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

Abstract

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 ( n 2 t ( 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 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 | 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 ≤ 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 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 ( n 2 t ( log n + 1 ) ) , but not in any proper time bound g ( n ) satisfying g ( 4 n ) = o ( t ( n ) / 1 6 n ) . 4. For every space-constructible, non-decreasing function s ( n ) = ω ( n 2 ) , 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 2 ) = 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.