可扩展自动机网络
Expansive Automata Networks
摘要：
自动机网络是一个映射${f:Q^n\rightarrow Q^n}$，其中$Q$是一个有限的字母表。它可以被看作是一个$n$实体的网络，每个实体都持有来自$Q$的状态，并根据确定性同步更新规则以这样一种方式进化，即每个实体只依赖于网络图（称为交互图）中的邻居。自动机网络理论的一个主要趋势是理解交互图如何影响$f$的动力学性质。在这项工作中，我们引入了以下属性称为膨胀性：在任何给定的节点上的状态序列的观察是足够的，以确定整个网络的初始配置。我们的主要结果是一个相互作用图的特征，允许膨胀。此外，我们表明，这个属性是通用的线性自动机网络在这样的图与足够大的字母表。然而，我们表明，当字母表是固定的相互作用图的大小独立的情况是更复杂的：没有字母表是足够的，以获得所有容许图的膨胀性，只有非线性的解决方案存在于某些情况下。最后，在其他结果中，我们考虑了一个更强的版本的膨胀性，我们要求从任何足够大的系统的观察，以确定初始配置。我们表明，它可以实现任何数量的节点，自然会产生最大距离可分码。
英文摘要：
An Automata Network is a map ${f:Q^n\rightarrow Q^n}$ where $Q$ is a finite alphabet. It can be viewed as a network of $n$ entities, each holding a state from $Q$, and evolving according to a deterministic synchronous update rule in such a way that each entity only depends on its neighbors in the network's graph, called interaction graph. A major trend in automata network theory is to understand how the interaction graph affects dynamical properties of $f$. In this work we introduce the following property called expansivity: the observation of the sequence of states at any given node is sufficient to determine the initial configuration of the whole network. Our main result is a characterization of interaction graphs that allow expansivity. Moreover, we show that this property is generic among linear automata networks over such graphs with large enough alphabet. We show however that the situation is more complex when the alphabet is fixed independently of the size of the interaction graph: no alphabet is sufficient to obtain expansivity on all admissible graphs, and only non-linear solutions exist in some cases. Finally, among other results, we consider a stronger version of expansivity where we ask to determine the initial configuration from any large enough observation of the system. We show that it can be achieved for any number of nodes and naturally gives rise to maximum distance separable codes.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
