We investigate the power of Non-commutative Arithmetic Circuits, which compute polynomials over the free non-commutative polynomial ring $${\mathbb{F}\langle{x_1,\ldots,x_N\rangle}}$$F⟨x1,…,xN⟩, where variables do not commute. We consider circuits that are restricted in the ways in which they can compute monomials: this can be seen as restricting the families of parse trees that appear in the circuit. Such restrictions capture essentially all non-commutative circuit models for which lower bounds are known. We prove several results about such circuits.1.We show exponential lower bounds for circuits with up to an exponential number of parse trees, strengthening the work of Lagarde et al. [Electronic Colloquium on Comput Complexity (ECCC) vol 23, no 94, 2016], who prove such a result for Unique Parse Tree (UPT) circuits which have a single parse tree. The polynomial we prove a lower bound for is in fact computable by a polynomial-sized non-commutative circuit.2.We show exponential lower bounds for circuits whose parse trees are rotations of a single tree. This simultaneously generalizes recent lower bounds of Limaye et al. (Theory Comput 12(1):1–38, 2016) and the above lower bounds of Lagarde et al. (2016), which are known to be incomparable. Here too, the hard polynomial is computable by a polynomial-sized non-commutative circuit.3.We make progress on a question of Nisan (STOC, pp 410–418, 1991) regarding separating the power of Algebraic Branching Programs (ABPs) and Formulas in the non-commutative setting by showing a tight lower bound of $${n^{\Omega(\log d)}}$$nΩ(logd) for any UPT formula computing the product of d$${n \times n}$$n×n matrices.When $${d \leq \log n}$$d≤logn, we can also prove superpolynomial lower bounds for formulas with up to $${2^{o(d)}}$$2o(d) many parse trees (for computing the same polynomial). Improving this bound to allow for $${2^{o(d)}}$$2o(d) trees would give an unconditional separation between ABPs and Formulas.4.We give deterministic whitebox PIT algorithms for UPT circuits over any field, strengthening a result of Lagarde et al. (2016), and also for sums of a constant number of UPT circuits with different parse trees.

中文翻译：

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具有受限解析树的非交换算术电路的下界和 PIT

我们研究了非交换算术电路的威力，它在自由非交换多项式环 $${\mathbb{F}\langle{x_1,\ldots,x_N\rangle}}$$F⟨x1 上计算多项式，... ,xN⟩，其中变量不交换。我们考虑在计算单项式的方式上受到限制的电路：这可以看作是限制出现在电路中的解析树的家族。这种限制基本上涵盖了所有已知下界的非交换电路模型。我们证明了关于此类电路的几个结果。 1. 我们展示了具有指数数量解析树的电路的指数下界，加强了 Lagarde 等人的工作。[Electronic Colloquium on Comput Complexity (ECCC) vol 23, no 94, 2016]，证明了具有单个解析树的唯一解析树 (UPT) 电路的这一结果。我们证明的多项式的下界实际上可以通过多项式大小的非交换电路计算。2. 我们展示了分析树是单棵树旋转的电路的指数下界。这同时概括了 Limaye 等人最近的下限。(Theory Comput 12(1):1–38, 2016) 和 Lagarde 等人的上述下限。（2016），众所周知，这是无与伦比的。在这里，硬多项式也可以通过多项式大小的非交换电路来计算。 3. 我们在 Nisan (STOC, pp 410–418, 1991) 的问题上取得了进展，关于分离代数分支程序 (ABP) 和对于计算 d$${n \times 乘积的任何 UPT 公式，通过显示 $${n^{\Omega(\log d)}}$$nΩ(logd) 的紧下界，在非交换设置中的公式n}$$n×n 个矩阵。当 $${d \leq \log n}$$d≤logn，我们还可以证明具有高达 $${2^{o(d)}}$$2o(d) 多个解析树（用于计算相同多项式）的公式的超多项式下界。改进这个界限以允许 $${2^{o(d)}}$$2o(d) 树将在 ABP 和公式之间无条件分离。 4.我们为任何领域的 UPT 电路提供确定性白盒 PIT 算法，加强拉加德等人的结果。(2016)，以及具有不同解析树的恒定数量的 UPT 电路的总和。