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).
This work fits into the research program of derandomizing PIT, in particular derandomizing black-box PIT. However, for many of the models of algebraic circuits studied, there are corresponding boolean circuit models for which derandomization questions can also be asked. In particular, for a class @math of boolean circuits, we can seek to construct a @math for @math , such that for any circuit @math on @math inputs, we have the @math -closeness of distributions @math , where @math denotes the uniform distribution on @math . Nisan @cite_15 studied pseudorandom generators for space-bounded computation, and for space @math computation gave a generator with seed length @math . Impagliazzo, Nisan and Wigderson @cite_5 later gave a different construction with the same seed length. Randomized space-bounded computation can be modeled with read-once oblivious (boolean) branching programs, and these generators apply to this model of computation as well.
@cite_5 @cite_15
2011438986 1982991092
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. The is the following problem: a coin is given, which lands on head with probability either @math or @math . We are given the outcome of @math independent tosses of this coin, and the goal is to guess which way the coin is biased, and to answer correctly with probability @math . When our computational model is unrestricted, the majority function is optimal, and succeeds when @math for a large enough constant @math . The coin problem is open and interesting in models that cannot compute the majority function. In this paper we study the coin problem in the model of . We prove that in order to succeed in this model, @math must be at least @math . For constant @math this is tight by considering the recursive tribes function, and for other values of @math this is nearly tight by considering other read-once AND-OR trees. We generalize this to a , where instead of independent tosses of a coin we are given independent tosses of one of two @math -sided dice. We prove that if the distributions are too close and the mass of each side of the dice is not too small, then the dice cannot be distinguished by small-width read-once branching programs. We suggest one application for this kind of theorems: we prove that Nisan's Generator fools width- @math read-once branching programs, using seed length @math . For @math , this seed length is @math . The coin theorem and its relatives might have other connections to PRGs. This application is related to the independent, but chronologically-earlier, work of Braver man, Rao, Raz and Yehudayoff BRRY . 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. We use randomness to exploit the potential sparsity of the Boolean matrix product in order to speed up the computation of the product. Our new fast output-sensitive algorithm for Boolean matrix product and its witnesses is randomized and provides the Boolean product and its witnesses almost certainly. Its worst-case time performance is expressed in terms of the input size and the number of non-zero entries of the product matrix. It runs in time @math , where the input matrices have size n×n, the number of non-zero entries in the product matrix is at most s, ? is the exponent of the fast matrix multiplication and @math denotes O(f(n)log? d n) for some constant d. By the currently best bound on ?, its running time can be also expressed as @math . Our algorithm is substantially faster than the output-sensitive column-row method for Boolean matrix product for s larger than n 1.232 and it is never slower than the fast @math -time algorithm for this problem. By applying the fast rectangular matrix multiplication, we can refine our upper bound further to the form @math , where ?(p,q,t) is the exponent of the fast multiplication of an n p ×n q matrix by an n q ×n t matrix. We also present a partial derandomization of our algorithm as well as its generalization to include the Boolean product of rectangular Boolean matrices. Finally, we show several applications of our output-sensitive algorithms.