistemic queries. Using this formalism, we can give meaning to epistemic queries in the context of nonmonotonic databases, including logic programs with negation as failure.

168 citations


Proceedings Article
14 Jul 1991
TL;DR: In this paper, the syntax and semantics of logic programs and deductive databases have been extended to allow for the correct representation of incomplete information in the presence of multiple extensions, including negation by failure and epistemic disjunction.
Abstract: The purpose of this paper is to expand the syntax and semantics of logic programs and deductive databases to allow for the correct representation of incomplete information in the presence of multiple extensions. The language of logic programs with classical negation, epistemic disjunction, and negation by failure is further expanded by a new modal operator K (where for the set of rules T and formula F, KF stands for "F is known to a reasoner with a set of premises T"). Theories containing such an operator will be called strongly introspective. We will define the semantics of such theories (which expands the semantics of deductive databases from [Gelfond and Lifschitz 1990bD and demonstrate the applicability of strongly introspective theories to formalization of some forms of commonsense reasoning.

85 citations


Journal ArticleDOI
Laurent Catach1
TL;DR: This work presents a general theorem proving system for propositional modal logics, called TABLEAUX, which provides an unified environment for various kinds of modal operators and for a wide class of modals, including usual temporal, epistemic or dynamic logics.
Abstract: We present a general theorem proving system for propositional modal logics, called TABLEAUX The main feature of the system is its generality, since it provides an unified environment for various kinds of modal operators and for a wide class of modal logics, including usual temporal, epistemic or dynamic logics We survey the modal languages covered by TABLEAUX, which range from the basic one L(□, ◊) through a complex multimodal language including several families of operators with their transitive-closure and converse The decision procedure we use is basically a semantic tableaux method, but with slight modifications compared to the traditional one We emphasize the advantages of such semantical proof methods for modal logics, since we believe that the models construction they provide represents perhaps the most attractive feature of these logics for possible applications in computer science and AI The system has been implemented in Prolog, and appears to be of reasonable efficiency for most current examples Experimental results are given in the paper, with two lists of test examples

65 citations


Proceedings Article
24 Aug 1991
TL;DR: In this article, a modal interpretation of default reasoning is presented, which yields a characterization of default extensions similar to the characterization of stable expansions by means of auto-epistemic interpretation.
Abstract: In the paper we study a new and natural modal interpretation of defaults. We show that under this interpretation there are whole families of modal nonmonotonic logics that accurately represent default reasoning. One of these logics is used in a definition of possible-worlds semantics for default logic. This semantics yields a characterization of default extensions similar to the characterization of stable expansions by means of autoepistemic interpretation. We also show that the disjunctive information can easily be handled if disjunction is represented by means of modal disjunctive defaults -- modal formulas that we use in our interpretation. Our results indicate that there is no single modal logic for describing default reasoning. On the contrary, there exist whole ranges of modal logics, each of which can be used in the embedding as a "host" logic.

53 citations


Journal ArticleDOI
TL;DR: This paper wants to trace the course of Prior's own struggles with the concepts and phenomena of modality, and the reasoning that led him to his own rather peculiar modal logic Q, and argues that those intuitions do not of themselves lead to Q, but that one must also accept a certain picture of what it is for a proposition to be possible.
Abstract: For Arthur Prior, the construction of a logic was a supremely philosophical task. As a logician, he could of course appreciate a finely crafted formal system for its own sake, independent of its "meaning" or philosophical significance. But a genuine logic a good one, at least lays bare the nature of those concepts that it purports to be a logic of: and this comes only by way of deep reflection and insightful philosophical analysis. This sort of reflection and analysis is no more evident than in Prior's own search for the "true" modal logic.' In this paper, I want to trace the course of Prior's own struggles with the concepts and phenomena of modality, and the reasoning that led him to his own rather peculiar modal logic Q. I find myself in almost complete agreement with Prior's intuitions and the arguments that rest upon them. However, I will argue that those intuitions do not of themselves lead to Q, but that one must also accept a certain picture of what it is for a proposition to be possible. That picture, though, is not inevitable. Rather, implicit in Prior's own account is an alternative picture that has already appeared in various guises, most prominently in the work of Adams [I], Fine [5], and more recently, Dcutsch [4]' and Almog [2]. I, too, will opt for this alternative, though I will spell it out rather differently than these philosophers. I will then show that, starting with the alternative picture, Prior's intuitions can lead instead to a much happier and more standard quantified modal logic than Q. The last section of the paper is devoted to the formal development of the logic and its metatheory. By way of preliminaries, then, let T be the basic propositional modal logic obtained by adding to the propositional calculus the

