Abstract
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.
Highlights
The problem of pattern occurrences in a random text has been widely studied for these last twenty years
The aim of the article is to introduce a symbolic method that enables the computation of the probability distribution associated to the number of pattern occurrences in a random text generated by a stochastic 0Lsystem
As far as random texts generated by a stochastic 0L-system are concerned, a semiring structure is established for finite or countable sets of weighted words
Summary
The problem of pattern occurrences in a random text has been widely studied for these last twenty years. We consider random texts that are generated by stochastic 0L-systems, which are generative parallel rewriting grammars This type of probabilistic model is widely used in development biology for the modelling of cellular organism development or for plant growth (see Lindenmayer (1968) and Prusinkiewicz and Lindenmayer (1990)). The aim of the article is to introduce a symbolic method that enables the computation of the probability distribution associated to the number of pattern occurrences in a random text generated by a stochastic 0Lsystem. As far as random texts generated by a stochastic 0L-system are concerned, a semiring structure is established for finite or countable sets of weighted words. This structure relies on binary operators which guarantee admissible constructions.
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