1. Survey on Polynomial Identity Testing Amir Shpilka Technion and MSR NE June 18, 2011 PIT Survey – CSR 1

2. Exam n=1. Is the following polynomial identically 0? June 18, 2011 2 Prove it! Will do so later.

3. Goal of talk Survey known results Explain some proof techniques Give an interesting set of `accessible’ open questions June 18, 2011 3

4. Talk outline Definition of the problem Connection to lower bounds (hardness) Survey of positive results Some proofs Connection to polynomial factorization June 18, 2011 4

5. Playground: Arithmetic Circuits Field:  Variables: X1,...,Xn Gates: ,  Every gate in the circuit computes a polynomial in [X1,...,Xn] Example:(X1 X2)  (X2 + 1) Size = number of wires Depth = length of longest input-output path Degree = max degree of internal gates June 18, 2011 5

6. Arithmetic Formulas Same, except underlying graph is a tree June 18, 2011 6

7. Bounded depth circuits  circuits: depth-2 circuits with + at the top and  at the bottom. Size s circuits compute s-sparse polynomials.  circuits: depth-3 circuits with + at the top,  at the middle and + at the bottom. Compute sums of products of linear functions. I.e. a sparse polynomial composed with a linear transformation.  circuits: depth-4 circuits. Compute sums of products of sparse polynomials. June 18, 2011 7

8. Why Arithmetic Circuits? Most natural model for computing polynomials For many problems (e.g. Matrix Multiplication, Det) best algorithm is an arithmetic circuit Great algorithmic achievements: Fourier Transform Matrix Multiplication Polynomial Factorization Structured model (compared to Boolean circuits) P vs. NP may be easier June 18, 2011 8

9. Polynomial Identity Testing Input:Arithmetic circuit computing f Problem: Is f0? f(x1,...,xn) C x1 x2 xn Randomized algorithm [Schwartz, Zippel, DeMillo-Lipton]: evaluate f at a random point Goal: A proof. I.e., a deterministic algorithm June 18, 2011 9

10. Black Box PIT Explicit Hitting Set Input: A Black-Box circuit computing f. Problem:Is f=0 ? f(x1,...,xn) (x1,...,xn) C Goal: deterministic algorithm (a.k.a. Hitting Set) S,Z,DM-L:  small Hitting Set (not explicit) June 18, 2011 10

11. Motivation Natural and fundamental problem Strong connection to circuit lower bounds Algorithmic importance: Primality testing [Agrawal-Kayal-Saxena] Parallel algorithms for finding matching [Karp-Upfal-Wigderson, Mulmuley-Vazirani-Vazirani] May help you solve exams! June 18, 2011 11

13. Hardness: PIT  lower bounds [Kabanets-Impagliazzo]: 2(n) lower bound for Permanent PIT in npolylog(n)time PIT  P  super-polynomial lower bounds: Boolean for NEXP or arithmetic for Permanent [Dvir-S-Yehudayoff]:(almost) same as K-I for bounded depth circuits [Heintz-Schnorr,Agrawal]: Polynomial time Black-Box PIT  Exponential lower bounds for arithmetic circuits Lesson: Derandomizing PIT essentially equivalent to proving lower bounds for arithmetic circuits June 18, 2011 13

14. Black-Box PIT  Lower Bounds [Heintz-Schnorr,Agrawal]: BB PIT for size s circuits in time poly(s) (i.e. poly(s) size hitting set)  exp. lower bounds for arithmetic circuits. Proof: Given ={pi}, find non-zero polynomial f inlog(||)variablse, such that f(pi)=0 for all i.  f does not have size s circuits  Gives lower bounds for f in EXP (PSPACE) Conjecture[Agrawal]: ={(y1,…, yn) : yi=yki mod r, k,r < s20} is a hitting set for size s circuits June 18, 2011 14

