On the Power of Border Width-2 ABPs over Fields of Characteristic 2

The celebrated result by Ben-Or and Cleve [SICOMP92] showed that algebraic formulas are polynomially equivalent to width-3 algebraic branching programs (ABP) for computing polynomials. i.e., VF = VBP3. Further, there are simple polynomials, such as ^P8<inf>i=1</inf> xiyi, that cannot be computed by width-2 ABPs [Allender and Wang, CC16]. Bringmann, Ikenmeyer and Zuiddam, [JACM18], on the other hand, studied these questions in the setting of approximate (i.e., border complexity) computation, and showed the universality of border width-2 ABPs, over fields of characteristic ≠ 2. In particular, they showed that polynomials that can be approximated by formulas can also be approximated (with only a polynomial blowup in size) by width-2 ABPs, i.e., VF = VBP2. The power of border width-2 algebraic branching programs when the characteristic of the field is 2 was left open. In this paper, we show that width-2 ABPs can approximate every polynomial irrespective of the field characteristic. We show that any polynomial f with ℓ monomials and with at most t odd-power indeterminates per monomial can be approximated by O(ℓ· (deg(f) + 2^t))-size width-2 ABPs. Since ℓ and t are finite, this proves universality of border width-2 ABPs. For univariate polynomials, we improve this upper-bound from O(deg(f)²) to O(deg(f)). Moreover, we show that, if a polynomial f can be approximated by small formulas, then the polynomial f^d, for some small power d, can be approximated by small width-2 ABPs. Therefore, even over fields of characteristic two, border width-2 ABPs are a reasonably powerful computational model. Our construction works over any field.