Decision Problems For Turing Machines

Olivier Finkel (ELM); Dominique Lecomte (IMJ)

arXiv:0909.0736·cs.LO·November 5, 2009

Decision Problems For Turing Machines

Olivier Finkel (ELM), Dominique Lecomte (IMJ)

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TL;DR

This paper precisely characterizes the computational complexity of two decision problems regarding the cardinalities of omega-languages of Turing machines, resolving questions posed by Castro and Cucker.

Contribution

It establishes the exact complexity classes for deciding whether a Turing machine's omega-language is countably infinite or uncountable.

Findings

01

Determined the problem of countably infinite omega-languages is D_2(Σ_1^1)-complete.

02

Established that the uncountability problem is Σ_1^1-complete.

03

Provided exact complexity classifications for these decision problems.

Abstract

We answer two questions posed by Castro and Cucker, giving the exact complexities of two decision problems about cardinalities of omega-languages of Turing machines. Firstly, it is $D_{2} (Σ_{1}^{1})$ -complete to determine whether the omega-language of a given Turing machine is countably infinite, where $D_{2} (Σ_{1}^{1})$ is the class of 2-differences of $Σ_{1}^{1}$ -sets. Secondly, it is $Σ_{1}^{1}$ -complete to determine whether the omega-language of a given Turing machine is uncountable.

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