Building The Ultimate Consistent Reader 1 Introduction l

Building The Ultimate Consistent Reader 1

Introduction l We’ve already built a consistent reader (cube-Vs. -point). . . l Except it had variables ranging over a set of polynomials instead of over the field. l In this lecture we’ll use that construction to build a perfected consistent reader. 2

Starting Point: Cube-Vs. -Point supposedly assigned the restriction of the polynomial to that cube var. point var. Each cube is actually a new domain! ? supposedly assigned the value of the polynomial in that point 3

General Framework: Sketch consistent reader repeat this process replacing the cube variables by variables with lower degree extension adding new readers composition “better” consistent reader 4

Extensions l We’ll introduce two extension procedures: – Power Substitution – Linearization Extension l Both embed the domains in new domains with higher dimension, but much lower degree. 5

Extensions l All our transformations, but the last one, will use the power substitution extension procedure. l The final reader will be created using the linearization extension. 6

Power Substitution - Example (b=3( Now, say the total degree of the polynomial deg satisfies r deg s. How many new variables do we need at most? i=0, 1, . . . t: = log 3(s+1) x 31 + x 15 = 0 i 3 xi: =x (31)3=1011 Extend this idea to general b (base) and d (dimension) (15)3=120 x 1 x 3 + x 12 x 2 = 0 Bound the degree: r/3 t deg 2 t 7

Embedding Extension Apply this transformation to every cube, where (the parameter s associated with the domain) (x 1, . . . , xd) 0 t t (x 1 b , . . , xdb ) d dt 8

Composition: How To Build A Consistent We only need to read one Reader For The New Domains? point! Hence the l For dimension of the new (having domains is constant each local reader variables from exactly one domain) generate a Cube-Vs. -Point reader. l Replace each occurrence of these variables in the original reader with the proper evaluation. l Put the new local test in conjunction with the existing local test. verify procedure Makethis sure you can takes polynomial time prove correctness. 9

What Next? l Repeat this process until (*) When will this happen? 1 - n 1/log l Recall that b=(s+1) (as long as this is not less than 2) t=log 1 - n new 1/log l s =dt(b-1)=polylog(n)s -i(1 - )n) O(log l Thus in the i’th iteration, s=2 (when -i(1 - )>0). l 10

Degree Decreases Rapidly l Let i 0: = /(1 - ). l 1 - -(i 0 -1)(1 - )>0 O(log s=2 l Hence in the (i 0 -1)’th iteration, -(i 0 -1)(1 - )n). l Then b=O(1). l Consequently, t=O(log 1 - n). l In the (i 0+1)’th iteration (*) should hold, since the dimension is constant. 11

Obtaining Linear Polynomials l When (s+dd) is small enough, we can apply another technique, called linearization extension, to obtain linear polynomials. l Which means our consistent reader relies on constant number of representation variables. 12

Linerization - Example How many new variables do we need at most? Now, say the total degree of the polynomial deg satisfies r deg s. The dimension of the polynomial is d. x 2 yz + xy 2 + z = 0 ux 2 yz : = x 2 yz uxy 2 : = xy 2 uz : = z Linear polynomial! ux 2 yz + uxy 2 + uz = 0 13

Linearization Embedding Apply this transformation to every cube (where m 1, . . . , m. M are all the degree-s dimension-d monomials): (x 1, . . . , xd) (m 1(x 1, . . , xd), . . , m. M(x 1, . . )) d M 14

Summary l Using the Cube-Vs. -Point consistent reader as a black box, we’ve managed to build an adequate consistent reader: – Each local reader depends on constant number of variables. – All variables range over the field. – The error probability is small. 15

Appendix 16

Linearization: How Many Degree-s, Dimension-d Monomials? Or equivalently - how many partitions of at most s balls into d boxes? +1 s+d objects arranged in a row d identical partitions, s identical balls d mobile partitions . . . s balls 17