Abstract
In this paper, we give subexponential size hitting sets for bounded depth multilinear arithmetic formulas. Using the known relation between black-box PIT and lower bounds, we obtain lower bounds for these models. For depth-3 multilinear formulas, of size exp $${(n^\delta)}$$ , we give a hitting set of size exp $${\left(\tilde{O}\left(n^{2/3 + 2\delta/3}\right) \right)}$$ . This implies a lower bound of exp $${(\tilde{\Omega}(n^{1/2}))}$$ for depth-3 multilinear formulas, for some explicit polynomial. For depth-4 multilinear formulas, of size exp $${(n^\delta)}$$ , we give a hitting set of size exp $${\left(\tilde{O}\left(n^{2/3 + 4\delta/3}\right) \right)}$$ . This implies a lower bound of exp $${(\tilde{\Omega}(n^{1/4}))}$$ for depth-4 multilinear formulas, for some explicit polynomial. A regular formula consists of alternating layers of $${+,\times}$$ gates, where all gates at layer i have the same fan-in. We give a hitting set of size (roughly) exp $${\left(n^{1- \delta}\right)}$$ , for regular depth-d multilinear formulas with formal degree at most n and size exp $${(n^\delta)}$$ , where $${\delta = O(1/{\sqrt{5}^d})}$$ . This result implies a lower bound of roughly exp $${(\tilde{\Omega}(n^{1/{\sqrt{5}^d}}))}$$ for such formulas. We note that better lower bounds are known for these models, but also that none of these bounds was achieved via construction of a hitting set. Moreover, no lower bound that implies such PIT results, even in the white-box model, is currently known. Our results are combinatorial in nature and rely on reducing the underlying formula, first to a depth-4 formula, and then to a read-once algebraic branching program (from depth-3 formulas, we go straight to read-once algebraic branching programs).
Highlights
Arithmetic circuits are the standard model for computing polynomials
Subexponential Size Hitting Sets for Bounded Depth Multilinear Formulas every polynomial computed by a ΣΠΣ formula of size s has the following form s di f=
Throughout this paper, we assume that for formulas of size 2nδ, the underlying field F is of size at least |F| ≥ 2n2δpoly log(n), and that if this is not the case we are allowed to query the formula on inputs from an extension field of the appropriate size
Summary
Arithmetic circuits are the standard model for computing polynomials. Roughly speaking, given a set of variables X = {x1, . . . , xn}, an arithmetic circuit uses additions and multiplications to compute a polynomial f in the set of variables X. An arithmetic circuit (or formula) is multilinear if the polynomial computed at each of its gates is multilinear (as a formal polynomial), that is, in each of its monomials the power of every input variable is at most one (see Section 1.1 for definition of the models studied in this paper). Two outstanding open problems in complexity theory are to prove exponential lower bounds on the size of arithmetic circuits, i.e., to prove a lower bound on the number of operations required to compute some polynomial f , and to give efficient deterministic polynomial identity testing (PIT for short) algorithms for them. It is known that solving any one of the problems (proving lower bound or deterministic PIT), with appropriate parameters, for small depth (multilinear) formulas, is equivalent to solving it in the general (multilinear) case [37, 6, 24, 15, 36]. Using the connection between explicit hitting sets and circuit lower bounds we get, as corollaries, subexponential lower bounds for these models
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