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

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. \r\nOur 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. \r\nThere 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. \r\nWe 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.