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Choice functions and well-orderings over the infinite binary tree

  • Research Article
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Central European Journal of Mathematics

Abstract

We give a new proof showing that it is not possible to define in monadic second-order logic (MSO) a choice function on the infinite binary tree. This result was first obtained by Gurevich and Shelah using set theoretical arguments. Our proof is much simpler and only uses basic tools from automata theory. We show how the result can be used to prove the inherent ambiguity of languages of infinite trees. In a second part we strengthen the result of the non-existence of an MSO-definable well-founded order on the infinite binary tree by showing that every infinite binary tree with a well-founded order has an undecidable MSO-theory.

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Authors and Affiliations

  1. Laboratoire d’Informatique Gaspard Monge, Université Paris-Est, Paris, France

    Arnaud Carayol

  2. Lehrstuhl Informatik 7, RWTH Aachen, Aachen, Germany

    Christof Löding

  3. Institute of Informatics, University of Warsaw, Warsaw, Poland

    Damian Niwinski

  4. LaBRI, Bordeaux University & CNRS, Bordeaux, France

    Igor Walukiewicz

Authors
  1. Arnaud Carayol
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  2. Christof Löding
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  3. Damian Niwinski
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  4. Igor Walukiewicz
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Correspondence to Arnaud Carayol.

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Carayol, A., Löding, C., Niwinski, D. et al. Choice functions and well-orderings over the infinite binary tree. centr.eur.j.math. 8, 662–682 (2010). https://doi.org/10.2478/s11533-010-0046-z

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  • DOI: https://doi.org/10.2478/s11533-010-0046-z

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