In contrast to deterministic or nondeterministic computation, it is
a fundamental open problem in randomized computation how to separate
different randomized time classes (at this point we do not even know
how to separate linear randomized time from ${\mathcal O}(n^{\log n})$
randomized time) or how to ... more >>>

This paper provides logspace and small circuit depth analogs
of the result of Valiant-Vazirani, which is a randomized (or
nonuniform) reduction from NP to its arithmetic analog ParityP.
We show a similar randomized reduction between the
Boolean classes NL and semi-unbounded fan-in Boolean circuits and
their arithmetic counterparts. These reductions ... more >>>

Locally testable codes are error-correcting codes that admit
very efficient codeword tests. Specifically, using a constant
number of (random) queries, non-codewords are rejected with
probability proportional to their distance from the code.

Locally testable codes are believed to be the combinatorial
core of PCPs. However, the relation is less ... more >>>

The closure of complexity classes is a elicate question and the answer varies depending on the type of reduction considered. The closure of most classes under many-to-one (Karp) reductions is clear, but the question becomes complicated when oracle (Cook) reductions are allowed, and even more so when the oracle reductions ... more >>>

We prove that the Shortest Vector Problem (SVP) on point lattices is NP-hard to approximate for any constant factor under polynomial time reverse unfaithful random reductions. These are probabilistic reductions with one-sided error that produce false negatives with small probability, but are guaranteed not to produce false positives regardless of ... more >>>