We study the problem of obtaining efficient, deterministic, black-box polynomial identity testing algorithms (PIT) for algebraic branching programs (ABPs) that are read-once and oblivious. This class has an efficient, deterministic, white-box polynomial identity testing algorithm (due to Raz and Shpilka [RS05]), 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 [Nis92] and Impagliazzo-Nisan-Wigderson [INW94] for read-once oblivious boolean branching programs. Thus our work can be seen as an algebraic analogue of these foundational results in boolean pseudorandomness. Our results are stronger for branching programs of bounded width, where we give a hitting set of size exp(lg 2 S/ lg lg S), corresponding to a seed length of lg 2 S/ lg lg 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. Thus, while our work is in some sense an algebraic analogue of existing boolean results, the two regimes seem to have non-trivial differences. In follow up work ([FS13]), we strengthened a result of Mulmuley [Mul12a], and showed that derandomizing a particular case of the Noether Normalization Lemma is reducible to blackbox 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 [Sax08]). For set-multilinear ABPs and non-commutative ABPs, we give quasi-polynomial-time black-box PIT algorithms, where the latter case involves evaluations over the algebra of (D + 1) × (D + 1) matrices, where D is the depth of the ABP. For (semi-)diagonal depth-4 circuits, we obtain a black-box PIT algorithm (over any characteristic) whose run-time is quasi-polynomial in the runtime of Saxena's white-box algorithm, matching the concurrent work of Agrawal, Saha, and Saxena [ASS12]. Finally, by combining our results with the reconstruction algorithm of Klivans and Shpilka [KS06], we obtain deterministic reconstruction algorithms for the above circuit classes. *
In comparison, the work of Forbes and Shpilka REF studied read-once oblivious (algebraic) branching programs, and achieved a quasi-polynomial-sized hitting set, which would correspond to a "seed length" of O(lg 2 s) for ABPs of size s. This result is analogous (in some sense) to the generator of Nisan for space bounded computations.
Quasipolynomial-time Identity Testing of Non-Commutative and Read-Once Oblivious Algebraic Branching Programs
54rd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2013,
Computer Science Mathematics