(67611) Advanced Topics in Complexity PCP Theory November 24, 2004 Lecture 6

(67611)Advanced Topics in Complexity:PCP Theory November24,2004Lecture6Lecturer:Irit Dinur Scribe:Sharon PeriAbstractIn this lecture we continue our discussion of locally testable and locally decodable codes.We introduce the family of Reed-M¨u ller Codes,which takes us diving into the geometry of m and the notion of Low Degree Testing.We further introduce the“lines-table”representation,and a mild yet powerful modification of local decodability,with“reject”as an optional output answer under cetain limitations1Locally Testable CodesIn previous lectures we have seen Hadamard Code,Quadratic Functions Code and Reed-Solomon Code.We have proven the Hadamard Code and Quadratic Functions Code to be both locally Testable and locally decodable.However,it was also noted that the Hadamard Code suffers from a major disadvantage in efficiency,being exponential in the input size(i.e.an input word of size n bits is encoded by2n bits).For the Quadratic Functions Code things are even a little worse,as it is exp(poly(n)).The Reed-Solomon Code on the other hand is considerably efficient,being polynomial in the input size.But is it locally testable/decodable as well?1.1Local Testability of Reed-Solomon CodeRecall the definition of Reed-Solomon Code:Definition1The Reed-Solomon code RS:k+1→n,wherein n=||,is defined for a given vector¯a=(a0,...,a k)∈k+1as the vector(P¯a(t))t∈∈n,where P¯a(x)=a0+a1x+a2x2+ ...+a k x k.An illustration of RS code word is presented in Figure1.Note that the distance of the RS code is n−k+1.How can we locally test the Reed-Solomon Code?We can no longer perform linearity testing, since it no longer applies for n>2and k>1.One idea is to query k+1entries in the code word(P¯a(t))t∈,reconstruct the polynomial P¯a by interpolation,evaluate it in k+2-th point i and compare with the i-th entry of the code word.Disadvantage:the number of queries is as the size of the input!Moreover,this lower bound of k+2queries is tight;note that there are k+1degrees of freedom for a polynomial of degree k.This excludes the Reed-Solomon Code from being locally testable, unless provided with an additional information on the input word.2Reed-M¨u ller CodeThe Reed-M¨u ller Code can be thought of as a generalization of the Hadamard Code and the Reed-Solomon Code,combining the advantages of each(efficiency of RS,local testability of Had)yet overcoming the exact same faults.Denote by P d:m→∈d[x1,x2...x m]a multivariate polynomial over m variables{x1,x2...x m}of degree≤d.The degree of a multivariate polynomiali1,i2,...i m a i1i2...i mx i11x i22...x i m m is de-fined to be the largest d for which there is a non-zero a i1i2...i m with i1+i2+...i m=d,i.e.themaximal degree of the monomials.For example,deg(x21x32)=5. Question:How many monomials are there of degree≤d over m variables?Given the definition above,one can easily see that there are d+md monomials in d[x1,x2...x m]which form a basis for this linear space.Figure 1:RS CodeDefinition 2The Reed-M¨u ller Code RM :(d +m d )→m is defined for a multivariate poly-nomial over m variables p ∈d [x 1,x 2...x m ](given by its d +m d coefficients)as RM(p )= p (x ):x ∈m ,the evaluation of p on each point of m .2.1The distance of the Reed-M¨u ller CodeProposition 3∀f,g ∈d [x 1,x 2...x m ],if f =g ,thenP rob x ∈m (f (x )=g (x ))≤dCorollary 4The distance of Reed-M¨u ller Code is ||m −d ||m −1since there are at most dlog log n).Also,for the code to have a large relative distance,we set ||=c ·d for a large enough c .Thecode word length is ||m =n O (1)which is reasonably efficient.The relative distance of the codeis 1−11Obviously,it must hold that ||>d ,otherwise the proposition is trivial.Figure2:Axis TestDefinition5Let A and P be two functions mapping a discrete domain D to a range R.Thendist(A,P)=P rob[A(x)=P(x)]x∈DIn other words,dist(A,P)is the portion of indices with different entries in A and P.Thus a low degree test,for some degree d,given a function/point-table A and a user-defined parameter ,tests whether it holds for some P of degree d that dist(A,P)< .3.1Locally testing the Reed-M¨u ller CodeOur goal is to perform a low degree testing of RM in O(d)queries to a given point-table A. For intuition purposes,wefirst outline a suggestive course of the way to solution and formalize later.The principal tool we have in hand when dealing with polynomials is interpolation.Generally, we require O(d m)points’evaluations in order to reconstruct a degree d polynomial over m variables.We wish to use only O(d)such points;so where do we start?Thefirst and probably most intuitive thing to do,is to consider an arbitrary axis i of m, algebraicly described as{x∈m:x i=t∈,x j=0∀j=i}.Thus thefirst attempt we have is3.1.1The Axis Test(See Figure2)•fix d+2random points on some arbitrary axis i•reconstruct the(restriction to the axis i of the)polynomial by interpolation using d+1of said points•evaluate on the remaining point and compare with the corresponding entry in the points tableProblem:there are only m such axes;what if most or all of their corresponding entries in the given points table are corrupted?Suppose that instead wefix some arbitrary point a=(a1,...,a m)∈F m and consider the i-th axis parallel line that goes through this point {(a1,a2,...,t,...a m)∈m:t∈}.This brings to the following most natural extension and our second attempt:Figure3:Axis-Parallel Line Test3.1.2The Axis-parallel Line Test:2(See Figure3)•pick some arbitrary axis i•pick some arbitrary point a=(a1,...,,a m)∈m•fix d+2random points on this axis-parallel line