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Exploring Multilinear Arithmetic Circuits: Complexity Classes and Key Results

This document provides a comprehensive overview of multilinear arithmetic circuits and their relationship with complexity classes such as NC1 and NC2. We discuss the nature of arithmetic circuits, including their size, degree, and depth. Notably, we present significant results proving that multilinear NC1 is distinct from multilinear NC2. The exploration of the rank of partial derivatives matrices associated with multilinear polynomials reveals intriguing insights into polynomial complexity. Our findings highlight the open questions surrounding the nature of arithmetic formulas versus circuits, as well as their implications in computational theory.

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Exploring Multilinear Arithmetic Circuits: Complexity Classes and Key Results

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  1. Multilinear NC1 Multilinear NC2 Ran Raz Weizmann Institute

  2. Arithmetic Circuits (and Formulas): • Field: F • Variables: X1,...,Xn • Gates: • Every gate in the circuit computes • a polynomial in F[X1,...,Xn] • Example:(X1¢ X1) ¢ (X2+ 1)

  3. Classes of Arithmetic Circuits: • NC1:Size: poly(n)Degree: poly(n) • Depth: O(log n) • (poly-size formulas) • NC2:Size: poly(n)Degree: poly(n) • Depth: O(log2n) • P:Size: poly(n) Degree: poly(n)

  4. Valiant Skyum Berkowitz Rackoff: • Arithmetic NC2 =Arithmetic P • [H]: poly-size arithmetic circuit ! • quasipoly-size arithmetic formula • Outstanding open problem: • Arithmetic NC1Arithmetic NC2 • Are arithmetic formulas weaker • than arithmetic circuits?

  5. Multilinear Circuits: • [NW]: • Every gate in the circuit computes • a multilinear polynomial • Example:(X1¢ X2) + (X2¢ X3) • (no high powers of variables)

  6. Motivation: • 1) For many functions, non-multilinear circuits are very counter-intuitive • 2) For many functions, most (or all) known circuits are multilinear • 3) Multilinear polynomials: interesting subclass of polynomials • 4)Multilinear circuits: strong subclass of circuits (contains other classes) • 5) Relations to quantum circuits[Aaronson]

  7. Previous Work : • [NW 95]:Lower bounds for a subclass of constant depth multilinear circuits • [Nis, NW, RS]: Lower bounds for other subclasses of multilinear circuits • [R 04]: Multilinear formulas for Determinant and Permanent are of size • [Aar 04]: Lower bounds for multilinear formulas for other functions

  8. Our Result: • Explicitf(X1,...,Xn), with coeff. • in{0,1},s.t., over any field: • 1) 9poly-size NC2 multilinear circuit forf • 2) Any multilinear formula forfis of size multilinear NC1 multilinear NC2

  9. Partial Derivatives Matrix [Nis]: • f=a multilinear polynomial over • {y1,...,ym} [ {z1,...,zm} • P=set of multilinear monomials in • {y1,...,ym}. |P|=2m • Q=set of multilinear monomials in • {z1,...,zm}. |Q|=2m

  10. Partial Derivatives Matrix [Nis]: • f=a multilinear polynomial over • {y1,...,ym} [ {z1,...,zm} • P=set of multilinear monomials in • {y1,...,ym}. |P|=2m • Q=set of multilinear monomials in • {z1,...,zm}. |Q|=2m • M = Mf = 2m dimensional matrix: • For every p 2 P, q 2 Q, • Mf(p,q)= coefficient ofpqinf

  11. Example: • f(y1,y2,z1,z2)=1+y1y2-y1z1z2 • Mf=

  12. Partial Derivatives Method [N,NW] • [Nis]: Iffis computed by a noncommutative formula of size s then Rank(Mf)= poly(s) • [NW,RS]:The same for other classes of formulas • Is the same true for multilinear formulas ?

  13. Counter Example: • Mfis a permutation matrix • Rank(Mf) =2m

  14. We Prove: • Partition(at random){X1,...,X2m} • ! {y1,...,ym} [ {z1,...,zm} • If f has poly-size multilinear • formula, then (w.h.p.): If for every partitionRank(Mf)=2m then any multilinear formula for f is of super-poly-size ( )

  15. High-Rank Polynomials: • Define:f(X1,..,X2m)isHigh-Rank • iffor every partitionRank(Mf)=2m f is High-Rank !any multilinear formula for f is of super-poly-size

  16. Our Result: Step 1 • Explicit f(X1,..,X2m)overC,s.t.: • 1)9poly-sizeNC2multilinearcircuit forf • 2)fis High-Rank • (coefficients different than 0,1) • (We use algebraicly independent • constants from C)

  17. Our Result: Step 2 • Explicit f(X1,..,X2m,X’1,..,X’r), with • coeff. in{0,1},and r=poly(m),s.t. • (over any field) • 1)9poly-sizeNC2multilinearcircuit forf • 2)a1,..,aralgeb. independent !f(X1,..,X2m,a1,..,ar)is High-Rank

  18. Our Result: Step 3 • If F is a finite field take F ½ G • of infinite transcendental dimension • (G contains an infinite number of • algeb. independent elements) • Step 2!lower bound overG • !lower bound overF

  19. The End

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