Showing papers on "Modal operator published in 1996"

Generalization of Rough Sets using Modal Logics

Abstract: The theory of rough sets is an extension of set theory with two additional unary set-theoretic operators defined based on a binary relation on the universe. These two operators are related to the modal operators in modal logics. By exploring the relationship between rough sets and modal logics, this paper proposes and examines a number of extended rough set models. By the properties satisfied by a binary relation, such as serial, reflexive, symmetric, transitive, and Euclidean, various classes of algebraic rough set models can be derived. They correspond to different modal logic systems. With respect to graded and probabilistic modal logics, graded and probabilistic rough set models are also discussed.

Multi-Dimensional Modal Logic

Abstract: We start with informally defining the subject matter of this book: multi-dimensional modal logic (MDML). First let us briefly consider what we understand by the notion of “modal logic”. The last decade has seen a development in modal logic towards a more abstract and technical approach. In this perspective of what one might call abstract modal logic, arbitrary relational structures can be seen as models for an (extended) modal language: any relation is a potential accessibility relation of some suitably defined modal operator. As the essentially modal aspect of the framework one could point out that the mechanism for evaluating formulas forces certain moves along the accessibility relations. Thus, for instance quantification over a model is restricted to an “accessible” part of the structure.

Power and Weakness of the Modal Display Calculus

Abstract: The present paper explores applications of Display Logic as defined in [1] to modal logic. Acquaintance with that paper is presupposed, although we will give all necessary definitions. Display Logic is a rather elegant proof-theoretic system that was developed to explore in depth the possibility of total Gentzenization of various propositional logics. By Gentzenization I understand the strategy to replace connectives by structures. Gentzenization is something of an ingenious optical trick because it uses a single symbol to mean different things depending on the place it occupies in the sequent. In the original Gentzen system it was the comma that had to be interpreted as and when to the left of the turnstile and as or when to the right. The interpretation of the structures oscillates between two logical symbols depending on whether it is in the antecedent or in the consequent. This is why we call symbols like comma Gentzen toggles. These two symbols between which this toggle switches are the Gentzen duals of each other. So, and and or are Gentzen duals. The strength of Display logic lies in a rather general cut-elimination theorem. In [10] and [9], Heinrich Wansing has refined these methods for modal logics; he showed that contrary to Belnap’s own Gentzenization of modal operators as binary structure operators, a unary one is more appropriate (not only from an esthetical point of view) and makes perfect sense semantically as well. The Gentzen dual of the modal operator □ is actually not — as one might expect — the. possibility operator ◇, but the backward looking possibility operator, denoted here by ◇. (To be consistent with that we write □ instead of □ and ◇ instead of ◇.) The corresponding toggle is denoted by •. The reason why this is so natural lies in the fact that it is the exact Display or Gentzen dual, for we have that the sequent •B ⊢ A and the sequent B ⊢ • A are equivalent if • is read as if in the antecedent and if it is read as 0 if in the consequent. Wansing uses this fact to display various modal and tense logics a la Belnap by providing some formula introduction rules and basic structural rules for K and Kt and then Gentzenizing the additional axioms. The benefit lies not only in the homogeneity with which all these systems are now handled and the rather clear intuitive background. The benefit lies in the possibility to use the general cut-elimination theorem of [1].

An Overview of Interpretability Logic.

Abstract: A miracle happens. In one hand we have a class of marvelously complex theories in predicate logic, theories with 'sufficient coding potential', like PA (Peano Arithmetic) or ZF (Zermelo Fraenkel Set Theory). In the other we have certain modal propositional theories of striking simplicity. We translate the modal operators of the modal theories to certain specic, fixed, defined predicates of the predicate logical theories. These special predicates generally contain an astronomical number of symbols. We interpret the propositional variables by arbitrary predicate logical sentences. And see: the modal theories are sound and complete for this interpretation. They codify precisely the schematic principles in their scope. Miracles do happen ....

Bi-Heyting algebras, toposes and modalities

Abstract: The aim of this paper is to introduce a new approach to the modal operators of necessity and possibility. This approach is based on the existence of two negations in certain lattices that we call bi-Heyting algebras. Modal operators are obtained by iterating certain combinations of these negations and going to the limit. Examples of these operators are given by means of graphs.

