# Limitations of the upward separation technique (preliminary version)

## Abstract

The upward separation technique was developed by Hartmanis, who used it to show that E=NE iff there is no sparse set in NP-P [Ha-83a]. 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 in NP-P, but NEE = EE; this is in contradiction to a result claimed in [Ha-83, HIS-85]. Thus, although the upward separation technique is useful in relating the existence of sets of polynomial (and greater) density in NP-P to the NTIME(T(n)) = DTIME(T(n)) problem, the existence of sets of very low density in NP-P can not be shown to have any bearing on this problem until proof techniques are developed which do not relativize.

The oracle construction is also of interest since it is the first example of an oracle relative to which EE = NEE and E ≠ NE. (The techniques of [BWX-82], [Ku-85], and [Li-87] 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 [Ha-83a].

## Preview

Unable to display preview. Download preview PDF.

## 6 References

- [Al-86]E. Allender, 1986
*The complexity of sparse sets in P*, Proc. 1st Structure in Complexity Theory Conference, Lecture Notes in Computer Science 223, Springer-Verlag, Berlin/New York, pp. 1–11.Google Scholar - [Al-89]E. Allender,
*The generalized Kolmogorov complexity of sets*, Proc. 4th Structure in Complexity Theory Conference.Google Scholar - [AR-88]E. Allender and R. Rubinstein, 1988
*P-printable sets*, SIAM J. Comput 17, 1193–1202.CrossRefGoogle Scholar - [AW-89]E. Allender and C. Wilson,
*Downward translations of equality*, submitted.Google Scholar - [BDG-88]J. Balcázar, J. Díaz, and J. Gabarró, 1988 Structural Complexity I, Springer-Verlag, Berlin/New York.Google Scholar
- [Be-89]R. Beigel,
*On the relativized power of additional accepting paths*, Proc. 4th Structure in Complexity Theory Conference.Google Scholar - [Bo-74]R. Book, 1974
*Tally Languages and Complexity Classes*, Information and Control 26, 186–193.CrossRefGoogle Scholar - [BWX-82]R. Book, C. Wilson, and Xu Mei-Rui, 1982
*Relativizing time, space, and time-space*, SIAM J. Comput. 11, 571–581.CrossRefGoogle Scholar - [CH-89]J. Cai and L. Hemachandra,
*On the power of parity polynomial time*, Proc. 6th Annual Symposium on Theoretical Aspects of Computer Science.Google Scholar - [De-76]M. Dekhtyar, 1976
*On the relativization of deterministic and nondeterministic complexity classes*, Proceedings of the 5th Symposium on Mathematical Foundations of Computer Science, Lecture Notes in Computer Science 45, Springer-Verlag, Berlin/New York, pp. 255–259.Google Scholar - [Ha-83]J. Hartmanis,
*Generalized Kolmogorov complexity and the structure of feasible computations*, Proc. 24th IEEE Symposium on Foundations of Computer Science, pages 439–445.Google Scholar - [Ha-83a]J. Hartmanis, 1983a
*On sparse sets in NP — P*, Information Processing Letters 16, 55–60.CrossRefGoogle Scholar - [HIS-85]J. Hartmanis, N. Immerman, and V. Sewelson. 1985
*Sparse sets in NP-P: EXPTIME versus NEXPTIME*, Information and Control, 65, 158–181.CrossRefGoogle Scholar - [HU-79]J. Hopcroft and J. Ullman, 1979 Introduction to Automata Theory, Languages, and Computation, Addison-Wesley, Reading, Massachusetts.Google Scholar
- [Ku-85]S. Kurtz, 1985
*Sparse sets in NP-P: relativizations*, SIAM J. Comput. 14, 113–119.CrossRefGoogle Scholar - [KSTT-89]J. Köbler, U. Schöning, S. Toda, and J. Torán,
*Turing machines with few accepting computations and low sets for PP*, Proc. 4th Structure in Complexity Theory Conference.Google Scholar - [LV-88]Ming Li and Paul Vitányi,
*Two decades of applied Kolmogorov complexity*, Proc. 3rd Structure in Complexity Theory Conference, pp. 80–101.Google Scholar - [Li-86]G. Lischke, 1986
*Oracle-constructions to prove all possible relationships between relativizations of P, NP, EL, NEL, EP and NEP*, Zeitschrift für mathematische Logic und Grundlagen der Mathematik 32, 257–270.Google Scholar - [Li-87]G. Lischke, 1987
*Relativizations of NP and EL, strongly separating, and sparse sets*, J. Information Processing and Cybernetics EIK 23, 99–112.Google Scholar - [Pa-82]Wolfgang Paul, 1982
*On-line simulation of k + 1 tapes by k tapes requires nonlinear time*, Information and Control 53, pp. 1–8.CrossRefGoogle Scholar - [Ru-88]R. Rubinstein,
*Structural complexity classes of sparse sets: intractability, data compression and printability*, doctoral dissertation, Northeastern University.Google Scholar - [Sc-85]U. Schöning, Complexity and Structure, Lecture Notes in Computer Science 211.Google Scholar
- [Wa-88]O. Watanabe, 1988
*On hardness of one-way functions*, Information Processing Letters 27, 151–157.CrossRefGoogle Scholar - [Wi-80]C. Wilson, 1980
*Relativization, reducibilities and the exponential hierarchy*, M.S. Thesis, University of Toronto.Google Scholar