Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



The halting problem relativized to complements

Author: Louise Hay
Journal: Proc. Amer. Math. Soc. 41 (1973), 583-587
MSC: Primary 02F30; Secondary 02F25
MathSciNet review: 0327495
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Abstract: Let $ {H^A} = \{ e\vert{\text{domain}}\{ e\} \cap A \ne \emptyset \} $. It is shown that there exists a set $ A$ of Turing degree $ a$ such that $ {H^A}$ is Turing-incomparable to $ {H^{\bar A}}$ whenever $ a$ is an r.e. degree with $ a' > 0'$, or $ a \geqq 0''$ or $ a \geqq 0'$ and $ a$ is r.e. in 0'. This contrasts with the fact that $ {H^A}$ is comparable to $ {H^{\bar A}}$ for almost all $ A$.

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Article copyright: © Copyright 1973 American Mathematical Society