skip to main content
10.1145/1374376.1374481acmconferencesArticle/Chapter ViewAbstractPublication PagesstocConference Proceedingsconference-collections
research-article

Algebrization: a new barrier in complexity theory

Published: 17 May 2008 Publication History

Abstract

Any proof of P!=NP will have to overcome two barriers: relativization and natural proofs. Yet over the last decade, we have seen circuit lower bounds (for example, that PP does not have linear-size circuits) that overcome both barriers simultaneously. So the question arises of whether there is a third barrier to progress on the central questions in complexity theory.
In this paper we present such a barrier, which we call algebraic relativization or algebrization. The idea is that, when we relativize some complexity class inclusion, we should give the simulating machine access not only to an oracle A, but also to a low-degree extension of A over a finite field or ring.
We systematically go through basic results and open problems in complexity theory to delineate the power of the new algebrization barrier. First, we show that all known non-relativizing results based on arithmetization -- both inclusions such as IP=PSPACE and MIP=NEXP, and separations such as MAEXP not in P/poly -- do indeed algebrize. Second, we show that almost all of the major open problems -- including P versus NP, P versus RP, and NEXP versus P/poly -- will require non-algebrizing techniques. In some cases algebrization seems to explain exactly why progress stopped where it did: for example, why we have superlinear circuit lower bounds for PromiseMA but not for NP.
Our second set of results follows from lower bounds in a new model of algebraic query complexity, which we introduce in this paper and which is interesting in its own right. Some of our lower bounds use direct combinatorial and algebraic arguments, while others stem from a surprising connection between our model and communication complexity. Using this connection, we are also able to give an MA-protocol for the Inner Product function with O(sqrt(n) log n) communication (essentially matching a lower bound of Klauck).

References

[1]
S. Aaronson. Oracles are subtle but not malicious. In Proc. IEEE Complexity, p. 340--354, 2006.
[2]
S. Arora, R. Impagliazzo, and U. Vazirani. Relativizing versus nonrelativizing techniques: the role of local checkability. Manuscript, 1992.
[3]
L. Babai, L. Fortnow, and C. Lund. Nondeterministic exponential time has two-prover interactive protocols. Computational Complexity, 1(1):3--40, 1991.
[4]
T. Baker, J. Gill, and R. Solovay. Relativizations of the P=?NP question. SIAM J. Comput., 4:431--442, 1975.
[5]
H. Buhrman, L. Fortnow, and T. Thierauf. Nonrelativizing separations. In Proc. IEEE Complexity, p. 8--12, 1998.
[6]
H. Buhrman, N. Vereshchagin, and R. de Wolf. On computation and communication with small bias. In Proc. IEEE Complexity, p. 24--32, 2007.
[7]
H. Buhrman and R. de Wolf. Complexity measures and decision tree complexity: a survey. Theoretical Comput. Sci., 288:21--43, 2002.
[8]
A. K. Chandra, D. Kozen, and L. J. Stockmeyer. Alternation. J. ACM, 28(1):114--133, 1981.
[9]
L. Fortnow. The role of relativization in complexity theory. Bulletin of the EATCS, 52:229--244, February 1994.
[10]
O. Goldreich, S. Micali, and A. Wigderson. Proofs that yield nothing but their validity or all languages in NP have zero-knowledge proof systems. J. ACM, 38(1):691--729, 1991.
[11]
J. Hartmanis, R. Chang, S. Chari, D. Ranjan, and P. Rohatgi. Relativization: a revisionistic perspective. Bulletin of the EATCS, 47:144--153, 1992.
[12]
J. Hartmanis and R. E. Stearns. On the computational complexity of algorithms. Transactions of the American Mathematical Society, 117:285--306, 1965.
[13]
J. E. Hopcroft, W. J. Paul, and L. G. Valiant. On time versus space. J. ACM, 24(2):332--337, 1977.
[14]
A. Juma, V. Kabanets, C. Rackoff, and A. Shpilka. The black-box query complexity of polynomial summation. ECCC TR07-125, 2007.
[15]
B. Kalyanasundaram and G. Schnitger. The probabilistic communication complexity of set intersection. SIAM J. Discrete Math, 5(4):545--557, 1992.
[16]
R. Kannan. Circuit-size lower bounds and non-reducibility to sparse sets. Information and Control, 55:40--56, 1982.
[17]
H. Klauck. Rectangle size bounds and threshold covers in communication complexity. In Proc. IEEE Complexity, p. 118--134, 2003.
[18]
A. Klivans and D. van Melkebeek. Graph nonisomorphism has subexponential size proofs unless the polynomial-time hierarchy collapses. SIAM J. Comput., 31:1501--1526, 2002.
[19]
C. Lund, L. Fortnow, H. Karloff, and N. Nisan. Algebraic methods for interactive proof systems. J. ACM, 39:859--868, 1992.
[20]
R. Raz. Exponential separation of quantum and classical communication complexity. In Proc. ACM STOC, p. 358--367, 1999.
[21]
R. Raz and A. Shpilka. On the power of quantum proofs. In Proc. IEEE Complexity, p. 260--274, 2004.
[22]
A. A. Razborov. Lower bounds for the size of circuits of bounded depth with basis łeft &,øø + ->. Mathematicheskie Zametki, 41(4):598--607, 1987.
[23]
A. A. Razborov. On the distributional complexity of disjointness. Theoretical Comput. Sci., 106:385--390, 1992.
[24]
A. A. Razborov. Quantum communication complexity of symmetric predicates. Izvestiya Math., 67(1):145--159, 2003.
[25]
A. A. Razborov and S. Rudich. Natural proofs. J. Comput. Sys. Sci., 55(1):24--35, 1997.
[26]
R. Santhanam. Circuit lower bounds for Merlin-Arthur classes. In Proc. ACM STOC, p. 275--283, 2007.
[27]
A. Shamir. IP=PSPACE. J. ACM, 39(4):869--877, 1992.
[28]
S. Toda. PP is as hard as the polynomial-time hierarchy. SIAM J. Comput., 20(5):865--877, 1991.
[29]
N. V. Vinodchandran. A note on the circuit complexity of PP. ECCC TR04-056, 2004.
[30]
C. B. Wilson. Relativized circuit complexity. J. Comput. Sys. Sci., 31(2):169--181, 1985.
[31]
A. C-C. Yao. How to generate and exchange secrets (extended abstract). In Proc. IEEE FOCS, p. 162--167, 1986.

