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Pseudorandom Generators for Read-Once Monotone Branching Programs

Authors Dean Doron, Raghu Meka, Omer Reingold, Avishay Tal, Salil Vadhan



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Author Details

Dean Doron
  • Department of Computer Science, Stanford University, CA, USA
Raghu Meka
  • Department of Computer Science, University of California at Los Angeles, CA, USA
Omer Reingold
  • Department of Computer Science, Stanford University, CA, USA
Avishay Tal
  • Department of Electrical Engineering and Computer Sciences, University of California at Berkeley, CA, USA
Salil Vadhan
  • John A. Paulson School of Engineering & Applied Sciences, Harvard University, Cambridge, MA, USA

Acknowledgements

We are grateful to Kristoffer Hansen for pointing us to [Barrington et al., 1998] and explaining how their results imply Theorem 3.

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Dean Doron, Raghu Meka, Omer Reingold, Avishay Tal, and Salil Vadhan. Pseudorandom Generators for Read-Once Monotone Branching Programs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 58:1-58:21, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2021.58

Abstract

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(log² n) seed length of Nisan’s classic construction [Noam Nisan, 1992] to the optimal O(log n). In this work, we construct an explicit PRG with seed length Õ(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 [Braverman et al., 2014; Koucký et al., 2011; De, 2011; Thomas Steinke, 2012], 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 AC⁰. 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 AC⁰, due to Doron, Hatami, and Hoza [Doron et al., 2019]. Our pseudorandom generator construction follows Ajtai and Wigderson’s approach of iterated pseudorandom restrictions [Ajtai and Wigderson, 1989; Gopalan et al., 2012]. 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 [Michael A. Forbes and Zander Kelley, 2018].

Subject Classification

ACM Subject Classification
  • Theory of computation → Pseudorandomness and derandomization
  • Theory of computation → Circuit complexity
Keywords
  • Branching programs
  • pseudorandom generators
  • constant depth circuits

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