Notre Dame Journal of Formal Logic

Immunity and Hyperimmunity for Sets of Minimal Indices

Frank Stephan and Jason Teutsch

Abstract

We extend Meyer's 1972 investigation of sets of minimal indices. Blum showed that minimal index sets are immune, and we show that they are also immune against high levels of the arithmetic hierarchy. We give optimal immunity results for sets of minimal indices with respect to the arithmetic hierarchy, and we illustrate with an intuitive example that immunity is not simply a refinement of arithmetic complexity. Of particular note here are the fact that there are three minimal index sets located in Π3 − Σ3 with distinct levels of immunity and that certain immunity properties depend on the choice of underlying acceptable numbering. We show that minimal index sets are never hyperimmune; however, they can be immune against the arithmetic sets. Lastly, we investigate Turing degrees for sets of random strings defined with respect to Bagchi's size-function s.

Article information

Source
Notre Dame J. Formal Logic Volume 49, Number 2 (2008), 107-125.

Dates
First available: 15 May 2008

Permanent link to this document
http://projecteuclid.org/euclid.ndjfl/1210859921

Digital Object Identifier
doi:10.1215/00294527-2008-001

Mathematical Reviews number (MathSciNet)
MR2402035

Zentralblatt MATH identifier
1142.03024

Subjects
Primary: 03D28: Other Turing degree structures
Secondary: 03D45: Theory of numerations, effectively presented structures [See also 03C57; for intuitionistic and similar approaches see 03F55]

Keywords
sets of minimal indices sets of random strings immune sets hyperimmune sets Goedel numberings Kolmogorov numberings

Citation

Stephan, Frank; Teutsch, Jason. Immunity and Hyperimmunity for Sets of Minimal Indices. Notre Dame Journal of Formal Logic 49 (2008), no. 2, 107--125. doi:10.1215/00294527-2008-001. http://projecteuclid.org/euclid.ndjfl/1210859921.


Export citation

References

  • [1] Arslanov, M. M., "Some generalizations of a fixed-point theorem", Izvestiya Vysshikh Uchebnykh Zavedeniĭ. Matematika, (1981), pp. 9--16.
  • [2] Blum, M., "On the size of machines", Information and Computation, vol. 11 (1967), pp. 257--65.
  • [3] Herrmann, E., "$1$"-reducibility inside an $\rm m$"-degree with a maximal set, The Journal of Symbolic Logic, vol. 57 (1992), pp. 1046--56.
  • [4] Jockusch, C. G., Jr., M. Lerman, R. I. Soare, and R. M. Solovay, "Recursively enumerable sets modulo iterated jumps and extensions of Arslanov's completeness criterion", The Journal of Symbolic Logic, vol. 54 (1989), pp. 1288--323.
  • [5] Kummer, M., "On the complexity of random strings (extended abstract)", pp. 25--36 in STACS 96 (Grenoble, 1996), vol. 1046 of Lecture Notes in Computer Science, Springer, Berlin, 1996.
  • [6] Lempp, S., and M. Lerman, "The decidability of the existential theory of the poset of recursively enumerable degrees with jump relations", Advances in Mathematics, vol. 120 (1996), pp. 1--142.
  • [7] Lerman, M., "The existential theory of the upper semilattice of Turing degrees with least element and jump is decidable", http://www.math.uconn.% edu/\~ lerman/eth-jump.pdf, draft, 2006.
  • [8] Lusin, N., "Sur l'existence d'un ensemble non dénombrable qui est de première catégorie dans tout ensemble parfait", Fundamenta Mathematicae, vol. 2 (1921), pp. 155--57.
  • [9] Mansfield, R., and G. Weitkamp, Recursive Aspects of Descriptive Set Theory, vol. 11 of Oxford Logic Guides, The Clarendon Press, New York, 1985.
  • [10] Medvedev, Y. T., "Degrees of difficulty of the mass problem", Doklady Akademii Nauk SSSR. N.S., vol. 104 (1955), pp. 501--504.
  • [11] Meyer, A. R., "Program size in restricted programming languages", Information and Computation, vol. 21 (1972), pp. 382--94.
  • [12] Odifreddi, P., Classical Recursion Theory. The Theory of Functions and Sets of Natural Numbers, vol. 125 of Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Co., Amsterdam, 1989.
  • [13] Sacks, G. E., "Recursive enumerability and the jump operator", Transactions of the American Mathematical Society, vol. 108 (1963), pp. 223--39.
  • [14] Schaefer, M., "A guided tour of minimal indices and shortest descriptions", Archive for Mathematical Logic, vol. 37 (1998), pp. 521--48.
  • [15] Schwarz, S., Quotient Lattices, Index Sets, and Recursive Linear Orderings, Ph.D. thesis, University of Chicago, Chicago, 1982.
  • [16] Soare, R. I., Recursively Enumerable Sets and Degrees. A Study of Computable Functions and Computably Generated Sets, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1987.
  • [17] Teutsch, J. R., Noncomputable Spectral Sets, Ph.D. thesis, Indiana University, Bloomington, 2007.
  • [18] Yates, C. E. M., "On the degrees of index sets", Transactions of the American Mathematical Society, vol. 121 (1966), pp. 309--28.