We prove an exponential lower bound on the lengths of resolution proofs of propositions expressing the finite Ramsey theorem for pairs.

We bound the minimum number $w$ of wires needed to compute any (asymptotically good) error-correcting code
$C:\{0,1\}^{\Omega(n)} \to \{0,1\}^n$ with minimum distance $\Omega(n)$,
using unbounded fan-in circuits of depth $d$ with arbitrary gates. Our main results are:

(1) If $d=2$ then $w = \Theta(n ({\log n/ \log \log n})^2)$.

(2) ... more >>>

We prove that the pseudorandom generator introduced in Impagliazzo et al. (1994) fools group products of a given finite group. The seed length is $O(\log n \log 1 / \epsilon)$, where $n$ is the length of the word and $\epsilon$ is the error. The result is equivalent to the statement ... more >>>

Khrapchenko's classical lower bound $n^2$ on the formula size of the
parity function~$f$ can be interpreted as designing a suitable
measure of subrectangles of the combinatorial rectangle
$f^{-1}(0)\times f^{-1}(1)$. Trying to generalize this approach we
arrived at the concept of \emph{convex measures}. We prove the
... more >>>

We give the first exponential separation between quantum and
classical multi-party
communication complexity in the (non-interactive) one-way and
simultaneous message
passing settings.
For every k, we demonstrate a relational communication problem
between k parties
that can be solved exactly by a quantum simultaneous message passing
protocol of
cost O(log ... more >>>

We define propositional quantum Frege proof systems and compare it
with classical Frege proof systems.

We shall prove a lower bound on the number of edges in some bounded
depth graphs. This theorem is stronger than lower bounds proved on
bounded depth superconcentrators and enables us to prove a lower bound
on certain bounded depth circuits for which we cannot use
superconcentrators: we prove that ... more >>>

In 1977 Valiant proposed a graph theoretical method for proving lower
bounds on algebraic circuits with gates computing linear functions.
He used this method to reduce the problem of proving
lower bounds on circuits with linear gates to to proving lower bounds
on the rigidity of a matrix, a ... more >>>

The rank of a matrix has been used a number of times to prove lower
bounds on various types of complexity. In particular it has been used
for the size of monotone formulas and monotone span programs. In most
cases that this approach was used, there is not a single ... more >>>

We consider some problems about pairs of disjoint $NP$ sets.
The theory of these sets with a natural concept of reducibility
is, on the one hand, closely related to the theory of proof
systems for propositional calculus, and, on the other, it
resembles the theory of NP completeness. Furthermore, such
more >>>

We show that an LK proof of size $m$ of a monotone sequent (a sequent

that contains only formulas in the basis $\wedge,\vee$) can be turned

into a proof containing only monotone formulas of size $m^{O(\log m)}$

and with the number of proof lines polynomial in $m$. Also we show

We consider computations of linear forms over {\bf R} by
circuits with linear gates where the absolute values
coefficients are bounded by a constant. Also we consider a
related concept of restricted rigidity of a matrix. We prove
some lower bounds on the size of such circuits and the
restricted ... more >>>

We consider the conjecture stating that a matrix with rank
$o(n)$ and ones on the main diagonal must contain nonzero
entries on a $2\times 2$ submatrix with one entry on the main
diagonal. We show that a slightly stronger conjecture implies
that ... more >>>

Razborov~\cite{Razborov96} recently proved that polynomial
calculus proofs of the pigeonhole principle $PHP_n^m$ must have
degree at least $\ceiling{n/2}+1$ over any field. We present a
simplified proof of the same result. The main
idea of our proof is the same as in the original proof
of Razborov: we want to describe ... more >>>

We prove an unexpected upper bound on a communication game proposed
by Jeff Edmonds and Russell Impagliazzo as an approach for
proving lower bounds for time-space tradeoffs for branching programs.
Our result is based on a generalization of a construction of Erdos,
Frankl and Rodl of a large 3-hypergraph with ... more >>>

We investigate the computational power of depth two circuits
consisting of $MOD^r$--gates at the bottom and a threshold gate at
the top (for short, threshold--$MOD^r$ circuits) and circuits with
two levels of $MOD$ gates ($MOD^{p}$-$MOD^q$ circuits.) In particular, we
will show the following results

(i) For all prime numbers $p$ ... more >>>

We prove lower bounds of the form $exp\left(n^{\epsilon_d}\right),$
$\epsilon_d>0,$ on the length of proofs of an explicit sequence of
tautologies, based on the Pigeonhole Principle, in proof systems
using formulas of depth $d,$ for any constant $d.$ This is the
largest lower bound for the strongest proof system, for which ... more >>>

We introduce a population genetics model in which the operators
are effectively computable -- computable in polynomial time on
Probabilistic Turing Machines. We shall show that in this model
a population can encode easily large amount of information
from enviroment into genetic code. Then it can process the
information as ... more >>>