Abstract
We consider sets that are high for the complexity class NP with respect to several operators on complexity classes. The notion of lowness and highness in the context of complexity theory was introduced by Schüning [34] and further investigated and generalized under several aspects (see [24, 7, 9, 37, 22, 20, 27, 4, 31, 25, 26]).
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References
L. Adleman. TWO theorems on random polynomial time., In Proceedings of the 19th IEEE Symposium on the Foundations of Computer Science, 75–83., IEEE Computer Society Press, 1978.
L. Adleman AND K. Manders., Reducibility, randomness, and intractibility., In Proceedings of the 9th ACM Symposium on Theory of Computing, 151–163. ACM Press, 1977
L. Adleman AND K. Manders., Reductions that lie. In Proceedings of the 20th IEEE Symposium on the Foundations of Computer Science, 397–410., IEEE Computer Society Press, 1979
E. Allender AND L. Hemachandra., Lower bounds for the low hierarchy., Journal of the ACM, 39: 234–250, 1992.
V. Arvind, J. Kobler, U. Schöning, AND R. Schuler., If NP has polynomial-size circuits, then MA=AM. Theoretical Computer Science, 137 (2): 279–282, 1995.
J. Balcázar. Self-reducibility structures and solutions of NP problems., In Revista Matematica, 175-184. Universidad Complutense de Madrid, 1989.
J Balcázar, R. Book, AND U. Schöning. Sparse sets, lowness and highness. SIAM Journal on Computing, 23: 679–688, 1986.
C.H. Bennett AND J. Gill., Relative to a random oracle A, P A ≠, NP A ≠ co-NP A with probability 1. SIAM Journal on Computing, 10: 96–113, 1981.
R. Book, P. Orponen, D. Russo, AND O. Watanabe., Lowness properties of sets in the exponential-time hierarchy., SIAM Journal on Computing, 17 (3): 504–516, 1988.
A. Borodin AND A. Demers., Some comments on functional self- reducibility and the NP hierarchy. Technical Report 76-284, Dept. Computer Science, Cornell University, 1976
N.Bshouty, R. Cleve, R. Gavaldà, S. Kannan, AND C. Tamon., Oracles and queries that are sufficient for exact learning. Technical Report TR 95-015, ECCC, 1995. To appear in Journal of Computer and System Sciences
J. L. Carter and M. N. Wegman., Universal classes of hash functions., Journal of Computer and System Sciences, 18: 143–154, 1979.
R. Chang, J. Kadin, AND P. Rohatgi., Connections between the complexity of unique satisfiability and the threshold behavior of randomized reductions., In Proceedings of the 6th Structure in Complexity Theory Conference, 255–269. IEEE Computer Society Press, 1991.
M. J. Chung AND B. Ravikumar., Strong nondeterministic Turing reduction-A technique for proving intractability., Journal of Computer and System Sciences, 39 (l): 2 - 20, 1989
SA. Cook., The complexity of theorem-proving procedures., In Proceedings of the 3rd ACM Symposium on Theory of Computing, 151–158. ACM Press, 1971
L. Fortnow. Complexity-Theoretic Aspects of Interactive Proof Systems, PhD thesis, MIT, 1989
M. Garey AND D. Johnson., Computers and Intractability - A Guide to the Theory of NP-Completeness. Freeman and Company, 1979.
S. Gupta. E-mail message on the TheoryNet, December 1995.
J. E. Hopcroft. Recent directions in algorithmic research., In P. Deussen, editor, Proceedings of the 5th Conference on Theoretical Computer Science, Lecture Notes in Computer Science #104, 123–134. Springer-Verlag, 1981
J. Kamper. Non-uniform proof systems: A new framework to describe non-uniform and probabilistic complexity classes. Theoretical Computer Science, 85 (2): 305–331, 1991.
RM. Karp AND RJ. Lipton., Some connections between nonuniform and uniform complexity classes., In Proceedings of the 12th ACM Symposium on Theory of Computing, 302–309. ACM Press, 1980
A. Klapper. Generalized lowness and highness and probabilistic complexity classes., Mathematical Systems Theory, 22: 37–45, 1989.
