In many practical situations, a set is (or several sets are) part of a natural description of the situation. For example, a black-and-white image can be naturally described as a set of all the black pixels. Another example is binary (yes/no) decision making: all possible alternatives need to be divided into either acceptable or not acceptable, and each alternative is characterized by the tuple consisting of the values of several real-valued parameters. In this example, a natural way to describe a user’s preferences is to describe the set of all the tuples that correspond to acceptable alternatives.

In practice, we rarely know the exact values of the corresponding parameters; we usually know these values with some degree of inaccuracy. Often, this inaccuracy can be described by dividing the whole domain into subsets (granules) so that, within the given accuracy, it is impossible to distinguish between different values within a granule. In this case, instead of knowing the exact set, we know the following: (a) the set of all the granules that definitely belong to the desired set, and (b) the set of all the granules that may contain elements of the desired set. The actual (unknown) set contains the lower approximation (a) and is contained in the upper approximation (b). This pair of lower and upper approximations is called a “rough set.” This book contains papers on the theory and applications of rough sets.

One of the main objectives of rough set theory is to find reasonable extensions for rough sets in known techniques and results about sets. For example, for regular sets A and B, A is either contained in B or it is not. For two rough sets, we can only say whether they can or cannot be contained; if they can, we can try to gauge how possible this containment is. In this book, Gomolińska’s paper analyzes methods for describing the degree of possibility.

Rough set theory was originally developed from scratch, based on the above approximation model. In the original development, many interesting results were proven. As it turns out, many related structures and results can be equivalently reformulated in terms of the more traditional mathematical theories. These reformulations enable researchers to apply known mathematical results and techniques to rough sets. Two of the book’s papers describe such reformulations: Wolski’s paper shows the equivalence to topological constructions, Scott approximations, pre-orders, and an abstract mathematical generalization of the logical entailment relation; in Cattaneo and Ciucci’s paper, the equivalence is with lattices and modal logics.

The use of rough sets goes beyond the approximate character of our knowledge. In many situations--for example, data mining--there is so much data that processing it all is not feasible. In such situations, a reasonable approach is to cut off irrelevant data details and introduce granularity--that is, partition data into clusters. Many such partitions are possible; to select an appropriate partition, it is important to compare how informative different partitions are and how much information we lose by introducing them. Bianucci and Cattaneo’s paper describes various ways of estimating the amount of information.

One of the main applications of rough sets is to related data mining problems, in which it is desirable not only to extract rules from the data, but also to gauge the degree of reasonableness of different rules. Szczȩch’s paper handles this topic.

Two papers deal with specific applications. Yue and Miao’s paper applies rough set techniques to situations where we need to recognize a character based on an approximately known black-and-white image--that is, an image described as a rough set. Maciocha and Kisielnicki’s paper applies rough sets to the evaluation of intangible assets in a Polish telecommunications sector.

I highly recommend this book to students and researchers who are already familiar with the basic ideas of rough set theory and want to learn about the latest developments. In principle, the book should also be accessible to other readers with a mathematical background, even if they are not very familiar with rough sets.