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Highly Undecidable Problems For Infinite Computations

Published online by Cambridge University Press:  14 February 2009

Olivier Finkel
Olivier Finkel*
Affiliation:
Equipe de Logique Mathématique, CNRS et Université Paris 7, France; [email protected]
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Abstract

We show that many classical decision problems about1-counter ω-languages, context free ω-languages, or infinitary rational relations, are Π½ -complete, hence located at the second level of the analytical hierarchy, and “highly undecidable”.In particular, the universalityproblem, the inclusion problem, the equivalence problem, the determinizability problem, the complementability problem, and theunambiguity problem are all Π½ -complete for context-free ω-languages or for infinitary rational relations. Topological and arithmetical properties of1-counter ω-languages, context free ω-languages, or infinitary rational relations, are also highly undecidable.These very surprising results provide the first examples of highly undecidable problems about the behaviour of verysimple finite machines like 1-counter automata or 2-tapeautomata.

Keywords

Infinite computations1-counter-automata2-tape automatadecision problemsarithmetical hierarchyanalytical hierarchycomplete setshighlyundecidable problems.
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 43 , Issue 2 , April 2009 , pp. 339 - 364
Copyright
© EDP Sciences, 2008

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