Quasi-polynomial Hitting Sets for Circuits with Restricted Parse Trees.

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. 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].