To-date computers are supposed to store and exploit knowledge. At least that is one of the aims of research fields such as Artificial Intelligence and Information Systems. However, the problem is to understand what knowledge means, to find ways of representing knowledge, and to specify automated machineries that can extract useful information from stored knowledge. Knowledge is something people have in their mind, and which they can express through natural language. Knowl edge is acquired not only from books, but also from observations made during experiments; in other words, from data. Changing data into knowledge is not a straightforward task. A set of data is generally disorganized, contains useless details, although it can be incomplete. Knowledge is just the opposite: organized (e.g. laying bare dependencies, or classifications), but expressed by means of a poorer language, i.e. pervaded by imprecision or even vagueness, and assuming a level of granularity. One may say that knowledge is summarized and organized data - at least the kind of knowledge that computers can store.

This book explores reasoning with rough sets by developing a granularity-based framework. It begins with a brief description of the rough set theory, then examines selected relations between rough set theory and non-classical logics including modal logic. In addition, it develops a granularity-based framework for reasoning in which various types of reasoning can be formalized. The book will be of interest to all researchers whose work involves Artificial Intelligence, databases and/or logic.

This book constitutes the refereed proceedings of the International Conference on Rough Sets and Emerging Intelligent Systems Paradigms, RSEISP 2007, held in Warsaw, Poland in June 2007 - dedicated to the memory of Professor Zdzislaw Pawlak. The 73 revised full papers papers presented together with 2 keynote lectures and 11 invited papers were carefully reviewed and selected from numerous submissions. The papers are organized in topical sections on foundations of rough sets, foundations and applications of fuzzy sets, granular computing, algorithmic aspects of rough sets, rough set applications, rough/fuzzy approach, information systems and rough sets, data and text mining, machine learning, hybrid methods and applications, multiagent systems, applications in bioinformatics and medicine, multimedia applications, as well as web reasoning and human problem solving.

In 1982, Professor Pawlak published his seminal paper on what he called "rough sets" - a work which opened a new direction in the development of theories of incomplete information. Today, a decade and a half later, the theory of rough sets has evolved into a far-reaching methodology for dealing with a wide variety of issues centering on incompleteness and imprecision of information - issues which playa key role in the conception and design of intelligent information systems. "Incomplete Information: Rough Set Analysis" - or RSA for short - presents an up-to-date and highly authoritative account of the current status of the basic theory, its many extensions and wide-ranging applications. Edited by Professor Ewa Orlowska, one of the leading contributors to the theory of rough sets, RSA is a collection of nineteen well-integrated chapters authored by experts in rough set theory and related fields. A common thread that runs through these chapters ties the concept of incompleteness of information to those of indiscernibility and similarity.

Rough Sets and Data Mining: Analysis of Imprecise Data is an edited collection of research chapters on the most recent developments in rough set theory and data mining. The chapters in this work cover a range of topics that focus on discovering dependencies among data, and reasoning about vague, uncertain and imprecise information. The authors of these chapters have been careful to include fundamental research with explanations as well as coverage of rough set tools that can be used for mining data bases. The contributing authors consist of some of the leading scholars in the fields of rough sets, data mining, machine learning and other areas of artificial intelligence. Among the list of contributors are Z. Pawlak, J Grzymala-Busse, K. Slowinski, and others. Rough Sets and Data Mining: Analysis of Imprecise Data will be a useful reference work for rough set researchers, data base designers and developers, and for researchers new to the areas of data mining and rough sets.

The objective of this book is two-fold. Firstly, it is aimed at bringing to gether key research articles concerned with methodologies for knowledge discovery in databases and their applications. Secondly, it also contains articles discussing fundamentals of rough sets and their relationship to fuzzy sets, machine learning, management of uncertainty and systems of logic for formal reasoning about knowledge. Applications of rough sets in different areas such as medicine, logic design, image processing and expert systems are also represented. The articles included in the book are based on selected papers presented at the International Workshop on Rough Sets and Knowledge Discovery held in Banff, Canada in 1993. The primary methodological approach emphasized in the book is the mathematical theory of rough sets, a relatively new branch of mathematics concerned with the modeling and analysis of classification problems with imprecise, uncertain, or incomplete information. The methods of the theory of rough sets have applications in many sub-areas of artificial intelligence including knowledge discovery, machine learning, formal reasoning in the presence of uncertainty, knowledge acquisition, and others. This spectrum of applications is reflected in this book where articles, although centered around knowledge discovery problems, touch a number of related issues. The book is intended to provide an important reference material for students, researchers, and developers working in the areas of knowledge discovery, machine learning, reasoning with uncertainty, adaptive expert systems, and pattern classification.

The monograph offers a view on Rough Mereology, a tool for reasoning under uncertainty, which goes back to Mereology, formulated in terms of parts by Lesniewski, and borrows from Fuzzy Set Theory and Rough Set Theory ideas of the containment to a degree. The result is a theory based on the notion of a part to a degree. One can invoke here a formula Rough: Rough Mereology : Mereology = Fuzzy Set Theory : Set Theory. As with Mereology, Rough Mereology finds important applications in problems of Spatial Reasoning, illustrated in this monograph with examples from Behavioral Robotics. Due to its involvement with concepts, Rough Mereology offers new approaches to Granular Computing, Classifier and Decision Synthesis, Logics for Information Systems, and are--formulation of well--known ideas of Neural Networks and Many Agent Systems. All these approaches are discussed in this monograph. To make the exposition self--contained, underlying notions of Set Theory, Topology, and Deductive and Reductive Reasoning with emphasis on Rough and Fuzzy Set Theories along with a thorough exposition of Mereology both in Lesniewski and Whitehead--Leonard--Goodman--Clarke versions are discussed at length. It is hoped that the monograph offers researchers in various areas of Artificial Intelligence a new tool to deal with analysis of relations among concepts.

Rough set approach to reasoning under uncertainty is based on inducing knowledge representation from data under constraints expressed by discernibility or, more generally, similarity of objects. Knowledge derived by this approach consists of reducts, decision or association rules, dependencies, templates, or classifiers. This monograph presents the state of the art of this area. The reader will find here a deep theoretical discussion of relevant notions and ideas as well as rich inventory of algorithmic and heuristic tools for knowledge discovery by rough set methods. An extensive bibliography will help the reader to get an acquaintance with this rapidly growing area of research.

Granular Computing is concerned with constructing and processing carried out at the level of information granules. Using information granules, we comprehend the world and interact with it, no matter which intelligent endeavor this may involve. The landscape of granular computing is immensely rich and involves set theory (interval mathematics), fuzzy sets, rough sets, random sets linked together in a highly synergetic environment. This volume is a first comprehensive treatment of this emerging paradigm and embraces its fundamentals, underlying methodological framework, and a sound algorithmic environment. The panoply of applications covered includes system identification, telecommunications, linguistics and music processing. Written by experts in the field, this volume will appeal to all developing intelligent systems, either working at the methodological level or interested in detailed system realization.

This book constitutes the thoroughly refereed post-proceedings of the Second International Conference on Rough Sets and Current Trends in Computing, RSCTC 2000, held in Banff, Canada in October 2000. The 80 revised papers presented together with an introduction and three keynote presentations have gone through two rounds of reviewing and revision. The papers are organized in topical sections on granual computing, rough sets and systems, fuzzy sets and systems, rough sets and data mining, nonclassical logics and reasoning, pattern recognition and image processing, neural networks and genetic algorithms, and current trends in computing.