## Thursday, April 23, 2009

### Lecture 26

Today we covered...

Proof Complexity and the NP vs. coNP problem. The "Resolution" proof system with the resolution rule ($C \vee x$, $C \vee \overline{x}$ $\vdash$ $C \vee C'$) and the weakening rule ($C \vdash C \vee x$). Completeness of Resolution with proof sketch. Various contradictions (tautologies, in fact): Pigeonhole Principle, Tseitin Tautologies, Random 3-CNF. Treelike Resolution. Resolution width. Short Proofs Are Narrow Theorem, proved for Treelike Resolution. Ben-Sasson & Wigderson Theorem: Unsatisfiable $k$-CNFs with "expansion" require wide Resolution proofs.

By the way, the question came up as to separating Treelike Resolution from General Resolution. A very strong result was proved for this problem by Ben-Sasson, Impagliazzo, and Wigderson: There is a natural family of contradictions with $n$ variables and $O(n)$ clauses, which have Resolution refutations of length $O(n)$ but requires Treelike Resolution refutations of length $2^{\Omega(n / \log n)}$.

The contradictions here are based on "pebbling" expander graphs; more specifically, the results in an old paper of Paul, Tarjan, and Celoni.