We show that disjointness requires randomized communication
Omega(\frac{n^{1/2k}}{(k-1)2^{k-1}2^{2^{k-1}}})
in the general k-party number-on-the-forehead model of complexity.
The previous best lower bound was Omega(\frac{log n}{k-1}). By
results of Beame, Pitassi, and Segerlind, this implies
2^{n^{Omega(1)}} lower bounds on the size of tree-like Lovasz-Schrijver
proof systems needed to refute certain unsatisfiable ... more >>>

We introduce the study of Kolmogorov complexity with error. For a
metric d, we define C_a(x) to be the length of a shortest
program p which prints a string y such that d(x,y) \le a. We
also study a conditional version of this measure C_{a,b}(x|y)
where the task is, given ... more >>>

The information contained in a string $x$ about a string $y$
is defined as the difference between the Kolmogorov complexity
of $y$ and the conditional Kolmogorov complexity of $y$ given $x$,
i.e., $I(x:y)=\C(y)-\C(y|x)$. From the well-known Kolmogorov--Levin Theorem it follows that $I(x:y)$ is symmetric up to a small ... more >>>

The language compression problem asks for succinct descriptions of
the strings in a language A such that the strings can be efficiently
recovered from their description when given a membership oracle for
A. We study randomized and nondeterministic decompression schemes
and investigate how close we can get to the information ... more >>>