Skip to main content Accessibility help

The universality of polynomial time Turing equivalence


We show that polynomial time Turing equivalence and a large class of other equivalence relations from computational complexity theory are universal countable Borel equivalence relations. We then discuss ultrafilters on the invariant Borel sets of these equivalence relations which are related to Martin's ultrafilter on the Turing degrees.

Hide All
Ambos-Spies, K. and Nies, A. (1992). The theory of the polynomial many-one degrees of recursive sets is undecidable. In: STACS 92 (Cachan, 1992), Lecture Notes in Comput. Sci. 577 Springer, Berlin 209–218. MR1255600 (94m:03068)
Baker, T., Gill, J. and Solovay, R. (1975). Relativizations of the ${\cal P} = ? {\cal N}{\cal P}$ question. SIAM Journal of Computational 4 (4) 431442. {MR{0395311 (52 #16108).
Bennett, C.H. and Gill, J. (1981). Relative to a random oracle A, P A NP A ≠ co − NP A with probability 1. SIAM Journal of Computationa 10 (1) 96113. MR605606 (83a:68044).
Blum, M. and Impagliazzo, R. (1987). Generic oracles and oracle classes (extended abstract). In: FOCS, 118–126.
Dougherty, R., Jackson, S. and Kechris, A.S. (1994). The structure of hyperfinite Borel equivalence relations. Transactions of the American Mathematical Society 341 (1) 193225.
Downey, R. and Nies, A. (2000). Undecidability results for low complexity time classes. Journal of Computer and System Sciences 60 (2) part 2 465479. Twelfth Annual IEEE Conference on Computational Complexity (Ulm, 1997). MR1785026 (2002d:03078).
Fortnow, L. (1994). The role of relativization in complexity theory. Bulletin of the European Association for Theoretical Computer Science 52 52229.
Jackson, S., Kechris, A.S. and Louveau, A. (2002). Countable Borel equivalence relations. Journal of Mathematical Logic 2 (1) 180. MR1900547 (2003f:03066).
Kechris, A.S. (1999). New directions in descriptive set theory. Bulletin of Symbolic Logic 5 (2) 161174. MR1791302 (2001h:03090).
Ladner, R.E. (1975). On the structure of polynomial time reducibility. Journal of the ACM 22 (1) 155171.
Marks, A., Slaman, T.A. and Steel, J.R. (2016). Martin's conjecture, arithmetic equivalence, and countable borel equivalence relations. In: A.S., Löwe, B. and Steel, J.R. (eds.) Ordinal Definability and Recursion Theory: The Cabal Seminar, Volume III, Lecture Notes in Logic 43, Cambridge University Press, 200219.
Martin, D.A. (1968). The axiom of determinateness and reduction principles in the analytical hierarchy. Bulletin of the American Mathematical Society 74 687689. MR0227022 (37 #2607).
Martin, D.A. (1975). Borel determinacy. Annals of Mathematics 102 (2) 363371. MR0403976 (53 #7785).
Rogers, H. Jr. (1967). Theory of Recursive Functions and Effective Computability, McGraw-Hill Book Co., New York. MR0224462 (37 #61).
Sacks, G.E. (1966). Degrees of Unsolvability, 2nd ed. Princeton University Press, Princeton, N.J.
Shinoda, J. and Slaman, T.A. (1990). On the theory of the PTIME degrees of the recursive sets. Journal of Computer and System Sciences 41 (3) 321366. MR1079470 (92b:03049).
Shore, R.A. (1979). The homogeneity conjecture. Proceedings of the National Academy of Sciences U.S.A. 76 (9) 42184219. MR543312 (81a:03046).
Shore, R.A. (1982). On homogeneity and definability in the first-order theory of the Turing degrees. Journal of Symbolic Logic 47 (1) 816. MR644748 (84a:03046).
Thomas, S. (2009). Martin's conjecture and strong ergodicity. Archive for Mathematical Logic 48 (8) 749759. MR2563815 (2011g:03120).
Williams, J. and Thomas, S. (2016). The bi-embeddability relation for finitely generated groups II Archive for Mathematical Logic 55 (3) 385396.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Mathematical Structures in Computer Science
  • ISSN: 0960-1295
  • EISSN: 1469-8072
  • URL: /core/journals/mathematical-structures-in-computer-science
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed