Skip to main content
Log in

Fourier Concentration from Shrinkage

  • Published:
computational complexity Aims and scope Submit manuscript

Abstract

For a class \({\mathcal{F}}\) of formulas (general de Morgan or read-once de Morgan), the shrinkage exponent \({\Gamma_{\mathcal{F}}}\) is the parameter measuring the reduction in size of a formula \({F\in\mathcal{F}}\) after \({F}\) is hit with a random restriction. A Boolean function \({f\colon \{0,1\}^n\to\{1,-1\}}\) is Fourier-concentrated if, when viewed in the Fourier basis, \({f}\) has most of its total mass on “low-degree” coefficients. We show a direct connection between the two notions by proving that shrinkage implies Fourier concentration: For a shrinkage exponent \({\Gamma_{\mathcal{F}}}\), a formula \({F\in\mathcal{F}}\) of size \({s}\) will have most of its Fourier mass on the coefficients of degree up to about \({s^{1/\Gamma_{\mathcal{F}}}}\). More precisely, for a Boolean function \({f\colon\{0,1\}^n\to\{1,-1\}}\) computable by a formula of (large enough) size \({s}\) and for any parameter \({r > 0}\),

$$\sum_{A\subseteq [n]\; :\; |A|\geq s^{1/\Gamma} \cdot r} \hat{f}(A)^2\leq s\cdot{\mathscr{polylog}}(s)\cdot exp\left(-\frac{r^{\frac{\Gamma}{\Gamma-1}}}{s^{o(1)}} \right),$$

where \({\Gamma}\) is the shrinkage exponent for the corresponding class of formulas: \({\Gamma=2}\) for de Morgan formulas, and \({\Gamma=1/\log_2(\sqrt{5}-1)\approx 3.27}\) for read-once de Morgan formulas. This Fourier concentration result is optimal, to within the \({o(1)}\) term in the exponent of \({s}\).

As a standard application of these Fourier concentration results, we get that subquadratic-size de Morgan formulas have negligible correlation with parity. We also show the tight \({\Theta(s^{1/\Gamma})}\) bound on the average sensitivity of read-once formulas of size \({s}\), which mirrors the known tight bound \({\Theta(\sqrt{s})}\) on the average sensitivity of general de Morgan \({s}\)-size formulas.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • A. Ambainis, A.M. Childs, B. Reichardt, R. Špalek & S. Zhang (2007). Any And-Or formula of size \({n}\) can be evaluated in time \({n^{1/2+o(1)}}\) on a quantum computer. In Proceedings of the Forty-Eighth Annual IEEE Symposium on Foundations of Computer Science, 363–372.

  • A.E. Andreev (1987). On a method of obtaining more than quadratic effective lower bounds for the complexity of \({\pi}\)-schemes. Vestnik Moskovskogo Universiteta. Matematika 42(1), 70–73. English translation in Moscow University Mathematics Bulletin.

  • Babai L., Fortnow L., Nisan N., Wigderson A. (1993) BPP has subexponential time simulations unless EXPTIME has publishable proofs. Computational Complexity 3: 307–318

    Article  MathSciNet  MATH  Google Scholar 

  • Beals R., Buhrman H., Cleve R., Mosca M., de Wolf R. (2001) Quantum lower bounds by polynomials. Journal of the Association for Computing Machinery 48(4): 778–797

    Article  MathSciNet  MATH  Google Scholar 

  • P. Beame, R. Impagliazzo & S. Srinivasan (2012). Approximating AC 0 by Small Height Decision Trees and a Deterministic Algorithm for #AC 0SAT. In Proceedings of the Twenty-Seventh Annual IEEE Conference on Computational Complexity, 117–125.

  • A. Bernasconi, C. Damm & I. Shparlinski (2000). The average sensitivity of square-freeness. Computational Complexity 9(1), 39–51. ISSN 1016-3328. URL http://dx.doi.org/10.1007/PL00001600.

  • Blum M., Micali S. (1984) How to generate cryptographically strong sequences of pseudo-random bits. SIAM Journal on Computing 13: 850–864

    Article  MathSciNet  MATH  Google Scholar 

  • R. Boppana (1989). Amplification of probabilistic Boolean formulas. In Randomness and Computation, S. Micali, editor, volume 5 of Advances in Computer Research, 27–45. JAI Press, Greenwich, CT. (preliminary version in FOCS’85).

  • Braverman M. (2010) Polylogarithmic independence fools \({AC^0}\) circuits. Journal of the Association for Computing Machinery 57(5): 28:1–28:10

    Article  MathSciNet  MATH  Google Scholar 

  • R. Chen, V. Kabanets, A. Kolokolova, R. Shaltiel & D. Zuckerman (2015a). Mining Circuit Lower Bound Proofs for Meta-Algorithms. Computational Complexity 24(2), 333–392. URL http://dx.doi.org/10.1007/s00037-015-0100-0.

  • R. Chen, V. Kabanets & N. Saurabh (2015b). An Improved Deterministic #SAT Algorithm for Small de Morgan Formulas. Algorithmica 1–20. ISSN 0178-4617. URL http://dx.doi.org/10.1007/s00453-015-0020-z.

