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Reviewer: William A Fahle

Although this book only contains 120 pages, it is dense with formulas and notation. Despite this, the authors find space for informal arguments, and even a little levity, to make the book more readable. The main purpose of the book is to review the research and literature to date in, and to unify the theory of, semi-feasible sets. The definition of a semi-feasible set is one for which there is a polynomial-time function that can decide, given two choices, which is the better one for potential membership in the set. (These terms, of course, are more rigorously defined in the text, but this is the gist of the definition.) These functions are also known as p -selective functions, and the sets as p -selective sets. Examples of these sets are presented, showing their existence, and the first chapter discusses why these sets are important. For example, a theorem is proved that, if there exists an NP-complete p -selective set, then P=NP . Chapter 2 discusses oracles and advice, as it relates to p -selective sets, and the third chapter defines and discusses lowness theory and its application to p -selectivity and nondeterminism. Chapter 4 covers hardness for complexity classes, and the question of whether p -selective sets can be hard for a given complexity class. Of course, these questions often hinge on other open complexity-class questions, such as "does coNP equal NP__?__" The fifth chapter discusses closure properties of p -selective sets, in the abstract algebra sense. For example, a theorem that p -selectivity is closed under complementation, that is, that a p -selective set's complement is also p -selective, is proven. This chapter also covers reductions and equivalence classes related to the central topic. Finally, chapter 6 discusses generalizations of p -selectivity, and other related notes. Not much time is spent on these topics, as they are not central to the book. The list of bibliographic references is not gigantic, but neither is the book itself. The index is rather comprehensive, given the size of the book; it is more than ten percent of the size of the text itself. It is easy to use, with dashes appearing before each entry on the second indent level. This makes it easy to follow the long second-level indents across page breaks. Within second-level topics, references to the first level keyword are always replaced with a tilde. The book, like so many others, claims to be accessible to readers who have only a basic understanding of computer science, perhaps at the level of a second-year undergraduate. I think this claim is more of an attempt by the publisher to widen the market for this book than it is a statement of fact. In fact, some knowledge of complexity classes, notation, counting, reductions, and so on is essential to reading this book. However, as far as it pertains to semi-feasible algorithms, the book is self-contained. No prior knowledge of the theory is assumed, but the book is definitely for those with advanced knowledge of complexity theory and its notation. There is an appendix that provides the definitions for the notation, but it is again dense, and assumes a breadth of theoretical computer science knowledge. In all, this is a well-executed book on an important subject, which has received little treatment of this kind. I share the authors' hope that this book will help to stimulate further research in this field. Online Computing Reviews Service