Abstract
Binary cube-free language and ternary square-free language are two “canonical” representatives of a wide class of languages defined by avoidance properties. Each of these two languages can be viewed as an infinite binary tree reflecting the prefix order of its elements. We study how “homogenious” these trees are, analysing the following parameter: the density of branching nodes along infinite paths. We present combinatorial results and an efficient search algorithm, which together allowed us to get the following numerical results for the cube-free language: the minimal density of branching points is between 3509/9120≈0.38476 and 13/29≈0.44828, and the maximal density is between 0.72 and 67/93≈0.72043. We also prove the lower bound 223/868≈0.25691 on the density of branching points in the tree of the ternary square-free language.
Highlights
A formal language, which is a subset of the set of all finite words over some alphabets, is one of the most common objects in discrete mathematics and computer science
To estimate the number of letters in a cube-free word that are fixed by small cubes, we analyze finite automata recognizing some approximations of the language CF
As we have seen in this paper, the branching density of particular infinite words in a typical power-free language of exponential growth can vary significantly
Summary
A formal language, which is a subset of the set of all finite words over some (usually finite) alphabets, is one of the most common objects in discrete mathematics and computer science. Polynomial-size binary power-free languages possess several distinctive properties (see, e.g., [9] Section 2.2); all properties stem from a close relation of all words from these languages to a single infinite word, called the Thue–Morse word [10]. Among these languages, the overlap-free language, avoiding all α-powers with α > 2, attracted the most attention; as a result, it is studied very well. The asymptotic order of growth for this language is computed exactly [11,12] It is decidable whether a subtree of the prefix tree, rooted at a given word w, is finite or infinite [13].
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