Linear Branching Programs and Directional Affine Extractors

TL;DR

This paper introduces directional affine extractors, a new class of functions that are hard for certain linear branching programs, and explores their implications for proof complexity and average-case hardness.

Contribution

The paper defines directional affine extractors, constructs explicit examples, and connects them to linear branching programs and proof systems, advancing understanding of computational hardness.

Findings

Directional affine extractors are hard on average for strongly read-once models.

Explicit constructions of directional affine extractors with good parameters are provided.

Weakly read-once linear BPs can be converted to Res[⊕] refutations with constant overhead.

Abstract

A natural model of read-once linear branching programs is a branching program where queries are $\mathbb{F}_2$ linear forms, and along each path, the queries are linearly independent. We consider two restrictions of this model, which we call weakly and strongly read-once, both generalizing standard read-once branching programs and parity decision trees. Our main results are as follows. - Average-case complexity. We define a pseudo-random class of functions which we call directional affine extractors, and show that these functions are hard on average for the strongly read-once model. We then present an explicit construction of such function with good parameters. This strengthens the result of Cohen and Shinkar (ITCS'16) who gave such average-case hardness for parity decision trees. Directional affine extractors are stronger than the more familiar class of affine extractors. Given the significance of these functions, we expect that our new class of functions might be of independent interest. - Proof complexity. We also consider the proof system $\text{Res}[\oplus]$ which is an extension of resolution with linear queries. A refutation of a CNF in this proof system naturally defines a linear branching program solving the corresponding search problem. Conversely, we show that a weakly read-once linear BP solving the search problem can be converted to a $\text{Res}[\oplus]$ refutation with constant blow up.