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Journal of Logic and Computation Advance Access originally published online on October 22, 2007
Journal of Logic and Computation 2007 17(6):1135-1151; doi:10.1093/logcom/exm038
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© The Author, 2007. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

Original Articles

Undecidability in the Homomorphic Quasiorder of Finite Labelled Forests

Oleg V. Kudinov

S.L. Sobolev Institute of Mathematics, Siberian Division of the Russian Academy of Sciences, Novosibirsk, Russia. E-mail: kud{at}math.nsc.ru

Victor L. Selivanov

A.P. Ershov Institute of Informatics Systems, Siberian Division of the Russian Academy of Sciences and Theoretische Informatik, Universität Würzburg, Wuerzburg, Germany. E-mail: vseliv{at}nspu.ru; selivanov{at}informatik.uni-wuerzburg.de

Received 9 October 2006.


  Abstract

We prove that the homomorphic quasiorder of finite k-labelled forests has a hereditary undecidable first-order theory for k ≥ 3, in contrast to the known decidability result for k = 2. We establish also hereditary undecidability (again for every k ≥ 3) of first-order theories of two other relevant structures: the homomorphic quasiorder of finite k-labelled trees, and of finite k-labelled trees with a fixed label of the root element. Finally, all three first-order theories are shown to be computably isomorphic to the first-order arithmetic.

Keywords: Labelled tree; forest; homomorphic quasiorder; undecidability; theory


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