Ewa Orlowska

This book presents the refereed proceedings of the Sixth European Workshop on Logics in Artificial Intelligence, JELIA '96, held in Evora, Portugal in September/October 1996. The 25 revised full papers included together with three invited papers were selected from 57 submissions. Many relevant aspects of AI logics are addressed. The papers are organized in sections on automated reasoning, modal logics, applications, nonmonotonic reasoning, default logics, logic programming, temporal and spatial logics, and belief revision and paraconsistency.

This paper contains a logic enabling us to reason in the presence of vague- ness phenomena. We consider an epistemological vagueness of concepts caused by the unavailability of total information about a continuous world which we describe in observational terms. Lack of information is manifested by the existence of borderline cases for concepts. Since we are unable to perceive concepts exactly, we cannot establish a sharp boundary between an extension of a concept and its complement. Some results for reasoning about vague concepts are already known in the literature [1], [2], [3], [4], [10]. Usually, a semantics of vague concepts is developed within the theory of fuzzy sets or the theory of supertruth [9]. Following the idea that reasoning about vague concepts requires a special logic, we introduce a propositional language with sentential operators en- abling us to dene positive, negative, and borderline regions of any vague concept. A semantics of the language is rened within a framework of the theory of rough sets . The admitted semantics provides means for dening extensions of concepts in a formal way. We show that these extensions re ect properly our intuition connected with vagueness. For ex- ample, the law of excluded middle is not valid on the level of extensions. Moreover, for any concept, the extensions of formulas representing its pos- itive, negative, and borderline regions provide a partition of a universe of discourse

In the following two sections we propose ordering relations for which the material for comparison of theories of concepts the theories deal with. The relations refer to what is called concept analysis. In the present section we introduce relations enabling us to compare theories from the point of view of the following aspect: How well the theories in question classify objects into positive and negative instances of concepts. Classifications of objects provided by a theory can be correct or incorrect from the point of view of some other theory. The respective relations reflect the following intuition: According to a theory H, a theory H2 is not worse than a theory H1 whenever the set of objects correctly classified by H2 includes that of H1, and the set of objects incorrectly classified by H1 includes that of H2. Let T HR be a family of theories, let CON be a set of concepts, and let OB be a set of objects to which the concepts from CON apply. For A ∈ CON let P) and N) be the sets of objects which satisfy A and do not satisfy A, respectively, according to the classification provided by a theory H ∈ T HR. In other words we consider sets of positive and negative instances of A determined in the respective theories.

One of the important issues in research on knowledge based computer systems is development of methods for reasoning about knowledge. In the present paper semantics for knowledge operators is introduced. The underlying logic is developed with epistemic operators relative to indiscernibility. Facts about knowledge expressible in the logic are discussed, in particular common knowledge and joint knowledge of n group of agents. Some paradoxes of epistemic logic are shown to be eliminated in the given system. A formal logical analysis of reasoning about knowledge is a subject of investigations both in logic and computer science, and several epistemic systems have been proposed to formalize the operator ‘an agent knows’. In the present paper we propose a formalization based on a semantic treatment of knowledge within the framework of rough set theory. The inspiration for the underlying epistemic logic came from the analysis of knowledge transfer in distributed systems developed in Orlowska and Sanders and from the author’s earlier work on indiscernibility and relative accessibility semantics.

The purpose of the present paper is to show that modal propositional logics can be interpreted in a logic based on relational calculus. We consider languages with necessity operators [R], where R is an accessibility relation expression representing an element of the algebra of binary relations with operations −,∪,∩, −1 , ◦. The relational logic is based on relational calculus enriched by operations of weakest prespecification and weakest postspecification introduced in Hoare and He Jifeng and investigated in He Jifeng et al . The logic is an extension of the system introduced in Or lowska and investigated in Buszkowski and Or lowska

The book presents logical foundations of dual tableaux together with a number of their applications both to logics traditionally dealt with in mathematics and philosophy (such as modal, intuitionistic, relevant, and many-valued logics) and to various applied theories of computational logic (such as temporal reasoning, spatial reasoning, fuzzy-set-based reasoning, rough-set-based reasoning, order-of magnitude reasoning, reasoning about programs, threshold logics, logics of conditional decisions). The distinguishing feature of most of these applications is that the corresponding dual tableaux are built in a relational language which provides useful means of presentation of the theories. In this way modularity of dual tableaux is ensured. We do not need to develop and implement each dual tableau from scratch, we should only extend the relational core common to many theories with the rules specific for a particular theory.

Interval temporal logics provide both an insight into a nature of time and a framework for temporal reasoning in various areas of computer science. In this paper we present sound and complete relational proof systems in the style of dual tableaux for relational logics associated with modal logics of temporal intervals and we prove that the systems enable us to verify validity and entailment of these temporal logics. We show how to incorporate in the systems various relations between intervals and/or various time orderings

A method of defining semantics of logics based on not necessarily distributive lattices is presented. The key elements of the method are representation theorems for lattices and duality between classes of lattices and classes of some relational systems . We suggest a type of duality referred to as a duality via truth which leads to Kripke-style semantics and three-valued semantics in the style of Allwein-Dunn. We develop two new representation theorems for lattices which, together with the existing theorems by Urquhart and Bimbo-Dunn, constitute a complete, in a sense, representation theory for lattices. As observed by Dunn and Hardegree, variations of Urquhart's duality arise by varying his disjointness assumption on the canonical frame. Four possible assumptions – disjoint, exhaustive, non-disjoint and non-exhaustive – are discussed in the paper. Each of the four corresponding representation theorems is expanded to a duality via truth. Based on these dualities we suggest four corresponding types of semantics for lattice-based logics. We also discuss a new topological representation of lattices