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Issue Theoret. Informatics Appl.
Volume 39, Number 1, January-March 2005
Imre Simon, the tropical computer scientist
Page(s) 161 - 174
DOI 10.1051/ita:2005010

Theoret. Informatics Appl. 39, 161-174 (2005)
DOI: 10.1051/ita:2005010

Some decompositions of Bernoulli sets and codes

Aldo de Luca

Dipartimento di Matematica e Applicazioni dell'Università di Napoli "Federico II", via Cintia, Complesso Universitario di Monte S. Angelo, 80126 Napoli, Italy and Istituto di Cibernetica "E. R. Caianiello" del CNR, 80078 Pozzuoli, Italy; aldo.deluca@dma.unina.it


Abstract
A decomposition of a set X of words over a d-letter alphabet $A=\{a_1,\ldots,a_d\}$ is any sequence $X_1,\ldots, X_d, Y_1,\ldots, Y_d$ of subsets of A* such that the sets Xi, $i=1,\ldots, d,$ are pairwise disjoint, their union is X, and for all i, $1\leq i\leq d$, $X_i\sim a_iY_i$, where ~ 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.


Mathematics Subject Classification. 94A45

Key words: Bernoulli sets -- codes -- decompositions -- commutative equivalence.


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