# Highly Undecidable Problems For Infinite Computations

@article{Finkel2009HighlyUP,
title={Highly Undecidable Problems For Infinite Computations},
author={Olivier Finkel},
journal={ArXiv},
year={2009},
volume={abs/0901.0373}
}
• O. Finkel
• Published 4 January 2009
• Mathematics
• ArXiv
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…
23 Citations
The Complexity of Infinite Computations In Models of Set Theory
• O. Finkel
• Computer Science
Log. Methods Comput. Sci.
• 2009
It is shown that the topological complexity of an \omega-language accepted by a 1-counter B\"uchi automaton or of an infinitary rational relation accepted by the 2-tape B\"uch automaton is not determined by the axiomatic system ZFC.
Three Applications to Rational Relations of the High Undecidability of the Infinite Post Correspondence Problem in a Regular ω-Language
• O. Finkel
• Mathematics
Int. J. Found. Comput. Sci.
• 2012
This work proves that the infinite Post Correspondence Problem in a regular ω-language is -complete, hence located beyond the arithmetical hierarchy and highly undecidable, and infer from this result that it is - complete to determine whether two given infinitary rational relations are disjoint.
Highly Undecidable Problems about Recognizability by Tiling Systems
It is shown here that these two decision problems are actually P$^{1}_{2}$-complete, hence located at the second level of the analytical hierarchy, and "highly undecidable".
Decision Problems for Recognizable Languages of Infinite Pictures
This paper reviews in this paper some recent results of [Fin09b] where the exact degree of numerous undecidable problems for Buchi-recognizable languages of infinite pictures were given, and proves here some more (high) undecidability results.
Infinite games specified by 2-tape automata
• O. Finkel
• Computer Science
Ann. Pure Appl. Log.
• 2016
On the expressive power of non-deterministic and unambiguous Petri nets over infinite words
• Computer Science
Fundam. Informaticae
• 2021
It is demonstrated that the ω-languages recognisable by unambiguous Petri nets are △30 sets, and it is shown that it is highly undecidable to determine the topological complexity of a Petri net ω -language.
Some problems in automata theory which depend on the models of set theory
• O. Finkel
• Mathematics
RAIRO Theor. Informatics Appl.
• 2011
It is proved that some fairly basic questions on automata reading infinite words depend on the models of the axiomatic system ZFC and that basic decision problems about 1-counter omega-languages or infinitary rational relations are actually located at the third level of the analytical hierarchy.
On the High Complexity of Petri Nets ømega-Languages
It is proved that the Borel and Wadge hierarchies of the class of $$\omega$$-languages of Petri nets and of (non-deterministic) Turing machines have the same topological complexity.
The Isomorphism Problem for omega-Automatic Trees
• Mathematics, Computer Science
CSL
• 2010
The main result of this paper is that the isomorphism for omega-automatic trees of finite height is at least has hard as second-order arithmetic and therefore not analytical. This strengthens a
An Effective Property of $\omega$-Rational Functions
We prove that $\omega$-regular languages accepted by B\"uchi or Muller automata satisfy an effective automata-theoretic version of the Baire property. Then we use this result to obtain a new

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