Complexity of Theorem-Proving Procedures

The concept of the complexity of theorem-proving procedures lies at the heart of computational complexity theory, a crucial field within computer science. This field examines the resources required to solve computational problems, typically in terms of time and space. The seminal work on this topic is often credited to Stephen Cook, who, in his 1971 paper titled "The Complexity of Theorem Proving Procedures", formally defined key aspects that are now foundational to our understanding of complexity classes, particularly NP-completeness.

Background and Significance

In the early 1970s, Cook's work laid the groundwork for understanding the P versus NP problem, one of the most critical unsolved problems in computer science. His publication introduced the notion that certain computational problems, such as the Boolean satisfiability problem (SAT), are NP-complete. This means that they are at least as hard to solve as any problem in the class NP (nondeterministic polynomial time), where solutions can be verified quickly by a deterministic algorithm.

The Cook-Levin Theorem

The Cook-Levin Theorem, also known as the NP-completeness theorem, is perhaps the most celebrated result from Cook's paper. It formally established that the Boolean satisfiability problem is NP-complete, thus demonstrating the existence of a problem within NP for which every problem in NP can be reduced to it in polynomial time. This theorem's implications are profound, as it suggests that if any NP-complete problem can be solved quickly (in polynomial time), then every problem in NP can be solved quickly.

Related Concepts

The complexity of theorem-proving is closely related to other significant topics in computer science and mathematics:

Graph Isomorphism: This problem involves determining whether two graphs are isomorphic, meaning they contain the same number of graph vertices connected in the same way.
Kolmogorov Complexity: While not directly linked, this concept involves the complexity of describing a dataset and has implications for understanding algorithmic complexity and randomness.
Gödel's Incompleteness Theorems: These theorems highlight inherent limitations in formal systems, suggesting that not all truths can be proven within the system itself, which aligns conceptually with the challenges of proving theorems computationally.
Four Color Theorem: An example of a complex theorem that was eventually proven using computational methods.

Related Figures

Richard Karp: Another pioneer in the field, Karp expanded upon Cook's work, identifying 21 additional NP-complete problems, further solidifying the foundation of complexity theory.
M. Davis and H. Putnam: Known for their work on the Davis-Putnam algorithm, which is used in automated theorem proving and contributes to understanding computational procedures.

Conclusion