Separating ABPs and Some Structured Formulas in the Non-Commutative Setting

TL;DR

This paper explores the separation between algebraic branching programs and formulas in the non-commutative setting by introducing abecedarian polynomials and establishing lower bounds for formulas computing these polynomials.

Contribution

It introduces abecedarian polynomials and models, and provides the first super-polynomial lower bounds for abecedarian formulas computing certain explicit polynomials, advancing the understanding of ABP and formula separation.

Findings

Explicit abecedarian polynomials computable by small ABPs

Super-polynomial lower bounds for abecedarian formulas computing certain polynomials

Potential to separate ABPs and formulas in the non-commutative setting

Abstract

The motivating question for this work is a long standing open problem, posed by Nisan (1991), regarding the relative powers of algebraic branching programs (ABPs) and formulas in the non-commutative setting. Even though the general question continues to remain open, we make some progress towards its resolution. To that effect, we generalise the notion of ordered polynomials in the non-commutative setting (defined by \Hrubes, Wigderson and Yehudayoff (2011)) to define abecedarian polynomials and models that naturally compute them. Our main contribution is a possible new approach towards separating formulas and ABPs in the non-commutative setting, via lower bounds against abecedarian formulas. In particular, we show the following. There is an explicit n-variate degree d abecedarian polynomial $f_{n,d}(x)$ such that 1. $f_{n, d}(x)$ can be computed by an abecedarian ABP of size O(nd); 2. any abecedarian formula computing $f_{n, \log n}(x)$ must have size that is super-polynomial in n. We also show that a super-polynomial lower bound against abecedarian formulas for $f_{\log n, n}(x)$ would separate the powers of formulas and ABPs in the non-commutative setting.