Computational complexity of problems for deterministic presentations of sofic shifts

TL;DR

This paper investigates the computational complexity of decision problems related to sofic shifts, revealing polynomial-time solutions for irreducible cases and PSPACE-completeness in the general case, with implications for automata theory.

Contribution

It provides a comprehensive complexity analysis of presentation problems for sofic shifts, including new algorithms and complexity classifications.

Findings

Problems are polynomial-time decidable for irreducible presentations.

General problems are PSPACE-complete for reducible presentations.

Most problems are polynomial-time solvable for synchronizing deterministic presentations.

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.