Multi-Agent Only Knowing

Abstract: Levesque introduced a notion of "only knowing", with the goal of capturing certain types of nonmonotonic reasoning. Levesque's logic dealt with only the case of a single agent. Recently, both Halpern and Lakemeyer independently attempted to extend Levesque's logic to the multi-agent case. Although there are a number of similarities in their approaches, there are some significant differences. In this paper, we reexamine the notion of only knowing, going back to first principles. In the process, we point out some problems with the earlier definitions. This leads us to reconsider what the properties of only knowing ought to be. We provide an axiom system that captures our desiderata, and show that it has a semantics that corresponds to it. The axiom system has an added feature of interest: it includes a modal operator for satisfiability, and thus provides a complete axiomatization for satisfiability in the logic K45.

A Simple Tableau System for the Logic of Elsewhere

Abstract: We analyze different features related to the mechanization of von Wright's logic of elsewhere e. First, we give a new proof of the NP-completeness of the satisfiability problem (giving a new bound for the size of models of the satisfiable formulae) and we show that this problem becomes linear-time when the number of propositional variables is bounded. Although e and the well-known propositional modal S5 share numerous common features we show that e is strictly more expressive than S5 (in a sense to be specified). Second, we present a prefixed tableau system for e and we prove both its soundness and completeness. Two extensions of this system are also defined, one related to the logical consequence relation and the other related to the addition of modal operators (without increasing the expressive power). An example of tableau proof is also presented. Different continuations of this work are proposed, one of them being to implement the defined tableau system, another one being to extend this system to richer logics that can be found in the literature.

Towards a computational treatment of deontic defeasibility

Abstract: In this paper we describe an algorithmic framework for a multi-modal logic arising from the combination of the system of modal (epistemic) logic devised by Meyer and van der Hoek for dealing with nonmonotonic reasoning with a deontic logic of the Jones and Porn-type. The idea behind this (somewhat eclectic) formal set-up is to have a modal framework expressive enough to model certain kinds of deontic defeasibility, in particular by taking into account preferences on norms. The appropriate inference mechanism is provided by a tableau-like modal theorem proving system which supports a proof method closely related to the semantics of modal operators. We argue that this system is particularly well-suited for mechanizing nonmonotonic forms of inference in a monotonic multimodal setting.

A Class of Information Logics with a Decidable Validity Problem

Abstract: For a class of prepositional information logics defined from Pawlak's information systems, the validity problem is proved to be decidable using a significant variant of the standard filtration technique. Actually the decidability is proved by showing that each logic has the strong finite model property and by bounding the size of the models. The logics in the scope of this paper are characterized by classes of Kripkestyle structures with interdependent equivalence relations and closed by the so-called restriction operation. They include Gargov's data analysis logic with local agreement and Nakamura's logic of graded modalities.

A Duplication and Loop Checking Free System for S4

Abstract: Most of the sequent/tableau based proof systems for the modal logic S4 need to duplicate formulas and thus are required to adopt some method of loop checking. In what follows we present a tableau-like proof system for S4, based on D'Agostino and Mondadori's classical KE, which is free of duplication and loop checking. The key feature of this system (let us call it KES4) consists in its use of (i) a label formalism which models the semantics of the modal operators according to the usual conditions for S4; and (ii) a label unification scheme which tells us when two labels "denote" the same world in the S4-model(s) generated in the course of proof search. Moreover, it uses special closure conditions to check models for putative contradictions.