K. Ko. Some observations on the probabilistic algorithms and NP-hard problems., Information Processing Letters, 14: 39–43, 1982.
K. Ko AND U. Schöning. On circuit-size complexity and the low hierarchy in NP., SIAM Journal on Computing, 14: 41–51, 1985.
J. Köbler. Locating P/poly optimally in the extended low hierarchy., Theoretical Computer Science, 134 (2): 263–285, 1994.
J. Köbler. On the structure of low sets., In Proceedings of the 10th Struc¬ture in Complexity Theory Conference, 246–261. IEEE Computer Society Press, 1995.
J. Köbler, U. Schöning, S. Toda, AND J. Torán., Turing machines with few accepting computations and low sets for PP. Journal of Computer and System Sciences, 44 (2): 272–286, 1992.
J. K Söbler, U. Schöning, AND J. Torán,. The Graph Isomorphism Problem: Its Structural Complexity. Birkhäuser, Boston, 1993.
J. Köbler AND O. Watanabe., New collapse consequences of NP having small circuits. In Proceedings of the 22nd International Colloquium on Automata, Languages, and Programming, Lecture Notes in Computer Science #944, 196–207. Springer-Verlag, 1995
T. Long. Strong nondeterministic polynomial-time reducibilities., Theoretical Computer Science, 21: 1–25, 1982.
T. Long AND M. Sheu., A refinement of the low and high hierarchies., Technical Report OSU-CISRC-2/91-TR6, The Ohio State University, 1991.
DA. Plaisted. New NP-hard and NP-complete polynomial and integer divisibility problems., In Proceedings of the 18th IEEE Symposium on the Foundations of Computer Science, 241–253., IEEE Computer Society Press, 1977.
DA. Plaisted. Complete divisibility problems for slowly utilized oracles. Theoretical Computer Science, 35: 245–260, 1985.
U. Schöning. A low and a high hierarchy within NP., Journal of Computer and System Sciences, 27: 14–28, 1983.
U. Schöning. Complexity and Structure, Lecture Notes in Computer Science #211., Springer-Verlag, 1986
U. Schöning. Robust oracle machines., In Proceedings of the 13th Symposium on Mathematical Foundations of Computer Science, Lecture Notes in Computer Science #324, 93–106. Springer-Verlag, 1988
U. Schöning. Probabilistic complexity classes and lowness., Journal of Computer and System Sciences, 39: 84–100, 1989.
U. Schöning. On random reductions from sparse sets to tally sets. Information Processing Letters, 46: 239–241, 1993.
A. Selman. Polynomial time enumeration reducibility., SIAM Journal on Computing, 7: 440–447, 1978.
M. Sipser. A complexity theoretic approach to randomness., In Proceedings of the 15th ACM Symposium on Theory of Computing, 330–335. ACM Press, 1983
L. Valiant AND V. Vazirani., NP is as easy as detecting unique solutions. Theoretical Computer Science, 47: 85–93, 1986.
U. Vazirani and V. Vazirani. A natural encoding scheme proved probabilistic polynomial complete. Theoretical Computer Science, 24: 291–300, 1983.
C. Wilson. Relativized circuit complexity., Journal of Computer and System Sciences, 31 (2): 169–181, 1985.
S. Zachos. Robustness of probabilistic computational complexity classes under definitional perturbations., Information and Control, 54: 143–154, 1982.
S. Zachos. Probabilistic quantifiers, adversaries, and complexity classes: an overview. In Proceedings of the 1st Structure in Complexity Theory Conference, Lecture Notes in Computer Science #223, 383–400., Springer- Verlag, 1986.
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© 1997 Kluwer Academic Publishers
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Köbler, J., Schöning, U. (1997). High Sets for NP. In: Du, DZ., Ko, KI. (eds) Advances in Algorithms, Languages, and Complexity. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-3394-4_6
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DOI: https://doi.org/10.1007/978-1-4613-3394-4_6
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