Showing papers on "Normal modal logic published in 1991"

A propositional modal logic of time intervals

Abstract: In certain areas of artificial intelligence there is need to represent continuous change and to make statements that are interpreted with respect to time intervals rather than time points. To this end, a modal temporal loglc based on time intervals is developed, a logic that can be viewed as a generalization of point-based modal temporal logic. Related loglcs are discussed, an intuitive presentation of the new logic is given, and its formal syntax and semantics are defined. No assumption is made about the underlying nature of time, allowing it to be discrete (such as the natural numbers) or continuous (such as the rationals or the reals), linear or branching, complete (such as the reals), or not (such as the rational). It is shown, however, that there are formulas in the logic that allow us to distinguish all these situations. A translation of our logic into first-order logic is given, which allows the application of some results on first-order logic to our modal logic. Finally. the difficulty of validity problems for the logic is considered. This turns out to depend critically, and in surprising ways, on our assumptions about time. For example, if our underlying temporal structure is the ratlonals, then, the validity problem is r. e .-complete; if it is the reals, then validity n II ~-hard: and if it is the natural numbers, then validity is fI ] -complete.

Many-valued modal logics

Abstract: Two families of many-valued modal logics are investigated. Semantically, one family is characterized using Kripke models that allow formulas to take values in a finite many-valued logic, at each possible world. The second family generalizes this to allow the accessibility relation between worlds also to be many-valued. Gentzen sequent calculi are given for both versions, and soundness and completeness are established.

An essay in combinatory dynamic logic

Abstract: We propose a refinement of Kripke modal logic, and in particular of PDL (propostional dynamic logic)

Semantics-Based Translation Methods for Modal Logics

Abstract: A general framework for translating logical formulae from one logic into another logic is presented. The framework is instantiated with two different approaches to translating modal logic formulae into predicate logic. The first one, the well known ‘relational’ translation makes the modal logic’s possible worlds structure explicit by introducing a distinguished predicate symbol to represent the accessibility relation. In the second approach, the ‘functional’ translation method, paths in the possible worlds structure are represented by compositions of functions which map worlds to accessible worlds. On the syntactic level this means that every flexible symbol is parametrized with particular terms denoting whole paths from the initial world to the actual world. The ‘target logic’ for the translation is a first-order many-sorted logic with built in equality. Therefore the ‘source logic’ may also be first-order many-sorted with built in equality. Furthermore flexible function symbols are allowed. The modal operators may be parametrized with arbitrary terms and particular properties of the accessibility relation may be specified within the logic itself.

Amalgamation and interpolation in normal modal logics

Abstract: This is a survey of results on interpolation in propositional normal modal logics. Interpolation properties of these logics are closely connected with amalgamation properties of varieties of modal algebras. Therefore, the results on interpolation are also reformulated in terms of amalgamation.

TABLEAUX: A general theorem prover for modal 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

Modal interpretations of default logic

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.

The Mckinsey axiom is not canonical

Abstract: The logic KM is the smallest normal modal logic that includes the McKinsey axiomIt is shown here that this axiom is not valid in the canonical frame for KM, answering a question first posed in the Lemmon-Scott manuscript [Lemmon, 1966].The result is not just an esoteric counterexample: apart from interest generated by the long delay in a solution being found, the problem has been of historical importance in the development of our understanding of intensional model theory, and is of some conceptual significance, as will now be explained.The relational semantics for normal modal logics first appeared in [Kripke, 1963], where a number of well-known systems were shown to be characterised by simple first-order conditions on binary relations (frames). This phenomenon was systematically investigated in [Lemmon, 1966], which introduced the technique of associating with each logic L a canonical frame which invalidates every nontheorem of L. If, in addition, each L-theorem is valid in , then L is said to be canonical. The problem of showing that L is determined by some validating condition C, meaning that the L-theorems are precisely those formulae valid in all frames satisfying C, can be solved by showing that satisfies C—in which case canonicity is also established. Numerous cases were studied, leading to the definition of a first-order condition Cφ associated with each formula φ of the formwhere Ψ is a positive modal formula.

