Sofic shifts are symbolic dynamical systems defined by the set of bi-infinite sequences on an edge-labeled directed graph, called a presentation. We study the computational complexity of an array of natural decision problems about presentations of sofic shifts, such as whether a given graph presents a shift of finite type, or an irreducible shift; whether one graph presents a subshift of another; and whether a given presentation is minimal, or has a synchronizing word. Leveraging connections to automata theory, we first observe that these problems are all decidable in polynomial time when the given presentation is irreducible (strongly connected), via algorithms both known and novel to this work. For the general (reducible) case, however, we show they are all PSPACE-complete. All but one of these problems (subshift) remain polynomial-time solvable when restricting to synchronizing deterministic presentations. We also study the size of synchronizing words and synchronizing deterministic presentations.

翻译： Sofic 转换是一组由边缘标签的定向图形(称为演示文稿)双无限制序列定义的象征性动态系统。我们研究了一系列自然决定问题的计算复杂性,这些自然决定问题涉及沉浮变化的演示,例如某个图形是显示一定类型的转变,还是不可减少的转变;一个图形是显示另一个的子转变;一个特定演示文稿是最小的,还是有同步的单词。利用自动磁盘连接理论,我们首先观察到,当给定的演示文稿不可复制(紧密连接)时,这些问题在多元时,通过已知的和对这项工作的新手算法都是可分解的。然而,对于一般(可减少的)案例,我们显示它们都是PSPACE的完整。在限制同步确定性演示时,除一个问题外,所有这些问题(次移)都仍然是多音时可溶解。我们还研究了同步的单词和同步确定性演示文稿的大小。