This advanced textbook presents a broad and up-to-date view of the computational complexity theory of boolean circuits. It combines the algorithmic and the automata-theoretic approaches, and includes extensive discussion of the literature to facilitate further study. It begins with efficient boolean circuits for problems with high practical relevance, then compares the computational model of boolean circuits with other models such as Turing machines and parallel machines. Examination of the complexity of specific problems leads to the definition of complexity classes. The theory of circuit complexity classes is then thoroughly developed, including the theory of lower bounds and advanced topics such as algebraic complexity and connections to finite model theory. This advanced textbook presents a broad and up-to-date view of the computational complexity theory of Boolean circuits. It combines the algorithmic and the computability-based approach, and includes extensive discussion of the literature to facilitate further study.
It begins with efficient Boolean circuits for problems with high practical relevance, e.g., arithmetic operations, sorting, and transitive closure, then compares the computational model of Boolean circuits with other models such as Turing machines and parallel machines. Examination of the complexity of specific problems leads to the definition of complexity classes. The theory of circuit complexity classes is then thoroughly developed, including the theory of lower bounds and advanced topics such as connections to algebraic structures and to finite model theory. 1. Complexity Measures and Reductions.- 2. Relations to Other Computation Models.- 3. Lower Bounds.- 4. The NC Hierarchy.- 5. Arithmetic Circuits.- 6. Polynomial Time and Beyond.- Appendix: Mathematical Preliminaries.- A1 Alphabets, Words, Languages.- A2 Binary Encoding.- A3 Asymptotic Behavior of Functions.- A4 Turing Machines.- A5 Logic.- A6 Graphs.- A7 Numbers and Functions.- A8 Algebraic Structures.- A9 Linear Algebra.- List of Figures.- Author Index.
