Proof Complexity Generators

Discover a state-of-the-art theory aiming to construct hard propositional tautologies needed to solve the NP vs. coNP problem.

Jan Krajíček (Author)

9781009611701, Cambridge University Press

Paperback / softback, published 26 June 2025

134 pages
22.8 x 15.2 x 0.8 cm, 0.21 kg

The P vs. NP problem is one of the fundamental problems of mathematics. It asks whether propositional tautologies can be recognized by a polynomial-time algorithm. The problem would be solved in the negative if one could show that there are propositional tautologies that are very hard to prove, no matter how powerful the proof system you use. This is the foundational problem (the NP vs. coNP problem) of proof complexity, an area linking mathematical logic and computational complexity theory. Written by a leading expert in the field, this book presents a theory for constructing such hard tautologies. It introduces the theory step by step, starting with the historic background and a motivational problem in bounded arithmetic, before taking the reader on a tour of various vistas of the field. Finally, it formulates several research problems to highlight new avenues of research.

1. Introduction
2. The dWPHP problem
3. τ-formulas and generators
4. The stretch
5. Nisan-Wigderson generator
6. Gadget generator
7. The case of ER
8. Consistency results
9. Contexts
10. Further research
Special symbols
References
Index.

Subject Areas: Mathematical foundations [PBC]