# Lower bounds for depth-three circuits with equals and mod-gates

## Abstract

We say an integer polynomial *p*, on Boolean inputs, weakly *m*-represents a Boolean function *f* if *p* is non-constant and is zero (mod *m*) whenever *f* is zero. In this paper we prove that if a polynomial weakly *m*-represents the Mod_{q} function on *n* inputs, where *q* and *m* are relatively prime and *m* is otherwise arbitrary, then the degree of the polynomial is *Ω(n)*. This generalizes previous results of Barrington, Beigel and Rudich [BBR] and Tsai [Tsai], which held only for constant (or slowly growing) *m*. The proof technique given here is quite different and somewhat simpler. We use a method in which the inputs are represented as complex *q*^{ th } roots of unity (following Barrington and Straubing [BS]). The representation is used to take advantage of a variant of the inverse Fourier transform and elementary properties of the algebraic integers. As a corollary of the main theorem and the proof of Toda's theorem, if *q, p* are distinct primes, any depth-three circuit which computes the Mod_{q} function, and consists of an equals gate at the output, Mod_{p}-gates at the next level, and AND-gates of small fan-in at the inputs, must be of exponential size. In terms of Turing machine complexity classes, there is an oracle *A* such that \(Mod_q P^A \nsubseteq C_ = P^{Mod_p P^A }\).

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## References

- [ABFR]J. Aspnes, R. Beigel, M. Furst, and S. Rudich, The expressive power of voting polynomials. In
*Proceedings of the 23rd Symposium on Theory of Computing*, (1991), 402–409.Google Scholar - [Al]E. Allender, A note on the power of threshold circuits. In
*Proceedings of the 30th Symposium on Foundations of Computer Science*, (1989), 580–584.Google Scholar - [Bar]D. Barrington, Bounded-width polynomial-size branching programs recognize exactly those languages in NC
^{1}, in*Journal of Computer and System Sciences***38**, (1989), 150–164.Google Scholar - [BBR]D. M. Barrington, R. Beigel, and S. Rudich, Representing Boolean functions as polynomials modulo composite numbers,
*Proceedings of the 24th ACM Symposium on Theory of Computing*(1992) 455–461.Google Scholar - [Bei]R. Beigel, When do extra majority gates help? polylog(
*n*) majority gates are equivalent to one, in*Proceedings of the 24th ACM Symposium on Theory of Computing*(1992) 450–454.Google Scholar - [BRS]R. Beigel, N. Reingold, and D. Spielman, PP is closed under intersection, in
*Proceedings of the 23rd ACM Symposium on the Theory of Computation*, (1991), 1–11. A journal version is to appear in*Journal of Computer and System Sciences*.Google Scholar - [BS]D. M. Barrington, H. Straubing, Complex polynomials and circuit lower bounds for modular counting, to appear in
*Computational Complexity*(also see LATIN '92).Google Scholar - [BT]R. Beigel, J. Tarui, On ACC. In
*Proceedings of the 32nd Symposium on Foundations of Computer Science*, (1991), 783–792.Google Scholar - [CGT]J.-Y. Cai, F. Green and T. Thierauf, On the correlation of symmetric functions, to appear in
*Mathematical Systems Theory*.Google Scholar - [FFK]S. Fenner, L. Fortnow and S. Kurtz, Gap-definable counting classes, in
*Proceedings of the Sixth Annual Conference on Structure in Complexity Theory*, IEEE Computer Society Press (1991) 30–42.Google Scholar - [GKRST]F. Green, J. Köbler, K. Regan, T. Schwentick, and J. Torán, The power of the middle bit of a #P function, to appear in
*Journal of Computer and System Sciences*. Preliminary versions appeared in*Proceedings of the 7th Structure in Complexity Theory Conference*, IEEE Computer Society Press, (1992), 111–117, and in*Proceedings of the Fourth Italian Conference on Theoretical Computer Science*, World-Scientific, Singapore, (1991), 317–329.Google Scholar - [Gre 91]F. Green, An oracle separating ⊕P from PP
^{PH},*Information Processing Letters***37**(1991) 149–153.Google Scholar - [Gre 93]F. Green, On the power of deterministic reductions to C=P, in
*Mathematical Systems Theory***26**(1993) 215–233.Google Scholar - [Has]J. Håstad, Computational limitations of small-depth circuits, the MIT press, Cambridge, 1987.Google Scholar
- [HMPST]A. Hajnal, W. Maass, P. Pudlák, M. Szegedy, and G. Turán, Threshold circuits of bounded depth, in
*Proceedings 28th Annual IEEE Symposium on Foundations of Computer Science*, IEEE Computer Society Press (1987) 99–110.Google Scholar - [IR]K. Ireland and M. Rosen, A classical introduction to modern number theory, Second Edition, Springer-Verlag, New York, 1990.Google Scholar
- [KP]M. Krause and P. Pudlák, On the computational power of depth 2 circuits with threshold and modulo gates, in
*Proceedings of the Twenty-Sixth Annual ACM Symposium on the Theory of Computing*, ACM Press (1994) 48–57.Google Scholar - [Raz]A. A. Razborov, Lower bounds on the size of bounded depth networks over a complete basis with logical addition,
*Matematicheskie Zametki***41**(1987) 598–607. English translation in*Mathematical Notes of the Academy of Sciences of the USSR***41**(1987) 333–338.Google Scholar - [Sm]R. Smolensky, Algebraic methods in the theory of lower bounds for Boolean circuit complexity, in
*Proceedings of the 19th Annual ACM Symposium on Theory of Computing*(1987) 77–82.Google Scholar - [Ta]J. Tarui, Degree complexity of Boolean functions and its applications to relativized separations, in
*Proceedings of the Sixth Annual Conference on Structure in Complexity Theory*, IEEE Computer Society Press (1991) 382–390.Google Scholar - [Tod]S. Toda. PP is as hard as the polynomial-time hierarchy. In
*SIAM Journal on Computing***20**, (1991) 865–877.Google Scholar - [Tsai]S.-C. Tsai, Lower bounds on representing Boolean functions as polynomials in
*Z*_{m}, in*Proceedings of the Eighth Annual Conference on Structure in Complexity Theory*, IEEE Computer Society Press (1993) 96–101.Google Scholar