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.

中文翻译：

---

非交换条件下的ABP和一些结构化公式的分离

这项工作的动机问题是由Nisan（1991）提出的一个长期存在的开放性问题，它涉及非交换条件下的代数分支程序（ABP）和公式的相对能力。尽管一般性问题仍然悬而未决，但我们在解决该问题上取得了一些进展。为此，我们在非交换设置（由\ Hrubes，Wigderson和Yehudayoff（2011）定义）中推广了有序多项式的概念，以定义自然的初学者多项式和模型。我们的主要贡献是可以通过对初学者公式的下限来在非交换环境中分离公式和ABP的一种可能的新方法。特别是，我们显示以下内容。存在一个显式的n阶d初学者多项式$ f_ {n，d}（x）$使得1. $ f_ {n，d}（x）$可以通过大小为O（nd）的初学者ABP来计算；2.计算$ f_ {n，\ log n}（x）$的任何初学者公式的大小都必须是n中的超多项式。我们还表明，针对$ f _ {\ log n，n}（x）$的初学者公式的超多项式下界将在非交换条件下分离公式和ABP的幂。