An Automata Network is a map $f:Q^n\to Q^n$ 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. 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. A major trend in automata network theory is to understand how the interaction graph affects dynamical properties of f. Our first 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. Besides, we show striking differences between the linear and the general non-linear case, in particular we prove that deciding expansivity is PSPACE-complete in the general case, while it can be done in polynomial time in the linear case. Finally, 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.

自动机网络是地图 $F：{问}^{ñ}\to {问}^{ñ}$其中Q是有限字母。可以将其视为由n个实体组成的网络，每个实体都具有Q的状态，并根据确定性同步更新规则进行演化，使得每个实体仅依赖于其在网络图中的邻居（称为交互图）。在这项工作中，我们引入了称为扩展性的以下属性：在任何给定节点上观察状态序列足以确定整个网络的初始配置。自动机网络理论的一个主要趋势是了解相互作用图如何影响f的动力学性质。。我们的第一个结果是表征允许扩展的交互图。此外，我们证明了在具有足够大字母的此类图上的线性自动机网络中，此属性是通用的。但是，我们表明，当字母独立于交互图的大小而固定时，情况会更加复杂：没有字母足以在所有可允许图上获得扩展性，并且在某些情况下仅存在非线性解。此外，我们显示了线性和一般非线性情况之间的惊人差异，特别是证明了在一般情况下决定扩张性是PSPACE完全的，而在线性情况下可以在多项式时间内完成。最后，我们考虑使用扩展性更强的版本，在该版本中，我们要求从对系统足够大的观察中确定初始配置。