Stephen Cook - Research

Research

Cook is considered one of the forefathers of computational complexity theory.

During his PhD, Cook worked on complexity of functions, mainly on multiplication. He received his PhD in 1966. A few years later, in his seminal 1971 paper "The Complexity of Theorem Proving Procedures", Cook formalized the notions of polynomial-time reduction (a.k.a. Cook reduction) and NP-completeness, and proved the existence of an NP-complete problem by showing that the Boolean satisfiability problem (usually known as SAT) is NP-complete. This theorem was proven independently by Leonid Levin in the Soviet Union, and has thus been given the name the Cook-Levin theorem. The paper also formulated the most famous problem in computer science, the P vs. NP problem. Informally, the "P vs. NP" question asks whether every optimization problem whose answers can be efficiently verified for correctness/optimality can be solved optimally with an efficient algorithm. Given the abundance of such optimization problems in everyday life, a positive answer to the "P vs. NP" question would likely have profound practical and philosophical consequences.

Cook conjectures that there are optimization problems (with easily checkable solutions) which cannot be solved by efficient algorithms, i.e., P is not equal to NP. This conjecture has generated a great deal of research in computational complexity theory, which has considerably improved our understanding of the inherent difficulty of computational problems and what can be computed efficiently. Yet, the conjecture remains open and is among the seven famous Millennium Prize Problems.

In 1982, Cook received the prestigious Turing award for his contributions to complexity theory. His citation reads:

For his advancement of our understanding of the complexity of computation in a significant and profound way. His seminal paper, The Complexity of Theorem Proving Procedures, presented at the 1971 ACM SIGACT Symposium on the Theory of Computing, laid the foundations for the theory of NP-Completeness. The ensuing exploration of the boundaries and nature of NP-complete class of problems has been one of the most active and important research activities in computer science for the last decade.

In his "Feasibly Constructive Proofs and the Propositional Calculus" paper published in 1975, he introduced the equational theory PV (standing for Polynomial-time Verifiable) to formalize the notion of proofs using only polynomial-time concepts. He made another major contribution to the field in his 1979 paper, joint with his student Robert A. Reckhow, "The Relative Efficiency of Propositional Proof Systems", in which they formalized the notions of p-simulation and efficient propositional proof system, which started an area now called propositional proof complexity. They proved that the existence of a proof system in which every true formula has a short proof is equivalent to NP = coNP. Cook co-authored a book with his student Phuong The Nguyen in this area titled "Logical Foundations of Proof Complexity".

His main research areas are complexity theory and proof complexity, with excursions into programming language semantics, parallel computation, and artificial intelligence. Other areas which he has contributed to include bounded arithmetic, bounded reverse mathematics, complexity of higher type functions, complexity of analysis, and lower bounds in propositional proof systems.