### 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).

Original language | English (US) |
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Pages (from-to) | 152-167 |

Number of pages | 16 |

Journal | Discrete Mathematics |

Volume | 342 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2019 |

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### Keywords

- Almost sure expressibility
- Existential monadic second order properties
- Finiteness of tree
- Galton–Watson tree

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

### Cite this

*Discrete Mathematics*,

*342*(1), 152-167. https://doi.org/10.1016/j.disc.2018.09.012