We provide another proof of the Sipser--Lautemann Theorem
by which $BPP\subseteq MA$ ($\subseteq PH$).
The current proof is based on strong
results regarding the amplification of $BPP$, due to Zuckerman.
Given these results, the current proof is even simpler than previous ones.
Furthermore, extending the proof leads to ... more >>>

For every $\epsilon>0$,
we present a {\em deterministic}\/ log-space algorithm
that correctly decides undirected graph connectivity
on all but at most $2^{n^\epsilon}$ of the $n$-vertex graphs.
The same holds for every problem in Symmetric Log-space (i.e., $\SL$).

Making no assumptions (and in particular not assuming the ... more >>>

We show that derandomizing Polynomial Identity Testing is,
essentially, equivalent to proving circuit lower bounds for
NEXP. More precisely, we prove that if one can test in polynomial
time (or, even, nondeterministic subexponential time, infinitely
often) whether a given arithmetic circuit over integers computes an
identically zero polynomial, then either ... more >>>

The notion of promise problems was introduced and initially studied
by Even, Selman and Yacobi
(Information and Control, Vol.~61, pages 159-173, 1984).
In this article we survey some of the applications that this
notion has found in the twenty years that elapsed.
These include the notion of ... more >>>

We study two quite different approaches to understanding the complexity
of fundamental problems in numerical analysis. We show that both hinge
on the question of understanding the complexity of the following problem,
which we call PosSLP:
Given a division-free straight-line program
producing an integer N, decide whether N>0.
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We prove several new results regarding the relationship between probabilistic time, BPTime(t), and alternating time, \Sigma_{O(1)} Time(t). Our main results are the following:

1) We prove that BPTime(t) \subseteq \Sigma_3 Time(t polylog(t)). Previous results show that BPTime(t) \subseteq \Sigma_2 Time(t^2 log t) (Sipser and Gacs, STOC '83; Lautemann, IPL '83) ... more >>>

We study class AvgBPP that consists of distributional problems that can be solved in average polynomial time (in terms of Levin's average-case complexity) by randomized algorithms with bounded error. We prove that there exists a distributional problem that is complete for AvgBPP under polynomial-time samplable distributions. Since we use deterministic ... more >>>