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The relation between tree size complexity and probability for Boolean functions generated by uniform random trees

Daxner Veronika (Sorbonne Universités, UPMC Univ Paris, CNRS, Paris, France)
Genitrini Antoine (Sorbonne Universités, UPMC Univ Paris, CNRS, Paris, France)
Gittenberger Bernhard (Technische Universität Wien, Wien, Austria)
Mailler Cécile (University of Bath, Department of Mathematical Sciences, BA AY Bath, UK)

An associative Boolean tree is a plane rooted tree whose internal nodes are labelled by and or or and whose leaves are labelled by literals taken from a fixed set of variables and their negations. We study the distribution induced on the set of Boolean functions by the uniform distribution on the set of associative trees of a large fixed size, where the size of a tree is defined as the number of its nodes. Using analytic combinatorics, we prove a relation between the probability of a given function and its tree size complexity.

Keywords: Boolean functions, probability distribution, Random Boolean formulas, tree size complexity, analytic combinatorics