A symbolic method to compute the probability distribution of the number of pattern occurences in random texts generated by stochastic 0L-systems

Cedric Loi (1, 2) , Paul-Henry Cournède (2, 1) , Jean Françon (3)

1 DIGIPLANTE - Modélisation de la croissance et de l'architecture des plantes
2 MAS - Mathématiques Appliquées aux Systèmes - EA 4037
3 Département d'informatique

The analysis of pattern occurrences has numerous applications, in particular in biology. In this article, a symbolic method is proposed to compute the distribution associated to the number of occurences of a specific pattern in a random text generated by a stochastic 0L-system. To that purpose, a semiring structure is set for combinatorial classes composed of weighted words. This algebraic structure relies on new union and concatenation operators which, under some assumptions, are admissible constructions. Decomposing the combinatorial classes of interest by using these binary operators enables the direct translation of specifications into a set of functional equations relating generating functions thanks to transformation rules. The article ends with two examples. The first one deals with unary patterns and the connection with multitype branching process is established. The second one is about a pattern composed of two letters and underlines the importance of writing a proper specification.

Mots clés: symbolic method; pattern occurrence; stochastic 0L-system; semiring; weighted words

Domaines: Algorithme et structure de données [cs.DS]; Combinatoire [math.CO]; Mathématique discrète [cs.DM]; Géométrie algorithmique [cs.CG]

Cedric Loi, Paul-Henry Cournède, Jean Françon. A symbolic method to compute the probability distribution of the number of pattern occurences in random texts generated by stochastic 0L-systems. 21st International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms (AofA'10), 2010, Vienna, Austria. pp.473-488, ⟨10.46298/dmtcs.2774⟩. ⟨inria-00529137v2⟩