15. Lower Bounds  Black-Box PIT [Kabanets-Impagliazzo]: arithmetic version of N-W generator: Combinatorial design: S1,…, Sn [m]s.t. |Si| = k and |Si Sj| < log n Assumef is a k-variate polynomial of arithmetic circuit complexity 2(k) For every ymgeneratewnas wi = f(y|Si) I.e. = {(f(y|S1),…, f(y|Sn) : ym} Proof Idea: C0 iff (roughly) for some i, xi – f(y|Si) is a factor of C. Gives circuit for f. June 18, 2011 15

16. Importance of  circuits [Agrawal-Vinay,Raz]: Exponential lower bounds for  circuitsimply exponential lower bounds for general circuits. Proof: 1. Depth reduction a-la P=NC2[Valiant-Skyum-Berkowitz-Rackoff] 2. Break the circuit in the middle and interpolate each part using  circuits. Cor[Agrawal-Vinay]: Polynomial time PIT of  circuits gives quasi-polynomial time PIT for general circuits. Proof: By [Heintz-Schnorr,Agrawal] polynomial time PIT  exponential lower bounds for  circuits. By [Agrawal-Vinay, Raz] exponential lower bounds for general circuits. Now use [K-I]. June 18, 2011 16

18. Connection to lower bounds (hardness)Survey of positive results Some proofs Connection to polynomial factorization June 18, 2011 17

19. Deterministic algorithms for PIT  circuits (a.k.a.,sparse polys) [BenOr-Tiwari, Grigoriev-Karpinski, Klivans-Spielman,…] Black-Box in polynomial time Non-commutative formulas [Raz-S] White-Box in polynomial time (k) circuits [Dvir-S,Kayal-Saxena,Karnin-S,Kayal-Saraf,Saxena-Seshadri] Black-Box in time nO(k) Read-k multilinear formulas [S-Volkovich, Anderson-van Melkebeek-Volkovich] White-Box in time nkO(k) Black-Box in nO(log(n)+kO(k)) Multilinear (k)[Karnin-Mukhopadhyay-S-Volkovich, Saraf-Volkovich] Black-Box in time npoly(k) June 18, 2011 18

20. Why study restricted models [Agrawal-Vinay] PIT for  circuits implies PIT for general depth. Gaining insight into more general questions: Intuitively:lower bounds imply PIT Multilinear formulas: super polynomial bounds [Raz,Raz-Yehudayoff] but no PIT algorithms Not even for Depth-3 multilinear formulas! Read-k, depth-3,4 multilinear formulas relaxations of the more general problem Interesting results: Structural theorems for (k) and (k) circuits. June 18, 2011 19

22. Connection to lower bounds (hardness)

23. Survey of positive resultsSome proofs: Depth-3 circuits Depth-4 circuits Connection to polynomial factorization June 18, 2011 20

24. Proofs – tailored for the model Proofs usually use `weakness’ inherent in model Depth 2: few monomials. Substituting yai to xiwe can control `collapses’ of different monomials. Non Commutative formulas: Polynomial has few linearly independent partial derivatives [Nisan]. Keep track of a basis for derivatives to do PIT. (k): setting a linear function to zero reduces top fan-in. If k=2 then multiplication gates must be the same. Calls for induction. Multilinear(k): in some sense `combination’ of sparse polynomials and multilnear(k). Read-k-Formulas: subformulas of root contain ½ of variables and are (roughly) read-(k-1). June 18, 2011 21

25. Solution to Exam n=1. Is the following polynomial identically 0? June 18, 2011 22 Prove it! Will do so later now

27. Depth 3 identities C = M1+ … + MkMi = j=1...diLi,j Rank: dimension of space spanned by {Li,j} In the exam: Rank=3 Turns out: this is (almost) the general case! Theorem [Dvir S]: If C  0 is a basic identity then dim(C) ≤ Rank(k,d) = (log(d))k White-Box Algorithm: find partition to sub-circuits of low dimension (after removal of g.c.d.) and brute force verify that they vanish. Improved nO(k) algorithm by [Kayal-Saxena]. June 18, 2011 25

28. Black-Box PIT Black-Box Algorithm[KarninS]:Intuitively, if we project the inputs to a `low’ dimensional space in a way that does not collapse the dimension below Rank(k,d) then the circuit should not become zero. Theorem [GabizonRaz]:  "small" explicit set of D-dimensional subspaces V1,...,Vm such that space of linear functions :dim(|Vi) = min(dim(),D)for most i June 18, 2011 26 In other words: the linear functions in  remain as independent as possible on Vi

