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

• Authors: Jakob Grue Simonsen
• Venue: Theoretical Computer Science
• Year: 2009

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)=@w(n), there is a symbolic dynamical system with language decidable in deterministic time O(n^2t(n)), but not in deterministic time o(t(n)). 2.For every space-constructible, non-decreasing function s(n)=@w(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^l^o^g^s- and @?"m^p-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 |@S|=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 @b-shifts, we prove that: 1.For all real numbers @b1, the language of the @b-shift is in P/poly. 2.If there exists a real number @b1 such that the language of the @b-shift is NP-hard under @?"T^p-reduction, then the polynomial hierarchy collapses to the second level. As NP-hardness under @?"m^p-reduction implies hardness under @?"T^p-reduction, this result implies that it is unlikely that a proof of existence of an NP-hard language of a @b-shift will be forthcoming. 3.For every time-constructible, non-decreasing function t(n)=n, there is a real number 1