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

TL;DR

This paper investigates how the size complexity of random Boolean expression trees influences the probability of generating specific Boolean functions, using combinatorial and analytical methods.

Contribution

It establishes a relation between tree size complexity and function probability by analyzing expansions of minimal trees through generating functions.

Findings

Derived a limiting distribution for Boolean functions from random trees

Connected tree size complexity with function probability

Applied combinatorial counting and singularity analysis techniques

Abstract

We consider a probability distribution on the set of Boolean functions in n variables which is induced by random Boolean expressions. Such an expression is a random rooted plane tree where the internal vertices are labelled with connectives And and OR and the leaves are labelled with variables or negated variables. We study limiting distribution when the tree size tends to infinity and derive a relation between the tree size complexity and the probability of a function. This is done by first expressing trees representing a particular function as expansions of minimal trees representing this function and then computing the probabilities by means of combinatorial counting arguments relying on generating functions and singularity analysis.