Computational complexity of problems for deterministic presentations of sofic shifts

Abstract

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.

The road problem and homomorphisms of directed graphs

“…However, it is quite natural from an automata-theoretic perspective, especially concerning computational complexity, to consider graphs that are not strongly connected. For instance, Eppstein [15] shows that it is NP-complete to determine the minimum length of a synchronizing word for a given synchronizing DFA, and his examples are not strongly connected; there are also several graph problems from symbolic dynamics [13] that are NP-complete in general but have polynomial-time algorithms in the strongly connected case.…”