On a question of McNaughton and Papert

In a recent book, McNaughton and Papert asked under what conditions a free submonoid of a free monoid is locally testable. The answer to this question is given here. The solution relates the concept of local testability with that of synchronization in a code and the algebraic notion of conjugacy in a monoid. The finiteness of the basis (or code) which generates the free submonoid plays an essential role in our result

Sorting suffixes of a text via its Lyndon Factorization

The process of sorting the suffixes of a text plays a fundamental role in Text Algorithms. They are used for instance in the constructions of the Burrows-Wheeler transform and the suffix array, widely used in several fields of Computer Science. For this reason, several recent researches have been devoted to finding new strategies to obtain effective methods for such a sorting. In this paper we introduce a new methodology in which an important role is played by the Lyndon factorization, so that the local suffixes inside factors detected by this factorization keep their mutual order when extended to the suffixes of the whole word. This property suggests a versatile technique that easily can be adapted to different implementative scenarios.Comment: Submitted to the Prague Stringology Conference 2013 (PSC 2013

Anti-Powers in Infinite Words

In combinatorics of words, a concatenation of $k$ consecutive equal blocks is called a power of order $k$. In this paper we take a different point of view and define an anti-power of order $k$ as a concatenation of $k$ consecutive pairwise distinct blocks of the same length. As a main result, we show that every infinite word contains powers of any order or anti-powers of any order. That is, the existence of powers or anti-powers is an unavoidable regularity. Indeed, we prove a stronger result, which relates the density of anti-powers to the existence of a factor that occurs with arbitrary exponent. As a consequence, we show that in every aperiodic uniformly recurrent word, anti-powers of every order begin at every position. We further show that every infinite word avoiding anti-powers of order $3$ is ultimately periodic, while there exist aperiodic words avoiding anti-powers of order $4$. We also show that there exist aperiodic recurrent words avoiding anti-powers of order $6$.Comment: Revision submitted to Journal of Combinatorial Theory Series

Combinatorics on words in information security: Unavoidable regularities in the construction of multicollision attacks on iterated hash functions

Classically in combinatorics on words one studies unavoidable regularities that appear in sufficiently long strings of symbols over a fixed size alphabet. In this paper we take another viewpoint and focus on combinatorial properties of long words in which the number of occurrences of any symbol is restritced by a fixed constant. We then demonstrate the connection of these properties to constructing multicollision attacks on so called generalized iterated hash functions.Comment: In Proceedings WORDS 2011, arXiv:1108.341

Minimal forbidden words and factor automata

International audienceLet L(M) be the (factorial) language avoiding a given antifactorial language M. We design an automaton accepting L(M) and built from the language M. The construction is eff ective if M is finite. If M is the set of minimal forbidden words of a single word v, the automaton turns out to be the factor automaton of v (the minimal automaton accepting the set of factors of v). We also give an algorithm that builds the trie of M from the factor automaton of a single word. It yields a non-trivial upper bound on the number of minimal forbidden words of a word

Unambiguously coded systems

We study the coded systems introduced by Blanchard and Hansel. We give several constructions which allow one to represent a coded system as a strongly unambiguous one

Text Compression Using Antidictionaries

International audienceWe give a new text compression scheme based on Forbidden Words ("antidictionary"). We prove that our algorithms attain the entropy for balanced binary sources. They run in linear time. Moreover, one of the main advantages of this approach is that it produces very fast decompressors. A second advantage is a synchronization property that is helpful to search compressed data and allows parallel compression. Our algorithms can also be presented as "compilers" that create compressors dedicated to any previously fixed source. The techniques used in this paper are from Information Theory and Finite Automata