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Reviewer: Arto Salomaa

Circuit complexity has been a significant field of research for more than half a century. The main goal is to establish lower bounds for specific problems such as matrix multiplication, computation of Boolean functions, communication protocols, or proof systems. Jukna, a well-known researcher in the field, has succeeded in producing an excellent comprehensive exposition on the field, starting from early results from the '40s and '50s and proceeding to the most recent achievements. The need for such a book is apparent from the importance of the field and because more than 20 years have elapsed since the previous such overview. The book is going to be very useful for researchers and graduate students in computer science and discrete mathematics. The exposition is largely self-contained, although some mathematical maturity is called for. The style of writing is pleasant, with enough (but not too much) motivation. I checked proofs here and there, and the arguments seem to be reliable. This area of complexity theory is known for not using unproven statements, such as P not being equal to NP. This book focuses on classical circuit models, rather than their algebraic or randomized versions. The first two chapters serve as an introduction to circuits and Boolean functions. The basics are very well presented, and one could use this part in a short course, or perhaps as reading material in a general course about complexity. The second part of the book deals with communication complexity, games on relations and on 0-1 matrices, and multiparty games. The book then proceeds to circuit complexity, monotone formulas, and circuits and span programs. Particularly impressive is the treatment of negation, clarifying when NOT gates are needed, and when they are useless. The next part deals with bounded depth circuits, with a special discussion on depth-3 circuits. Branching programs, decision trees, and some fragments of proof complexity are presented in the last part. The book ends with a set of short tutorials on linear algebra, graph theory, and probability theory. Obviously, this short review can only hint at the contents of this massive 600-page book. The bibliography contains some 600 items. The index is useful because the reader will inevitably forget some of the numerous notions. An index of names would have been helpful, particularly because the book covers many of the results by early Russian authors such as Lupanov, Yablonskii, Khrapchenko, and Nechiporuk. For a long time, these results were unknown in the West. An index of names would help a historically minded reader. In this respect, the remark on pages 563-564 and the comparison with the Russian notion of "perebor" are both quite interesting. The many exercises and research problems round out the highlights of this recommendable book. Online Computing Reviews Service