We consider the problem of estimating the average of a huge set of values.
That is,
given oracle access to an arbitrary function $f:\{0,1\}^n\mapsto[0,1]$,
we need to estimate $2^{-n} \sum_{x\in\{0,1\}^n} f(x)$
upto an additive error of $\epsilon$.
We are allowed to employ a randomized algorithm which may err ... more >>>

A Uniform Generation procedure for $NP$ is an
algorithm which given any input in a fixed NP-language, outputs a uniformly
distributed NP-witness for membership of the input in the language.
We present a Uniform Generation procedure for $NP$ that runs in probabilistic
polynomial-time with an NP-oracle. This improves upon ... more >>>

We introduce "resource-bounded betting games", and propose
a generalization of Lutz's resource-bounded measure in which the choice
of next string to bet on is fully adaptive. Lutz's martingales are
equivalent to betting games constrained to bet on strings in lexicographic
order. We show that if strong pseudo-random number generators exist,
more >>>

We present a weaker variant of the PCP Theorem that admits a
significantly easier proof. In this
variant the prover only has $n^t$ time to compute each
bit of his answer, for an arbitray but fixed constant
$t$, in contrast to
being all powerful. We show that
3SAT ... more >>>

We present a novel technique, based on the Jensen-Shannon divergence
from information theory, to prove lower bounds on the query complexity
of sampling algorithms that approximate functions over arbitrary
domain and range. Unlike previous methods, our technique does not
use a reduction from a binary decision problem, but rather ... more >>>

We initiate a general study of the randomness complexity of
property testing, aimed at reducing the randomness complexity of
testers without (significantly) increasing their query complexity.
One concrete motovation for this study is provided by the
observation that the product of the randomness and query complexity
of a tester determine ... more >>>

We give new evidence that quantum computers -- moreover, rudimentary quantum computers built entirely out of linear-optical elements -- cannot be efficiently simulated by classical computers. In particular, we define a
model of computation in which identical photons are generated, sent through a linear-optical network, then nonadaptively measured to count ... more >>>

We obtain the first deterministic extractors for sources generated (or sampled) by small circuits of bounded depth. Our main results are:

(1) We extract $k (k/nd)^{O(1)}$ bits with exponentially small error from $n$-bit sources of min-entropy $k$ that are generated by functions $f : \{0,1\}^\ell \to \{0,1\}^n$ where each output ... more >>>