On the Size of Depth-Three Boolean Circuits for Computing Multilinear Functions

Abstract

We propose that multi-linear functions of relatively low degree over GF(2) may be good candidates for obtaining exponential 1 lower bounds on the size of constant-depth Boolean circuits (computing explicit functions). Specifically, we propose to move gradually from linear functions to multilinear ones, and conjecture that, for any t ≥ 2, some explicit t-linear functions F : ({0, 1} n ) t → {0, 1} require depth-three circuits of size exp(Ω(tn t/(t+1) )).Towards studying this conjecture, we suggest to study two frameworks for the design of depththree Boolean circuits computing multilinear functions, yielding restricted models for which lower bounds may be easier to prove. Both correspond to constructing a circuit by expressing the target polynomial as a composition of simpler polynomials. The first framework corresponds to a direct composition, whereas the second (and stronger) framework corresponds to nested composition and yields depth-three Boolean circuits via a "guess-and-verify" paradigm. The corresponding restricted models of circuits are called D-canonical and ND-canonical, respectively.Our main results are (1) a generic upper bound on the size of depth-three D-canonical circuits for computing any t-linear function, and (2) a lower bound on the size of any depththree ND-canonical circuits for computing some (in fact, almost all) t-linear functions. These bounds match the foregoing conjecture (i.e., they have the form of exp(tn t/(t+1) )). Another important result is a separation of the two models: We prove that ND-canonical circuits can be super-polynomially smaller than their D-canonical counterparts. We also reduce proving lower bounds for the ND-model to Valiant's matrix rigidity problem (for parameters that were not the focus of previous works).The study of the foregoing (Boolean) models calls for an understanding of new types of arithmetic circuits, which we define in this paper and may be of independent interest. These circuits compute multilinear polynomials by using arbitrary multilinear gates of some limited arity. It turns out that a GF(2)-polynomial is computable by such circuits with at most s gates of arity at most s if and only if it can be computed by ND-canonical circuits of size exp(s). A similar characterization holds for D-canonical circuits if we further restrict the arithmetic circuits to have depth two. We note that the new arithmetic model makes sense over any field, and indeed all our results carry through to all fields. Moreover, it raises natural arithmetic complexity problems which are independent of our original motivation.

The composition complexity of majority

“…In the same way, our model serves as a sanity check for Open Problem 1any proof that depth-3 circuits for Maj n require size 2 Ω( √ n log n) must prove Theorem 1 as well. Strong lower bounds have indeed been proven in the model introduced by [GW20]. Goldreich and Tal [GT18] proved a lower bound of 2 Ω(n 2/3 ) .…”

“…Lower bounds for canonical boolean circuits. Goldreich and Wigderson [GW20] recently introduced a new restricted of model of "canonical" boolean circuits, with the hope of proving 2 ω( √ n) lower bounds for this model. Their model is inspired by the structure of optimal depth-3 circuits for Parity n , which dissects the computation into disjoint parities of smaller arity (over √ n variables).…”

“…Their model is inspired by the structure of optimal depth-3 circuits for Parity n , which dissects the computation into disjoint parities of smaller arity (over √ n variables). [GW20] proposes a generalization of this construction, where one aims to represent a multi-linear function as a depth-2 circuit where the gates are arbitrary multi-linear functions of small arity. They propose proving strong lower bounds in this model as a "sanity check" for proving better general depth-3 circuit bounds (Open Problem 1)-to do the latter, one must necessarily do the former as well.…”

Improved upper bounds for the rigidity of Kronecker products

“…Since then, several other problems in complexity theory have been reduced to proving rigidity lower bounds for explicit families of matrices (see e.g. [Raz89,GW20] and a survey [Lok09]).…”