Cited By

View all

Recommendations

Comments

Information & Contributors

Information

Published In

cover image ACM Conferences
STOC '08: Proceedings of the fortieth annual ACM symposium on Theory of computing
May 2008
712 pages
ISBN:9781605580470
DOI:10.1145/1374376
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: 17 May 2008

Permissions

Request permissions for this article.

Check for updates

Author Tags

  1. arithmetization
  2. communication complexity
  3. interactive proofs
  4. low-degree polynomials
  5. oracles
  6. query complexity

Qualifiers

  • Research-article

Conference

STOC '08
Sponsor:
STOC '08: Symposium on Theory of Computing
May 17 - 20, 2008
British Columbia, Victoria, Canada

Acceptance Rates

STOC '08 Paper Acceptance Rate 80 of 325 submissions, 25%;
Overall Acceptance Rate 1,469 of 4,586 submissions, 32%

Upcoming Conference

STOC '25
57th Annual ACM Symposium on Theory of Computing (STOC 2025)
June 23 - 27, 2025
Prague , Czech Republic

Contributors

Other Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

  • Downloads (Last 12 months)11
  • Downloads (Last 6 weeks)2
Reflects downloads up to 18 Feb 2025

Other Metrics

Citations

Cited By

View all
  • (2024)Lower Bounds for Levin–Kolmogorov ComplexityTheory of Cryptography10.1007/978-3-031-78011-0_7(191-221)Online publication date: 2-Dec-2024
  • (2017)An adaptivity hierarchy theorem for property testingProceedings of the 32nd Computational Complexity Conference10.5555/3135595.3135622(1-25)Online publication date: 9-Jul-2017
  • (2017)Complexity-theoretic foundations of quantum supremacy experimentsProceedings of the 32nd Computational Complexity Conference10.5555/3135595.3135617(1-67)Online publication date: 9-Jul-2017
  • (2017)Arithmetic CryptographyJournal of the ACM10.1145/304667564:2(1-74)Online publication date: 15-Apr-2017
  • (2017)Block-symmetric polynomials correlate with parity better than symmetricComputational Complexity10.1007/s00037-017-0153-326:2(323-364)Online publication date: 1-Jun-2017
  • (2016)Average-case lower bounds and satisfiability algorithms for small threshold circuitsProceedings of the 31st Conference on Computational Complexity10.5555/2982445.2982446(1-35)Online publication date: 29-May-2016
  • (2016)Cell-probe lower bounds for dynamic problems via a new communication modelProceedings of the forty-eighth annual ACM symposium on Theory of Computing10.1145/2897518.2897556(362-374)Online publication date: 19-Jun-2016
  • (2016)Amortized Dynamic Cell-Probe Lower Bounds from Four-Party Communication2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS.2016.41(305-314)Online publication date: Oct-2016
  • (2015)Arithmetic CryptographyProceedings of the 2015 Conference on Innovations in Theoretical Computer Science10.1145/2688073.2688114(143-151)Online publication date: 11-Jan-2015
  • (2014)Annotations in Data StreamsACM Transactions on Algorithms10.1145/263692411:1(1-30)Online publication date: 25-Aug-2014
  • Show More Cited By

View Options

Login options

View options

PDF

View or Download as a PDF file.

PDF

eReader

View online with eReader.

eReader

Figures

Tables

Media

Share

Share

Share this Publication link

Share on social media