### Abstract

In the regime of Galton-Watson trees, first order logic statements are roughly equivalent to examining the presence of specific finite subtrees. We consider the space of all trees with Poisson offspring distribution and show that such finite subtrees will be almost surely present when the tree is infinite. Introducing the notion of universal trees, we show that all first order sentences of quantifier depth k depend only on local neighbourhoods of the root of sufficiently large radius depending on k. We compute the probabilities of these neighbourhoods conditioned on the tree being infinite. We give an almost sure theory for infinite trees.

Original language | English (US) |
---|---|

Title of host publication | A Journey through Discrete Mathematics |

Subtitle of host publication | A Tribute to Jiri Matousek |

Publisher | Springer International Publishing |

Pages | 711-734 |

Number of pages | 24 |

ISBN (Electronic) | 9783319444796 |

ISBN (Print) | 9783319444789 |

DOIs | |

State | Published - Jan 1 2017 |

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### ASJC Scopus subject areas

- Computer Science(all)
- Mathematics(all)
- Economics, Econometrics and Finance(all)
- Business, Management and Accounting(all)

### Cite this

*A Journey through Discrete Mathematics: A Tribute to Jiri Matousek*(pp. 711-734). Springer International Publishing. https://doi.org/10.1007/978-3-319-44479-6_29

**First order probabilities for Galton-Watson trees.** / Podder, Moumanti; Spencer, Joel.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

*A Journey through Discrete Mathematics: A Tribute to Jiri Matousek.*Springer International Publishing, pp. 711-734. https://doi.org/10.1007/978-3-319-44479-6_29

}

TY - CHAP

T1 - First order probabilities for Galton-Watson trees

AU - Podder, Moumanti

AU - Spencer, Joel

PY - 2017/1/1

Y1 - 2017/1/1

N2 - In the regime of Galton-Watson trees, first order logic statements are roughly equivalent to examining the presence of specific finite subtrees. We consider the space of all trees with Poisson offspring distribution and show that such finite subtrees will be almost surely present when the tree is infinite. Introducing the notion of universal trees, we show that all first order sentences of quantifier depth k depend only on local neighbourhoods of the root of sufficiently large radius depending on k. We compute the probabilities of these neighbourhoods conditioned on the tree being infinite. We give an almost sure theory for infinite trees.

AB - In the regime of Galton-Watson trees, first order logic statements are roughly equivalent to examining the presence of specific finite subtrees. We consider the space of all trees with Poisson offspring distribution and show that such finite subtrees will be almost surely present when the tree is infinite. Introducing the notion of universal trees, we show that all first order sentences of quantifier depth k depend only on local neighbourhoods of the root of sufficiently large radius depending on k. We compute the probabilities of these neighbourhoods conditioned on the tree being infinite. We give an almost sure theory for infinite trees.

UR - http://www.scopus.com/inward/record.url?scp=85016159421&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85016159421&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-44479-6_29

DO - 10.1007/978-3-319-44479-6_29

M3 - Chapter

AN - SCOPUS:85016159421

SN - 9783319444789

SP - 711

EP - 734

BT - A Journey through Discrete Mathematics

PB - Springer International Publishing

ER -