Lower Bounds and Separations for Constant Depth Multilinear Circuits

We prove an exponential lower bound for the size of constant depth multilinear arithmetic circuits computing either the determinant or the permanent (a circuit is called multilinear, if the polynomial computed by each of its gates is multilinear). We also prove a super-polynomial separation between the size of product-depth d and product-depth d+1 multilinear circuits (where d is constant). That is, there exists a polynomial f such that (1) There exists a multilinear circuit of product-depth d+1 and of polynomial size computing f and (2) Every multilinear circuit of product-depth d computing f has super-polynomial size.

[1]  J. Håstad.  Computational limitations of small-depth circuits , 1987 .

[2]  Roman Smolensky,et al.  Algebraic methods in the theory of lower bounds for Boolean circuit complexity , 1987, STOC.

[3]  V. Strassen.  Die Berechnungskomplexität von elementarsymmetrischen Funktionen und von Interpolationskoeffizienten , 1973 .

[4]  Ran Raz,et al.  Multi-linear formulas for permanent and determinant are of super-polynomial size , 2004, STOC '04.

[5]  Noam Nisan,et al.  Lower bounds on arithmetic circuits via partial derivatives , 2005, computational complexity.

[6]  Eden Chlamtác,et al.  Approximation Algorithms Using Hierarchies of Semidefinite Programming Relaxations , 2007, 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07).

[7]  Michael Sipser,et al.  Parity, circuits, and the polynomial-time hierarchy , 1981, 22nd Annual Symposium on Foundations of Computer Science (sfcs 1981).

[8]  Marek Karpinski,et al.  An exponential lower bound for depth 3 arithmetic circuits , 1998, STOC '98.

[9]  Miklós Ajtai,et al.  ∑11-Formulae on finite structures , 1983, Ann. Pure Appl. Log..

[10]  Noam Nisan,et al.  Lower bounds for non-commutative computation , 1991, STOC '91.

[11]  Avi Wigderson,et al.  Depth-3 arithmetic formulae over fields of characteristic zero , 1999, Proceedings. Fourteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat.No.99CB36317).

[12]  Victor Shoup,et al.  Lower bounds for polynomial evaluation and interpolation problems , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[13]  A. Yao.  Separating the polynomial-time hierarchy by oracles , 1985 .

[14]  Alexander A. Razborov,et al.  Exponential Lower Bounds for Depth 3 Arithmetic Circuits in Algebras of Functions over Finite Fields , 2000, Applicable Algebra in Engineering, Communication and Computing.

[15]  Ran Raz.  Elusive functions and lower bounds for arithmetic circuits , 2008, STOC '08.

[16]  Ran Raz,et al.  Separation of Multilinear Circuit and Formula Size , 2006, Theory Comput..

[17]  Ran Raz,et al.  Balancing Syntactically Multilinear Arithmetic Circuits , 2008, computational complexity.

[18]  Avi Wigderson,et al.  Depth-3 arithmetic circuits over fields of characteristic zero , 2002, computational complexity.

[19]  Ran Raz,et al.  A Lower Bound for the Size of Syntactically Multilinear Arithmetic Circuits , 2007, 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07).

[20]  R. Raz.  Multilinear-NC 1 != Multilinear-NC 2 , 2004, FOCS 2004.

[21]  A. Razborov.  Lower bounds on the size of bounded depth circuits over a complete basis with logical addition , 1987 .