Pattern Avoidability with Involution

An infinte word w avoids a pattern p with the involution t if there is no substitution for the variables in p and no involution t such that the resulting word is a factor of w. We investigate the avoidance of patterns with respect to the size of the alphabet. For example, it is shown that the pattern a t(a) a can be avoided over three letters but not two letters, whereas it is well known that a a a is avoidable over two letters.Comment: In Proceedings WORDS 2011, arXiv:1108.341

The Ehrenfeucht–Silberger problem

AbstractWe consider repetitions in words and solve a longstanding open problem about the relation between the period of a word and the length of its longest unbordered factor (where factor means uninterrupted subword). A word u is called bordered if there exists a proper prefix that is also a suffix of u, otherwise it is called unbordered. In 1979 Ehrenfeucht and Silberger raised the following problem: What is the maximum length of a word w, w.r.t. the length τ of its longest unbordered factor, such that τ is shorter than the period π of w. We show that, if w is of length 73τ or more, then τ=π which gives the optimal asymptotic bound

On the Pseudoperiodic Extension of u^l = v^m w^n

We investigate the solution set of the pseudoperiodic extension of the classical Lyndon and Sch"utzenberger word equations. Consider u_1 ... u_l = v_1 ... v_m w_1 ... w_n, where u_i is in {u, theta(u)} for all 1 = 12 or m,n >= 5 and either m and n are not both even or not all u_i\u27s are equal, all solutions are pseudoperiodic

An Optimal Bound on the Solution Sets of One-Variable Word Equations and its Consequences

We solve two long-standing open problems on word equations. Firstly, we prove that a one-variable word equation with constants has either at most three or an infinite number of solutions. The existence of such a bound had been conjectured, and the bound three is optimal. Secondly, we consider independent systems of three-variable word equations without constants. If such a system has a nonperiodic solution, then this system of equations is at most of size 17. Although probably not optimal, this is the first finite bound found. However, the conjecture of that bound being actually two still remains open

Testing Generalised Freeness of Words

Pseudo-repetitions are a natural generalisation of the classical notion of repetitions in sequences: they are the repeated concatenation of a word and its encoding under a certain morphism or antimorphism (anti-/morphism, for short). We approach the problem of deciding efficiently, for a word w and a literal anti-/morphism f, whether w contains an instance of a given pattern involving a variable x and its image under f, i.e., f(x). Our results generalise both the problem of finding fixed repetitive structures (e.g., squares, cubes) inside a word and the problem of finding palindromic structures inside a word. For instance, we can detect efficiently a factor of the form xx^Rxxx^R, or any other pattern of such type. We also address the problem of testing efficiently, in the same setting, whether the word w contains an arbitrary pseudo-repetition of a given exponent