  • Chernoff H. (1952) A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations. Annals of Mathematical Statistics 23: 493–509

    Article  MathSciNet  MATH  Google Scholar 

  • Dubiner M., Zwick U. (1994) How Do Read-Once Formulae Shrink? Combinatorics, Probability & Computing 3: 455–469

    Article  MathSciNet  MATH  Google Scholar 

  • Fahri E., Goldstone J., Gutmann S. (2008) A quantum algorithm for the hamiltonian NAND tree. Theory of Computing 4: 169–190

    Article  MathSciNet  MATH  Google Scholar 

  • Furst M., Saxe J.B., Sipser M. (1984) Parity, Circuits, and the Polynomial-Time Hierarchy. Mathematical Systems Theory 17(1): 13–27

    Article  MathSciNet  MATH  Google Scholar 

  • A. Ganor, I. Komargodski, T. Lee & R. Raz (2012). On the Noise Stability of Small De Morgan Formulas. Electronic Colloquium on Computational Complexity TR12-174.

  • P. Gopalan, R. Meka, O. Reingold, L. Trevisan & S. Vadhan (2012). Better pseudorandom generators via milder pseudorandom restrictions. In Proceedings of the Fifty-Third Annual IEEE Symposium on Foundations of Computer Science, 120–129.

  • J. Håstad (1986). Almost optimal lower bounds for small depth circuits. In Proceedings of the Eighteenth Annual ACM Symposium on Theory of Computing, 6–20.

  • Håstad J. (1998) The Shrinkage Exponent Of De Morgan Formulae Is 2. SIAM Journal on Computing 27: 48–64

    Article  MathSciNet  Google Scholar 

  • J. Håstad (2014). On the Correlation of Parity and Small-Depth Circuits. SIAM J. Comput. 43(5), 1699–1708. URL http://dx.doi.org/10.1137/120897432.

  • Håstad J., Impagliazzo R., Levin L., Luby M. (1999) A pseudorandom generator from any one-way function. SIAM Journal on Computing 28: 1364–1396

    Article  MathSciNet  MATH  Google Scholar 

  • Håstad J., Razborov A.A., Yao A.C. (1995) On the Shrinkage Exponent for Read-Once Formulae. Theoretical Computer Science 141(1&2): 269–282

    Article  MathSciNet  MATH  Google Scholar 

  • Hoeffding W. (1963) Probability inequalities for sums of bounded random variables. Journal of the American Statistical Association 58(301): 13–30

    Article  MathSciNet  MATH  Google Scholar 

  • R. Impagliazzo & V. Kabanets (2014). Fourier concentration from shrinkage. In Proceedings of the Twenty-Ninth IEEE Annual Conference on Computational Complexity, 321–332.

  • R. Impagliazzo, W. Matthews & R. Paturi (2012a). A satisfiability algorithm for AC0. In Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, 961–972.

  • R. Impagliazzo, R. Meka & D. Zuckerman (2012b). Pseudorandomness from shrinkage. In Proceedings of the Fifty-Third Annual IEEE Symposium on Foundations of Computer Science, 111–119.

  • R. Impagliazzo & A. Wigderson (1997). P=BPP if E requires exponential circuits: Derandomizing the XOR Lemma. In Proceedings of the Twenty-Ninth Annual ACM Symposium on Theory of Computing, 220–229.

  • Kabanets V., Impagliazzo R. (2004) Derandomizing polynomial identity tests means proving circuit lower bounds. Computational Complexity 13(1–2): 1–46

    Article  MathSciNet  MATH  Google Scholar 

  • J. Kahn, G. Kalai & N. Linial (1988). The influence of variables on Boolean functions (extended abstract). In Proceedings of the Twenty-Ninth Annual IEEE Symposium on Foundations of Computer Science, 68–80.

  • Kannan R. (1982) Circuit-size lower bounds and non-reducibility to sparse sets. Information and Control 55: 40–56

    Article  MathSciNet  MATH  Google Scholar 

  • Karp R.M., Lipton R.J. (1982) Turing machines that take advice. L’Enseignement Mathématique 28(3-4): 191–209

    MathSciNet  MATH  Google Scholar 

  • V.M. Khrapchenko (1971). A method of determining lower bounds for the complexity of \({\pi}\)-schemes. Matematicheskie Zametki 10(1), 83–92. English translation in Mathematical Notes of the Academy of Sciences of the USSR.

  • I. Komargodski & R. Raz (2013). Average-case lower bounds for formula size. In Proceedings of the Forty-Fifth Annual ACM Symposium on Theory of Computing, 171–180.

  • I. Komargodski, R. Raz & A. Tal (2013). Improved Average-Case Lower Bounds for DeMorgan Formula Size. Electronic Colloquium on Computational Complexity 20(58).

  • T. Lee (2009). A note on the sign degree of formulas. CoRR arXiv:0909.4607.

