Some decompositions of Bernoulli sets and codes

Luca, Aldo de

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 39 (2005) no. 1, pp. 161-174

Résumé

A decomposition of a set $X$ of words over a $d$-letter alphabet $A=\{a_1,...,a_d\}$ is any sequence $X_1,...,X_d,Y_1,...,Y_d$ of subsets of $A^{*}$ such that the sets $X_i$, $i=1,...,d,$ are pairwise disjoint, their union is $X$, and for all $i$, $1\le i\le d$, $X_i\sim a_i{Y}_i$, where $\sim$ denotes the commutative equivalence relation. We introduce some suitable decompositions that we call good, admissible, and normal. A normal decomposition is admissible and an admissible decomposition is good. We prove that a set is commutatively prefix if and only if it has a normal decomposition. In particular, we consider decompositions of Bernoulli sets and codes. We prove that there exist Bernoulli sets which have no good decomposition. Moreover, we show that the classical conjecture of commutative equivalence of finite maximal codes to prefix ones is equivalent to the statement that any finite and maximal code has an admissible decomposition.

Détail

DOI : 10.1051/ita:2005010

Classification : 94A45
Keywords: Bernoulli sets, codes, decompositions, commutative equivalence