On ≤1−ttp-sparseness and nondeterministic complexity classes

  • Osamu Watanabe
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 317)


For any reduction r, a set is called “≤ r p -sparse” if it is ≤ r p -reducible to a sparse set. The difficulty of sets in nondeterministic complexity classes is investigated in terms of non-≤ 1−tt p -sparseness, i.e., not being ≤ 1−tt p -reducible to any sparse set. In particular, nondeterministic complexity classes used to specify various types of one-way functions are mainly considered, i.e., UP, N(poly), UBPP, and UP. For each such class, we prove that it contains a non-≤ 1−tt p -sparse set unless it is included in P. Since these classes are included in more general nondeterministic complexity classes such as NP, easy consequences of our observations show the nonsparseness of ≤ 1−tt p -hard sets for such classes.


Polynomial Time Complexity Class Boolean Formula Pseudo Polynomial Time Promise Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [Ad78]
    L. Adleman, Two theorems on random polynomial time, in “Proc. 17th Ann. Sympos. on Foundations of Computer Science”, IEEE, New York (1978), 75–83.Google Scholar
  2. [Al85]
    E. Allender, “Invertible Functions”, Ph.D. Dissertation, Georgia Institute of Technology (1985).Google Scholar
  3. [Al86]
    E. Allender, The complexity of sparse sets in P, in “Proc. of 1st Structure in Complexity Theory Conference”, Lecture Notes in Computer Science 223, Springer-Verlag, Berlin (1986), 1–11.Google Scholar
  4. [Ba85]
    L. Babai, Trading group theory for randomness, in “Proc. 17th ACM Sympos. on Theory of Computing”, ACM (1985), 421–429.Google Scholar
  5. [BK88]
    R. Book and K. Ko, On sets reducible to sparse sets, SIAM J. Comput. 17 (1988), to appear.Google Scholar
  6. [ESY84]
    S. Even, A. Selman and Y. Yacobi, The complexity of promise problems with applications to public-key cryptography, Inform. and Control, 61 (1984), 159–173.Google Scholar
  7. [Fo79]
    S. Fortune, A note on sparse complete sets, SIAM J. Comput. 8 (1979), 431–433.Google Scholar
  8. [GJ79]
    M. Garey and D. Johnson, “Computers and Intractability, A Guide to the Theory of NP-Completeness”, Freeman, San Francisco (1979).Google Scholar
  9. [GS88]
    J. Grollman and A. Selman, Complexity measures for public-key cryptosystems, SIAM J. Comput. 11 (1988), to appear.Google Scholar
  10. [Ha78]
    J. Hartmanis, “Feasible Computations and Provable Complexity Properties”, SIAM, Philadelphia (1978).Google Scholar
  11. [HU79]
    J. Hopcroft and J. Ullman, “Introduction to Automata Theory, Language, and Computation”, Addison-Wesley, Reading (1979).Google Scholar
  12. [JY86]
    D. Joseph and P. Young, Some remarks on witness functions for nonpolynomial and noncomplete sets in NP, Theoret. Comput. Sci. 39 (1985), 225–237.Google Scholar
  13. [KLD85]
    K. Ko, T. Long and D. Du, On one-way functions and polynomial time isomorphisms, Theoret. Comput. Sci. 47 (1986), 263–276.Google Scholar
  14. [LLS76]
    R. Ladner, N. Lynch and A. Selman, A comparison of polynomial time reducibilities, Theoret. Comput. Sci. 1 (1975), 103–123.Google Scholar
  15. [Ma82]
    S. Mahaney, Sparse complete sets for NP: solution of a conjecture of Berman and Hartmanis, J. Comput. Syst. Sci. 25 (1982), 130–143.Google Scholar
  16. [Ma86]
    S. Mahaney, Sparse and reducibility, in “Studies in Complexity Theory (R. Book ed.)”, Pitman, London (1986).Google Scholar
  17. [MP79]
    A. Meyer and M. Paterson, With what frequency are apparently intractable problems difficult?, Technical Report, Massachusetts Institute of Technology, Cambridge TM-126 (1979).Google Scholar
  18. [PY82]
    C. Papadimitriou and M. Yannakakis, The complexity of facets (and some facets of complexity), J. Comput. Syst. Sci. 28 (1984), 244–259.Google Scholar
  19. [RSA78]
    R. Rivest, A. Shamir and L. Adleman, A method for obtaining digital signatures and public-key cryptosystems, Comm. ACM 21 (1978), 120–126.Google Scholar
  20. [Sc86]
    U. Schöning, Complete sets and closeness to complexity classes, Math. Syst. Theory 19 (1986), 29–41.Google Scholar
  21. [Se87]
    A. Selman, Promise problems complete for complexity classes, Technical Report, Northeastern University, Boston NU-CCS-87-9 (1987).Google Scholar
  22. [Se88]
    A. Selman, Complexity issues in cryptography, in “Computational Complexity Theory (J. Hartmanis ed.)”, Proc. Sympos. in Applied Math., American Math. Society (1988), to appear.Google Scholar
  23. [Uk83]
    E. Ukkonen, Two results on polynomial time Turing reductions to sparse sets, SIAM J. Comput. 12 (1983), 580–587.Google Scholar
  24. [Va76]
    L. Valiant, Relative complexity of checking and evaluating, Inform. Process. Lett. 5 (1) (1976), 20–23.Google Scholar
  25. [VV85]
    L. Valiant and V. Vazirani, NP is as easy as detecting unique solutions, Theoret. Comput. Sci. 47 (1986), 85–93.Google Scholar
  26. [Wa85]
    O. Watanabe, On one-one polynomial time equiyalence relations, Theoret. Comput. Sci. 38 (1985), 157–165.Google Scholar
  27. [Wa87]
    O. Watanabe, Polynomial time reducibility to a set of small density, in “Proc. 2nd Structure in Complexity Theory Conference”, IEEE, New York (1987), 138–146.Google Scholar
  28. [Wa88]
    O. Watanabe, On the difficulty of one-way functions, Inform. Process. Lett., to appear.Google Scholar
  29. [Ye83]
    Y. Yesha, On certain polynomial-time truth-table reductions to sparse sets, SIAM J. Comput. 12 (1983), 411–425.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1988

Authors and Affiliations

  • Osamu Watanabe
    • 1
  1. 1.Dept. of MathematicsUniversity of California, Santa BarbaraSanta BarbaraUSA

Personalised recommendations