This largely self-contained monograph presents a systematic, exhaustive, and up-to-date overview of formal methods and theories related to data analysis and inference that are inspired by the concept of rough sets.
The book consists of 15 chapters. Chapter 1 is an introduction, presenting the basic mathematical notions and facts used in the book: set-theoretical notions, relations, strings, trees and orders, algebras, lattices, Boolean algebras, and algebras of relations. The remaining 14 chapters each begin with an introduction and an outline of the chapter, and end with notes that provide a historical perspective on the subject of the chapter with references to the literature. Some open problems are also mentioned in the notes. Chapters 2 through 14 are grouped into five parts. Part 2 contains two chapters; each other part contains three.
In Part 1, structures with incomplete information are considered. In chapter 2, structures of information are considered. The most fundamental of these structures is an information system with incomplete information. The notion of an information system is extended to the notion of a fuzzy information system. Every information system contains implicit information that can be derived from the content of a given information system, or fuzzy information system. Chapter 3 studies derived information reflecting relationships among objects, referred to as information relations. In chapter 4, the author discusses implicit information of another kind: how an information system can induce various families of operators acting on sets of objects. This class of operators is referred to as “information operators,” and includes knowledge operators, generalized approximation operators, and fuzzy information operators. The later parts of the monograph are devoted to the study of formal methods for the analysis of data from information systems, based on the processing of information relations and operators.
Part 2 includes chapters 5 and 6, and presents the basic notions of information logics. The term “information logic” refers to any logical system developed both for the representation of, and for reasoning with, data presented in an information system. Classical logic and the general notion of modal logic are presented, along with some special modal logics, namely those with accessibility relations, rare-logics, and fuzzy modal logics. The techniques underlying the methodology of information logics, and some basic notions of computational complexity, are described.
In Part 3, proof systems for information logics are studied. The logics presented in this part are modal logics with rather rich languages. The deductive systems of these logics have the forms of Hilbert-style or Rasiowa-Sikorski-style systems. In chapter 7, a method of reasoning about similarity is considered. Specifically, a logic for reasoning about nondeterministic information, an information logic, and a logic for reasoning about relative similarity are presented. In chapter 8, logics for reasoning abou t indiscernibility, namely of relative indiscernibility and fuzzy logic of graded modalities, are presented. In chapter 9, reasoning about knowledge is considered. A basic logic with knowledge operators, and a logic with relative knowledge operators, are described.
In Part 4, the computational aspects of the formal systems developed in Part 3 are studied. Most of the information logics are shown to be decidable. In chapter 10, relationships between the rare-logics and standard modal logics, introduced in chapter 5, are considered. In chapter 11, the decidability of the satisfiability problem of information logics presented in Part 3 is studied. Two standard methods, the method of finite structures and the method of translation, are used. Chapter 12 contains a systematic study of the complexity of formal systems developed in Part 3.
Algebraic approaches to information systems with incomplete information and some methodological issues that rely on relationships between algebraic and logical systems are considered in Part 5. In chapter 13, the general notion of informational representability and a method of proving informational representability are presented. Informational interpretation of standard algebraic structures is considered in chapter 14. Relationships between some structures with information operators are presented in chapter 4, and various standard algebraic systems of algebraic logic are discussed. In chapter 15, information algebras are considered. The algebraic systems that are the counterparts to the information frames defined in chapter 3 are presented.
The book is well written, and can be used by specialists in computer science and artificial intelligence, as well as by graduate students interested in studying information systems with incomplete information and formal methods for data analysis and inference inspired by the concept of rough sets.
