Matrix identities and the pigeonhole principle
- 47 Downloads
We show that short bounded-depth Frege proofs of matrix identities, such as PQ=I⊃QP=I (over the field of two elements), imply short bounded-depth Frege proofs of the pigeonhole principle. Since the latter principle is known to require exponential-size bounded-depth Frege proofs, it follows that the propositional version of the matrix principle also requires bounded-depth Frege proofs of exponential size.
KeywordsPigeonhole Principle Matrix Identity Exponential Size Propositional Version Frege Proof
Unable to display preview. Download preview PDF.
- 1.Miklós Ajtai.: The complexity of the pigeonhole principle. In Proceedings of the 29th Annual IEEE Symposium on the Foundations of Computer Science. 346–355 (1988)Google Scholar
- 2.Paul Beame, Russell Impagliazzo, Jan Krajícek, Toniann Pitassi, Pavel Pudlák, Alan Woods.: Exponential lower bounds for the pigeonhole principle. In Proceedings of the 24th Annual ACM Symposium on theory of computing. 200–220 (1992)Google Scholar
- 3.Peter Clote, Evangelos Kranakis.: Boolean Functions and Computation Models. Springer-Verlag, 2002Google Scholar
- 4.Stephen~A. Cook, Michael Soltys.: The proof complexity of linear algebra. In Seventeenth Annual IEEE Symposium on Logic in Computer Science (LICS 2002). 2002Google Scholar
- 6.Toniann Pitassi, Paul Beame, Russell: Impagliazzo. Exponential lower bounds for the pigeonhole principle. Computational Complexity. 3, 97–140 (1993)Google Scholar
- 7.Michael Soltys.: The Complexity of Derivations of Matrix Identities. PhD thesis, University of Toronto, 2001Google Scholar
- 8.Michael Soltys.: Extended Frege and Gaussian elimination. Bulletin of the Section of Logic, Polish Academy of Sciences. 31~(4), 189–206 (2002)Google Scholar