Choice functions and well-orderings over the infinite binary tree

Arnaud Carayol (1) , Christof Loeding (2) , Damian Niwinski (3) , Igor Walukiewicz (4)

1 LIGM - Laboratoire d'Informatique Gaspard-Monge
2 Lehrstuhl für Informatik VII
3 MIMUW - Faculty of Mathematics, Informatics, and Mechanics [Warsaw]
4 LaBRI - Laboratoire Bordelais de Recherche en Informatique

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.

Mots clés: monadic second-order logic; definability; choice function

Domaines: Logique en informatique [cs.LO]

Arnaud Carayol, Christof Loeding, Damian Niwinski, Igor Walukiewicz. Choice functions and well-orderings over the infinite binary tree. Central European Journal of Mathematics, 2010, 8 (6), pp.662-682. ⟨10.2478/s11533-010-0046-z⟩. ⟨hal-00733452⟩