Boolean Functions and Computation Models

1. Front Matter

   Pages I-XIV

   PDF

2. Boolean Functions and Circuits

   • Peter Clote, Evangelos Kranakis

   Pages 1-60

3. Circuit Lower Bounds

   • Peter Clote, Evangelos Kranakis

   Pages 61-154

4. Circuit Upper Bounds

   • Peter Clote, Evangelos Kranakis

   Pages 155-205

5. Randomness and Satisfiability

   • Peter Clote, Evangelos Kranakis

   Pages 207-245

6. Propositional Proof Systems

   • Peter Clote, Evangelos Kranakis

   Pages 247-412

7. Machine Models and Function Algebras

   • Peter Clote, Evangelos Kranakis

   Pages 413-495

8. Higher Types

   • Peter Clote, Evangelos Kranakis

   Pages 497-567

9. Back Matter

   Pages 569-602

   PDF

The foundations of computational complexity theory go back to Alan Thring in the 1930s who was concerned with the existence of automatic procedures deciding the validity of mathematical statements. The first example of such a problem was the undecidability of the Halting Problem which is essentially the question of debugging a computer program: Will a given program eventu­ ally halt? Computational complexity today addresses the quantitative aspects of the solutions obtained: Is the problem to be solved tractable? But how does one measure the intractability of computation? Several ideas were proposed: A. Cobham [Cob65] raised the question of what is the right model in order to measure a "computation step" , M. Rabin [Rab60] proposed the introduction of axioms that a complexity measure should satisfy, and C. Shannon [Sha49] suggested the boolean circuit that computes a boolean function. However, an important question remains: What is the nature of computa­ tion? In 1957, John von Neumann [vN58] wrote in his notes for the Silliman Lectures concerning the nature of computation and the human brain that . . . logics and statistics should be primarily, although not exclusively, viewed as the basic tools of 'information theory'. Also, that body of experience which has grown up around the planning, evaluating, and coding of complicated logical and mathematical automata will be the focus of much of this information theory. The most typical, but not the only, such automata are, of course, the large electronic computing machines.

• Beweissysteme
• Boolean function
• Boolesche Funktionen
• Erfüllbarkeitsproblem der Aussagenlogik
• Komplexitätstheorie
• Paralleles Rechnen
• combinatorics
• complexity
• complexity theory
• proof theory

From the reviews:

"The monograph gives the most recent and complete description of lower bounds for depth-restricted circuits, and propositional proof systems. … the authors present a research monograph on important subjects and provide many very recent results. I would recommend it for any university library and also for researchers." (Ingo Wegener, The Computer Journal, Vol. 46 (3), 2003)

• Department of Computer Science and Department of Biology, Boston College, Chestnut Hill, USA

  Peter Clote

• School of Computer Science, Carleton University, Ottawa, Canada

  Evangelos Kranakis

Policies and ethics