# Pseudorandom generators for read-once monotone branching programs

### Citation:

Doron, Dean, Raghu Meka, Omer Reingold, Avishay Tal, and Salil Vadhan. “Pseudorandom generators for read-once monotone branching programs.” Electronic Colloquium on Computational Complexity (ECCC) 2021, no. 18 (2021).
 ECCC 2021.pdf 861 KB ECCC 2021 rev1.pdf 793 KB

### Abstract:

Version History: Originally published as "Monotone branching programs: pseudorandomness and circuit complexity".

Motivated by the derandomization of space-bounded computation, there has been a long line of work on constructing pseudorandom generators (PRGs) against various forms of read-once branching programs (ROBPs), with a goal of improving the $$O(\log^2n)$$ seed length of Nisan’s classic construction to the optimal $$O(\log n)$$.

In this work, we construct an explicit PRG with seed length $$\tilde{O}(\log n)$$ for constant-width ROBPs that are monotone, meaning that the states at each time step can be ordered so that edges with the same labels never cross each other. Equivalently, for each fixed input, the transition functions are a monotone function of the state. This result is complementary to a line of work that gave PRGs with seed length $$O(\log n)$$ for (ordered) permutation ROBPs of constant width, since the monotonicity constraint can be seen as the “opposite” of the permutation constraint.

Our PRG also works for monotone ROBPs that can read the input bits in any order, which are strictly more powerful than read-once $$\mathsf{AC^0}$$. Our PRG achieves better parameters (in terms of the dependence on the depth of the circuit) than the best previous pseudorandom generator for read-once $$\mathsf{AC^0}$$, due to Doron, Hatami, and Hoza.

Our pseudorandom generator construction follows Ajtai and Wigderson’s approach of iterated pseudorandom restrictions. We give a randomness-efficient width-reduction process which proves that the branching program simplifies to an $$O(\log n)$$-junta after only $$O(\log \log n)$$ independent applications of the Forbes-Kelley pseudorandom restrictions.

Publisher's Version

Last updated on 03/13/2021