Minimal pairs for polynomial time reducibilities

  • Klaus Ambos-Spies
Part of the Lecture Notes in Computer Science book series (LNCS, volume 270)


Two recursive sets A and B form a minimal pair with respect to some polynomial time reducibility notion ≤pr if neither A nor B can be computed in polynomial time but every set which reduces to both A and B is polynomial time computable. We show that for every recursive set A∉P there is a recursive set B such that A and B form a minimal pair. Moreover, similar results for pairs without greatest predecessors are proved.


Polynomial Time Turing Machine Minimal Pair Polynomial Hierarchy Polynomial Time Reducibility 
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4. References

  1. 1.
    L.Adleman, Two theorems on random polynomial time, 19th IEEE Sympos. Foundations of Comput. Sci., 1978, 75–83.Google Scholar
  2. 2.
    K. Ambos-Spies, On the structure of the polynomial time degrees, in "STACS 84, Symposium on Theoretical Aspects of Computer Science", Lecture Notes Comput. Sc. 166 (1984) 198–208, Springer-Verlag.Google Scholar
  3. 3.
    K. Ambos-Spies, Inhomogeneities in the polynomial time degrees: the degrees of super sparse sets, Inform. Proc. Letters 22 (1986) 113–117.Google Scholar
  4. 4.
    K. Ambos-Spies, A note on complete problems for complexity classes, Inform. Proc. Letters 23 (1986) 227–230.Google Scholar
  5. 5.
    K. Ambos-Spies, Randomness, relativizations, and polynomial reducibilities, in "Structure in Complexity Theory", Lecture Notes Comput. Sc. 223 (1986) 23–34, Springer-VerlagGoogle Scholar
  6. 6.
    K.Ambos-Spies, Polynomial time degrees of NP-sets, in "Current Trends in Theoretical Computer Science" (E.Börger, ed.), Computer Science Press (to appear).Google Scholar
  7. 7.
    K.Ambos-Spies, On the relative complexity of hard problems for complexity classes without complete problems (submitted for publication).Google Scholar
  8. 8.
    P. Chew and M. Machtey, A note on structure and looking back applied to the relative complexity of computable functions. J. Comput. System Sci. 22 (1981) 53–59.Google Scholar
  9. 9.
    S.A.Cook, The complexity of theorem proving procedures, Third Annual ACM Sympos. Theory Comput., 1971, 151–158.Google Scholar
  10. 10.
    J. Gill, Computational complexity of probabilistic Turing machines, SIAM J. Computing 6 (1977) 675–695.Google Scholar
  11. 11.
    R.M.Karp, Reducibility among combinatorial problems, in "Complexity of Computer Computations" (R.E.Miller and J.W. Thatcher, eds.), Plenum Press, 1972, 85–103.Google Scholar
  12. 12.
    R.E.Ladner, On the structure of polynomial time reducibility, J. ACM (1975) 155–171.Google Scholar
  13. 13.
    L.H. Landweber, R.J. Lipton and E.L. Robertson, On the structure of sets in NP and other complexity classes, Theoret. Comput. Sci. 15 (1981) 181–200.Google Scholar
  14. 14.
    M. Machtey, Minimal pairs of polynomial degrees with subexponential complexity, Theoret. Comput. Sci. 2 (1976) 73–76.Google Scholar
  15. 15.
    K. Mehlhorn, The "almost all" theory of subrecursive degrees is decidable, in "Automata, Languages and Programming, 2nd Colloquium", Lecture Notes Comput. Sc. 15 (1971) 317–325, Springer-Verlag.Google Scholar
  16. 16.
    K. Mehlhorn, Polynomial and abstract subrecursive classes, J. Comput. System Sci. 12 (1976) 147–178.Google Scholar
  17. 17.
    U. Schöning, A uniform approach to obtain diagonal sets in complexity classes, Theoret. Comput. Sci. 18 (1982) 95–103.Google Scholar
  18. 18.
    U. Schöning, Minimal pairs for P, Theoret. Comput. Sci. 31 (1984) 41–48.Google Scholar
  19. 19.
    L.J. Stockmeyer, The polynomial-time hierarchy, Theoret. Comput. Sci. 3 (1977) 1–22.Google Scholar
  20. 20.
    L. Valiant, Relative complexity of checking and evaluating, Inform. Proc. Letters 5 (1976) 20–23.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1987

Authors and Affiliations

  • Klaus Ambos-Spies
    • 1
  1. 1.Fachbereich InformatikUniversität DortmundGermany

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