Ten Conferences WORDS: Open Problems and Conjectures

In connection to the development of the field of Combinatorics on Words, we present a list of open problems and conjectures that were stated during the ten last meetings WORDS. We wish to continually update the present document by adding informations concerning advances in problems solving

Completing circular codes in regular submonoids

AbstractLet M be an arbitrary submonoid of the free monoid A∗, and let X⊆M be a variable length code (for short a code). X is weakly M-complete iff any word in M is a factor of some word in X∗ [J. Néraud, C. Selmi, Free monoid theory: Maximality and completeness in arbitrary submonoids, Internat. J. Algebra Comput. 13 (5) (2003) 507–516]. Given a regular submonoid M, and given an arbitrary code X⊆M, we are interested in the existence of a weakly M-complete code Xˆ that contains X. Actually, in [J. Néraud, Completing a code in a regular submonoid, in: Acts of MCU’2004, Lect. Notes Comput. Sci. 3354 (2005) 281–291; J. Néraud, Completing a code in a submonoid of finite rank, Fund. Inform. 74 (2006) 549–562], by presenting a general formula, we have established that, in any case, such a code Xˆ exists. In the present paper, we prove that any regular circular code X⊆M may be embedded into a weakly M-complete one iff the minimal automaton with behavior M has a synchronizing word. As a consequence of our result an extension of the famous theorem of Schützenberger is stated for regular circular codes in the framework of regular submonoids. We study also the behaviour of the subclass of uniformly synchronous codes in connection with these questions

Embedding a θ\theta-invariant code into a complete one

Let A be a finite or countable alphabet and let $\theta$ be a literal (anti-)automorphism onto A * (by definition, such a correspondence is determinated by a permutation of the alphabet). This paper deals with sets which are invariant under $\theta$ ($\theta$-invariant for short) that is, languages L such that $\theta$ (L) is a subset of L.We establish an extension of the famous defect theorem. With regards to the so-called notion of completeness, we provide a series of examples of finite complete $\theta$-invariant codes. Moreover, we establish a formula which allows to embed any non-complete $\theta$-invariant code into a complete one. As a consequence, in the family of the so-called thin $\theta$--invariant codes, maximality and completeness are two equivalent notions.Comment: arXiv admin note: text overlap with arXiv:1705.0556

Topologies for Error-Detecting Variable-Length Codes

Given a finite alphabet $A$, a quasi-metric $d$ over $A^*$, and a non-negative integer $k$, we introduce the relation $\tau_{d,k}\subseteq A^*\times A^*$ such that $(x,y)\in\tau_{d,k}$ holds whenever $d(x,y)\le k$. The error detection capability of variable-length codes is expressed in term of conditions over $\tau_{d,k}$. With respect to the prefix metric, the factor one, and any quasi-metric associated with some free monoid (anti-)automorphism, we prove that one can decide whether a given regular variable-length code satisfies any of those error detection constraints.Comment: arXiv admin note: text overlap with arXiv:2208.1468

Conferences WORDS, years 1997-2017: Open Problems and Conjectures

International audienceIn connection with the development of the field of Combinatorics on Words, we present a list of open problems and conjectures which were stated in the context of the eleven international meetings WORDS, which held from 1997 to 2017

Loopless Algorithms to Generate Maximum Length Gray Cycles wrt. k-Character Substitution

Given a binary word relation $\tau$ onto $A^*$ and a finite language $X\subseteq A^*$, a $\tau$-Gray cycle over $X$ consists in a permutation $\left(w_{[i]}\right)_{0\le i\le |X|-1}$ of $X$ such that each word $w_{[i]}$ is an image under $\tau$ of the previous word $w_{{[i-1]}}$. We define the complexity measure $\lambda_{A,\tau}(n)$, equal to the largest cardinality of a language $X$ having words of length at most $n$, and s.t. some $\tau$-Gray cycle over $X$ exists. The present paper is concerned with $\tau=\sigma_k$, the so-called $k$-character substitution, s.t. $(u,v)\in\sigma_k$ holds if, and only if, the Hamming distance of $u$ and $v$ is $k$. We present loopless (resp., constant amortized time) algorithms for computing specific maximum length $$\sigma_k$-Gray cycles.Comment: arXiv admin note: text overlap with arXiv:2108.1365

A generalization of Girod's bidirectional decoding method to codes with a finite deciphering delay

International audienceGirod"s encoding method has been introduced in order to efficiently decode from both directions messages encoded by using prefix codes. In the present paper, we generalize this method to codes with a finite deciphering delay. In particular, we show that our decoding algorithm can be realized by a deterministic finite transducer. We also investigate some properties of the corresponding unlabeled graph

Complete Variable-Length Codes: An Excursion into Word Edit Operations

International audienceGiven an alphabet A and a binary relation τ ⊆ A * × A * , a language X ⊆ A * is τ-independent if τ (X) ∩ X = ∅; X is τ-closed if τ (X) ⊆ X. The language X is complete if any word over A is a factor of some concatenation of words in X. Given a family of languages F containing X, X is maximal in F if no other set of F can stricly contain X. A language X ⊆ A * is a variable-length code if any equation among the words of X is necessarily trivial. The study discusses the relationship between maximality and completeness in the case of τ-independent or τ-closed variable-length codes. We focus to the binary relations by which the images of words are computed by deleting, inserting, or substituting some characters

Gray Cycles of Maximum Length Related to k-Character Substitutions

Given a word binary relation τ we define a τ-Gray cycle over a finite language X to be a permutation w [i] 0≤i≤|X|−1 of X such that each word wi is an image of the previous word wi−1 by τ. In that framework, we introduce the complexity measure λ(n), equal to the largest cardinality of a language X having words of length at most n, and such that a τ-Gray cycle over X exists. The present paper is concerned with the relation τ = σ k , the so-called k-character substitution, where (u, v) belongs to σ k if, and only if, the Hamming distance of u and v is k. We compute the bound λ(n) for all cases of the alphabet cardinality and the argument n