Showing papers on "Modal operator published in 1990"

Bilattices and Modal Operators

Abstract: A bilattice is a set equipped with two partial orders and a negation operation that inverts one of them while leaving the other unchanged; it has been suggested that the truth values used by inference systems should be chosen from such a structure instead of the two-point set {t, f}. Given such a choice, we redefine a modal operator to be a function on the bilattice selected, and show that this definition generalizes both Kripke's possible worlds approach and Moore's autoepistemic logic. Extensions to causal and temporal reasoning are also discussed.

Half-order modal logic: how to prove real-time properties

Abstract: We introduce a novel extension of propositional modal logic that is interpreted over Kripke structures in which a value is associated with every possible woxld. These values are, however, not treated as full first-order objects; they can be accessed only by a very restricted form of quantification: the “freeze” quantifier binds a variable to the value of the current world. We present a complete proof system for this (“hulf_o4er”) modal logic. As a special case, we obtain the real-time temporal logic TPTL of [AH89]: the models are restricted to infinite sequences of states, whose values are monotonically increasing natural numbers. The ordering relation between states is interpreted as temporal precedence, while the value associated with a state is interpreted as its “real” time. We extend our proof system to be complete for TPTL, and demonstrate how it can be used to derive real-time properties.

Effective domains and intrinsic structure

Abstract: Topos theory is the categorical analog of constructive set theory; and conveniently, PERs (partial equivalence relations) do sit inside a topos-the category of PERs can be (loosely speaking) identified with the full subcategory of modest sets in Hyland's effective topos. (The effective topos is the topos-theoretic version of recursive realizability.) Working in the effective topos is especially attractive since not only can set-theoretic reasoning be used, but one also has a lot of category-theoretic and topos-theoretic machinery at one's disposal. That is the point of view taken in this research. The basic theory of Sigma -spaces is discussed. A convex power domain is also presented. Modal operators are outlined. Parallelism and sheaves are examined. Finally, the fixed-point classifier is presented. >

Bilattices and modal operators

Abstract: A bilattice is a set equipped with two partial orders and a negation operation that inverts one of them while leaving the other unchanged; it has been suggested that the truth values used by inference systems should be chosen from such a structure instead of the two-point set {t, f}. Given such a choice, we redefine a modal operator to be a function on the bilattice selected, and show that this definition generalizes both Kripke's possible worlds approach and Moore's autoepistemic logic. Extensions to causal and temporal reasoning are also discussed.

A general approach for determining the validity of commonsense assertions using conditional logics

Abstract: An approach to theorem proving for the class of normal conditional logics is presented. These logics have been shown to be appropriate for representing a wide variety of commonsense assertions, including default and prototypical properties, counterfactuals, notions of obligation, and others. the logics are based on a possible worlds semantics but unlike the better-known modal logics of necessity and possibility, they contain a binary “variable conditional” operator, ⟹, rather than a unary modal operator. the truth of a statement A ⟹ B depends both on the accessibility relation between worlds and on the proposition expressed by the antecedent A. The approach develops an extension of the semantic tableaux approach to theorem proving. Basically, it consists in attempting to find an interpretation which will falsify a sentence or set of sentences. If successful, then a specific falsifying truth assignment is obtained; if not, then the sentence is valid. Since this method is based directly on the notion of truth, it is arguably more natural and intuitive than those based on proof-theoretic methods. the approach has been proven correct for the class of normal conditional logics. In addition, it has been implemented and tested on a number of different logics. Various heuristics have been incorporated, and the implementation, while exponential in the worst case, is shown to be reasonably efficient for a large set of test cases.

Modal logics of higher-order probability

Abstract: This paper discusses the relationship between probability and modal logic. We show that it is both natural and useful to think of probability as a modal operator. Contrary to popular belief in AI, a probability ranging between 0 and 1 represents a range between impossibility and necessity, not between simple falsity and truth. We examine two classes of probability models: flat and staged. The flat models are straightforward generalizations of models for alethic logic. We show that one of the more interesting constraints relating higher- and lower-order probabilities forces all higher-order probabilities in flat models to be either zero or one. We introduce staged models as a means of avoiding this problem. Constraints on the two types of models define various classes of probability logics. We relate some of these probability logics to alethic logic.

Conversion principles and the basis of aristotle's modal logic

Abstract: Aristotle founds his modal syllogistic, like his plain syllogistic, on a small set of ‘perfect’ or obviously valid sylligisms. The rest he reduces to those, usually by means of modal conversion principles. These principles are open to more than one reading, however, and they are in fact invalid on one traditional reading (de re), valid on the other (de dicto). It is argued here that this way of framing the contrast is not Aristotelian, and that an interpretation involving modal copulae allows us to see how these principles, and the modal system as a whole, are to be understood in light of close and precise connections to Aristotle's essentialist metaphysics.

Modal Logic Programming

Abstract: Modal logic offers a rich paradigm for programming. Several executable languages based on various types of modal logic are presented. Modality is not only useful for programming on domains for which the notion of time is essential but also can be used as structural concepts. In particular, it can be applied as a high level executable specification language especially in such domains as real-time, process control, distributed programming. This paper surveys several works in this direction which have been done in recent years at Kyoto University.

Logical Semantics of Multiple World Mechanism for Knowledge Representation in Logic Programming

Abstract: Summary A logic-based knowledge representation system called Uranus uses a multiple world mechanism. The heart of the mechanism is making programs context (situation) sensitive. A program transported into another world context may behave differently from the original context. This polymorphism of programs has turned out to be a powerful tool in knowledge representation. For example, the mechanism may be used for 1) representing a concept hierarchy, 2) representing a time sequence, and 3) common sense reasoning. By defining a visibility relation among worlds from the inheritee (upper class, in the case of a class hierarchy, and past in the case of a time sequence) to the inheriter (lower class / future), which is the opposite direction of usual cases, a simple semantics is given to assertions: The semantics of (assert P Q ) is given as □( P ← Q ) ← WP where □ is a modal operator meaning that the formula following it is true within visible worlds, and W is a modal operator meaning that the formula following it is not explicitly denied within visible worlds. By following this semantics, we can also treat common sense reasoning in a simple manner.

Knowledge Revision and Multiple Extensions

Abstract: We present a tense logic, ZK, for representing changes within knowledge based systems. ZK has tense operators for the future and for the past, and a modal operator for describing consistency. A knowledge based system consists of a set K of general laws (described by first order formulae) and of a set of states, each described by a first order formula (called descriptive formula). Changes are represented by pairs of formulae (P,R) (precondition, result). A change can occur within a state whenever the preconditions are true. The descriptive formula of the resulting new state is the conjunction of R with the maximal subformula of the descriptive formula of the old state which is consistent with K and R. Generally, a change will yield more than one new state (multiple extensions).

Logics with Structured Contexts.

Abstract: The concept of structured context is developed. It generalizes in a natural way the modal operators known in modal logics.