Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
skip to main content
10.1145/1060590.1060609acmconferencesArticle/Chapter ViewAbstractPublication PagesstocConference Proceedingsconference-collections
Article

Beyond NP: the work and legacy of Larry Stockmeyer

Published: 22 May 2005 Publication History

Abstract

Shortly after Steve Cook and Richard Karp showed the ex-istence of many natural NP-complete languages, researchers started to realize the great importance of the P versus NP problem and the difficulty of settling it. One graduate student at the Massachusetts Institute of Technology started to look beyond NP, asking what problems have a higher complexity and how do we classify them. Larry Stockmeyer discovered an amazing structure of complexity classes that continues to direct the research in complexity to this day. Stockmeyer passed away on July 31, 2004 at the age of 55 and in this paper we review some of his research and the legacy he has left on the community.

References

[1]
S. Aaronson. The complexity zoo. http://complexityzoo.com.
[2]
S. Arora, C. Lund, R. Motwani, M. Sudan, and M. Szegedy. Proof verification and the hardness of approximation problems. Journal of the ACM, 45(3):501--555, May 1998.
[3]
S. Arora and S. Safra. Probabilistic checking of proofs: A new characterization of NP. Journal of the ACM, 45(1):70--122, January 1998.
[4]
L. Babai. Trading group theory for randomness. In Proceedings of the 17th ACM Symposium on the Theory of Computing, pages 421-429. ACM, New York, 1985.
[5]
L. Babai and S. Moran. Arthur-Merlin games: a randomized proof system, and a hierarchy of complexity classes. Journal of Computer and System Sciences, 36(2):254-276, 1988.
[6]
T. Baker, J. Gill, and R. Solovay. Relativizations of the P = NP question. SIAM Journal on Computing, 4(4):431-442, 1975.
[7]
M. Bellare, O. Goldreich, and E. Petrank. Uniform generation of NP-witnesses using an NP-oracle. Information and Computation, 163:510-526, 2000.
[8]
R. Boppana, J. Hastad, and S. Zachos. Does co-NP have short interactive proofs? Information Processing Letters, 25(2):127--132, 1987.
[9]
R. Boppana and M. Sipser. The complexity of finite functions. In J. van Leeuwen, editor, Handbook of Theoretical Computer Science, chapter 14, pages 757--804. North-Holland, 1990.
[10]
R. Britt. Universe measured: We're 156 billion light-years wide!, May 2004. Space.com.
[11]
N. Bshouty, R. Cleve, R. Gavald'a, S. Kannan, and C. Tamon. Oracles and queries that are sufficient for exact learning. Journal of Computer and System Sciences, 52(3):421--433, June 1996.
[12]
J. Cai. Sp2⊆ZPPNP. In Proceedings of the 42nd IEEE Symposium on Foundations of Computer Science, pages 620--628. IEEE, New York, 2001.
[13]
A. Chandra, D. Kozen, and L. Stockmeyer. Alternation. Journal of the ACM, 28(1):114--133, 1981.
[14]
A. Chandra and L. Stockmeyer. Alternation. In Proceedings of the 17th IEEE Symposium on Foundations of Computer Science, pages 98--109. IEEE, New York, 1976.
[15]
S. Cook. The complexity of theorem-proving procedures. In Proceedings of the 3rd ACM Symposium on the Theory of Computing, pages 151--158. ACM, New York, 1971.
[16]
C. Dwork and L. Stockmeyer. Finite state verifiers I: The power of interaction. Journal of the ACM, 39(4):800--828, October 1992.
[17]
D. Eppstein. Computational complexity of games and puzzles. http://www.ics.uci.edu/~eppstein/cgt/hard.html.
[18]
S. Even and R. Tarjan. A combinatorial problem which is complete in polynomial space. Journal of the ACM, 23(4):710--719, October 1976.
[19]
R. Fagin. Generalized first-order spectra and polynomial-time recognizable sets. In R. Karp, editor, Complexity of computation, volume 7 of SIAM-AMS Proceedings, pages 43--73. Society for Industrial and Applied Mathematics, 1974.
[20]
U. Feige, S. Goldwasser, L. Lovasz, S. Safra, and M. Szegedy. Interactive proofs and the hardness of approximating cliques. Journal of the ACM, 43(2):268--292, March 1996.
[21]
A. Fraenkel, M. Garey, D. Johnson, T. Schaefer, and Y. Yesha. The complexity of checkers on an N-N board. In Proceedings of the 19th IEEE Symposium on Foundations of Computer Science, pages 55--64. IEEE, New York, 1978.
[22]
M. Furst, J. Saxe, and M. Sipser. Parity, circuits and the polynomial-time hierarchy. Mathematical Systems Theory, 17:13--27, 1984.
[23]
M. Garey and D. Johnson. Computers and Intractability. A Guide to the theory of NP-completeness. W. H. Freeman and Company, New York, 1979.
[24]
O. Goldreich, S. Micali, and A. Wigderson. Proofs that yield nothing but their validity or all languages in NP have zero-knowledge proof systems. Journal of the ACM, 38(3):691--729, 1991.
[25]
S. Goldwasser, S. Micali, and C. Rackoff. The knowledge complexity of interactive proof-systems. SIAM Journal on Computing, 18(1):186--208, 1989.
[26]
S. Goldwasser and M. Sipser. Private coins versus public coins in interactive proof systems. In S. Micali, editor, Randomness and Computation, volume 5 of Advances in Computing Research, pages 73--90. JAI Press, Greenwich, 1989.
[27]
J. Hartmanis and R. Stearns. On the computational complexity of algorithms. Transactions of the American Mathematical Society, 117:285--306, 1965.
[28]
J. H&229;stad. Almost optimal lower bounds for small depth circuits. In S. Micali, editor, Randomness and Computation, volume 5 of Advances in Computing Research, pages 143--170. JAI Press, Greenwich, 1989.
[29]
N. Immerman. Nondeterministic space is closed under complementation. SIAM Journal on Computing,17(5):935-938, 1988.
[30]
M. Jerrum, A. Sinclair, and E. Vigoda. A polynomial-time approximation algorithm for the permanent of a matrix with non-negative entries. In Proceedings of the 33rd ACM Symposium on the Theory of Computing, pages 712--721. ACM, New York, 2001.
[31]