29. Black-Box PIT If C’ has the same rank as C’|V, then C’ and C’|V are isomorphic. Hence, C’|V  0 C  0 Corollary: i, C|Vihas low "rank“ Chas low "rank" Corollary: if i, C|Vi 0 then Chas structure (i.e. Cis sum of circuits of low "rank") Theorem: if i, C|Vi 0 then C  0. Algorithm: For every i, brute force compute C|Vi Time: poly(n)ddim(Vi) = dO(Rank(k,d)) If Chas high rank then by Gabizon-Raz, for some i,C|Vi has high rank. If Cis not a sum of low rank circuits then for some i, C|Viis not a sum of low rank circuits. This contradicts the structural theorem. Cis sum of low rank subcircuitsVis.t. rank of subcircuits remain the same. C|Viis zero each subcircuit vanishes on Vi.  subcircuitscompute the zero polynomial. 27

30. Depth 3 identities Lesson 1: depth 3 identities arevery structured! Lesson 2: Rank is an important invariant to study. Improvements [Kayal-Saraf,Saxena-Seshadri]: finite, klog(d) < Rank(k,d) < k3log(d) over , k < Rank(k,d) < k2log(k) Improves [Dvir-S] + [Karnin-S] (plug and play) NEW: [Saxena-Seshadri] BB-PIT in time nO(k) June 18, 2011 28

31. Bounding the rank Basic observation: Consider C = M1 + M2 M1 = M2= Fact: linear functions are irreducible polynomial. Corollary: C  0 then M1, M2 have same factors. Corollary: matching i (i) s.t. Li ~ L'(i)

33. (2m)functions in span0

34. Black-Box PIT for multilinear (k) C = T1+ … + Tk Ti = j=1...diPi,j where Ti are multilinear and Pi,j are sparse Rank: sparsity of Ti Theorem [Saraf-Volkovich]: If C  0 is a basic identity then #monom(C) ≤ spoly(k) White-Box Algorithm: find partition to sub-circuits of low dimension (after removal of g.c.d.) and brute force verify that they vanish. Black-Box Algorithm: If circuit not sparse can fix many variables and get sparse but not too sparse circuit. Use interpolation for  circuits. June 18, 2011 31

36. Connection to lower bounds (hardness)

37. Survey of positive results

38. Some proofs:

39. Depth-3 circuits

40. Depth-4 circuitsConnection to polynomial factorization June 18, 2011 32

41. PIT and Factoring f is composed if f(X) = g(X|S)h(X|T) where S and T are disjoint [S-Volkovich]: PIT is equivalent to factoring to decomposable factors. : f  0 ifff+yz has two decomposable factors. : Claim: If we have a PIT for all circuits of the form C1 + C2C3, whereCi  Mthen given C  Mwe can deterministically output all decomposable factors of C. June 18, 2011 33

42. PIT and Factoring Deterministic decomposable factoring is equivalent to lower bounds: Deterministic factoring implies NEXP does not have small arithmetic circuits Lower bounds imply Deterministic decomposable factoring PIT  factoring formultilinear polynomials Deterministic decomposable factoring for depth-2, (k), sum of read-once… Open problem: is PIT equivalent to general factorization? June 18, 2011 34

44. Connection to lower bounds (hardness)

45. Survey of positive results

46. Some proofs:

47. Depth-3 circuits

48. Depth-4 circuits

49. Connection to polynomial factorizationJune 18, 2011 35

50. Some `accessible’ open problems Give a Black-Box PIT algorithm for non-commutative formulas Solve PIT for depth-3 circuits Solve PIT for multilinear depth-3 circuits Black-Box PIT for set-multilinear depth-3 circuits (degree d tensors) Polynomial time BB-PIT for read-k formulas PIT for (non-multilinear) depth-4 with restricted fan-in Is PIT equivalent to general factorization? June 18, 2011 36

51. Thank You! June 18, 2011 37