TR05-057 Authors: Venkatesan Guruswami, Valentine Kabanets

Publication: 27th May 2005 14:55

Downloads: 3213

Keywords:

We prove a version of the derandomized Direct Product Lemma for

deterministic space-bounded algorithms. Suppose a Boolean function

$g:\{0,1\}^n\to\{0,1\}$ cannot be computed on more than $1-\delta$

fraction of inputs by any deterministic time $T$ and space $S$

algorithm, where $\delta\leq 1/t$ for some $t$. Then, for $t$-step

walks $w=(v_1,\dots, v_t)$ in some explicit $d$-regular expander graph

on $2^n$ vertices, the function

$g'(w)\stackrel{\mathrm{def}}{=}g(v_1)\dots g(v_t)$ cannot be computed

on more than $1-\Omega(t\delta)$ fraction of inputs by any

deterministic time $\approx T/d^t-\poly(n)$ and space $\approx

S-O(t)$. As an application, by iterating this construction, we get a

deterministic linear-space ``worst-case to constant average-case''

hardness amplification reduction, as well as a family of logspace

encodable/decodable error-correcting codes that can correct up to a

constant fraction of errors. Logspace encodable/decodable codes (with

linear-time encoding and decoding) were previously constructed by

Spielman~\cite{Spi96}. Our codes have weaker parameters (encoding

length is polynomial, rather than linear), but have a conceptually

simpler construction. The proof of our Direct Product Lemma is

inspired by Dinur's remarkable recent proof of the PCP theorem by gap

amplification using expanders~\cite{Din05}.