Lecture 26

Today we covered...

Proof Complexity and the NP vs. coNP problem. The "Resolution" proof system with the resolution rule ($C\vee x$, $C\vee \overline{x}$ $⊢$ $C\vee Cʹ$) and the weakening rule ($C⊢C\vee x$). 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 $O\left(n\right)$ clauses, which have Resolution refutations of length $O\left(n\right)$ 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.