M. Jerrum, L. Valiant, and V. Vazirani. Random generation of combinatorial structures from a uniform distribution. Theoretical Computer Science, 43:169--188, 1986.
[32]
J. Kadin. The polynomial time hierarchy collapses if the Boolean hierarchy collapses. SIAM Journal on Computing, 17(6):1263--1282, December 1988.
[33]
R. Karp. Reducibility among combinatorial problems. In R. Miller and J. Thatcher, editors, Complexity of Computer Computations, pages 85--103. Plenum Press, 1972.
[34]
R. Karp and R. Lipton. Some connections between nonuniform and uniform complexity classes. In Proceedings of the 12th ACM Symposium on the Theory of Computing, pages 302-309. ACM, 1980.
[35]
A. Klivans and D. van Melkebeek. Graph nonisomorhism has subexponential size proofs unless the polynomial-time hierarchy collapses. SIAM Journal on Computing, 31(5):1501-1526, 2002.
[36]
K. Ko. Relativized polynomial time hierarchies having exactly k levels. SIAM Journal on Computing, 18:392--408, 1989.
[37]
J. Köbler and O. Watanabe. New collapse consequences of NP having small circuits. SIAM Journal on Computing, 28(1):311-324, February 1999.
[38]
D. Kozen. On parallelism in Turing machines. In Proceedings of the 17th IEEE Symposium on Foundations of Computer Science, pages 89-97. IEEE, New York, 1976.
[39]
L. Levin. Universal'nyie perebornyie zadachi (Universal search problems: in Russian). Problemy Peredachi Informatsii, 9(3):265--266, 1973. Corrected English translation in {64}.
[40]
C. Lund, L. Fortnow, H. Karloff, and N. Nisan. Algebraic methods for interactive proof systems. Journal of the ACM, 39(4):859--868, 1992.
[41]
A. Meyer and L. Stockmeyer. The equivalence problem for regular expressions with squaring requires exponential space. In Proceedings of the 13th IEEE Symposium on Switching and Automata Theory, pages 125--129. IEEE, New York, 1972.
[42]
L. Meyer. Weak monadic second-order theory of successor is not elementary recursive. In Proceedings of the Boston University Logic Colloquium, volume 453 of Lecture Notes in Mathematics, pages 132--154. Springer, 1975.
[43]
C. Papadimitriou. Games against nature. Journal of Computer and System Sciences, 31:288--301, 1985.
[44]
F. Pohl. Beyond the Blue Event Horizon. Ballantine Books, 1980. Chapter 12.
[45]
J. Robson. N - N checkers is EXPTIME complete. SIAM Journal on Computing, 13(2):252--267, May 1984.
[46]
A. Russell and R. Sundaram. Symmetric alternation captures BPP. Computational Complexity, 7(2):152--162, 1998.
[47]
W. Ruzzo. On uniform circuit complexity. Journal of Computer and System Sciences, pages 365--383, 1981.
[48]
W. Savitch. Relationship between nondeterministic and deterministic tape classes. Journal of Computer and System Sciences, 4:177--192, 1970.
[49]
M. Schaefer. Deciding the vapnik-chervonenkis dimension is -complete. Journal of Computer and System Sciences, 58:177-182, 1999.
[50]
M. Schaefer and C. Umans. Complete in the polynomial-time hierarchy: a compendium. SIGACT News, 33(3):32--49, September 2002.
[51]
T. Schaefer. On the complexity of some two-person perfect information games. Journal of Computer and System Sciences, 16:185--225, 1978.
[52]
R. Shaltiel and C. Umans. Pseudorandomness for approximate counting and sampling. In Proceedings of the 20th IEEE Conference on Computational Complexity. IEEE Computer Society, Los Alamitos, 2005. To appear.
[53]
A. Shamir. IP = PSPACE. Journal of the ACM, 39(4):869--877, 1992.
[54]
M. Sipser. Borel sets and circuit complexity. In Proceedings of the 15th ACM Symposium on the Theory of Computing, pages 61-69. ACM, New York, 1983.
[55]
M. Sipser. A complexity theoretic approach to randomness. In Proceedings of the 15th ACM Symposium on the Theory of Computing, pages 330--335. ACM, New York, 1983.
[56]
L. Stockmeyer. The complexity of decision problems in automata theory and logic. PhD thesis, Massachusetts Institute of Technology, June 1974.
[57]
L. Stockmeyer. The polynomial-time hierarchy. Theoretical Computer Science, 3:1--22, 1977.
[58]
L. Stockmeyer. On approximation algorithms for #P. SIAM Journal on Computing, 14(4):1--13, November 1985.
[59]
L. Stockmeyer and A. Chandra. Provably difficult combinatorial games. SIAM Journal on Computing, 8(2):151--174, May 1979.
[60]
L. Stockmeyer and A. Meyer. Word problems requiring exponential time. In Proceedings of the 5th ACM Symposium on the Theory of Computing, pages 1--9. ACM, New York, 1973.
[61]
L. Stockmeyer and A. Meyer. Cosmological lower bound on the circuit complexity of a small problem in logic. Journal of the ACM, 49(6):753-784, November 2002.
[62]
R. Szelepcsényi. The method of forced enumeration for nondeterministic automata. Acta Informatica, 26:279--284, 1988.
[63]
S. Toda. PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing, 20(5):865--877, 1991.
[64]
R. Trakhtenbrot. A survey of Russian approaches to Perebor (brute-force search) algorithms. Annals of the History of Computing, 6(4):384--400, 1984.
[65]
C. Umans. Hardness of approximating Σp2 minimization problems. In Proceedings of the 40th IEEE Symposium on Foundations of Computer Science, pages 465--474. IEEE, New York, 1999.
[66]
C. Umans. The minimum equivalent DNF problem and shortest implicants. Journal of Computer and System Sciences, 63(4):597--611, December 2001.
[67]
L. Valiant. The complexity of computing the permanent. Theoretical Computer Science, 8:189--201, 1979.
[68]
L. Valiant. The complexity of reliability and enumeration problems. SIAM Journal on Computing, 8:410--421, 1979.
[69]
C. Wrathall. Complete sets and the polynomial-time hierarchy. Theoretical Computer Science, 3:23--33, 1977.
[70]
A. Yao. Separating the polynomial-time hierarchy by oracles. In Proceedings of the 26th IEEE Symposium on Foundations of Computer Science, pages 1--10. IEEE, New York, 1985.

