$\mathsf{VNP}=\mathsf{VP}$ in the multilinear world

Highlights

• We study exponential sums of polynomials in different models of computation.

• We show that polynomial-size multilinear circuits are closed under exponential sums.

• We need to use varying techniques based on the characteristic of the field.

• We show some essential differences between general circuits and multilinear circuits.

Abstract

In this note, we show that over fields of any characteristic, exponential sums of Boolean instantiations of polynomials computed by multilinear circuits can be computed by multilinear circuits with polynomial blow-up in size. In particular, multilinear-VNP equals multilinear-VP. Our result showing closure under exponential sums also holds for other restricted multilinear classes – polynomials computed by multilinear (bounded-width) algebraic branching programs and formulas. Furthermore, it holds even if the circuit class is not fully multilinear but computes a polynomial that is multilinear in the summation variables.

Introduction

Valiant [1], [2] introduced algebraic complexity theory to study the complexity of polynomial families. One of the most fundamental problems in algebraic complexity theory is to decide whether the classes VP and VNP are distinct. These classes were first defined by Valiant in [1], [2] as the algebraic analogues of the Boolean complexity classes P and NP (however, a closer look at the definitions reveals that they are really an analogue of LOGCFL and #P; LOGCFL is a subclass of P). For basics and detailed treatment of the subject we refer the reader to [3], [4]. We borrow notations from [4], [5]. VP is the class of sequences of polynomials with polynomially bounded degree that are computable by polynomial sized arithmetic circuits. Moreover, a sequence of polynomials $(f_n)$ belongs to VNP if and only if there exist polynomials p and q, and a sequence $(g_n)\in$ VP such that for all n,

$$\begin{array}{c}{f}_n(x_1,\dots ,x_{q(n)})\hfill \\ =\sum _{\overline{y}\in {\{0,1\}}^{p(\overline{x})}}{g}_n(x_1,\dots ,x_{q(n)},y_1,\dots ,y_{p(n)}).\hfill \end{array}$$

So, in other words, one can think of VNP as exponential sums of polynomial sized circuits; $\mathsf{VNP}=\sum \cdot \mathsf{VP}$. Hence the VP versus VNP question can also be thought of as understanding the power of exponential sums. In the foundational paper [2], Valiant observed that exponential sums of polynomial sized formulas $(\sum \cdot \mathsf{VF})$ or $(\sum \cdot \mathsf{V}{\mathsf{P}}_{\mathsf{e}})$ exactly capture exponential sums of polynomial sized circuits $(\sum \cdot \mathsf{VP})$. That is, ${\mathsf{VNP}}_e=\mathsf{VNP}$ (see also [6]). He used this observation crucially to show that the permanent polynomial is VNP-hard. VBP is the class of polynomial families computed by algebraic branching programs of polynomial size; equivalently, families that can be represented as the determinant of a polynomially large matrix, where the entries are either field elements or variables. It is known that $\mathsf{VF}\subseteq \mathsf{VBP}\subseteq \mathsf{VP}$; hence from Valiant's observation, it follows that $\sum \cdot \mathsf{VF}=\sum \cdot \mathsf{VBP}=\sum \cdot \mathsf{VP}=\mathsf{VNP}$.

Valiant's observation raises a natural question to study: How powerful are exponential sums of restricted circuit classes?

A natural restriction on arithmetic circuits is multilinearity. A polynomial is called multilinear if each variable in the polynomial has degree at most 1. An arithmetic circuit is called multilinear if every gate in it computes a multilinear polynomial. Furthermore, if for every product gate, the sub-circuits rooted at the left and right child are variable-disjoint, then the circuit is called syntactic multilinear.

The exponential summation under the restriction of syntactic multilinearity was studied by Jansen et al. [7], [8], [5]. They showed that syntactic multilinear classes are closed under exponential sums. In particular, exponential summation does not add any power to syntactic multilinear formulas. Contrast this with the case of general formulas, where it become as powerful as VNP. Exponential summations of polynomials were also studied by Juma et al. [9]. Their motivation was to obtain query algorithms for #SAT that are better than brute-force. They proved that over fields of characteristic different from 2, multilinear polynomials are closed under exponential sums [9, Observation 1.3].

In this note we study the exponential summation under the restriction of multilinearity (not necessarily syntactic). Using techniques different from those used in [5], [9], we extend their results by showing that over any field, exponential summation does not add power to multilinear circuit classes. In particular, since in the multilinear world we know that VF is strictly weaker than VBP [10], [11], our result implies that in the multilinear world we do not have an analogue of the collapse $\sum \cdot \mathsf{VF}=\sum \cdot \mathsf{VBP}=\sum \cdot \mathsf{VP}$ that holds in the general world. A corollary of our result is that $\mathsf{VNP}=\mathsf{VP}$ in the multilinear setting, whereas we do not believe that a similar thing holds in non-multilinear setting. Thus our result highlights essential differences between the general and multilinear worlds, and indicates that separations/collapses in the restricted multilinear world may have no bearing on the true state of affairs in the general world.

We obtain our result (Theorem 1) by considering summations of general polynomials, but the summation is over variables that have degree at most 1 in the polynomial. We show that such a summation over multilinear variables is as good as evaluating the polynomial at one or a small number of points.