# Some observations about *NP* complete sets

Conference paper

First Online:

## Abstract

In this paper, we summarize and extend some recent results about the properties of *NP* complete sets and related results about the structure of feasible computations.

## Keywords

Polynomial Time Turing Machine Complexity Class Polynomial Time Hierarchy Polynomial Size Circuit
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.

## Preview

Unable to display preview. Download preview PDF.

## References

- [BE 78]P. Berman. Relationships between density and deterministic complexity of NP-complete languages.
*Proceedings of the 5th International Colloquium on Automata, Languages, and Programming,*Springer-Verlag*Lecture Notes in Computer Science*, 62, pp. 63–71, 1978.Google Scholar - [Bet 78]R.V. Book, et al, ”Inclusion Complete Tally Languages and the Hartmanis-Berman Conjecture”, Math. Systems Theory 11, pp. 1–8, 1978.Google Scholar
- [BBL *84]J. Balcazar, R. Book, T. Long, U. Schoening, and A. Selman. Sparse oracles and uniform complexity classes. In
*Proceedings IEEE Symposium on Foundations of Computer Science*, pp. 308–313, 1984.Google Scholar - [BGS 75]T. Baker, J. Gill, and R. Solovay. Relativizations of the P=? NP question.
*SIAM Journal on Computing*4, pp. 431–442, 1975.Google Scholar - [BH 77]L. Berman and J. Hartmanis. On isomorphisms and density of
*NP*and other complete sets. SIAM Journal on Computing 6, pp. 305–322, 1977.Google Scholar - [Bo 74]R.V. Book. Tally Languages and Complexity classes.
*Information and Control 26*, pp. 186–193, 1974.Google Scholar - [CH 86]J. Cai and L. Hemachandra. The Boolean Hierarchy: hardware over
*NP*. In*Structure in Complexity Theory*Springer-Verlag*Lecture Notes in Computer Science #233*, pp. 105–124, 1986.Google Scholar - [GJ 79]M. Garey and D. Johnson.
*Computers and Intractability: A Guide to the Theory of NP-Completeness*. W.H. Freeman and Company, 1979.Google Scholar - [GJ 86]J. Goldsmith and D. Joseph. Three results on the polynomial isomorphism of complete sets. In
*Proceedings IEEE Symposium on Foundations of Computer Science*, pp. 390–397, 1986.Google Scholar - [GS 84]J. Grollmann and A. Selman. Complexity measures for public-key cryptosystems. In
*Proceedings IEEE Symposium on Foundations of Computer Science*, pp. 495–503, 1984.Google Scholar - [Har 83a]J. Hartmanis. On Sparse Sets in
*NP-P: Information Processing Letters 16*, pp. 55–60, 1983.Google Scholar - [Har 83b]J. Hartmanis. Generalized Kolmogorov complexity and the structure of feasible computations. In
*Proceedings IEEE Symposium on Foundations of Computer Science*, pp. 439–445, 1983.Google Scholar - [Har 85]J. Hartmanis. Solvable problems with conflicting relativizations.
*Bulletin of the European Association for Theoretical Computer Science*, pp. 40–49, Oct. 1985.Google Scholar - [HH 86a]J. Hartmanis and L. Hemachandra. Complexity classes without machines: on complete languages for UP. In
*Automata, Languages, and Programming (ICALP 1986)*Springer-Verlag*Lecture Notes in Computer Science #226*, pp. 123–135, 1986.Google Scholar - [HH 86b]J. Hartmanis and L. Hemachandra. On Sparse Oracles Separating Feasible Complexity Classes. In
*STACS: 3rd Annual Symposium on Theoretical Aspects of Computer Science*, Springer-Verlag*Lecture Notes in Computer Science #210*, pp. 321–333, 1986.Google Scholar - [HH 87]J. Hartmanis and L. Hemachandra. One-Way Functions, Robustness, and the Non-Isomorphism of NP Complete Sets. Department of Computer Science, Cornell University, Ithaca, New York. Technical Report TR 86–796. To be presented at
*Structure in Complexity Theory*, 1987.Google Scholar - [HI 85]J. Hartmanis and N. Immerman. On complete problems for NP ∩ coNP. In
*Automata, Languages, and Programming (ICALP 1985)*, Springer-Verlag*Lecture Notes in Computer Science #194*, pp. 250–259, 1985.Google Scholar - [HIS 85]J. Hartmanis, N. Immerman, and V. Sewelson. Sparse sets in NP-P: EXPTIME versus NEXPTIME.
*Information and Control*, 65, pp. 159–181, May/June 1985.Google Scholar - [HU 79]J. Hopcroft and J. Ullman.
*Introduction to Automata Theory, Languages, and Computation*. Addison-Wesley, 1979.Google Scholar - [HY 84]J. Hartmanis and Y. Yesha. Computation times of NP sets of different densities.
