Highly Undecidable Problems For Infinite Computations

Olivier Finkel (1, 2)

1 ELM - Équipe de Logique Mathématique
2 LIP - Laboratoire de l'Informatique du Parallélisme

We show that many classical decision problems about 1-counter omega-languages, context free omega-languages, or infinitary rational relations, are $\Pi_2^1$-complete, hence located at the second level of the analytical hierarchy, and ``highly undecidable". In particular, the universality problem, the inclusion problem, the equivalence problem, the determinizability problem, the complementability problem, and the unambiguity problem are all $\Pi_2^1$-complete for context-free omega-languages or for infinitary rational relations. Topological and arithmetical properties of 1-counter omega-languages, context free omega-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 very simple finite machines like 1-counter automata or 2-tape automata.

Mots clés: Infinite computations; 1-counter-automata; 2-tape automata; decision problems; arithmetical hierarchy; analytical hierarchy; complete sets; highly undecidable problems.; highly undecidable problems

Domaines: Logique en informatique [cs.LO]; Complexité [cs.CC]; Logique [math.LO]

Olivier Finkel. Highly Undecidable Problems For Infinite Computations. RAIRO - Theoretical Informatics and Applications (RAIRO: ITA), 2009, 43 (2), pp.339-364. ⟨hal-00349761⟩