Second order logic on random rooted trees

TL;DR

This paper investigates the expressibility of finiteness and infiniteness in random rooted trees using second order logic, showing limitations of EMSO logic in distinguishing finite from infinite trees in broad models.

Contribution

It proves that finiteness cannot be characterized by EMSO logic in broad classes of random rooted trees, extending understanding of logical expressibility in probabilistic structures.

Findings

Finiteness is not EMSO-expressible in broad classes of random trees.

Infiniteness is EMSO-expressible, but finiteness is not, in these models.

Constructs a finite tree and infinite trees indistinguishable by EMSO logic.

Abstract

We address questions of logic and expressibility in the context of random rooted trees. Infiniteness of a rooted tree is not expressible as a first order sentence, but is expressible as an existential monadic second order sentence (EMSO). On the other hand, finiteness is not expressible as an EMSO. For a broad class of random tree models, including Galton-Watson trees with offspring distributions that have full support, we prove the stronger statement that finiteness does not agree up to a null set with any EMSO. We construct a finite tree and a non-null set of infinite trees that cannot be distinguished from each other by any EMSO of given parameters. This is proved via set-pebble Ehrenfeucht games (where an initial colouring round is followed by a given number of pebble rounds).