Cost and dimension of words of zero topological entropy

TL;DR

This paper introduces a new measure called cost and cost dimension to analyze the complexity of languages generated by infinite words with zero topological entropy, providing a characterization of linear factor complexity.

Contribution

It defines the concepts of cost and cost dimension for languages and characterizes words with linear factor complexity using these measures.

Findings

Languages of linear complexity have cost 0 and cost dimension 2.

Languages with sub-linear complexity can have infinite cost dimension.

The measures reveal deep combinatorial structures in language complexity.

Abstract

Let $A^*$ denote the free monoid generated by a finite nonempty set $A.$ In this paper we introduce a new measure of complexity of languages $L\subseteq A^*$ defined in terms of the semigroup structure on $A^*.$ For each $L\subseteq A^*,$ we define its {\it cost} $c(L)$ as the infimum of all real numbers $\alpha$ for which there exist a language $S\subseteq A^*$ with $p_S(n)=O(n^\alpha)$ and a positive integer $k$ with $L\subseteq S^k.$ We also define the {\it cost dimension} $d_c(L)$ as the infimum of the set of all positive integers $k$ such that $L\subseteq S^k$ for some language $S$ with $p_S(n)=O(n^{c(L)}).$ We are primarily interested in languages $L$ given by the set of factors of an infinite word $x=x_0x_1x_2\cdots \in A^\omega$ of zero topological entropy, in which case $c(L)<+\infty.$ We establish the following characterisation of words of linear factor complexity: Let $x\in A^\omega$ and $L=$Fac$(x)$ be the set of factors of $x.$ Then $p_x(n)=\Theta(n)$ if and only $c(L)=0$ and $d_c(L)=2.$ In other words, $p_x(n)=O(n)$ if and only if Fac$(x)\subseteq S^2$ for some language $S\subseteq A^+$ of bounded complexity (meaning $\limsup p_S(n)<+\infty).$ In general the cost of a language $L$ reflects deeply the underlying combinatorial structure induced by the semigroup structure on $A^*.$ For example, in contrast to the above characterisation of languages generated by words of sub-linear complexity, there exist non factorial languages $L$ of complexity $p_L(n)=O(\log n)$ (and hence of cost equal to $0)$ and of cost dimension $+\infty.$ In this paper we investigate the cost and cost dimension of languages defined by infinite words of zero topological entropy.