A Near-Optimal Depth-Hierarchy Theorem for Small-Depth Multilinear Circuits

Abstract

We study the size blow-up that is necessary to convert an algebraic circuit of productdepth ∆ + 1 to one of product-depth ∆ in the multilinear setting.We show that for every positive ∆ = ∆(n) = o(log n/ log log n), there is an explicit multilinear polynomial P (∆) on n variables that can be computed by a multilinear formula of product-depth ∆ + 1 and size O(n), but not by any multilinear circuit of product-depth ∆ and size less than exp(n Ω(1/∆) ). This result is tight up to the constant implicit in the double exponent for all ∆ = o(log n/ log log n).This strengthens a result of Raz and Yehudayoff (Computational Complexity 2009) who prove a quasipolynomial separation for constant-depth multilinear circuits, and a result of Kayal, Nair and Saha (STACS 2016) who give an exponential separation in the case ∆ = 1.Our separating examples may be viewed as algebraic analogues of variants of the Graph Reachability problem studied by Chen, Oliveira, Servedio and Tan (STOC 2016), who used them to prove lower bounds for constant-depth Boolean circuits. ΣΠ · · · O d for some d = d(N ) where d is a slow growing function of the number of variables N . We consider such a representation of a polynomial f as a computation of f .The efficiency of such a computation is captured by the following two complexity measures: the size of the corresponding expression, which captures the number of operations used in computing the polynomial; and the product-depth of the circuit, which is the number of Πs in the expression for the circuit class (e.g. 1 for ΣΠΣ circuits, 2 for ΣΠΣΠ circuits etc.) and which measures in some sense the inductive complexity of the corresponding computation. 2 This paper is motivated by questions in a general body of results in the area that go by the name of Depth-reduction. Informally, the question is: what is the worst case blow-up in size required to convert a circuit of product-depth ∆ to one of product-depth ∆ ′ < ∆? This is an important question, since circuits of smaller depth are frequently easier to analyze and understand, and such results allow us to transfer this understanding to more complex (i.e. higher depth) classes of circuits. Consequently, there have been many results addressing this general question for various models of Algebraic (and also Boolean) computation including Algebraic and Boolean formulas [Bre74, Spi73], Boolean circuits [HPV77], and Algebraic circuits [VSBR83, AV08, Koi12, Tav15, GKKS16]. Recently, Tavenas [Tav15] and Gupta, Kamath, Kayal and Saptharishi [GKKS16] (building on [VSBR83, AV08, Koi12]) showed that a strong enough exponential lower bound for ΣΠΣ circuits would imply a superpolynomial lower bound for general algebraic circuits. Interesting impossibility results are also known in this direction: for instance, it is known that the result of Tavenas [Tav15], which converts a general circuit to a ΣΠΣΠ circuit of subexponential size, cannot be improved in the restricted model of homogeneous 3 ΣΠΣΠ circuits [GKKS14, FLMS14, KLSS14, KS14].Here, we study a more fine-grained vers...

Superpolynomial Lower Bounds Against Low-Depth Algebraic Circuits

“…Informally, we show that for any constant Γ, circuits of depth Γ are superpolynomially more powerful than circuits of depth Γ ´1. This parallels a similar body of work in Boolean circuit complexity [22,23] and also in the setting of multilinear circuits [63,9]. Specifically, we prove the following result.…”

“…´,π 2 ,τ 2 q where τ 2 is defined as in (9). Plugging in (14) and using the formulas F i,τ i constructed by induction, we see that P pS,J `,J ´,π,τ q has a set-multilinear formula of depth at most 2δ and size at most This proves the induction hypothesis and hence completes the proof of the claim.…”

“…´, π 2 q as in (P3) above. Then by (9), we see that for any τ P t0, 1u k `´k ´, we get P pS,J `,J ´,π,τ q " ÿ τ 1 extends τ P pS 1 ,J 1 `,J 1…”

Observations on Symmetric Circuits

“…An approach to this question is to prove lower bounds in restricted models and try to extend these. This includes results in the non-commutative setting by Nisan [8], lower bounds for multilinear circuits such as [2,10] or the recent VP vs VNP separation in the monotone world by Yehudayoff [16]. The latest model, and our focus of this paper, was given by Dawar and Wilsenach [3].…”

Limitations of Sums of Bounded-Read Formulas

“…Following Raz's work, there has been significant interest in proving lower bounds on the size of syntactic multilinear circuits. Exponential separation of constant depth multilinear circuits is known [11], while the best known lower bound for unbounded depth syntactic multilinear circuits is only almost quadratic [2].…”