8 research outputs found
Pseudorandomness for Regular Branching Programs via Fourier Analysis
We present an explicit pseudorandom generator for oblivious, read-once,
permutation branching programs of constant width that can read their input bits
in any order. The seed length is , where is the length of the
branching program. The previous best seed length known for this model was
, which follows as a special case of a generator due to
Impagliazzo, Meka, and Zuckerman (FOCS 2012) (which gives a seed length of
for arbitrary branching programs of size ). Our techniques
also give seed length for general oblivious, read-once branching
programs of width , which is incomparable to the results of
Impagliazzo et al.Our pseudorandom generator is similar to the one used by
Gopalan et al. (FOCS 2012) for read-once CNFs, but the analysis is quite
different; ours is based on Fourier analysis of branching programs. In
particular, we show that an oblivious, read-once, regular branching program of
width has Fourier mass at most at level , independent of the
length of the program.Comment: RANDOM 201
Better Pseudorandom Generators from Milder Pseudorandom Restrictions
We present an iterative approach to constructing pseudorandom generators,
based on the repeated application of mild pseudorandom restrictions. We use
this template to construct pseudorandom generators for combinatorial rectangles
and read-once CNFs and a hitting set generator for width-3 branching programs,
all of which achieve near-optimal seed-length even in the low-error regime: We
get seed-length O(log (n/epsilon)) for error epsilon. Previously, only
constructions with seed-length O(\log^{3/2} n) or O(\log^2 n) were known for
these classes with polynomially small error.
The (pseudo)random restrictions we use are milder than those typically used
for proving circuit lower bounds in that we only set a constant fraction of the
bits at a time. While such restrictions do not simplify the functions
drastically, we show that they can be derandomized using small-bias spaces.Comment: To appear in FOCS 201
Pseudorandom Generators for Width-3 Branching Programs
We construct pseudorandom generators of seed length that -fool ordered read-once branching programs
(ROBPs) of width and length . For unordered ROBPs, we construct
pseudorandom generators with seed length . This is the first improvement for pseudorandom
generators fooling width ROBPs since the work of Nisan [Combinatorica,
1992].
Our constructions are based on the `iterated milder restrictions' approach of
Gopalan et al. [FOCS, 2012] (which further extends the Ajtai-Wigderson
framework [FOCS, 1985]), combined with the INW-generator [STOC, 1994] at the
last step (as analyzed by Braverman et al. [SICOMP, 2014]). For the unordered
case, we combine iterated milder restrictions with the generator of
Chattopadhyay et al. [CCC, 2018].
Two conceptual ideas that play an important role in our analysis are: (1) A
relabeling technique allowing us to analyze a relabeled version of the given
branching program, which turns out to be much easier. (2) Treating the number
of colliding layers in a branching program as a progress measure and showing
that it reduces significantly under pseudorandom restrictions.
In addition, we achieve nearly optimal seed-length
for the classes of: (1) read-once polynomials on
variables, (2) locally-monotone ROBPs of length and width
(generalizing read-once CNFs and DNFs), and (3) constant-width ROBPs of length
having a layer of width in every consecutive
layers.Comment: 51 page
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Pseudorandomness for Width-2 Branching Programs
We show that pseudorandom generators that fool degree-k polynomials over 2 also fool branching programs of width-2 and polynomial length that read k bits of input at a time. This model generalizes polynomials of degree k over 2 and includes some other interesting classes of functions, for instance, k-DNFs.
The proof essentially follows by a new decomposition theorem for width-2 branching programs. The theorem states that if f can be computed by a width-2 branching program that reads k bits of input at a time, then f can be (roughly) written as a sum f=∑iαifi where each fi is a degree-k polynomial and ∑i|αi| is small.
Bogdanov and Viola (FOCS 2007) constructed a pseudorandom generator that fools degree-k polynomials over 2 for arbitrary k. Their construction consists of summing k independent copies of a generator that ϵ-fools linear functions. Our second result investigates the limits of such constructions: We show that, in general, such a construction is not pseudorandom against bounded fan-in circuits of depth O((log(klog1/ϵ))2)