And/or trees: A local limit point of view

TL;DR

This paper introduces a universal framework for analyzing random and/or trees, showing how the induced distribution on Boolean functions converges under local conditions, with different behaviors depending on the local limit shape.

Contribution

It unifies various models of random trees and characterizes the limiting distribution of Boolean functions derived from them, including novel cases.

Findings

Distribution converges under weak local conditions.

Two behaviors depending on local limit shape: degenerate and non-degenerate.

Relationship between Boolean function probability and complexity in the non-degenerate case.

Abstract

We present here a new and universal approach for the study of random and/or trees, unifying in one framework many different models, including some novel ones not yet understood in the literature. An and/or tree is a Boolean expression represented in (one of) its tree shapes. Fix an integer $k$, take a sequence of random (rooted) trees of increasing size, say $(t_n)_{n\ge 1}$, and label each of these random trees uniformly at random in order to get a random Boolean expression on $k$ variables. We prove that, under rather weak local conditions on the sequence of random trees $(t_n)_{n\ge 1}$, the distribution induced on Boolean functions by this procedure converges as $n$ tends to infinity. In particular, we characterise two different behaviours of this limit distribution depending on the shape of the local limit of $(t_n)_{n\ge 1}$: a degenerate case when the local limit has no leaves; and a non-degenerate case, which we are able to describe in more details under stronger conditions. In this latter case, we provide a relationship between the probability of a given Boolean function and its complexity. The examples covered by this unified framework include trees that interpolate between models with logarithmic typical distances (such as random binary search trees) and other ones with square root typical distances (such as conditioned Galton--Watson trees).