We study the problem of obtaining deterministic black-box polynomial identity testing algorithms (PIT) for algebraic branching programs (ABPs) that are read-once and oblivious. This class has an deterministic white-box polynomial identity testing algorithm (due to Raz and Shpilka), but prior to this work there was no known such black-box algorithm. The main result of this work gives the first quasi-polynomial sized hitting sets for size S circuits from this class, when the order of the variables is known. As our hitting set is of size exp(lg^2 S), this is analogous (in the terminology of boolean pseudorandomness) to a seed-length of lg^2 S, which is the seed length of the pseudorandom generators of Nisan and Impagliazzo-Nisan-Wigderson for read-once oblivious boolean branching programs. Our results are stronger for branching programs of bounded width, where we give a hitting set of size exp(lg^2 S lglg S), corresponding to a seed length of lg^2 S lglg S. This is in stark contrast to the known results for read-once oblivious boolean branching programs of bounded width, where no pseudorandom generator (or hitting set) with seed length o(lg^2 S) is known. In follow up work, we strengthened a result of Mulmuley, and showed that derandomizing a particular case of the Noether Normalization Lemma is reducible to black-box PIT of read-once oblivious ABPs. Using the results of the present work, this gives a derandomization of Noether Normalization in that case, which Mulmuley conjectured would difficult due to its relations to problems in algebraic geometry. We also show that several other circuit classes can be black-box reduced to read-once oblivious ABPs, including set-multilinear ABPs (a generalization of depth-3 set-multilinear formulas), non-commutative ABPs (generalizing non-commutative formulas), and (semi-)diagonal depth-4 circuits (as introduced by Saxena).
However, despite the similarities to the boolean regime, our results improve when the branching programs have bounded width. Specifically, our hitting sets ( thm:small width ) achieve a seed length of @math , and it has been a long-standing open problem (see [Open Problem 8.6] Vadhan12 ) to achieve a seed length (for pseudorandom generators, or even hitting sets) of @math for boolean read-once oblivious branching programs of constant width. Despite much recent work (see @cite_31 @cite_22 @cite_41 @cite_49 @cite_7 @cite_10 @cite_2 @cite_27 ), such seed-lengths are only known for branching programs that are restricted even further, such as regular or permutation branching programs. It is an interesting question as to whether any insights of this paper can achieve a similar seed length in the boolean regime, or in general whether there are any formal connections between algebraic and boolean pseudorandomness for read-once oblivious branching programs.
@cite_22 @cite_7 @cite_41 @cite_27 @cite_49 @cite_2 @cite_31 @cite_10
2099601775 2950183285 1652911975 2022644058
We prove the existence of a polynomial time computable pseudorandom generator that @math -fools constant width regular read-once branching programs of length @math using a seed of length @math . The previous best pseudorandom generator for regular branching programs used a seed of length @math , and was due to Braverman et. al. and Brody and Verbin (FOCS 2010). Our pseudorandom generator is the INW generator due to Impagliazzo et. al. (STOC 1994). Our work shares some similarity with the recent work of Kouck ' y et. al. (STOC 2011)who get similar seed length for permutation branching programs. However, our work proceeds by analyzing the eigenvalues of the stochastic matrices that arise in the transitions of the branching program which arguably makes the technique more robust. As a corollary of our techniques, we present new results on the small biased spaces for group products'' problem by Meka and Zuckerman (RANDOM 2009). We get a pseudorandom generator with seed length @math . Previously, using the result of Kouck ' y et. al., it was possible to get a seed length of @math for this problem. 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( ^ 3 2 n) or O( ^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. Recently, an interest in constructing pseudorandom or hitting set generators for restricted branching programs has increased, which is motivated by the fundamental problem of derandomizing space bounded computations. Such constructions have been known only in the case of width 2 and in very restricted cases of bounded width. In our previous work, we have introduced a so-called richness condition which is, in a certain sense, sufficient for a set to be a hitting set for read-once branching programs of width 3. In this paper, we prove that, for a suitable constant C, any almost C log n-wise independent set satisfies this richness condition. Hence, we achieve an explicit polynomial time construction of a hitting set for read-once branching programs of width 3 with the acceptance probability greater than √12 13 by using the result due to (1992). We study the natural question of constructing pseudorandom generators (PRGs) for low-degree polynomial threshold functions (PTFs). We give a PRG with seed-length log n eO(d) fooling degree d PTFs with error at most e. Previously, no nontrivial constructions were known even for quadratic threshold functions and constant error e. For the class of degree 1 threshold functions or halfspaces, we construct PRGs with much better dependence on the error parameter e and obtain the following results. A PRG with seed length O(log n log(1 e)) for error e ≥ 1 poly(n). A PRG with seed length O(log n) for e ≥ 1 poly(log n). Previously, only PRGs with seed length O(log n log2(1 e) e2) were known for halfspaces. We also obtain PRGs with similar seed lengths for fooling halfspaces over the @math dimensional unit sphere. The main theme of our constructions and analysis is the use of invariance principles to construct pseudorandom generators. We also introduce the notion of monotone read-once branching programs, which is key to improving the dependence on the error rate e for halfspaces. These techniques may be of independent interest.