Existential monadic second order logic on random rooted trees

Alexander E. Holroyd, Avi Levy, Moumanti Podder, Joel Spencer

Research output: Contribution to journalArticlepeer-review


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 languageEnglish (US)
Pages (from-to)152-167
Number of pages16
JournalDiscrete Mathematics
Issue number1
StatePublished - Jan 2019


  • 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


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