Logical analysis requires precision in the definition and understanding of concepts and in the classification of data. However, many problems have inherently imprecise and vague components. For example, the fundamental concept of a set assumes that an element of the set is either in it or out of it. Often, especially in scientific analysis and philosophical argument, this distinction is unclear and uncertain. The concepts of fuzzy sets (where membership has a value between 0 and 1) and rough sets (where there is a boundary region between being in the set and out of the set) are two complementary ways of dealing with vagueness.

Rough sets were first proposed by Zdzisław Pawlak in a paper [1] in 1982. Pawlak was one of the major figures in the development of the Polish schools of computer science and artificial intelligence. This volume is the second of two volumes of papers published in his memory after his death in 2006. It contains 30 papers covering theory and the application of rough sets contributed by colleagues, former students, and collaborators around the world. The first paper is a personal recollection by Anita Wasilewska of her collaboration with Pawlak. The remaining 29 are technical.

The preface should be read first. It lists the titles of all the chapters in both volumes, with a few sentences or a short paragraph briefly summarizing the subject of each. The table of contents follows, providing more detail, including the titles of each section of each paper. There is no apparent grouping of chapters according to larger themes. Topics include the theory of rough sets, logic on rough sets, algorithms, databases, applications in image processing and feature selection, medical data interpretation, music, and hardware applications, among many others.

Although it is impractical to discuss each paper even briefly, a selection will give a picture of the diversity of topics. The second chapter, “Knowledge Algebras and Their Discrete Duality,” by Ewa Orlowska and Anna Maria Radzikowska, uses rough sets to create algebras to represent and operate upon knowledge frames. Chapter 5, “Game-Theoretic Rough Sets for Feature Selection,” by Nouman Azam and Jing Tao Yao, applies game theory to rough sets and concludes with an application to textual analysis. In chapter 8, “Incorporating Rough Data in Database Design for Imprecise Information Representation,” authors Theresa Beaubouef and Frederick E. Petry discuss relational databases from a rough set viewpoint, including modifications needed for entity-relationship models and changes that must be made to normal forms. Chapter 13, “Providing Feedback in Ukrainian Sign Language Tutoring Software,” by M. V. Davydov and others, describes the use of rough sets for the evaluation of the skill of sign language interpreters by extracting features of their movements when precisely forming the words and phonemes. In chapter 16, “Hardware Implementation of Rough Set Methods in Programmable Logic Devices,” authors Maciej Kopczynski and Jaroslaw Stepaniuk recount recent experiments in the implementation of rough set processing in field-programmable gate array (FPGA) logic devices. Chapter 25, “Rough Set-Based Identification of Heart Valve Diseases Using Heart Sounds,” by Moustafa A. Salama and others, is a thorough example of applying rough sets for feature extraction and rule generation, in this case for the diagnosis of heart valve diseases.

Each of the chapters has its own list of references. The style of writing is at a very sophisticated level. Like its companion, this volume is written for specialists in the theory and applications of rough sets.