Improved Pseudorandom Generators for Depth 2 Circuits

Abstract

We prove the existence of a poly(n, m)-time computable pseudorandom generator which "1/poly(n, m)-fools" DNFs with n variables and m terms, and has seed length O(log 2 nm · log log nm). Previously, the best pseudorandom generator for depth-2 circuits had seed length O(log 3 nm), and was due to Bazzi (FOCS 2007).It follows from our proof that a 1/mÕ (log mn) -biased distribution 1/poly(nm)-fools DNFs with m terms and n variables. For inverse polynomial distinguishing probability this is nearly tight because we show that for every m, δ there is a 1/m Ω(log 1/δ) -biased distribution X and a DNF φ with m terms such that φ is not δ-fooled by X.For the case of read-once DNFs, we show that seed length O(log mn · log 1/δ) suffices, which is an improvement for large δ.It also follows from our proof that a 1/m O(log 1/δ) -biased distribution δ-fools all read-once DNF with m terms. We show that this result too is nearly tight, by constructing a 1/mΩ (log 1/δ) -biased distribution that does not δ-fool a certain m-term read-once DNF.

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“…Section 5. The [12] result is rediscovered by De, Etesami, Trevisan, and Tulsiani [14], who also show that it is essentially tight by constructing a distribution which is n −Ω(log(1/δ )/ log log(1/δ )) -biased yet does not δ -fool a read-once DNF. In particular, fooling with polynomial error requires super-polynomial bias.…”

“…We also remark that since O(log 2 n)-wise independence suffices to fool DNF formulas [4], one must consider linear codes with dual distance less than log 2 n in our construction, and so D e has bias at least (1 − log 2 n/n) e = 2 −O(log 2 n) . On the other hand, [14] shows that 2 −O(log 2 n log log n) -bias fools DNF formulas.…”

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“…This notion of sandwiching is in fact a tight characterisation of small bias [15,Proposition 2.7]. That is, any function f fooled by all small bias generators has sandwiching polynomials satisfying the hypotheses of Proposition 7.1.…”

“…We use the following extension of a standard result [40,Theorem 4]. 15 Technically, we must pad U with zeros in the locations specified by T (i. e., U i = 0 for i ∈ T ) to obtain the right length.…”

Deterministic Search for CNF Satisfying Assignments in Almost Polynomial Time

“…We further observe that only an ε-hitting set for CNFs rather than an ε-PRG is required for this oblivious approach, but the best known explicit construction of hitting sets for general CNFs is simply the [DETT10] PRG. We recall that a seemingly-modest improvement of the [DETT10] PRG's seed length fromÕ(log 2 (M/ε)) to O(log 1.99 (M/ε)), even for ε-hitting sets, would improve state-of-theart lower bounds against depth-three circuits, breaking a longstanding barrier in circuit complexity. (For the special case of read-once CNF formulas, Síma and Zák [SZ10] have given an ε-hitting set of poly(n) size for ε > 5/6, and Gopalan et al [GMR + 12] have given an ε-PRG with seed length O(log(n/ε)).…”