Quasi-polynomial Hitting Sets for Circuits with Restricted Parse Trees

TL;DR

This paper develops quasipolynomial and polynomial size hitting sets for non-commutative UPT and FewPT circuits, extending previous results and enabling efficient polynomial identity testing in these models.

Contribution

It introduces explicit hitting sets for UPT and FewPT circuits, generalizing prior work from commutative models and advancing non-commutative polynomial identity testing.

Findings

Quasipolynomial size hitting sets for UPT circuits

Quasipolynomial size hitting sets for FewPT circuits

Polynomial size hitting sets for UPT circuits with bounded preimage-width

Abstract

We study the class of non-commutative Unambiguous circuits or Unique-Parse-Tree (UPT) circuits, and a related model of Few-Parse-Trees (FewPT) circuits (which were recently introduced by Lagarde, Malod and Perifel [LMP16] and Lagarde, Limaye and Srinivasan [LLS17]) and give the following constructions: (1) An explicit hitting set of quasipolynomial size for UPT circuits, (2) An explicit hitting set of quasipolynomial size for FewPT circuits (circuits with constantly many parse tree shapes), (3) An explicit hitting set of polynomial size for UPT circuits (of known parse tree shape), when a parameter of preimage-width is bounded by a constant. The above three results are extensions of the results of [AGKS15], [GKST15] and [GKS16] to the setting of UPT circuits, and hence also generalize their results in the commutative world from read-once oblivious algebraic branching programs (ROABPs) to UPT-set-multilinear circuits. The main idea is to study shufflings of non-commutative polynomials, which can then be used to prove suitable depth reduction results for UPT circuits and thereby allow a careful translation of the ideas in [AGKS15], [GKST15] and [GKS16].