Highly Undecidable Problems For Infinite Computations

TL;DR

This paper demonstrates that many classical decision problems related to infinite computations, such as universality and equivalence, are highly undecidable, residing at the second level of the analytical hierarchy, even for simple automata.

Contribution

It establishes the $oldsymbol{ ext{Pi}}_2^1$-completeness of key decision problems for 1-counter omega-languages and related models, revealing their high undecidability.

Findings

Many decision problems are $oldsymbol{ ext{Pi}}_2^1$-complete.

Problems include universality, inclusion, equivalence, and determinizability.

Results apply to simple automata like 1-counter and 2-tape automata.

Abstract

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.