Quasi-polynomial hitting-set for set-depth-Δ formulas

Manindra Agrawal, Chandan Saha, Nitin Saxena
2013 Proceedings of the 45th annual ACM symposium on Symposium on theory of computing - STOC '13  
We call a depth-4 formula C set-depth-4 if there exists a (unknown) partition X1 • • • X d of the variable indices [n] that the top product layer respects, i.e. C(x) = k i=1 d j=1 fi,j(xX j ), where fi,j is a sparse polynomial in F[xX j ]. Extending this definition to any depth -we call a depth-∆ formula C (consisting of alternating layers of Σ and Π gates, with a Σ-gate on top) a set-depth-∆ formula if every Π-layer in C respects a (unknown) partition on the variables; if ∆ is even then the
more » ... duct gates of the bottom-most Π-layer are allowed to compute arbitrary monomials. In this work, we give a hitting-set generator for set-depth-∆ formulas (over any field) with running time polynomial in exp((∆ 2 log s) ∆−1 ), where s is the size bound on the input set-depth-∆ formula. In other words, we give a quasi-polynomial time blackbox polynomial identity test for such constant-depth formulas. Previously, the very special case of ∆ = 3 (also known as set-multilinear depth-3 circuits) had no known sub-exponential time hitting-set generator. This was declared as an open problem by Shpilka & Yehudayoff (FnT-TCS 2010); the model being first studied by Nisan & Wigderson (FOCS 1995). Our work settles this question, not only for depth-3 but, up to depth log s/ log log s, for a fixed constant < 1. The technique is to investigate depth-∆ formulas via depth-(∆ − 1) formulas over a Hadamard algebra, after applying a 'shift' on the variables. We propose a new algebraic conjecture about the low-support rank-concentration in the latter formulas, and manage to prove it in the case of set-depth-∆ formulas. • depth-2 formulas [KS01], • depth-3 formulas with bounded top fanin [ASSS12, SS11], • depth-4 (bounded depth) constant-occur formulas [ASSS12], and a quasi-polynomial time hitting-set generator for • multilinear constant-read formulas [AvMV11],
doi:10.1145/2488608.2488649 dblp:conf/stoc/AgrawalSS13 fatcat:5ajt6l7etneghgq2kqxyssukmu