that contains a,as defined earlier•reconstruct the(restriction to the line of the)polynomial by interpolation•evaluate on the remaining point and compare with the corresponding entry in the points tableAlthough this suggestion seems fairly better than the previous,it still appears as if we are not yet exhausting all possibilities;indeed,why restricting ourselves to axis-parallel lines,when we can choose any arbitrary line in m?Let us then introduce the parametric representation of a line in the followingDefinition6A line in m is defined x,h={x+t·h}t∈wherein x∈m,h∈m\{0}.A line x,h can also be considered as a univariate function x,h:→m,defined x,h(t)= x+t·h.Consider the restriction of P to the line denoted by P| .This is a univariate function P( (t))=P(x+t·h)of degree≤d in t.Examplem=3d=4P(x,y,z)=xyz2+3z4+2x=(0,0,0)+t·(1,1,1)=(t,t,t)P( (t))=t·t·t2+3t4+2t=4t4+2tRemark Since the geometry of m might be very different from the more intuitive m it would not be a bad idea to perform some cautious checks:Figure4:Point and Line Test•Is there really exists a line between each two points of m?•Are there really d+2different points on such line?The answer to both questions is positive and easily verified:For each two different points x,y∈m set h=x−y and scale by the desired factor t∈\{0}.To see that a line contains||different points,note that ={x+th}t∈.Assume that for somet1,t2it holds that x+t1h=x+t2h.It follows that h(t1−t2)=0,and since h=¯0it must holdsthat(t1−t2)=0,and thus t1=t2.We might also like to exploit for our needs the followingLemma7deg(P)=d iff∀ ∈m deg(P| )=dProof Idea One direction is immediate;if P is of degree d then so is its restriction to anyline ∈m(which yields a univariate polynimal of degree d).The opposite direction is shownby induction,and using interpolation.3This test was implicit in[GLR+91]and further analyzed by[RS92,ALM+92,AS97].Note that it overcomes the dependence of on m we had in the axis-parallel line test.The analysis by[GLR+91,RS92]yielded =Ω(1/d). [ALM+92]combined the analysis of[RS92,AS98]to obtain =Ω(1).Using more powerful techniques,[AS97]showed the test achieves =1−o(1).[RS97]proposed an even more general test,where a random line is replaced by a random plane,and achieves =1−o(1).We will discuss this plane test in detail in the following lectures.3.2Local Decoding of Reed-M¨u ller CodeGiven a code word(table)A which only a certain portion of it is spoiled,we wish to decode a point x.Thus,our promise is that there exists a low degree polynomial P such that dist(A,P)≤ .For which settings of we can guarantee correct local decoding with high probabilty?What is the correlation/dependence of in the other parameters?Lemma8Given a table A:m→and provided that∃P∈d[x1,...x m]such thatdist(A,P)<1||m <1In summary,we have a procedure for local decoding using d+1=Θ(d=log n)queries,with dist(A,P)=Θ(14The lines tableWe have already seen that the degree d gives us a tight lower bound on the number of queries (since there are d degrees of freedom).The idea now is to extend the RM code.Instead of evaluation on each point,we add information about the restriction of the polynomial P to each of the lines x,h∈m.The code word is thus a table with an entry per line,altogether||m(||m−1)Figure5:Folding the lines table5Local-decoding with rejectWe have seen a local-decoding procedure that makes O(d)queries,and can handle words of distance<1||By applying the union bound on the probability to err(either A( )or A( )are inconsistent with the global polynomial P yet the test A( (t ))?=A( (t))passes)we have in total that P rob[M succeeds]≥1−das asserted(wherein the event of success of M is either3responding reject or decoding P(x)correctly).Part3of the theorem is harder to prove and requires somewhat heavier tools we are yet to develop in the following lectures.References[ALM+92]Sanjeev Arora,Carsten Lund,Rajeev Motwani,Madhu Sudan,and Mario Szegedy.Proof verification and hardness of approximation problems.In IEEE Symposium onFoundations of Computer Science,pages14–23,1992.[AS97]Sanjeev Arora and Madhu Sudan.Improved low-degree testing and its applications.pages485–495,1997.[AS98]Sanjeev Arora and Shmuel Safra.Probabilistic checking of proofs:a new character-ization of NP.Journal of the ACM,45(1):70–122,1998.[BFL90]L Babai,Lance Fortnow,and Carsten Lund.Non-deterministic exponential time has two-prover interactive protocols.In IEEE Symposium on Foundations of ComputerScience,pages16–25,1990.[BFLS91]L´a szl´o Babai,Lance Fortnow,Leonid A.Levin,and Mario Szegedy.Checking com-putations in polylogarithmic time.pages21–32,1991.[FGL+96]Uriel Feige,ShafiGoldwasser,Laszlo Lov´a sz,Shmuel Safra,and Mario Szegedy.Interactive proofs and the hardness of approximating cliques.Journal of the ACM,43(2):268–292,1996.[GLR+91]Peter Gemmell,Richard J.Lipton,Ronitt Rubinfeld,Madhu Sudan,and Avi Wigderson.Self-testing/correcting for polynomials and for approximate functions.In ACM Symposium on Theory of Computing,pages32–42,1991.[PS94]Alexander Polishchuk and Daniel A.Spielman.Nearly-linear size holographic proofs.In STOC’94:Proceedings of the twenty-sixth annual ACM symposium on Theoryof computing,pages194–203.ACM Press,1994.[RS92]Rubinfeld and Sudan.Self-testing polynomial functions efficiently and over rational domains.In SODA:ACM-SIAM Symposium on Discrete Algorithms(A Confer-ence on Theoretical and Experimental Analysis of Discrete Algorithms),pages23–32,1992.[RS97]Ran Raz and Shmuel Safra.A sub-constant error-probability low-degree test,and a sub-constant error-probability PCP characterization of NP.pages475–484,1997.。