Some decompositions of Bernoulli sets and codes

Aldo de Luca

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications (2005)

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

Abstract

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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 ≤ i ≤ d , X i ∼ a i Y 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.

How to cite

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Luca, Aldo de. "Some decompositions of Bernoulli sets and codes." RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications 39.1 (2005): 161-174. <http://eudml.org/doc/245194>.

@article{Luca2005,
abstract = {A decomposition of a set $X$ of words over a $d$-letter alphabet $A=\lbrace a_1,\ldots ,a_d\rbrace $ is any sequence $X_1,\ldots , X_d, Y_1,\ldots , Y_d$ of subsets of $A^*$ such that the sets $X_i$, $i=1,\ldots , d,$ are pairwise disjoint, their union is $X$, and for all $i$, $1\le i\le d$, $X_i\sim a_iY_i$, 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.},
author = {Luca, Aldo de},
journal = {RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications},
keywords = {Bernoulli sets; codes; decompositions; commutative equivalence; Bernoulli distributions; prefix set},
language = {eng},
number = {1},
pages = {161-174},
publisher = {EDP-Sciences},
title = {Some decompositions of Bernoulli sets and codes},
url = {http://eudml.org/doc/245194},
volume = {39},
year = {2005},
}

TY - JOUR
AU - Luca, Aldo de
TI - Some decompositions of Bernoulli sets and codes
JO - RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications
PY - 2005
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=\lbrace a_1,\ldots ,a_d\rbrace $ is any sequence $X_1,\ldots , X_d, Y_1,\ldots , Y_d$ of subsets of $A^*$ such that the sets $X_i$, $i=1,\ldots , d,$ are pairwise disjoint, their union is $X$, and for all $i$, $1\le i\le d$, $X_i\sim a_iY_i$, 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.
LA - eng
KW - Bernoulli sets; codes; decompositions; commutative equivalence; Bernoulli distributions; prefix set
UR - http://eudml.org/doc/245194
ER -

References

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  4. [4] A. de Luca, Some combinatorial results on Bernoulli sets and codes. Theor. Comput. Sci. 273 (2002) 143–165. Zbl0996.68146
  5. [5] G. Hansel, Baïonnettes et cardinaux. Discrete Math. 39 (1982) 331–335. Zbl0486.20032
  6. [6] 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. Zbl0483.94028
  7. [7] D. Perrin and M.P. Schützenberger, A conjecture on sets of differences on integer pairs. J. Combin. Theory Ser. B 30 (1981) 91–93. Zbl0468.05033
  8. [8] J.E. Pin and I. Simon, A note on the triangle conjecture. J. Comb. Theory Ser. A 32 (1982) 106–109. Zbl0472.05010
  9. [9] P. Shor, A counterexample to the triangle conjecture. J. Comb. Theory Ser. A 38 (1985) 110–112. Zbl0558.20032

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