AI 中文总结
本文研究代数通信复杂度，提出通用下界框架，为多项式求值和集合识别问题证明紧或近紧的概率下界，并应用于代数扫描器和BSS类模型。
AI 中文摘要
通信复杂度研究当问题的输入被分配给多个参与方时，必须交换多少信息才能解决问题。经典设置处理的是在两个参与方之间分配的布尔输入。我们研究一种代数变体，其中输入是域 $\mathbb{F} \in \{\mathbb{R}, \mathbb{C}\}$ 上的向量。Alice 和 Bob 分别拥有输入 $X\in \mathbb{F}^n$ 和 $Y\in \mathbb{F}^n$。我们考虑两类任务：多项式求值问题（计算多项式 $g\in \mathbb{F}[X,Y]$ 的值）和集合识别问题（判断 $(X,Y)$ 是否属于 $S$，其中 $S\subseteq \mathbb{F}^{n} \times \mathbb{F}^n$）。在这两种设置中，Alice 和 Bob 发送仅依赖于各自输入的多项式的求值结果。在集合识别问题中，裁判接收消息，并可以对迄今收到的消息应用多项式测试；这些测试的结果决定接受或拒绝。协议可以是确定性的或概率性的。我们研究：- 上界和归约：我们为一系列自然的多项式求值和集合识别问题给出了非平凡的上界，并证明了不同问题之间的归约，这有助于组织该模型的整体图景。- 下界框架和紧下界：我们的主要技术贡献是一个用于证明代数集合识别问题下界的通用框架。我们为自然问题证明了几个概率下界，给出了其代数通信的紧或近紧刻画。- 框架的应用：最后，我们给出了框架的两个应用：证明一类从左到右的代数算法（代数扫描器）的下界，以及一个受 BSS 模型启发的更一般的代数计算设置的下界。
英文摘要
Communication complexity studies how much information must be exchanged to solve a problem whose input is split among several parties. The classical setting deals with Boolean inputs split between two parties. We study an algebraic variant, where the inputs are vectors over a field $\mathbb{F} \in \{\mathbb{R}, \mathbb{C}\}$. Alice and Bob have inputs $X\in \mathbb{F}^n$ and $Y\in \mathbb{F}^n$, respectively. We consider two kinds of tasks: the polynomial evaluation problem (compute the value of a polynomial $g\in \mathbb{F}[X,Y]$), and the set-recognition problem (decide whether (X,Y) is in $S$, for $S\subseteq \mathbb{F}^{n} \times \mathbb{F}^n$). In both settings, Alice and Bob send evaluations of polynomials depending only on their own inputs. In the set-recognition problem, a referee receives the messages and may apply polynomial tests to the messages received so far; the outcomes of these tests determine acceptance or rejection. The protocols may be deterministic or probabilistic. We study: - Upper bounds and reductions: We give non-trivial upper bounds for a range of natural polynomial evaluation and set-recognition problems and prove reductions between different problems, which help organize the landscape of the model. - A lower bound framework and tight lower bounds: Our main technical contribution is a general framework for proving lower bounds for algebraic set-recognition problems. We prove several probabilistic lower bounds for natural problems, giving tight or near-tight characterizations of their algebraic communication. - Applications of the framework: Finally, we give two applications of our framework: proving lower bounds for a class of left-to-right algebraic algorithms (algebraic scanners) and a more general algebraic computational setting inspired by the BSS model.