39 citations


Book ChapterDOI
01 Jan 1991
TL;DR: Ghilardi and Meloni as discussed by the authors show that by imposing a restriction on the topos itself, they may define lax modal operators on predicates of all objects of the topology.
Abstract: In the last few years, two topes-theoretic approaches to modal logic have been developed simultaneously but independently: one started by Reyes [9] and further developed in Lavendhomme, Lucas and Reyes [7], and the other due to Ghilardi and Meloni [2]. At first sight, these approaches appear quite unrelated and the formal systems to which they lead are different. It may come, therefore, as a surprise that both are particular cases of a more general one which we shall describe in detail in this paper. There is a fundamental difference between modal operators in a topos such as 0 (necessity) and other logical operators such as -, (negation): while -, is functorial in the sense that it commutes with pull-backs (and hence it defines a map -,: n n), this is not so for O. Indeed, the only operator 0: n n such that Op P and oT = T is the identity. In Reyes [9], this difficulty is circumvented by considering a topos over a base topos and restricting the domain of application of modal operators to predicates of "constant" objects only, but keeping functoriality with respect to "constant" maps. Ghilardi and Meloni [2] on the other hand, restrict the domain of application as in Reyes [9] but relax functoriality to lax functoriality for constant maps. This feature of the approach of Ghilardi and Meloni [2] will be kept in our work. Indeed, it has to be kept in any context which aims to generalize their approach. In a sequel to this paper, we shall show that by imposing a restriction on the topos itself, we may define lax modal operators on predicates of all objects of the topos.

28 citations


Proceedings ArticleDOI
18 Jun 1991
TL;DR: The authors propose a new semantics for the R/sub B/ modal operator, such that the definition of security would allow a certain number of dependencies (called secure dependencies) between objects of the system.
Abstract: In the context of the modal logic of security, confidentiality is defined by the formula K/sub B/ phi to R/sub B/ phi that could be read 'If B knows phi then B should have the permission to know phi '. The authors propose a new semantics for the R/sub B/ modal operator, such that the definition of security would allow a certain number of dependencies (called secure dependencies) between objects of the system. They formally compare this definition of security with non-interference, non-deducibility and generalized non-interference, especially with respect to assumptions on the systems as non-determinism and input-totalness. >

23 citations


Journal ArticleDOI
TL;DR: In this paper, the structural properties of modal operators were investigated at the purely structural level, and a notion of sequent was proposed to distinguish between modal logic systems of different types of operators.
Abstract: This paper was drafted in 1981—2 when the first author was lecturing on modal logic for the Philosophy Subfaculty at Oxford University and the second author was visiting Oxford on study leave; it was revised the following year^ Both of us had been graduate students of D S Scott at Oxford in the 1970's and were impressed by his emphasis on the desirability of isolating the structural properties of a (logical) consequence relation — such as are encoded in the principles (jR), (M), and (T) of [11], [12] —from principles relating to specific connectives Extending this idea to the case of the modal operators, we found that distinctions between several well-known systems of (normal) modal logic could be reflected at the purely structural level, if an appropriate notion of sequent was adopted Actually, we work with one notion of sequent in §§1—4 and consider a somewhat more refined version in §5 On later finding that sequents of the latter type had already been used by M Sato, who, in §34 of [10], credits the idea to O Sonobe, we had some misgivings about publishing the material at full length That anticipation notwithstanding, however, it appears to us still worth proceeding with a somewhat abridged version of the paper, both so as to highlight the original motivation and also because our treatment and Sato's differ on many points of detail We should mention that K Dosen, in [4], also advocates a variation on the traditional idea of what a sequent should look like for the case of modal logic Though the framework he sets up is quite different from our own, he is in part motivated by similar consideratons (eg, the concern with 'unique characterization' — see §4 below) Some aspects of our own way of proceeding may be seen (again, in retrospect) as steps in the execution of Belnap's "Display Logic' programme (see [2]), in that the rules (a) of §4 serve

22 citations


Journal ArticleDOI
TL;DR: Provability logics with many modal operators for progressions of theories obtained by iterating their consistency statements are introduced and the corresponding arithmetical completeness theorem is proved.
Abstract: Provability logics with many modal operators for progressions of theories obtained by iterating their consistency statements are introduced. The corresponding arithmetical completeness theorem is proved.

