Abstract
We prove that the topological entropy of subshifts having decidable language is uncomputable in the following sense: For no error bound less than 1/4 does there exists a program that, given a decision procedure for the language of a subshift as input, will approximate the entropy of the subshift within the error bound. In addition, we prove that not only is the topological entropy of sofic shifts computable to arbitary precision (a well-known fact), but all standard comparisons of the topological entropy with rational numbers are decidable.
Highlights
Dynamical systems are among the most studied objects in branches of mathematics, computer science, and physics
For two well-known species of dynamical systems, namely cellular automata [9] and iterated piecewise affine functions [12], the topological entropy is known to be computably unapproximable in the sense that for sufficiently low ǫ ∈ R+, no program can yield a rational number p/q in finite time such that the topological entropy differs from p/q by at most ǫ
We study computability and decidability issues related to another standard class of dynamical systems, the subshifts
Summary
Dynamical systems are among the most studied objects in branches of mathematics, computer science, and physics. 2. is the topological entropy of sofic subshifts computable in the sense that there is a program that, when given a right-resolving graph representation of the shift as input, the program will approximate the topological entropy to any desired precision (a well-known fact), but comparison under of the topological entropy with any rational number is decidable. As the input may be any program in a (Turing-complete) programming language, the class of “allowed” inputs is clearly undecidable: Namely the class of total programs that decide languages of shift spaces Contrast this with certain other classes of dynamical systems for which the entropy is known to be uncomputable, e.g. cellular automata, where it is decidable whether the input (a finite description of a cellular automaton) is of the correct form. We further believe that future efforts directed at demarcating the limits of computability for shift spaces should focus on suitable subrecursive classes, e.g. shifts with primitive recursive languages
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