Subshift of finite type4/20/2023 We call A a (two-sided) shift of finite type. Ais called a subshift of nite type (or usually just a shift of nite type) if there exists a nite set of words Fsuch that X X(F). We also study the size of synchronizing words and synchronizing deterministic presentations. denote the set of all bi-infinite sequences of symbols subject to the same conditions. All but one of these problems (subshift) remain polynomial-time solvable when restricting to synchronizing deterministic presentations. For the general (reducible) case, however, we show they are all PSPACE-complete. 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. 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. Sofic shifts are symbolic dynamical systems defined by the set of bi-infinite sequences on an edge-labeled directed graph, called a presentation. Theorem 3.2 Let (, ) be a subshift of nite type determined by the 0, 1-matrix A.Denote A A by B the -algebra consisting of all Borel subsets of, s dim ( )(having known that A A H A s log ,where is the spectral radius of A).
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