18 citations


Journal ArticleDOI
TL;DR: Rough polyadic modal logics contain modal operators of many arguments with a relational semantics, based on the Pawlak's rough set theory, and has many applications in different branches in Artificial Intelligence and theoretical computer science.
Abstract: Rough polyadic modal logics, introduced in the paper, contain modal operators of many arguments with a relational semantics, based on the Pawlak's rough set theory. Rough set approach is developed as an alternative to the fuzzy set philosophy, and has many applications in different branches in Artificial Intelligence and theoretical computer science

Proceedings Article
24 Aug 1991
TL;DR: In this paper the application of the idea of parametrized modal operators is extended in in two ways: first of all a modified neighbourhood semantics is defined which permits among others the interpretation of the parameters as probability values.
Abstract: The parameters of the parameterized modal operators [p] and 〈p〉 usually represent agents (in the epistemic interpretation) or actions (in the dynamic logic interpretation) or the like. In this paper the application of the idea of parametrized modal operators is extended in in two ways: First of all a modified neighbourhood semantics is defined which permits among others the interpretation of the parameters as probability values. A formula [5] F may for example express the fact that in at least 50% of all cases (worlds) F holds. These probability values can be numbers, qualitative descriptions and even arbitrary terms. Secondly a general theory of the parameters and in particular of the characteristic operations on the parameters is developed which unifies for example the multiplication of numbers in the probabilistic interpretation of the parameters and the sequencing of actions in the dynamic logic interpretation.

Journal ArticleDOI
TL;DR: A formal approach to user models in data retrieval applications, with an emphasis on retrieval systems using a browsing interaction style, and the extension of the method to deal with non-browsing systems is discussed.
Abstract: The interaction between a data base system and its users searching for interesting information can be improved by applying user modelling techniques. Currently existing techniques are, however, hard to compare and evaluate, because the models they generate do not have rigorously defined semantics. This paper contains a formal approach to user models in data retrieval applications, with an emphasis on retrieval systems using a browsing interaction style. A first-order logic is defined formalizing interests of a user in the availability of data meeting certain constraints. Next, the logic is augmented with a modal operator, analogous to the possibility operator of the standard Kripke possible-worlds semantics, to express the predictive qualities of a user model. Applications of the formalism are included to illustrate the viability of the approach, and the extension of the method to deal with non-browsing systems is discussed.

Journal ArticleDOI
Thomas Magnell1
TL;DR: In this article, it is argued that no such claims are warranted, since Russell did not advance an identifiable modal logic or anything more than a modest modal theory and his view of naming involves a notion of guaranteed reference.
Abstract: Russell has recently been held to have had a modal logic, a full modal theory and a view of naming that anticipates Kripke's intuitions on rigid designation. It is argued here that no such claims are warranted. While Russell was not altogether silent on matters modal, he did not advance an identifiable modal logic or anything more than a modest modal theory. His view of naming involves a notion of guaranteed reference. But what Kripke's intuitions about rigidity primarily pertain to is fixed reference, something demonstrably different.

Proceedings ArticleDOI
28 Aug 1991
TL;DR: A modal logic called distribued logic (DL) is formed by using ideas from both the interleaving and partial ordering approach for representing real-timed concurrency in temporal logic.
Abstract: Temporal logic is widely acclaimed to be a highly successful tool for analyzing non-real-time properties of programs. However, a few fundamental problems arise while designing temporal logic-based-techniques to verify real-time properties of programs. in this context, we formulate a modal logic called distribued logic (DL) by using ideas from both the interleaving and partial ordering approach. This logic uses spatial modal operators in addition to temporal operators for representing real-timed concurrency. In addition to the syntax and semantics of the logic, programming model and a formal proof technique based on the logic are also presented. Finally, use of the proof method is illustrated through the analysis of the real-time properties of a generic multiproceess producer/consumer program.

Book ChapterDOI
29 Oct 1991
TL;DR: In this introduction to temporal logic, temporal logic has been investigated in the artificial intelligence field in order to build a theory of events and actions and the use of standard first-order logic and of the intensional one is seen.
Abstract: In this introduction we start speaking about temporal logic. In fact, while a wide corpus of studies about temporal logic exists, it is obvious that in a universe where modem relativistic physics has been written down in the book space and time are related both in conceptualization and in formal theories about them. Besides and most important spatial and temporal logic a r e placed in the same category from the fact of being both topologic logics, that is logics in which a particular collocation of the entities of the universe in an abstract space has a particular relevance. In the philosophy tradition there are two approaches to the representation of time in logic, the first-order and the modal, or intensional, one. First-order supporters sustain the theory that time, while being an important one, is just yet another variable, and so we do not need a particular logic built around time in order to expose the temporal properties of the entities described. The theoretical background of such scientists is generally the study of the logic foundations of mathematics. In opposition when logic is used to analyze the formal properties of the language the tense structure becomes a relevant topic. The first modem philosopher to propose a modal logic for the analysis of the language was Prior, in its Tense Logic [Prior 1955]. In his work, from the proposition p the modal assertions Fp (it will be the case that p) Pp (it has been the case that p) and other ones can be derived, where F and P are the modal operators. In the eighties temporal logic has been investigated in the artificial intelligence field in order to build a theory of events and actions. Here too we can see the use of standard first-order logic and of the intensional one. The intensional temporal logic that we can find in

Journal Article
TL;DR: This paper introduces an extension of Montague's intensional logic that is equipped with a well-defined 'Boolean semantics', analogous with the Keenan-Faltz semantics.