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Limitations of the upward separation technique

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Abstract

The upward separation technique was developed by Hartmanis, who used it to show thatE=NE iff there is no sparse set inNP -P [15]. This paper shows some inherent limitations of the technique. The main result of this paper is the construction of an oracle relative to which there are extremely sparse sets inNP-P, butNEE=EE; this is in contradiction to a result claimed in [14] and [16]. Thus, although the upward separation technique is useful in relating the existence of sets of polynomial (and greater) density inNP-P to the NTIME(T(n))=DTIME(T(n)) problem, the existence of sets ofvery low density inNP-P cannot be shown to have any bearing on this problem using proof techniques that relativize.

The oracle construction is also of interest since it is the first example of an oracle relative to whichEE=NEE andE ≠NE. (The techniques of [10], [17], [21], and [23] do not suffice to construct such an oracle.) The construction is novel and the techniques may be useful in other settings.

In addition, this paper also presents a number of new applications of the upward separation technique, including some new generalizations of the original result of [15].

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References

  1. E. Allender, The complexity of sparse sets inP, Proc. 1st Structure in Complexity Theory Conference, Lecture Notes in Computer Science, Vol. 223, Springer-Verlag, Berlin, 1986, pp. 1–11.

    Google Scholar 

  2. E. Allender, The generalized Kolmogorov complexity of sets,Proc. 4th Structure in Complexity Theory Conference, 1989, pp. 186–194.

  3. E. Allender, Limitations of the upward separation technique,Proc. 16th International Colloquium on Automata, Languages, and Programming, Lecture Notes in Computer Science, Vol. 372, Springer-Verlag, Berlin, 1989, pp. 18–30.

    Google Scholar 

  4. E. Allender and R. Rubinstein,P-printable sets,SIAM Journal on Computing,17 (1988), 1193–1202.

    Google Scholar 

  5. E. Allender and C. Wilson, Downward translations of equality, to appear inTheoretical Computer Science.

  6. J. Balcázar, J. Díaz, and J. Gabarró,Structural Complexity, Vol. I, Springer-Verlag, Berlin, 1988.

    Google Scholar 

  7. R. Beigel, On the relativized power of additional accepting paths,Proc. 4th Structure in Complexity Theory Conference, 1989, pp. 216–224.

  8. A. Blass and Y. Gurevich, On the unique satisfiability problem,Information and Control,55 (1982), 80–88.

    Google Scholar 

  9. R. Book, Tally languages and complexity classes,Information and Control,26 (1974), 186–193.

    Google Scholar 

  10. R. Book, C. Wilson, and Xu Mei-Rui, Relativizing time, space, and time-space,SIAM Journal on Computing,11 (1982), 571–581.

    Google Scholar 

  11. J. Cai and L. Hemachandra, On the power of parity polynomial time,Proc. 6th Annual Symposium on Theoretical Aspects of Computer Science, Lecture Notes in Computer Science, Vol. 349, Springer-Verlag, Berlin, 1989, pp. 229–239.

    Google Scholar 

  12. M. Dekhtyar, On the relativization of deterministic and nondeterministic complexity classes,Proc. 5th Symposium on Mathematical Foundations of Computer Science, Lecture Notes in Computer Science, Vol. 45, Springer-Verlag, Berlin, 1976, pp. 255–259.

    Google Scholar 

  13. R. Gavaldà, L. Torenvliet, O. Watanabe, and J. Balcázar, Generalized Kolmogorov complexity in relativized separations,Proc. 15th International Symposium on Mathematical Foundations of Computer Science, Lecture Notes in Computer Science, Vol. 452, Springer-Verlag, Berlin, 1990, pp. 269–276.

    Google Scholar 

  14. J. Hartmanis, Generalized Kolmogorov complexity and the structure of feasible computations,Proc. 24th IEEE Symposium on Foundations of Computer Science, 1983, pp. 439–445.

  15. J. Hartmanis, On sparse sets inNP-P, Information Processing Letters,16 (1983), 55–60.

    Google Scholar 

  16. J. Hartmanis, N. Immerman, and V. Sewelson. Sparse sets inNP-P: EXPTIME versus NEXPTIME,Information and Control,65 (1985), 158–181.

    Google Scholar 

  17. H. Heller, On relativized exponential and probabilistic complexity classes,Information and Control,71 (1986), 231–243.

    Google Scholar 

  18. J. Hopcroft and J. Ullman,Introduction to Automata Theory, Languages, and Computation, Addison-Wesley, Reading, Massachusetts, 1979.

    Google Scholar 

  19. R. Impagliazzo and G. Tardos Decision versus search in super-polynomial time,Proc. 30th Annual IEEE Symposium on Foundations of Computer Science, (1989), pp. 222–227.

  20. J. Köbler, U. Schoning, S. Toda, and J. Torán, Turing machines with few accepting computations and low sets for PP,Proc. 4th Structure in Complexity Theory Conference, 1989, pp. 208–215.

  21. S. Kurtz, Sparse sets inNP-P: relativizations,SIAM Journal on Computing,14 (1985), 113–119.

    Google Scholar 

  22. G. Lischke, Oracle-constructions to prove all possible relationships between relativizations ofP, NP, EL, NEL, EP andNEP, Zeitschrift für mathematische Logic und Grundlagen der Mathematik,32 (1986), 257–270.

    Google Scholar 

  23. G. Lischke, Relativizations ofNP andEL, strongly separating, and sparse sets,J. Information Processing and Cybernetics EIK,23 (1987), 99–112.

    Google Scholar 

  24. Ming Li and P. Vitányi, Two decades of applied Kolmogorov complexity,Proc. 3rd Structure in Complexity Theory Conference, 1988, pp. 80–101.

  25. W. Paul, On-line simulation ofk + 1 tapes byk tapes requires nonlinear time,Information and Control,53 (1982), 1–8.

    Google Scholar 

  26. R. Rubinstein, Structural Complexity Classes of Sparse Sets: Intractability, Data Compression and Printability, Doctoral dissertation, Northeastern University, 1988.

  27. U. Schöning,Complexity and Structure, Lecture Notes in Computer Science, Vol. 211, Springer-Verlag, Berlin, 1985.

    Google Scholar 

  28. O. Watanabe, On hardness of one-way functions,Information Processing Letters,27 (1988), 151–157.

    Google Scholar 

  29. C. Wilson, Relativization, Reducibilities and the Exponential Hierarchy, M.S. thesis, University of Toronto, 1980.

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Authors and Affiliations

  1. Department of Computer Science, Rutgers University, 08903, New Brunswick, NJ, USA

    Eric Allender

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  1. Eric Allender
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Additional information

A preliminary version of this paper was presented at the 16th International Colloquium on Automata, Languages, and Programming [3]. The author was supported in part by National Science Foundation Research Initiation Grant Number CCR-8810467.

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Allender, E. Limitations of the upward separation technique. Math. Systems Theory 24, 53–67 (1991). https://doi.org/10.1007/BF02090390

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  • DOI: https://doi.org/10.1007/BF02090390

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