In this paper, we prove the first \emph{super-polynomial} and, in fact, \emph{exponential} lower bound for the model of \emph{sum of read-once oblivious algebraic branching programs} (ROABPs). In particular, we give an explicit polynomial such that any sum of ROABPs (equivalently, sum of \emph{ordered} set-multilinear branching programs, each with a possibly different ordering) computing it must have exponential size. This result generalizes the seminal work of Nisan (STOC 1991), which proved an exponential lower bound for a single ROABP. It also strengthens the work of Arvind and Raja (Chic. J. Theor. Comput. Sci., 2016), as well as the work of Bhargav, Dwivedi, and Saxena (2023), both of which established lower bounds against certain restricted versions of this model, and strongly answers an open question from both papers that asked to prove super-polynomial lower bounds for the corresponding \emph{unrestricted} model. The significance of our lower bounds is underscored by the recent work of Bhargav, Dwivedi, and Saxena (2023), which showed that super-polynomial lower bounds against a sum of ordered set-multilinear branching programs -- for a polynomial of sufficiently low degree -- would imply super-polynomial lower bounds against general ABPs, thereby resolving Valiant's longstanding conjecture that the permanent polynomial can not be computed efficiently by ABPs. More precisely, their work shows that if one could obtain such lower bounds when the degree is bounded by $O(\log n/ \log \log n)$, then it would imply super-polynomial lower bounds against general ABPs. In this paper, we show super-polynomial lower bounds against this model for a polynomial whose degree is as small as $\omega(\log n)$. Prior to our work, showing such lower bounds was open \emph{irrespective} of the assumption on the degree.

翻译：本文证明了首次针对“一次读取无感代数分支程序（ROABP）”和式模型的**超多项式**（实际上是指数级）下界。具体地，我们给出一个显式多项式，使得计算它的任意ROABP和式（等价于任意有序集合多线性分支程序的和式，每个程序可能具有不同的顺序）规模必须为指数级。该结果推广了Nisan（STOC 1991）的开创性工作——其证明了单个ROABP的指数级下界，同时强化了Arvind与Raja（Chic. J. Theor. Comput. Sci., 2016）以及Bhargav、Dwivedi与Saxena（2023）的工作——这两项工作均证明了该模型某些受限版本的指数级下界，并有力回应了两篇论文中提出的开放性问题：要求证明对应**无限制**模型的超多项式下界。我们的下界意义因近期Bhargav、Dwivedi与Saxena（2023）的工作而更加凸显：该工作表明，针对低次多项式，若能证明有序集合多线性分支程序和式的超多项式下界，则可推导出一般代数分支程序（ABP）的超多项式下界，从而解决Valiant的长期猜想——即永久多项式无法被ABP高效计算。确切而言，其工作指出：若能在次数不超过$O(\log n/ \log \log n)$时获得此类下界，则意味着一般ABP存在超多项式下界。本文中，我们针对次数低至$\omega(\log n)$的多项式证明了该模型的超多项式下界。在本工作之前，无论对次数作何种假设，此类下界的证明均为开放问题。