Michael Forbes : Pseudorandomness for Multilinear Read-Once Algebraic Branching Programs, in any Order

It is an important open question whether we can derandomize small space computation, that is, whether RL equals L. One version of this question is to construct pseudorandom generators for read-once oblivious branching programs. There are well-known results in this area (due to Nisan, and Impagliazzo-Nisan-Wigderson), but they fail to achieve optimal seed-length. Further, it has been observed `that these pseudorandom generators depend strongly on the "order" of the "reads" of the branching program. When this order is allowed to vary, only much weaker results are known.

In this work, we consider an "algebraic" version of this question. That is, we seek to fool read-once algebraic branching programs, regardless of the variable order. By rephrasing and improving the techniques of Agrawal-Saha-Saxena, we are able to construct hitting sets for multilinear polynomials in this unknown-order model that have polylogarithmic "seed-length". This constitutes the first quasipolynomial-time, deterministic, black-box polynomial identity testing (PIT) algorithm for this model.

Joint work with Ramprasad Saptharishi (MSR India) and Amir Shpilka (Technion).