# Some decompositions of Bernoulli sets and codes

RAIRO - Theoretical Informatics and Applications (2010)

- Volume: 39, Issue: 1, page 161-174
- ISSN: 0988-3754

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topde Luca, Aldo. "Some decompositions of Bernoulli sets and codes." RAIRO - Theoretical Informatics and Applications 39.1 (2010): 161-174. <http://eudml.org/doc/92753>.

@article{deLuca2010,

abstract = {
A decomposition of a set X of words over a d-letter alphabet
A = \{a1,...,ad\} is any sequence X1,...,Xd,Y1,...,Yd
of subsets of A* such that the sets Xi, i = 1,...,d, are pairwise disjoint,
their union is X, and for all i, 1 ≤ i ≤ d, Xi ~ aiYi, 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.
},

author = {de Luca, Aldo},

journal = {RAIRO - Theoretical Informatics and Applications},

keywords = {Bernoulli sets; codes; decompositions; commutative equivalence.; Bernoulli distributions; prefix set},

language = {eng},

month = {3},

number = {1},

pages = {161-174},

publisher = {EDP Sciences},

title = {Some decompositions of Bernoulli sets and codes},

url = {http://eudml.org/doc/92753},

volume = {39},

year = {2010},

}

TY - JOUR

AU - de Luca, Aldo

TI - Some decompositions of Bernoulli sets and codes

JO - RAIRO - Theoretical Informatics and Applications

DA - 2010/3//

PB - EDP Sciences

VL - 39

IS - 1

SP - 161

EP - 174

AB -
A decomposition of a set X of words over a d-letter alphabet
A = {a1,...,ad} is any sequence X1,...,Xd,Y1,...,Yd
of subsets of A* such that the sets Xi, i = 1,...,d, are pairwise disjoint,
their union is X, and for all i, 1 ≤ i ≤ d, Xi ~ aiYi, 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.

LA - eng

KW - Bernoulli sets; codes; decompositions; commutative equivalence.; Bernoulli distributions; prefix set

UR - http://eudml.org/doc/92753

ER -

## References

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- A. de Luca, Some combinatorial results on Bernoulli sets and codes. Theor. Comput. Sci.273 (2002) 143–165.
- G. Hansel, Baïonnettes et cardinaux. Discrete Math.39 (1982) 331–335.
- D. Perrin and M.P. Schützenberger, Un problème élémentaire de la théorie de l'Information, in Theorie de l'Information, Colloq. Internat. du CNRS No. 276, Cachan (1977) 249–260.
- D. Perrin and M.P. Schützenberger, A conjecture on sets of differences on integer pairs. J. Combin. Theory Ser. B30 (1981) 91–93.
- J.E. Pin and I. Simon, A note on the triangle conjecture. J. Comb. Theory Ser. A32 (1982) 106–109.
- P. Shor, A counterexample to the triangle conjecture. J. Comb. Theory Ser. A38 (1985) 110–112.

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