Notre Dame Journal of Formal Logic

On the Symmetric Enumeration Degrees

Charles M. Harris

Abstract

A set A is symmetric enumeration (se-) reducible to a set B (A ≤\sb se B) if A is enumeration reducible to B and \barA is enumeration reducible to \barB. This reducibility gives rise to a degree structure (D\sb se) whose least element is the class of computable sets. We give a classification of ≤\sb se in terms of other standard reducibilities and we show that the natural embedding of the Turing degrees (D\sb T) into the enumeration degrees (D\sb e) translates to an embedding (ι\sb se) into D\sb se that preserves least element, suprema, and infima. We define a weak and a strong jump and we observe that ι\sb se preserves the jump operator relative to the latter definition. We prove various (global) results concerning branching, exact pairs, minimal covers, and diamond embeddings in D\sb se. We show that certain classes of se-degrees are first-order definable, in particular, the classes of semirecursive, Σ\sb n ⋃ Π\sb n, Δ\sb n (for any n \in ω), and embedded Turing degrees. This last result allows us to conclude that the theory of D\sb se has the same 1-degree as the theory of Second-Order Arithmetic.

Article information

Source
Notre Dame J. Formal Logic, Volume 48, Number 2 (2007), 175-204.

Dates
First available in Project Euclid: 16 May 2007

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1179323263

Digital Object Identifier
doi:10.1305/ndjfl/1179323263

Mathematical Reviews number (MathSciNet)
MR2306392

Zentralblatt MATH identifier
1139.03030

Subjects
Primary: 03D30: Other degrees and reducibilities

Keywords
symmetric enumeration reducibility symmetric enumeration degrees

Citation

Harris, Charles M. On the Symmetric Enumeration Degrees. Notre Dame J. Formal Logic 48 (2007), no. 2, 175--204. doi:10.1305/ndjfl/1179323263. https://projecteuclid.org/euclid.ndjfl/1179323263


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References

  • [1] Arslanov, M. M., I. S. Kalimullin, and S. B. Cooper, "Splitting properties of total enumeration degrees", Algebra and Logic, vol. 42 (2003), pp. 1--13.
  • [2] Ershov, Y. L., "A certain hierarchy of sets. III", Algebra i Logika, vol. 9 (1970), pp. 34--51. Reprinted in Algebra and Logic, vol. 9 (1970), pp. 20--31.
  • [3] Harris, C., "Symmetric enumeration reducibility", pp. 196--208 in New Computational Paradigms: First Conference on Computablitiy in Europe, CiE 2005, Amsterdam, edited by S. B. Cooper and B. Löwe, vol. 3526 of Lecture Notes in Computer Science, Springer, 2005.
  • [4] Jockusch, C. G., Jr., "Semirecursive sets and positive reducibility", Transactions of the American Mathematical Society, vol. 131 (1968), pp. 420--36.
  • [5] Kalimullin, I. S., "Definability of the jump operator in the enumeration degrees", Journal of Mathematical Logic, vol. 3 (2003), pp. 257--67.
  • [6] Kleene, S. C., and E. L. Post, "The upper semi-lattice of degrees of recursive unsolvability", Annals of Mathematics. Second Series, vol. 59 (1954), pp. 379--407.
  • [7] Lachlan, A. H., "Lower bounds for pairs of recursively enumerable degrees", Proceedings of the London Mathematical Society. Third Series, vol. 16 (1966), pp. 537--69.
  • [8] Lachlan, A. H., "Embedding nondistributive lattices in the recursively enumerable degrees", pp. 149--77 in Conference in Mathematical Logic-London, 1970, edited by W. Hodges, vol. 255 of Springer Lecture Notes in Mathematics, Springer, Berlin, 1972.
  • [9] Lacombe, D., "Sur le semi-réseau constitué par les degrés d'indécidabilité récursive", Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences (Paris), vol. Series A-B 239 (1954), pp. 1108--9.
  • [10] Lerman, M., "On suborderings of the $\alpha $"-recursively enumerable $\alpha $-degrees, Annals of Pure and Applied Logic, vol. 4 (1972), pp. 369--92.
  • [11] McEvoy, K., "Jumps of quasi-minimal enumeration degrees", The Journal of Symbolic Logic, vol. 50 (1985), pp. 839--48.
  • [12] McEvoy, K., and S. B. Cooper, "On minimal pairs of enumeration degrees", The Journal of Symbolic Logic, vol. 50 (1985), pp. 983--1001.
  • [13] Medvedev, Y. T., "On nonisomorphic recursively enumerable sets", Doklady Akademii Nauk SSSR, vol. 102 (1955), pp. 211--14.
  • [14] Odifreddi, P. G., Classical Recursion Theory. Vol. II, vol. 143 of Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Co., Amsterdam, 1999.
  • [15] Rozinas, M. G., "The semilattice of e-degrees", pp. 71--84 in Recursive Functions (Russian), edited by E. A. Polyakov, Ivanovskij Gosudarstvennyj Universitet, Ivanovo, 1978.
  • [16] Sacks, G. E., Degrees of Unsolvability, Princeton University Press, Princeton, 1963.
  • [17] Sacks, G. E., "The recursively enumerable degrees are dense", Annals of Mathematics. Second Series, vol. 80 (1964), pp. 300--312.
  • [18] Selman, A. L., "Arithmetical reducibilities. II", Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 18 (1972), pp. 83--92.
  • [19] Simpson, S. G., "First-order theory of the degrees of recursive unsolvability", Annals of Mathematics. Second Series, vol. 105 (1977), pp. 121--39.
  • [20] Spector, C., "On degrees of recursive unsolvability", Annals of Mathematics. Second Series, vol. 64 (1956), pp. 581--92.
  • [21] Yates, C. E. M., "A minimal pair of recursively enumerable degrees", The Journal of Symbolic Logic, vol. 31 (1966), pp. 159--68.