We prove a quasi-polynomial lower bound on the size of bounded-depth Frege proofs of the pigeonhole principle PHP n m where m = (1 + 1/polylog n)n. This lower bound qualitatively matches the known quasi-polynomial-size bounded-depth Frege proofs for these principles. Our technique, which uses a switching lemma argument like other lower bounds for bounded-depth Frege proofs, is novel in that the tautology to which this switching lemma is applied remains random throughout the argument.
All Science Journal Classification (ASJC) codes
- Computer Science(all)
- Pigeonhole principle
- Propositional proof complexity