Abstract
During the last few years, unprecedented progress has been made in structural complexity theory; class inclusions and relativized separations were discovered, and hierarchies collapsed. We survey this progress, highlighting the central role of counting techniques. We also present a new result whose proof demonstrates the power of combinatorial arguments: there is a relativized world in which UP has no Turing complete sets.
Supported by NSF grant CCR-8809174 and a Hewlett-Packard Corporation equipment grant.
This is a preview of subscription content, log in via an institution.
Preview
Unable to display preview. Download preview PDF.
References
A. Amir and W. Gasarch. Polynomial terse sets. In Proceedings 2nd Structure in Complexity Theory Conference, pages 22–27, 1987.
E. Allender. Invertible functions. 1985. Ph.D. thesis, Georgia Institute of Technology.
E. Allender. The complexity of sparse sets in P. In Proceedings 1st Structure in Complexity Theory Conference, pages 1–11, Springer-Verlag Lecture Notes in Computer Science #223, June 1986.
K. Ambos-Spies. A note on complete problems for complexity classes. Information Processing Letters, 23:227–230, 1986.
A. Borodin, S. Cook, W. Ruzzo, and M. Tompa. Two applications of complementation via induction counting. In Proceedings 3rd Structure in Complexity Theory Conference, IEEE Computer Society Press, June 1988. To appear.
R. Beigel. Bounded Queries to SAT and the Boolean Hierarchy. Technical Report TR-7, Johns Hopkins Department of Computer Science, Baltimore, MD, June 1987.
T. Baker, J. Gill, and R. Solovay. Relativizations of the P=?NP question. SIAM Journal on Computing, 4(4):431–442, 1975.
L. Berman and J. Hartmanis. On isomorphisms and density of NP and other complete sets. SIAM Journal on Computing, 6(2):305–322, 1977.
M. Blum and R. Impagliazzo. Generic oracles and oracle classes. In 28th Annual IEEE Symposium on Foundations of Computer Science, October 1987.
J. Cai. With probability one, a random oracle separates PSPACE from the polynomial-time hierarchy. In 18th ACM Symposium on Theory of Computing, pages 21–29, 1986.
J. Cai, T. Gundermann, J. Hartmanis, L. Hemachandra, V. Sewelson, K. Wagner, and G. Wechsung. The boolean hierarchy I: Structural properties. To appear in SIAM Journal on Computing.
J. Cai, T. Gundermann, J. Hartmanis, L. Hemachandra, V. Sewelson, K. Wagner, and G. Wechsung. The boolean hierarchy II: Applications. To appear in SIAM J. on Computing.
J. Cai and L. Hemachandra. The boolean hierarchy: Hardware over NP. In Proceedings 1st Structure in Complexity Theory Conference, pages 105–124, Springer-Verlag Lecture Notes in Computer Science #223, June 1986.
J. Cai and L. Hemachandra. On the Power of Parity Polynomial Time. Technical Report CUCS 274-87, Columbia Computer Science Department, New York, NY, December 1987.
J. Cai and L. Hemachandra. Enumerative counting is hard. In Proceedings 3rd Structure in Complexity Theory Conference, IEEE Computer Society Press, June 1988. To appear.
M. Furst, J. Saxe, and M. Sipser. Parity, circuits, and the polynomial-time hierarchy. Mathematical Systems Theory, 17:13–27, 1984.
M. Garey and D. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman and Company, 1979.
J. Goldsmith and D. Joseph. Three results on the polynomial isomorphism of complete sets. In Proceedings 27th IEEE Symposium on Foundations of Computer Science, pages 390–397, 1986.
J. Grollmann and A. Selman. Complexity measures for public-key cryptosystems. In Proceedings 25th IEEE Symposium on Foundations of Computer Science, pages 495–503, 1984.
T. Gundermann and G. Wechsung. Counting classes with finite acceptance types. To appear.
L. Hemachandra. Can P and NP Manufacture Randomness? Technical Report TR86-795, Cornell Computer Science Department, Ithaca, NY, December 1986.
L. Hemachandra. Counting in Structural Complexity Theory. PhD thesis, Cornell University, Ithaca, NY, May 1987. Available as Cornell Department of Computer Science Technical Report TR87-840.
L. Hemachandra. On ranking. In Proceedings 2nd Structure in Complexity Theory Conference, pages 103–117, IEEE Computer Society Press, June 1987.
L. Hemachandra. The strong exponential hierarchy collapses. In 19th ACM Symposium on Theory of Computing, pages 110–122, May 1987.
J. Hartmanis and L. Hemachandra. Complexity classes without machines: On complete languages for UP. In Automata, Languages, and Programming (ICALP 1986), pages 123–135, Springer-Verlag Lecture Notes in Computer Science #226, July 1986. To appear in Theoretical Computer Science.
J. Hartmanis and L. Hemachandra. One-way functions, robustness, and the non-isomorphism of NP-complete sets. In Proceedings 2nd Structure in Complexity Theory Conference, pages 160–174, IEEE Computer Society Press, June 1987.
J. Hartmanis and N. Immerman. On complete problems for NP∩coNP. In Automata, Languages, and Programming (ICALP 1985), pages 250–259, Springer-Verlag Lecture Notes in Computer Science #194, 1985.
N. Immerman. Nondeterministic Space is Closed under Complement. Technical Report YALEU/DCS/TR 552, Yale University, Department of Computer Science, New Haven, CT, July 1987. To appear in STRUCTURES 1988.
