Ewa Orlowska

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

We present an approach to nondeterministic time based on a concept of nondeterministic world. A nondeterministic world is a set of some equipos- sible states. Such a world is not treated as whole, it is neither atomic nor indivisible. Given a world representing the present, the past worlds are its subsets and the future worlds are sets in which it is contained. Thus we assume that worlds involve change and the way they change is expansion. In our approach indeterminism occurs in the two levels. First, we admit nondeterministic worlds, consisting of possibly many instances. Second, we treat the future as nondeterministically reachable. As a consequence, for any world there are many past worlds from which the world might be reached and many future worlds which might be reached from the world

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