Bringing about rationality: incorporating plans into a BDI agent architecture

Abstract: Agents attempt to achieve their intentions through the use of plans, leading to further intentions corresponding to the actions and subgoals of those plans We extend Rao and Georgeff's logic of belief, desire and intention with a logical representation of plans This representation allows the specification of subgoals, using Segerberg's "bringing it about" modal operator By using a plan library to semantically restrict an agent's intention-worlds, we define a framework that models the reasoning process of an intention-based autonomous agent We show that this framework supports several desirable properties involving an agent's commitment to future intentions, based on its available plans

Proof Realization of Intuitionistic and Modal Logics

Abstract: Logic of Proofs (LP) has been introduced in [2] as a collection of all valid formulas in the propositional language with labeled logical connectives [[t]]( ) where t is a proof term with the intended reading of [[t]]F as \t is a proof of F". LP is supplied with a natural axiom system, completeness and decidability theorems. LP may express some constructions of logic which have been formulated or/and interpreted in an informal metalanguage involving the notion of proof, e.g. the intuitionistic logic and its Brauwer-Heyting-Kolmogorov semantics, classical modal logic S4, etc (cf. [2]). In the current paper we demonstrate how the intuitionistic propositional logic Int can be directily realized into the Logic of Proofs. It is shown, that the proof realizability gives a fair semantics for Int.

A Complete Set of Satisfaction Rules for Property Detection in Distributed Computations

Abstract: This paper presents a general framework for specification and detection of properties in distributed computations. A property of states in distributed computation in progress is defined by predicates (called behavioral patterns) and satisfaction rules (called modal operators). A behavioral pattern is obtained by combining basic predicates that are defined over either local states or consistent global states of the computation. In both cases, we model a distributed computation by a directed acyclic graph in which vertices represent (local or global) states and edges represent causal relation over the states. Specification and verification of behavioral patterns are formulated as instances of the language recognition problem, and basic concepts used in formal language theory are applied. A basic predicate is assimilated to a symbol, and a behavioral pattern is specified as a language (i.e., a set of words) defined over an alphabet of symbols. Based on this model and given a behavioral pattern, we define four modal operators (i.e., four different satisfaction rules). Three of them are equivalent to modal operators previously introduced in related works. Finally, we present algorithms to verify each of the four modal operators over a class of behavioral patterns, called regular patterns.

On Epistemic Modal Predicate Logic

Abstract: In their earlier investigations Frege and Russell have specially turned their attention to logical analysis of complex sentences with subordinate clauses, which are introduced by the connective that and the verbs know, believe, doubt and so on (e.g. ‘Columbus believed that he had paved a new nautical way to India’). Following Russell and Ducass, relations, which are expressed by these complex sentences, are called propositional or epistemic attitudes. The truth of those sentences does not always depend on truth values of the subordinate clauses, it is not their function. Therefore, replacement of the subordinate clause by any of the equivalent sentences can change the truth value of the complex one. For a long time there was no satisfactory theory of logical analysis of these sentences. Unsuccessful was, ultimately, also the search of such criteria of identity of senses and synonymity of sentences, which guarantee permissibility of the above mentioned replacement.

Modal Tree-Sequents

Abstract: We develop cut-free calculi of sequents for normal modal logics by using treesequents, which are trees of sequences of formulas. We introduce modal operators corresponding to the ways we move formulas along the branches of such trees, only considering fixed distance movements. Finally, we exhibit syntactic cut-elimination theorems for all the main normal modal logics. Mathematics Subject Classification: 03B45, 03F05.

Auxiliary modal operators and the characterization of modal frames

Abstract: In modal logics we are interested in classes of frames which determine the logic under consideration. Such classes are usually distinguished by their respective frame properties, often also called the modal logic's background theory. In general these characterizations are not unique and it is desireable (and that not only from a theorem prover's perspective) to nd a strongest possible. In this paper an approach is presented which helps in this respect. It allows us to transform a given background theory into one which is more general and which modal logics cannot distinguish from the former because of their syntactic and semantic restrictions. The underlying technique is based on the idea to nd conservative extensions (of a given logic) whose determining properties serve as a starting point from which it is possible to extract signi cantly stronger characterizations of the original logic.

Logical Foundations for Modal Interpretations of Quantum Mechanics

Abstract: This paper proposes a logic, motivated by modal interpretations, in which every quantum mechanics propositions has a truth-value. This logic is completely classical, hence violates the conditions of the Kochen-Specker theorem. It is shown how the violation occurs, and it is argued that this violation is a natural and acceptable consequence of modal interpretations. It is shown that despite its classicality, the proposed logic is empirically indistinguishable from quantum logic.