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].
Supported in part by National Science Foundation Research Initiation Grant number CCR-8810467.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
6 References
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.
E. Allender, The generalized Kolmogorov complexity of sets, Proc. 4th Structure in Complexity Theory Conference.
E. Allender and R. Rubinstein, 1988 P-printable sets, SIAM J. Comput 17, 1193–1202.
E. Allender and C. Wilson, Downward translations of equality, submitted.
J. Balcázar, J. Díaz, and J. Gabarró, 1988 Structural Complexity I, Springer-Verlag, Berlin/New York.
R. Beigel, On the relativized power of additional accepting paths, Proc. 4th Structure in Complexity Theory Conference.
R. Book, 1974 Tally Languages and Complexity Classes, Information and Control 26, 186–193.
R. Book, C. Wilson, and Xu Mei-Rui, 1982 Relativizing time, space, and time-space, SIAM J. Comput. 11, 571–581.
J. Cai and L. Hemachandra, On the power of parity polynomial time, Proc. 6th Annual Symposium on Theoretical Aspects of Computer Science.
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.
J. Hartmanis, Generalized Kolmogorov complexity and the structure of feasible computations, Proc. 24th IEEE Symposium on Foundations of Computer Science, pages 439–445.
J. Hartmanis, 1983a On sparse sets in NP — P, Information Processing Letters 16, 55–60.
J. Hartmanis, N. Immerman, and V. Sewelson. 1985 Sparse sets in NP-P: EXPTIME versus NEXPTIME, Information and Control, 65, 158–181.
J. Hopcroft and J. Ullman, 1979 Introduction to Automata Theory, Languages, and Computation, Addison-Wesley, Reading, Massachusetts.
S. Kurtz, 1985 Sparse sets in NP-P: relativizations, SIAM J. Comput. 14, 113–119.
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.
Ming Li and Paul Vitányi, Two decades of applied Kolmogorov complexity, Proc. 3rd Structure in Complexity Theory Conference, pp. 80–101.
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.
G. Lischke, 1987 Relativizations of NP and EL, strongly separating, and sparse sets, J. Information Processing and Cybernetics EIK 23, 99–112.
Wolfgang Paul, 1982 On-line simulation of k + 1 tapes by k tapes requires nonlinear time, Information and Control 53, pp. 1–8.
R. Rubinstein, Structural complexity classes of sparse sets: intractability, data compression and printability, doctoral dissertation, Northeastern University.
U. Schöning, Complexity and Structure, Lecture Notes in Computer Science 211.
O. Watanabe, 1988 On hardness of one-way functions, Information Processing Letters 27, 151–157.
C. Wilson, 1980 Relativization, reducibilities and the exponential hierarchy, M.S. Thesis, University of Toronto.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1989 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Allender, E. (1989). Limitations of the upward separation technique (preliminary version). In: Ausiello, G., Dezani-Ciancaglini, M., Della Rocca, S.R. (eds) Automata, Languages and Programming. ICALP 1989. Lecture Notes in Computer Science, vol 372. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0035749
Download citation
DOI: https://doi.org/10.1007/BFb0035749
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-51371-1
Online ISBN: 978-3-540-46201-9
eBook Packages: Springer Book Archive