The Proof Puzzle: Exploring Theorem Complexity

This podcast is on paper that explores the computational complexity of theorem-proving procedures, focusing on the problem of determining whether a propositional formula is a tautology. It introduces the concept of polynomial reducibility, demonstrating that many seemingly difficult problems, such as the subgraph isomorphism problem, can be reduced to the tautology problem. The paper then investigates the complexity of proof procedures for the predicate calculus, proposing a measure of efficiency based on the number of substitution instances needed to reach a contradiction. A key finding suggests that determining tautologyhood is computationally very hard, potentially providing a significant breakthrough in complexity theory if proven. The work also considers implications for the time required for deterministic versus non-deterministic Turing machines.