A general framework for modal deduction

Abstract: A general method of automated modal logic theorem proving is discussed and illustrated. This method is based on the substitutional framework for the development of systems for hybrid reasoning. Sentences in modal logic are translated into a constraint logic in which the constraints represent the connections between worlds in the possible world semantics for modal logic. Deduction in the constraint logic is performed by a non-modal deductive system which has been systematically enhanced with special-purpose constraint processing mechanisms. The result is a modal logic theorem prover, whose soundness and completeness is an immediate consequence of the correctness of the non-modal deductive system and some general results on constraint deduction. The framework achieves significant generality in that it provides for the extension of a wide range of non-modal systems to corresponding modal systems and that can be done for a wide range of modal logics.

The true modal logic

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

The disjunction property of intermediate propositional logics

Abstract: This paper is a survey of results concerning the disjunction property, Hallden-completeness, and other related properties of intermediate prepositional logics and normal modal logics containing S4.

Modal Logics for Mobile Processes

Abstract: In process algebras, bisimulation equivalence is typically defined directly in terms of the operational rules of action; it also has an alternative characterisation in terms of a simple modal logic (sometimes called Hennessy-Milner logic. This paper first defines two forms of bisimulation equivalence for the π-calculus, a process algebra which allows dynamic reconfiguration among processes; it then explores a family of possible logics, with different modal operators. It is proven that two of these logics characterise the two bisimulation equivalences. Also, the relative expressive power of all the logics is exhibited as a lattice.

Quantification in autoepistemic logic

Abstract: : Quantification in modal logic is interesting from a technical and philosophical standpoint. Here we look at quantification in autoepistemic logic, which is a modal logic of self-knowledge. We propose several different semantics, all based on the idea that having beliefs about an individual amounts to having a belief using a certain type of name for the individual.

A Perspective on Modal Sequent Logic

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

Nonmonotonic default modal logics

Abstract: Conclusions by failure to prove the opposite are frequently used in reasoning about an incompletely specified world. This naturally leads to logics for default reasoning which, in general, are nonmonotonic, i.e., introducing new facts can invalidate previously made conclusions. Accordingly, a nonmonotonic theory is called (nonmonotonically) degenerate, if adding new axioms does not invalidate already proved theorems. We study nonmonotonic logics based on various sets of defaults and present a necessary and sufficient condition for a nonmonotonic modal theory to be degenerate. In particular, this condition provides several alternative descriptions of degenerate theories. Also we establish some closure properties of sets of defaults defining a nonmonotonic modal logic.

Modal Logic Should Say More Than It Does.

Abstract: First-order modal logics, as traditionally formulated, are not expressive enough. It is this that is behind the difficulties in formulating a good analog of Herbrand's Theorem, as well as the well-known problems with equality, non-rigid designators, definite descriptions, and non-designating terms. We show how all these problems disappear when modal language is made more expressive in a simple, natural way. We present a semantic tableaux system for the enhanced logic, and (very) briefly discuss implementation issues.

Rough polyadic modal logics

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

Relational proof systems for some AI logics

Abstract: Relational methodology of defining automated proof systems has been applied to a modal logic for reasoning with incomplete information and to an epistemic logic for reasoning about partial knowledge of groups of agents.

Modal companions of superintuitionistic logics: syntax, semantics, and preservation theorems

Abstract: This paper studies the class of superintuitionistic logics and the class of normal extensions of the modal system S4, and the syntactic and semantic connections between the two classes, given by the mapping (which assigns to every modal logic its superintuitionistic fragment) and by the mappings and (which assign to every superintuitionistic logic its smallest and its greatest companion, respectively). It is shown that from classes of relational models with respect to which a logic is complete, one can construct a class of models with respect to which the logics and are complete. The relationship of inference (of canonical formulas) in logics , , and is also described. As a consequence, preservation theorems are obtained for finite approximability, for Kripke completeness and for the disjunction property at the transition from to , and also for decidability at the transition to and .Bibliography: 21 titles.

Parameter structures for parametrized modal operators

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.

A constraint logic approach to modal deduction

Abstract: A general method of automated modal logic theorem proving is discussed and illustrated. Sentences in modal logic are translated into a constraint logic in which the constraints represent the connections between worlds in the possible world semantics for modal logic. Deduction in the constraint logic is performed by a non-modal deductive system which has been systematically enhanced with special-purpose constraint processing mechanisms. The result is a modal logic theorem prover, whose soundness and completeness is an immediate consequence of the correctness of the non-modal deductive system and some general results on constraint deduction. The framework achieves significant generality in that it provides for the extension of a wide range of non-modal systems to corresponding modal systems and that this can be done for a wide range of modal logics. A number of existing modal logic theorem proving systems can be rationally reconstructed as particular instances of this general method.