*Theoretical Computer Science*, 34, pp. 17–32, 1984.Google Scholar - [JY 85]D. Joseph and P. Young. Some remarks on witness functions for non-polynomial and non-complete sets in NP.
*Theoretical Computer Science*, 39, pp. 225–237, 1985.Google Scholar - [Kad 86a]J. Kadin.
*Deterministic Polynomial Time with O(log(n)) Queries*. Technical Report TR-86-771, Cornell University, August 1986.Google Scholar - [Kad 86b]J. Kadin.
*P*^{NP[log]}and sparse Turing complete sets for NP. 1986. Accepted for*Structure in Complexity Theory*, 1987.Google Scholar - [KL 80]R. Karp and R. Lipton. Some connections between nonuniform and uniform complexity classes. In
*ACM Symposium on Theory of Computing*, pp. 302–309, 1980.Google Scholar - [KMR 86]S. Kurtz, S. Mahaney, and J. Royer. Collapsing degrees. In
*Proceedings IEEE Symposium on Foundations of Computer Science*, pp. 380–389, 1986.Google Scholar - [Kol 65]A. Kolmogorov. Three approaches for defining the concept of information quantity.
*Prob. Inform. Trans.*, 1, pp. 1–7, 1965.Google Scholar - [Koz 80]D. Kozen. Indexing of Subrecursive classes.
*Theoretical Computer Science*, pp. 277–301, 1980.Google Scholar - [Kur 83]S. Kurtz. A relativized failure of the Berman-Hartmanis conjecture. 1983. Unpublished manuscript.Google Scholar
- [Kur 85]S. Kurtz. Sparse Sets in
*NP-P*: Relativizations.*SIAM Journal of Computing 14*, pp. 113–119, 1983.Google Scholar - [La 75]R. Ladner. On the Structure of polynomial time reducibility.
*Journal of the Association for Computing Machinery*22, pp. 155–171, 1975.Google Scholar - [Li 85]Ming Li. Lower Bounds in Computational Complexity, Ph.D. Thesis, Cornell University, Ithaca, New York, 1985.Google Scholar
- [Lon 82]T.J. Long. A note on sparse oracles for NP.
*Journal of Computer and System Sciences*, 24, pp. 224–232, 1982.Google Scholar - [Long 85]T.J. Long. On restricting the size of oracles compared with restricting access to oracles.
*SIAM Journal on Computing*, 14, pp. 585–597, 1985.Google Scholar - [Mah 80]S. Mahaney. Sparse complete sets for NP: solution of a conjecture of Berman and Hartmanis. In
*Proceedings IEEE Symposium on Foundations of Computer Science*, pp. 54–60, 1980.Google Scholar - [Mah 82]S. Mahaney. Sparse complete sets for NP: solution of a conjecture of Berman and Hartmanis.
*Journal of Computer and Systems Sciences*, 25, pp. 130–143, 1982.Google Scholar - [Mah 83]S. Mahaney. On the number of p-isomorphism classes of NP-complete sets.
*Proceedings of the 22nd IEEE Symposium on Foundations of Computer Science*, pp. 130–143, 1981.Google Scholar - [Mah 86]S. Mahaney. Sparse sets and reducibilities. In
*Studies in Complexity Theory*, ed. R.V. Book, John Wiley and sons, Inc., New York, pp. 63–118, 1986.Google Scholar - [MY 85]S. Mahaney and P. Young. Reductions among polynomial isomorphism types.
*Theoretical Computer Science*, pp. 207–224, 1985.Google Scholar - [Rac 82]C. Rackoff. Relativized questions involving probabilistic algorithms.
*Journal of ACM*, 29, pp. 261–268, 1982.Google Scholar - [Re 83]K. Regan. On diagonalization methods and the structure of language classes.
*Proceedings FCT '83*, Springer-Verlag*Lecture Notes in Computer Science #158*, pp. 368–380, 1983.Google Scholar - [Re 86]K. Regan. On the separation of complexity classes. Ph.D. dissertation, Oxford University, August 1986.Google Scholar
- [Ro 67]H. Rogers, Jr. ”Theory of Recursive Functions and Effective Computability”, McGraw-Hill, New York, NY, 1967.Google Scholar
- [Sch 86]U. Schoening.
*Complexity and Structure*. Springer-Verlag*Lecture Notes in Computer Science #211*, 1986.Google Scholar - [Sip 82]M. Sipser. On relativization and the existence of complete sets. In
*Automata, Language, and Programming (ICALP 1982)*, Springer-Verlag*Lecture Notes in Computer Science #140*, pp. 523–531, 1982.Google Scholar - [SG 84]A.L. Selman and J. Grollmann. Complexity measures for public-key cryptosystems.
*Proceedings of the 25th IEEE Symposium on Foundations of Computer Science*, pp. 495–503, 1984.Google Scholar - [Sto 77]L. Stockmeyer. The polynomial-time hierarchy.
*Theoretical Computer Science*, 3, pp. 1–22, 1977.Google Scholar - [Yao 85]A. Yao. Separating the Polynomial-time hierarchy by oracles. In
*Proceedings IEEE Symposium on Foundations of Computer Science*, pp. 1–10, 1985.Google Scholar

## Copyright information

© Springer-Verlag Berlin Heidelberg 1987