The part of the book on first-order topics ends with a chapter on logical characterizations of resource bounds for parallel random-access machines and circuit complexity.
Chapter eleven concerns uniformity in circuit complexity (the distinction between the existence of circuits for solving a problem, and their algorithmic constructibility), and chapter twelve concerns the role of ordering and counting predicates in logical characterizations of complexity classes.
[1][2][3] The book is primarily aimed as a reference to researchers in this area,[1] but it could also be used as the basis of a graduate course, and comes equipped with exercises for this purpose.
Although it claims to be self-contained, reviewer W. Klonowski writes that its readers need to already understand both classical complexity and the basics of mathematical logic.
[2] Reviewer Anuj Dawar writes that some of the early promise of descriptive complexity has been dampened by its inability to bring logical tools to bear on the core problems of complexity theory, and by the need to add computation-like additions to logical languages in order to use them to characterize computation.