Binary Patterns in Binary Cube-Free Words: Avoidability and Growth

The avoidability of binary patterns by binary cube-free words is investigated and the exact bound between unavoidable and avoidable patterns is found. All avoidable patterns are shown to be D0L-avoidable. For avoidable patterns, the growth rates of the avoiding languages are studied. All such languages, except for the overlap-free language, are proved to have exponential growth. The exact growth rates of languages avoiding minimal avoidable patterns are approximated through computer-assisted upper bounds. Finally, a new example of a pattern-avoiding language of polynomial growth is given.Comment: 18 pages, 2 tables; submitted to RAIRO TIA (Special issue of Mons Days 2012

Every Binary Pattern of Length Greater Than 14 Is Abelian-2-Avoidable

We show that every binary pattern of length greater than 14 is abelian-2-avoidable. The best known upper bound on the length of abelian-2-unavoidable binary pattern was 118, and the best known lower bound is 7. We designed an algorithm to decide, under some reasonable assumptions, if a morphic word avoids a pattern in the abelian sense. This algorithm is then used to show that some binary patterns are abelian-2-avoidable. We finally use this list of abelian-2-avoidable pattern to show our result. We also discuss the avoidability of binary patterns on 3 and 4 letters

Avoidability of formulas with two variables

In combinatorics on words, a word $w$ over an alphabet $\Sigma$ is said to avoid a pattern $p$ over an alphabet $\Delta$ of variables if there is no factor $f$ of $w$ such that $f=h(p)$ where $h:\Delta^*\to\Sigma^*$ is a non-erasing morphism. A pattern $p$ is said to be $k$-avoidable if there exists an infinite word over a $k$-letter alphabet that avoids $p$. We consider the patterns such that at most two variables appear at least twice, or equivalently, the formulas with at most two variables. For each such formula, we determine whether it is $2$-avoidable, and if it is $2$-avoidable, we determine whether it is avoided by exponentially many binary words

Avoidability index for binary patterns with reversal

For every pattern $p$ over the alphabet $\{x,y,x^R,y^R\}$, we specify the least $k$ such that $p$ is $k$-avoidable.Comment: 15 pages, 1 figur

Binary patterns in the Prouhet-Thue-Morse sequence

We show that, with the exception of the words $a^2ba^2$ and $b^2ab^2$, all (finite or infinite) binary patterns in the Prouhet-Thue-Morse sequence can actually be found in that sequence as segments (up to exchange of letters in the infinite case). This result was previously attributed to unpublished work by D. Guaiana and may also be derived from publications of A. Shur only available in Russian. We also identify the (finitely many) finite binary patterns that appear non trivially, in the sense that they are obtained by applying an endomorphism that does not map the set of all segments of the sequence into itself

A family of formulas with reversal of high avoidability index

We present an infinite family of formulas with reversal whose avoidability index is bounded between 4 and 5, and we show that several members of the family have avoidability index 5. This family is particularly interesting due to its size and the simple structure of its members. For each k ∈ {4,5}, there are several previously known avoidable formulas (without reversal) of avoidability index k, but they are small in number and they all have rather complex structure.http://dx.doi.org/10.1142/S021819671750024

Binary patterns in binary cube-free words: Avoidability and growth

The avoidability of binary patterns by binary cube-free words is investigated and the exact bound between unavoidable and avoidable patterns is found. All avoidable patterns are shown to be D0L-avoidable. For avoidable patterns, the growth rates of the avoiding languages are studied. All such languages, except for the overlap-free language, are proved to have exponential growth. The exact growth rates of languages avoiding minimal avoidable patterns are approximated through computer-assisted upper bounds. Finally, a new example of a pattern-avoiding language of polynomial growth is given. © 2014 EDP Sciences

Link-time smart card code hardening

This paper presents a feasibility study to protect smart card software against fault-injection attacks by means of link-time code rewriting. This approach avoids the drawbacks of source code hardening, avoids the need for manual assembly writing, and is applicable in conjunction with closed third-party compilers. We implemented a range of cookbook code hardening recipes in a prototype link-time rewriter and evaluate their coverage and associated overhead to conclude that this approach is promising. We demonstrate that the overhead of using an automated link-time approach is not significantly higher than what can be obtained with compile-time hardening or with manual hardening of compiler-generated assembly code