J. Kadin. PNP[log n] and sparse Turing-complete sets for NP. In Proceedings 2nd Structure in Complexity Theory Conference, pages 33–40, IEEE Computer Society Press, June 1987.
R. Karp and R. Lipton. Some connections between nonuniform and uniform complexity classes. In 12th ACM Sym. on Theory of Computing, pages 302–309, 1980.
S. Kurtz, S. Mahaney, and J. Royer. Collapsing degrees. In Proceedings 27th IEEE Symposium on Foundations of Computer Science, pages 380–389, 1986.
K. Ko. Relativized polynomial time hierarchies having exactly k levels. In 20th ACM Symposium on Theory of Computing, ACM Press, May 1988.
W. Kowalczyk. Some connections between representability of complexity classes and the power of formal reasoning systems. In Mathematical Foundations of Computer Science, pages 364–369, Springer-Verlag Lecture Notes in Computer Science #176, 1984.
J. Köbler, U. Schöning, and K. Wagner. The Difference and Truth-Table Hierarchies for NP. Technical Report, Fachberichte Informatik, EWH Rheinland-Pfalz, Koblenz, West Germany, July 1986.
S. Kurtz. A Relativized Failure of the Berman-Hartmanis Conjecture. Technical Report TR83-001, University of Chicago Department of Computer Science, Chicago, IL, 1983.
K. Lange, B. Jenner, and B. Kirsig. The logarithmic alternation hierarchy collapses: AΣ L2 =AΠ L2 . In Automata, Languages, and Programming (ICALP 1987), Springer-Verlag Lecture Notes in Computer Science, 1987.
R. Ladner, N. Lynch, and A. Selman. A comparison of polynomial time reducibilities. Theoretical Computer Science, 1(2):103–124, 1975.
T. Long. A note on sparse oracles for NP. Journal of Computer and System Sciences, 24:224–232, 1982.
S. Mahaney. Sparse complete sets for NP: Solution of a conjecture of Berman and Hartmanis. Journal of Computer and System Sciences, 25(2):130–143, 1982.
C. Papadimitriou and S. Zachos. Two remarks on the power of counting. In Proceedings 6th GI Conference on Theoretical Computer Science, pages 269–276, Springer-Verlag Lecture Notes in Computer Science #145, 1983.
H. Rogers, Jr. The Theory of Recursive Functions and Effective Computability. McGraw-Hill, 1967.
U. Schöning. A low and a high hierarchy in NP. Journal of Computer and System Sciences., 27:14–28, 1983.
U. Schöning. Robust algorithms: A different approach to oracles. Theoretical Computer Science, 40:57–66, 1985.
U. Schöning. Complexity and Structure. Springer Verlag Lecture Notes in Computer Science #211, 1986.
U. Schöning. The power of counting. In Proceedings 3rd Structure in Complexity Theory Conference, IEEE Computer Society Press, June 1988. To appear.
J. Simon. On the difference between one and many. In Automata, Languages, and Programming (ICALP 1977), pages 480–491, Springer-Verlag Lecture Notes in Computer Science #52, 1977.
M. Sipser. On relativization and the existence of complete sets. In Automata, Languages, and Programming (ICALP 1982), Springer-Verlag Lecture Notes in Computer Science #140, 1982.
L. Stockmeyer. The polynomial-time hierarchy. Theoretical Computer Science, 3:1–22, 1977.
U. Schöning and K. Wagner. Collapsing oracle hierarchies, census functions, and logarithmically many queries. In STACS 1988: 5th Annual Symposium on Theoretical Aspects of Computer Science, Springer-Verlag Lecture Notes in Computer Science, February 1988.
R. Szelepcsényi. The method of forcing for nondeterministic automata. Bulletin of the EATCS, (33):96–99, 1987.
G. Tardos. Query complexity, or why is it difficult to separate NPA ∩ coNPA from PA by random oracles A. July 1987. Manuscript.
S. Toda. Σ2SPACE[n] is closed under complement. Journal of Computer and System Sciences, 35:145–152, 1987.
Jacobo Torán. Structural Properties of the Counting Hierarchies. PhD thesis, Universitat Politècnica de Catalunya, Barcelona, Spain, 1988.
L. Valiant. The relative complexity of checking and evaluating. Information Processing Letters, 5:20–23, 1976.
K. Wagner. Some Observations on the Connection Between Counting and Recursion. Technical Report MIP-8611, Universität Passau, Fakultät für Mathematik und Informatik, June 1986.
Osamu Watanabe. On hard one-way functions (Abstract). Bulletin of the European Association for Theoretical Computer Science, June 1986.
G. Wechsung. On the boolean closure of NP. In Proceedings of the 1985 International Conference on Fundamentals of Computation Theory, pages 485–493, Lecture Notes in Computer Science, Springer-Verlag, 1985.
A. Yao. Separating the polynomial-time hierarchy by oracles. In Proceedings 26th IEEE Symposium on Foundations of Computer Science, pages 1–10, 1985.
S. Zachos. Probabilistic quantifiers, adversaries, and complexity classes: An overview. In Proceedings 1st Structure in Complexity Theory Conference, pages 383–400, IEEE Computer Society Press, June 1986.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1988 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Hemachandra, L.A. (1988). Structure of complexity classes: Separations, collapses, and completeness. In: Chytil, M.P., Koubek, V., Janiga, L. (eds) Mathematical Foundations of Computer Science 1988. MFCS 1988. Lecture Notes in Computer Science, vol 324. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0017131
Download citation
DOI: https://doi.org/10.1007/BFb0017131
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-50110-7
Online ISBN: 978-3-540-45926-2
eBook Packages: Springer Book Archive