Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-1973-0327495-X
MathSciNet review: 0327495
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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$.


References [Enhancements On Off] (What's this?)

  • [1] J. C. E. Dekker and J. Myhill, Retraceable sets, Canad. J. Math. 10 (1958), 357-373. MR 20 #5733. MR 0099292 (20:5733)
  • [2] L. Hay, The class of recursively enumerable subsets of a recursively enumerable set, Pacific J. Math. 46 (1973), 167-183. MR 0392525 (52:13342)
  • [3] C. G. Jockusch, Jr., The degrees of bi-immune sets, Z. Math. Logik Grundlagen Math. 15 (1969), 135-140. MR 39 #5360. MR 0244043 (39:5360)
  • [4] R. W. Robinson, Interpolation and embedding in the recursively enumerable degrees, Ann. of Math. (2) 93 (1971), 285-314. MR 43 #51. MR 0274286 (43:51)
  • [5] H. Rogers, Jr., Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967. MR 37 #61. MR 0224462 (37:61)
  • [6] A. L. Selman, Applications of forcing to the degree-theory of the arithmetical hierarchy, Proc. London. Math. Soc. 25 (1972), 586-602. MR 0314604 (47:3155)
  • [7] -, Relativized halting problems (to appear).
  • [8] J. R. Schoenfield, On degrees of unsolvability, Ann. of Math. (2) 69 (1959), 644-653. MR 21 #4097. MR 0105355 (21:4097)
  • [9] J. Stillwell, Decidability of the ``almost all'' theory of degrees, J. Symbolic Logic 37 (1972), 501-506. MR 0349369 (50:1863)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 02F30, 02F25

Retrieve articles in all journals with MSC: 02F30, 02F25


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1973-0327495-X
Article copyright: © Copyright 1973 American Mathematical Society

American Mathematical Society