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THE COMPLEXITY OF ORDER-COMPUTABLE SETS

Part of: Theory of computing Computability and recursion theory

Published online by Cambridge University Press:  24 January 2025

HUISHAN WU
HUISHAN WU*
Affiliation:
SCHOOL OF INFORMATION SCIENCE BEIJING LANGUAGE AND CULTURE UNIVERSITY 15 XUEYUAN ROAD, HAIDIAN DISTRICT BEIJING 100083 CHINA
*
E-mail: huishanwu@blcu.edu.cn
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Abstract

This paper studies the conjecture of Hirschfeldt, Miller, and Podzorov in [13] on the complexity of order-computable sets, where a set A is order-computable if there is a computable copy of the structure $(\mathbb {N}, <,A)$ in the language of linear orders together with a unary predicate. The class of order-computable sets forms a subclass of $\Delta ^{0}_{2}$ sets. Firstly, we study the complexity of computably enumerable (c.e.) order-computable sets and prove that the index set of c.e. order-computable sets is $\Sigma ^{0}_{4}$-complete. Secondly, as a corollary of the main result on c.e. order-computable sets, we obtain that the index set of general order computable sets is $\Sigma ^{0}_{4}$-complete within the index set of $\Delta ^{0}_{2}$ sets. Finally, we continue to study the complexity of more general $\Delta ^{0}_{2}$ sets and prove that the index set of $\Delta ^{0}_{2}$ sets is $\Pi ^{0}_{3}$-complete.

Keywords

computability theoryorder-computable setsΔ02 sets

MSC classification

Primary: 03D80: Applications of computability and recursion theory 68Q15: Complexity classes (hierarchies, relations among complexity classes, etc.) 68Q17: Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.)

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Type
Article
Information
The Journal of Symbolic Logic , First View , pp. 1 - 24
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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