In this paper we study program checking (in the
sense of Blum and Kannan) using constant-depth circuits as
checkers. Our focus is on the number of queries made by the
checker to the program being checked and we term this as the
query complexity of the checker for the given ... more >>>

The essential idea in the fast parallel computation of division and
related problems is that of Chinese remainder representation
(CRR) -- storing a number in the form of its residues modulo many
small primes. Integer division provides one of the few natural
examples of problems for which ... more >>>

Integer division has been known to lie in P-uniform TC^0 since
the mid-1980's, and recently this was improved to DLOG-uniform
TC^0. At the time that the results in this paper were proved and
submitted for conference presentation, it was unknown whether division
lay in DLOGTIME-uniform TC^0 (also known as ... more >>>

We construct a randomness-efficient averaging sampler that is computable by uniform constant-depth circuits with parity gates (i.e., in AC^0[mod 2]). Our sampler matches the parameters achieved by random walks on constant-degree expander graphs, allowing us to apply a variety expander-based techniques within NC^1. For example, we obtain the following results:

Program checking, program self-correcting and program self-testing
were pioneered by [Blum and Kannan] and [Blum, Luby and Rubinfeld] in
the mid eighties as a new way to gain confidence in software, by
considering program correctness on an input by input basis rather than
full program verification. Work in ... more >>>

We prove an n^{Omega(1)}/2^{O(k)} lower bound on the randomized k-party communication complexity of read-once depth 4 AC^0 functions in the number-on-forehead (NOF) model for O(log n) players. These are the first non-trivial lower bounds for general NOF multiparty communication complexity for any AC^0 function for omega(log log n) players. For ... more >>>

Given two $n$-variable boolean functions $f$ and $g$, we study the problem of computing an $\varepsilon$-approximate isomorphism between them. I.e.\ a permutation $\pi$ of the $n$ variables such that $f(x_1,x_2,\ldots,x_n)$ and $g(x_{\pi(1)},x_{\pi(2)},\ldots,x_{\pi(n)})$ differ on at most an $\varepsilon$ fraction of all boolean inputs $\{0,1\}^n$. We give a randomized $2^{O(\sqrt{n}\log(n)^{O(1)})}$ algorithm ... more >>>

We study the locality of an extension of first-order logic that captures graph queries computable in AC$^0$, i.e., by families of polynomial-size constant-depth circuits. The extension considers first-order formulas over relational structures which may use arbitrary numerical predicates in such a way that their truth value is independent of the ... more >>>