ABSTRACT
We prove super-linear lower bounds for the number of edges in constant depth circuits with n inputs and up to n outputs. Our lower bounds are proved for all types of constant depth circuits, e.g., constant depth arithmetic circuits and constant depth Boolean circuits with arbitrary gates. The bounds apply for several explicit functions, and, most importantly, for matrix product. In particular, we obtain the following results:
We show that the number of edges in any constant depth arithmetic circuit for matrix product (over any field is super-linear in m^2 (where m \times m is the size of each matrix). That is, the lower bound is super-linear in the number of input variables. Moreover, if the circuit is bilinear the result applies also for the case where the circuit gets for free any product of two linear functions.
We show that the number of edges in any constant depth arithmetic circuit for the trace of the product of 3 matrices (over fields with characteristic~0) is super-linear in m^2. (Note that the trace is a single-output function).
We give explicit examples for n Boolean functions f_1,\dots,f_ , such that any constant depth Boolean circuit with arbitrary gates for f_1,...,f_n has a super-linear number of edges. The lower bound is proved also for circuits with arbitrary gates over any finite field. The bound applies for matrix product over finite fields as well as for several other explicit functions.
- 1.M.Ajti.S11 -formul e on .nite structures.Annals of Pure and Applied Logic 1983.Google Scholar
- 2.W.B ur nd V.Strassen.The complexity of p rti l deriv tives.Theoretical Computer Science 22:317 -330, 1983.Google Scholar
- 3.M.Blaser.A 52 lower bound for the rank of non matrix multiplication over rbitrary .elds.In FOCS pages 45-50,1999. Google Scholar
- 4.N.H.Bshouty.A lower bound for matrix multiplication.SIAM Journal on Computing 18:759 -765,1989. Google Scholar
- 5.D.Dolev,C.Dwork,N.J.Pippenger,nd A.Wigderson.Superconcentrators,generalizer and generalized connectors.In STOC pages 42 -51,1983. Google Scholar
- 6.M.L.Furst,J.B.Sxe,ndM.Sipser.Prity,circuits, and the polynomial-time hierarchy.In FOCS pages 260 -270,1981.Google Scholar
- 7.J.v.z.G then.Algebraic complexity theory.Ann. Rev. Computer Science pages 317 -347,1988. Google Scholar
- 8.D.Grigoriev nd M.K rpinski.An exponential lower bound for depth 3 rithmetic circuits.In STOC pages 577 -582,1998. Google Scholar
- 9.D.Grigoriev nd A.A.R zborov.Exponential complexity lower bounds for depth 3 rithmetic circuits in algebras of functions over .nite .elds.In FOCS pages 269 -278,1998. Google Scholar
- 10.A.H jnal,W.Maass,P.Pudlak,M.Szegedy,nd G.Turan.Threshold circuits of bounded depth.In FOCS pages 99 -110,1987.Google Scholar
- 11.J.Hostad.Almost optimal lower bounds for small depth circuits.In STOC pages 6 -20,1986. Google Scholar
- 12.R.Impagliazzo,R.Paturi,and M.Saks.Size-depth trade-o .s for threshold circuits.SIAM Journal on Computing 26:693 -707,1997. Google Scholar
- 13.P.Pudl k.Communic tion in bounded depth circuits. Combinatorica 14(2):203 -216,1994.Google Scholar
- 14.A.A.Razborov.Lower bounds for the size of circuits with bounded depth with basis {., .}.Mat. Zametki 1987.Google Scholar
- 15.A.Shpilk nd A.Wigderson.Depth-3 rithmetic formul e over .elds of characteristic zero.In CCC volume 14,pages 87 -96,1999. Google ScholarDigital Library
- 16.R.Smolensky.Algebraic methods in the theory of lower bounds for Boole n circuit complexity.In STOC pages 77 -82,1987. Google Scholar
- 17.V.Strassen.Gaussian elimination is not optimal. Numer. Math 13:354 -356,1969.Google Scholar
- 18.V.Strassen.Die berechnungskomplexiat von elementarsymmetrischen funktionen und von interpolationskoe .zienten.Numer. Math 20:238 -251, 1973.Google Scholar
- 19.L.G.V liant.Graph-theoretic rguments in low-level complexity.In Lecture notes in Computer Science volume 53,pages 162 -176.1977.Google Scholar
- 20.A.C.Yao.Sep rating the polynomial hierarchy by oracles.In FOCS pages 1 -10,1985. Google Scholar
Index Terms
- Lower bounds for matrix product, in bounded depth circuits with arbitrary gates
Recommendations
Lower Bounds for Matrix Product in Bounded Depth Circuits with Arbitrary Gates
We prove superlinear lower bounds for the number of edges in constant depth circuits with n inputs and up to n outputs. Our lower bounds are proved for all types of constant depth circuits, e.g., constant depth arithmetic circuits and constant depth ...
Lower bounds for depth-2 and depth-3 Boolean circuits with arbitrary gates
CSR'08: Proceedings of the 3rd international conference on Computer science: theory and applicationsWe consider depth-2 and 3 circuits over the basis consisting of all Boolean functions. For depth-3 circuits, we prove a lower bound Ω(n log n) for the size of any circuit computing the cyclic convolution. For depth-2 circuits, a lower bound Ω(n3/2) for ...
Exponential lower bound for bounded depth circuits with few threshold gates
We prove an exponential lower bound on the size of bounded depth circuits with O(logn) threshold gates computing an explicit function (namely, the parity function). Previously exponential lower bounds were known only for circuits with one threshold ...
Comments