On the computational complexity of the languages of general symbolic dynamical systems and beta-shifts

We consider the computational complexity of languages of symbolic dynamical systems. In particular, we study complexity hierarchies and membership of the non-uniform class P/poly. We prove: 1. For every time-constructible, non-decreasing function t(n) = ω(n), there is a symbolic dynamical system with language decidable in deterministic time O(nt(n)), but not in deterministic time o(t(n)). 2. For every space-constructible, non-decreasing function s(n) = ω(n), there is a symbolic dynamical system with language decidable in deterministic space O(s(n)), but not in deterministic space o(s(n)). 3. There are symbolic dynamical systems having hard and complete languages under ≤ m and ≤pm-reduction for every complexity class above LOGSPACE in the backbone hierarchy (hence, P-complete, NP-complete, coNP-complete, PSPACE-complete, and EXPTIME-complete sets). 4. There are decidable languages of symbolic dynamical systems in P/poly for every alphabet of size |Σ| ≥ 1. 5. There are decidable languages of symbolic dynamical systems not in P/poly iff the alphabet size is > 1. For the particular class of symbolic dynamical systems known as β-shifts, we prove that: 1. For all real numbers β > 1, the language of the β-shift is in P/poly. 2. If there exists a real number β > 1 such that the language of the β-shift is NP-hard under ≤pT -reduction, then the polynomial hierarchy collapses to the second level. As NP-hardness under ≤pm-reduction implies hardness under ≤ p T -reduction, this result implies that it is unlikely that a proof of existence of an NP-hard language of a β-shift will be forthcoming. 3. For every time-constructible, non-decreasing function t(n) ≥ n, there is a real number 1 < β < 2 such that the language of the β-shift is decidable in time O(nt(log n+1)), but not in any proper time bound g(n) satisfying g(4) = o(t(n)/16). 4. For every space-constructible, non-decreasing function s(n) = ω(n), there is a real number 1 < β < 2 such that the language of the β-shift is decidable in space O(s(n)), but not in space g(n) where g is any function satisfying g(n) = o(s(n)). 5. There exists a real number 1 < β < 2 such that the language of the β-shift is recursive, but not context-sensitive.