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

We propose that multi-linear functions of relatively low degree over GF(2) may be goodcandidates for obtaining exponential lower bounds on the size of constant-depth Boolean cir-cuits (computing explicit functions). Specifically, we propose to move gradually from linearfunctions to multilinear ones, and conjecture that, for any t ≥ 2, some explicit t-linear functionsF : ({0, 1}n)t → {0, 1} require depth-three circuits of size exp(Ω(tn)).Towards studying this conjecture, we suggest to study two frameworks for the design of depth-three Boolean circuits computing multilinear functions, yielding restricted models for whichlower bounds may be easier to prove. Both correspond to constructing a circuit by expressingthe target polynomial as a composition of simpler polynomials. The first framework correspondsto a direct composition, whereas the second (and stronger) framework corresponds to nestedcomposition and yields depth-three Boolean circuits via a ”guess-and-verify” paradigm in thestyle of Valiant. The corresponding restricted models of circuits are called D-canonical andND-canonical, respectively.Our main results are (1) a generic upper bound on the size of depth-three D-canonicalcircuits for computing any t-linear function, and (2) a lower bound on the size of any depth-three ND-canonical circuits for computing some (in fact, almost all) t-linear functions. Thesebounds match the foregoing conjecture (i.e., they have the form of exp(tn)). Anotherimportant result is separating the two models: We prove that ND-canonical circuits can besuper-polynomially smaller than their D-canonical counterparts. We also reduce proving lowerbounds for the ND-model to Valiant’s matrix rigidity problem (for parameters that were notthe focus of previous works).The study of the foregoing (Boolean) models calls for an understanding of new types ofarithmetic circuits, which we define in this paper and may be of independent interest. Thesecircuits compute multilinear polynomials by using arbitrary multilinear gates of some limitedarity. It turns out that a GF(2)-polynomial is computable by such circuits with at most s gatesof 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 arithmeticcircuits 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 arithmeticcomplexity problems which are independent of our original motivation.