*(English)*Zbl 0833.68049

The book contains an introduction to the complexity theory at the graduate level. It describes in a comprehensive way such issues as: problems and algorithms, Turing machines as a universal model of computations, logic (Boolean, first-order and second-order) as the most important application to be studied. Then complexity classes are discussed in detail, starting with relations between them and including the concept of completeness (in class P and NP). Later NP-complete problems are described together with proof methodology. Class of co-NP problems, randomized computations and cryptography are discusses in a sequel. This part of the book is complemented by a discussion of approximability and the basic question “N$=$NP?” including intermediate degrees, isomorphism and density, and oracles.

The next part of the book discusses the detailed structure of class P, concentrating on parallel algorithms and logarithmic space complexity.

Finally, several complexity issues concerned with problems beyond class NP are analyzed. They include polynomial hierarchy, polynomial space and a discussion of exponential time complexity. Summing up, the book is well written and requires almost no mathematical prerequisites. Its main drawback is that it leaves the main question “P$=$NP?” still open.