Abstract
We consider the problem of bounding the correlation between parity and modular polynomials over ℤ q , for arbitrary odd integer q ≥3. We prove exponentially small upper bounds for classes of polynomials with certain linear algebraic properties. As a corollary, we obtain exponential lower bounds on the size necessary to compute parity by depth-3 circuits of certain form. Our technique is based on a new representation of the correlation using exponential sums.
Our results include Goldmann’s result [Go] on the correlation between parity and degree one polynomials as a special case. Our general expression for representing correlation can be used to derive the bounds of Cai, Green, and Thierauf [CGT] for symmetric polynomials, using ideas of the [CGT] proof. The classes of polynomials for which we obtain exponentially small upper bounds include polynomials of large degree and with a large number of terms, that previous techniques did not apply to.
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References
Ajtai, M.: \(\Sigma^1_1\)-formulae on finite structures. Ann. Pure Appl. Logic 24, 1–48 (1983)
Allender, E.: A note on the power of threshold circuits. In: Proceedings of the 30th Annual IEEE Symposium on Foundations of Computer Science, pp. 580–584 (1989)
Alon, N., Beigel, R.: Lower bounds for approximations by low degree polynomials over ℤ m . In: Proceedings of the 16th Annual IEEE Conference on Computational Complexity, pp. 184–187 (2001)
Barrington, D.: Bounded-width polynomial size branching programs recognize exactly those languages in NC 1. Journal of Computer and System Sciences 38, 150–164 (1989)
Beigel, R., Maciel, A.: Upper and lower bounds for some depth-3 circuit classes. In: Proceedings of the 30th Symposium on Foundations of Computer Science, pp. 580–584 (1989)
Bourgain, J.: Estimation of certain exponential sums arising in complexity theory. C.R. Acad. Sci. Paris I 340, 627–631 (2005)
Cai, J.-Y., Green, F., Thierauf, T.: On the correlation of symmetric functions. Mathematical Systems Theory 29, 245–258 (1996)
Furst, M., Saxe, J., Sipser, M.: Parity, circuits, and the polynomial hierarchy. Mathematical Systems Theory 17, 13–27 (1984)
Green, F.: Exponential sums and circuits with single threshold gate and mod-gates. Theory Computing Systems 32, 453–466 (1999)
Green, F.: The Correlation between parity and quadratic polynomials mod 3. In: Proceedings of the 17th Annual IEEE Conference on Computational Complexity, pp. 65–72 (2002)
Green, F., Roy, A., Straubing, H.: Bounds on an exponential sum arising in boolean circuit complexity. C.R. Acad. Sci. Paris I 340 (2005)
Goldmann, M.: A note on the power of majority gates and modular gates. Information Processing Letters 53, 321–327 (1995)
Grolmusz, V.: A weight-size tradeoff for circuits with mod m gates. In: Proceedings of the 26th ACM Symposium on the Theory of Computing, pp. 68–74 (1994)
Grolmusz, V., Tardos, G.: Lower bounds for MOD p – MOD m circuits. SIAM J. Comput. 29(4), 1209–1222 (2000)
Hajnal, A., Maass, W., Pudlák, P., Szegedy, M., Turán, G.: Threshold Circuits of bounded depth. In: Proceedings of the 28th Annual IEEE Symposium on Foundations of Computer Science, pp. 99–110 (1987)
Hansen, K., Miltersen, P.: Some meet-in-the-middle circuit lower bounds. In: Fiala, J., Koubek, V., Kratochvíl, J. (eds.) MFCS 2004. LNCS, vol. 3153, pp. 334–345. Springer, Heidelberg (2004)
Håstad, J.: Computational Limitations of Small-Depth Circuits. MIT Press, Cambridge (1986)
Håstad, J., Goldmann, M.: On the power of small depth threshold circuits. Computational Complexity 1(2), 113–129 (1991)
Krause, M., Pudlák, P.: On the computational power of depth 2 circuits with threshold and modulo gates. In: Proceedings of the 26th ACM Symposium on the Theory of Computing, pp. 48–57 (1994)
Lidl, R., Niederreiter, H.: Introduction to Finite Fields and Their Applications. Cambridge University Press, Cambridge
Razborov, A., Wigderson, A.: nΩ(logn) Lower bounds on the size of depth-3 threshold circuits with AND gates at the bottom. Information Processing Letters 45, 303–307 (1993)
Smolensky, R.: Algebraic methods in the theory of lower bounds for boolean circuit complexity. In: Proceedings of the 19th Annual ACM Symposium on Theory of Computing, pp. 77–82 (1987)
Szegö, G.: Orthogonal Polynomials. American Mathematical Society, Providence (1939)
Yao, A.: Separating the polynomial hierarchy by oracles. In: Proceedings of the 26th Annual IEEE Symposium on Foundations of Computer Science, pp. 1–10 (1985)
Yao, A.: On ACC and threshold circuits. In: Proceedings of the 31th Annual IEEE Symposium on Foundations of Computer Science, pp. 619–627 (1990)
Zabek, S.: Sur la périodicité modulo m des suites de nombres \({n\choose k}\). Ann. Univ. Mariae Curie Sklodowska A10, 37–47 (1956)
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Gál, A., Trifonov, V. (2006). On the Correlation Between Parity and Modular Polynomials. In: Královič, R., Urzyczyn, P. (eds) Mathematical Foundations of Computer Science 2006. MFCS 2006. Lecture Notes in Computer Science, vol 4162. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11821069_34
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DOI: https://doi.org/10.1007/11821069_34
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