We study the diagonalization in the context of proof
complexity. We prove that at least one of the
following three conjectures is true:

1. There is a boolean function computable in E
that has circuit complexity $2^{\Omega(n)}$.

2. NP is not closed under the complement.

3. There is no ... more >>>

Recently Schuler \cite{Sch03} presented a randomized algorithm that
solves SAT in expected time at most $2^{n(1-1/\log_2(2m))}$ up to a
polynomial factor, where $n$ and $m$ are, respectively, the number of
variables and the number of clauses in the input formula. This bound
is the best known ... more >>>

We analyze the efficiency of the random walk algorithm on random 3CNF instances, and prove em linear upper bounds on the running time
of this algorithm for small clause density, less than 1.63. Our upper bound matches the observed running time to within a multiplicative factor. This is the ... more >>>