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And/or trees: A local limit point of view
Author(s) -
Broutin Nicolas,
Mailler Cécile
Publication year - 2018
Publication title -
random structures and algorithms
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.314
H-Index - 69
eISSN - 1098-2418
pISSN - 1042-9832
DOI - 10.1002/rsa.20758
Subject(s) - mathematics , random tree , random binary tree , limit (mathematics) , sequence (biology) , tree (set theory) , combinatorics , discrete mathematics , binary tree , boolean model , binary search tree , logarithm , limit of a function , computer science , mathematical analysis , motion planning , artificial intelligence , biology , robot , genetics
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 ≥ 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 ≥ 1, the distribution induced on Boolean functions by this procedure converges as n tends to infinity. In particular, we characterize two different behaviors of this limit distribution depending on the shape of the local limit of( t n ) n ≥ 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).

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