5 research outputs found
Transition Property For Cube-Free Words
We study cube-free words over arbitrary non-unary finite alphabets and prove
the following structural property: for every pair of -ary cube-free
words, if can be infinitely extended to the right and can be infinitely
extended to the left respecting the cube-freeness property, then there exists a
"transition" word over the same alphabet such that is cube free. The
crucial case is the case of the binary alphabet, analyzed in the central part
of the paper.
The obtained "transition property", together with the developed technique,
allowed us to solve cube-free versions of three old open problems by Restivo
and Salemi. Besides, it has some further implications for combinatorics on
words; e.g., it implies the existence of infinite cube-free words of very big
subword (factor) complexity.Comment: 14 pages, 5 figure
Transition Property for Cube-Free Words
We study cube-free words over arbitrary non-unary finite alphabets and prove the following structural property: for every pair (u, v) of d-ary cube-free words, if u can be infinitely extended to the right and v can be infinitely extended to the left respecting the cube-freeness property, then there exists a “transition” word w over the same alphabet such that uwv is cube free. The crucial case is the case of the binary alphabet, analyzed in the central part of the paper. The obtained “transition property”, together with the developed technique, allowed us to solve cube-free versions of three old open problems by Restivo and Salemi. Besides, it has some further implications for combinatorics on words; e.g., it implies the existence of infinite cube-free words of very big subword (factor) complexity. © 2020, Springer Science+Business Media, LLC, part of Springer Nature.E.A. Petrova — Supported by the Russian Science Foundation, grant 18-71-00043
Transition Property for -Power Free Languages with and Letters
In 1985, Restivo and Salemi presented a list of five problems concerning
power free languages. Problem states: Given -power-free words
and , decide whether there is a transition from to . Problem
states: Given -power-free words and , find a transition word
, if it exists.
Let denote an alphabet with letters. Let denote
the -power free language over the alphabet , where
is a rational number or a rational "number with ". If is a "number
with " then suppose and . If is "only" a
number then suppose and or and . We show
that: If is a right extendable word in and
is a left extendable word in then there is a
(transition) word such that . We also show a
construction of the word
Branching densities of cube-free and square-free words
Binary cube-free language and ternary square-free language are two “canonical” represen-tatives 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. © 2021 by the authors. Licensee MDPI, Basel, Switzerland.This research was funded by Ministry of Science and Higher Education of the Russian Federation (Ural Mathematical Center project No. 075-02-2020-1537/1)