We study an important specialization of the general problem of broadcasting on directed acyclic graphs, namely, that of broadcasting on two-dimensional (2D) regular grids. Consider an infinite directed acyclic graph with the form of a 2D regular grid, which has a single source vertex $X$ at layer 0, and $k + 1$ vertices at layer $k \geq 1$ , which are at a distance of $k$ from $X$ . Every vertex of the 2D regular grid has outdegree 2, the vertices at the boundary have indegree 1, and all other non-source vertices have indegree 2. At time 0, $X$ is given a uniform random bit. At time $k \geq 1$ , each vertex in layer $k$ receives transmitted bits from its parents in layer $k-1$ , where the bits pass through independent binary symmetric channels with common crossover probability $\delta \in \left({0,\frac {1}{2}}\right)$ during the process of transmission. Then, each vertex at layer $k$ with indegree 2 combines its two input bits using a common deterministic Boolean processing function to produce a single output bit at the vertex. The objective is to recover $X$ with probability of error better than $\frac {1}{2}$ from all vertices at layer $k$ as $k \rightarrow \infty $ . Besides their natural interpretation in the context of communication networks, such broadcasting processes can be construed as one-dimensional (1D) probabilistic cellular automata, or discrete-time statistical mechanical spin-flip systems on 1D lattices, with boundary conditions that limit the number of sites at each time $k$ to $k+1$ . Inspired by the literature surrounding the “positive rates conjecture” for 1D probabilistic cellular automata, we conjecture that it is impossible to propagate information in a 2D regular grid regardless of the noise level $\delta $ and the choice of common Boolean processing function. In this paper, we make considerable progress towards establishing this conjecture, and prove using ideas from percolation and coding theory that recovery of $X$ is impossible for any $\delta \in \left({0,\frac {1}{2}}\right)$ provided that all vertices with indegree 2 use either AND or XOR for their processing functions. Furthermore, we propose a detailed and general martingale-based approach that establishes the impossibility of recovering $X$ for any $\delta \in \left({0,\frac {1}{2}}\right)$ when all NAND processing functions are used if certain structured supermartingales can be rigorously constructed. We also provide strong numerical evidence for the existence of these supermartingales by computing several explicit examples for different values of $\delta $ via linear programming.

中文翻译：

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在二维规则网格上广播

我们研究了在有向无环图上广播的一般问题的一个重要专业，即在二维（2D）规则网格上广播的问题。考虑一个二维规则网格形式的无限有向无环图，它有一个源顶点 $X$在第 0 层，并且 $k + 1$层的顶点 $k \geq 1$ ，它们的距离为 $k$从 $X$ . 二维规则网格的每个顶点的出度为 2，边界处的顶点的入度为 1，所有其他非源顶点的入度为 2。在时间 0， $X$给定一个统一的随机位。当时 $k \geq 1$ , 层中的每个顶点 $k$在层中接收来自其父级的传输比特 $k-1$ ，其中比特通过具有共同交叉概率的独立二进制对称通道 $\delta \in \left({0,\frac {1}{2}}\right)$在传输过程中。然后，层的每个顶点 $k$indegree 2 使用通用的确定性布尔处理函数组合其两个输入位，以在顶点处产生单个输出位。目标是恢复 $X$错误概率优于 $\frac {1}{2}$从层的所有顶点 $k$作为 $k \rightarrow \infty $ . 除了它们在通信网络环境中的自然解释外，这种广播过程还可以解释为一维 (1D) 概率元胞自动机，或一维晶格上的离散时间统计机械自旋翻转系统，其边界条件限制了每次的站点 $k$至 $k+1$ . 受到围绕一维概率元胞自动机的“正率猜想”的文献的启发，我们推测无论噪声水平如何，都不可能在二维规则网格中传播信息 $\三角洲$以及常用布尔处理函数的选择。在本文中，我们在建立这一猜想方面取得了相当大的进展，并利用渗透和编码理论的思想证明了 $X$任何人都不可能 $\delta \in \left({0,\frac {1}{2}}\right)$前提是所有入度为 2 的顶点对它们的处理功能使用 AND 或 XOR。此外，我们提出了一种详细且通用的基于鞅的方法，该方法确定了恢复的不可能性 $X$对于任何 $\delta \in \left({0,\frac {1}{2}}\right)$如果可以严格构造某些结构化的超鞅，则当使用所有 NAND 处理功能时。我们还通过计算不同值的几个显式示例，为这些超鞅的存在提供了强有力的数值证据。 $\三角洲$通过线性规划。