Cited By

View all
  • (2021)Quantum advantage from energy measurements of many-body quantum systemsQuantum10.22331/q-2021-06-02-4655(465)Online publication date: 2-Jun-2021
  • (2018)Anticoncentration theorems for schemes showing a quantum speedupQuantum10.22331/q-2018-05-22-652(65)Online publication date: 22-May-2018
  • (2011)Efficient probabilistically checkable debatesProceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniques10.5555/2033252.2033298(519-529)Online publication date: 17-Aug-2011
  • Show More Cited By

Recommendations

Comments

Information & Contributors

Information

Published In

cover image ACM Conferences
STOC '05: Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
May 2005
778 pages
ISBN:1581139608
DOI:10.1145/1060590
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

Sponsors

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 22 May 2005

Permissions

Request permissions for this article.

Check for updates

Author Tags

  1. Larry Stockmeyer
  2. aternation
  3. polynomial-time hierarchy

Qualifiers

  • Article

Conference

STOC05
Sponsor:
STOC05: Symposium on Theory of Computing
May 22 - 24, 2005
MD, Baltimore, USA

Acceptance Rates

Overall Acceptance Rate 1,469 of 4,586 submissions, 32%

Contributors

Other Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

  • Downloads (Last 12 months)7
  • Downloads (Last 6 weeks)1
Reflects downloads up to 12 Sep 2024

Other Metrics

Citations

Cited By

View all
  • (2021)Quantum advantage from energy measurements of many-body quantum systemsQuantum10.22331/q-2021-06-02-4655(465)Online publication date: 2-Jun-2021
  • (2018)Anticoncentration theorems for schemes showing a quantum speedupQuantum10.22331/q-2018-05-22-652(65)Online publication date: 22-May-2018
  • (2011)Efficient probabilistically checkable debatesProceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniques10.5555/2033252.2033298(519-529)Online publication date: 17-Aug-2011
  • (2011)Efficient Probabilistically Checkable DebatesApproximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques10.1007/978-3-642-22935-0_44(519-529)Online publication date: 2011

View Options

Get Access

Login options

View options

PDF

View or Download as a PDF file.

PDF

eReader

View online with eReader.

eReader

Media

Figures

Other

Tables

Share

Share

Share this Publication link

Share on social media