  • Linial N., Mansour Y., Nisan N. (1993) Constant Depth Circuits, Fourier Transform and Learnability. Journal of the Association for Computing Machinery 40(3): 607–620

    Article  MathSciNet  MATH  Google Scholar 

  • Y. Mansour (1995). An \({O(n^{\log\log n})}\) Learning Algorithm for DNF under the Uniform Distribution. J. Comput. Syst. Sci. 50(3), 543–550. URL http://dx.doi.org/10.1006/jcss.1995.1043.

  • Nisan N., Wigderson A. (1994) Hardness vs. Randomness. Journal of Computer and System Sciences 49: 149–167

    Article  MathSciNet  MATH  Google Scholar 

  • R. O’Donnell (2014). Analysis of Boolean Functions. Cambridge University Press. ISBN 978-1-10-703832-5.

  • Paterson M., Zwick U. (1993) Shrinkage of de Morgan Formulae under Restriction. Random Structures and Algorithms 4(2): 135–150

    Article  MathSciNet  MATH  Google Scholar 

  • B. Reichardt (2009). Span programs and quantum query complexity: The general adversary bound is nearly tight for every Boolean function. In Proceedings of the Fiftieth Annual IEEE Symposium on Foundations of Computer Science, 544–551.

  • B. Reichardt (2011). Reflections for quantum query algorithms. In Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA ’11, 560–569.

  • B. Reichardt & R. Špalek (2008). Span-program-based quantum algorithms for evaluating formulas. In Proceedings of the Fortieth Annual ACM Symposium on Theory of Computing, 103–112.

  • R. Santhanam (2010). Fighting Perebor: New and Improved Algorithms for Formula and QBF Satisfiability. In Proceedings of the Fifty-First Annual IEEE Symposium on Foundations of Computer Science, 183–192.

  • R.E. Schapire (1994). Learning probabilistic read-once formulas on product distributions. Machine Learning 14(1), 47–81. ISSN 0885-6125. URL http://dx.doi.org/10.1007/BF00993162.

  • K. Seto & S. Tamaki (2012). A Satisfiability Algorithm and Average-Case Hardness for Formulas over the Full Binary Basis. In Proceedings of the Twenty-Seventh Annual IEEE Conference on Computational Complexity, 107–116.

  • B.A. Subbotovskaya (1961). Realizations of linear function by formulas using \({\vee}\), \({\&}\), \({^-}\). Doklady Akademii Nauk SSSR 136(3), 553–555. English translation in Soviet Mathematics Doklady.

  • A. Tal (2014). Shrinkage of De Morgan Formulae by Spectral Techniques. In 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, Philadelphia, PA, USA, October 18-21, 2014, 551–560. URL http://dx.doi.org/10.1109/FOCS.2014.65.

  • A. Tal (2015). #SAT Algorithms from Shrinkage. Electronic Colloquium on Computational Complexity (ECCC) 22, 114. URL http://eccc.hpi-web.de/report/2015/114.

  • L. Trevisan & T. Xue (2013). A Derandomized Switching Lemma and an improved Derandomization of AC\({^0}\). In Proceedings of the Twenty-Eighth Annual IEEE Conference on Computational Complexity, 242–247.

  • Umans C. (2003) Pseudo-random generators for all hardnesses. Journal of Computer and System Sciences 67(2): 419–440

    Article  MathSciNet  MATH  Google Scholar 

  • Valiant L.G. (1984a) Short Monotone Formulae for the Majority Function. Journal of Algorithms 5(3): 363–366

    Article  MathSciNet  MATH  Google Scholar 

  • Valiant L.G. (1984b) A theory of the learnable. Communications of the ACM 27(11): 1134–1142

    Article  MATH  Google Scholar 

  • R. Williams (2013). Improving Exhaustive Search Implies Superpolynomial Lower Bounds. SIAM J. Comput. 42(3), 1218–1244. URL http://dx.doi.org/10.1137/10080703X.

  • R. Williams (2014). Nonuniform ACC Circuit Lower Bounds. J. ACM 61(1), 2:1–2:32. URL http://doi.acm.org/10.1145/2559903.

  • R. de Wolf (2008). A Brief Introduction to Fourier Analysis on the Boolean Cube. Number 1 in Graduate Surveys. Theory of Computing Library, 1–20. URL http://www.theoryofcomputing.org/library.html.

  • A.C. Yao (1982). Theory and applications of trapdoor functions. In Proceedings of the Twenty-Third Annual IEEE Symposium on Foundations of Computer Science, 80–91.

  • A.C. Yao (1985). Separating the polynomial-time hierarchy by oracles. In Proceedings of the Twenty-Sixth Annual IEEE Symposium on Foundations of Computer Science, 1–10.

  • F. Zane (1998). Circuits, CNFs, and Satisfiability. Ph.D. thesis, UCSD.

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Valentine Kabanets.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Impagliazzo, R., Kabanets, V. Fourier Concentration from Shrinkage. comput. complex. 26, 275–321 (2017). https://doi.org/10.1007/s00037-016-0134-y

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00037-016-0134-y

Keywords

Subject classification

Navigation