1. Zeev Dvir Weizmann Institute of Science Amir Shpilka Technion Locally decodable codes with 2 queries and polynomial identity testing for depth 3 circuits

2. This talk Explaining the title:Explaining the title: –Locally Decodable codes –Polynomial identity testing –depth 3 circuits Results:Results: –Improved bounds for 2-queries LDC's –Getting 2-LDC's from identically zero depth 3 circuits. –Structural theorem for identically zero depth 3 circuits –PIT for depth 3 circuits

3. Locally decodable codes x1x1x1x1... xkxkxkxk... xnxnxnxn y1y1y1y1 y2y2y2y2... yiyiyiyi... yjyjyjyj... ymymymym xkxkxkxk Def: E: F n ! F m is q-LDC if x k can be recovered from q entries of E(x). Even if E(x) is corrupted in  m coordinates. With high probability. Algorithm k w.h.p

4. Main questions: constructing LDC's, proving lower bounds on their length. Known constructions: q-LDC E: F n ! F m with m = exp(n loglog(q)/q ¢ log(q) ) [BIKR02]. Lower bounds: [KT00]: m =  (n 1 + 1/q-1 ) [GKST01]: In linear 2-LDC over F m = exp(  (n)- log| F |) [KdW03]: In 2-LDC over { 0,1 } m = exp(  (n)). Our result: In linear 2-LDC m = exp(  (n)). Works for every field size, i.e. F = R.

5. Assumption: f has succinct representation. Motivation: Natural problem, many applications: primality testing, finding matching... Schwartz-Zippel: Evaluate f(x) at a random point. Long Term Goals: Deterministic algorithm. Short Term Goals: Restricted Models. Polynomial identity testing f(x 1,...,x n )  0 0 ?

6. General circuits: Randomized algorithms [S80],[Z79],[CK97],[LV98],[AB03]: poly( 1 / ,size) time, n ¢ log(d/  ) random bits Hardness vs. Randomness trade-off: [KI03] PIT 2 P ) arithmetic lower bound for NEXP –NEXP * P/poly or –PERM  arithmetic P/poly Lower bounds for arithmetic circuits imply sub-exponential time deterministic algs.  look for PIT where l.b. are known!

7. Non-commutative formulas: (vars do not commute) [N91] exponential lower bound on formula size [RS04] PIT determ. poly-time in size of formula “Depth 2” circuits: (sparse polynomials) [BoT88],[GKS90],...,[KS01]: deterministic poly time. No sub-exp time deterministic algs. for depth > 2 Open [KS01]: depth 3 circuits w. top fan-in = 3. This paper: depth 3 circuits with small top fan-in: deterministic: quasi-polynomial time PIT alg. (poly time for multilinear circuits). randomized: polynomial time polylog random bits. New result: [KS06] polynomial time algorithm.

8. Depth 3 circuits -  (k) circuits + XXXX +++ top fan-in x1x1 xnxn ckck c1c1 a1a1 anan 1 a0a0 L i,j =  t=1...n a t ¢ x t + a 0 M i =  j=1...d i L i,j... M1M1 MkMk L 1,1 C(x) =  i=1...k c i ¢ M i =  i c i ¢  j L i,j L 1,d 1

9. What's next: Sketch of lower bound for 2-LDC.Sketch of lower bound for 2-LDC. PIT for depth 3 circuits with top fan-in = 2.PIT for depth 3 circuits with top fan-in = 2. PIT for depth 3 circuits with top fan-in = 3.PIT for depth 3 circuits with top fan-in = 3. General depth 3 circuits (sketch)General depth 3 circuits (sketch) Structural theorem for identically zero depth 3 circuits.Structural theorem for identically zero depth 3 circuits. PIT algorithms for depth 3 circuitsPIT algorithms for depth 3 circuits

10. Thm 1 [ GKST01 ]: For any linear 2-LDC over {0,1} of length m, m = exp(  (n)). Proof: Isoperimetric inequality. Thm 2 [ GKST01 ]: For any linear 2-LDC over F of length m, m = exp(  (n) – log| F |). Proof: combine next lemma with theorem 1. Lemma [ GKST01 ]: If 9 linear 2-LDC over F of length m then 9 linear 2-LDC over {0,1} of length | F | ¢ m. Proof: randomly map all multiples of all coordinates to {0,1}. New Lemma: If 9 linear 2-LDC over F of length m then 9 linear 2-LDC over {0,1} of length m. Proof: randomly map well chosen multiple of each coordinate to {0,1}.

11. What's next: Sketch of lower bound for 2-LDC. Sketch of lower bound for 2-LDC. PIT for depth 3 circuits with top fan-in = 2.PIT for depth 3 circuits with top fan-in = 2. PIT for depth 3 circuits with top fan-in = 3.PIT for depth 3 circuits with top fan-in = 3. General depth 3 circuits (sketch)General depth 3 circuits (sketch) Structural theorem for identically zero depth 3 circuits.Structural theorem for identically zero depth 3 circuits. PIT algorithms for depth 3 circuitsPIT algorithms for depth 3 circuits

12. Identically zero  (2) circuits Reminder: C(x) = c 1 ¢ M 1 + c 2 ¢ M 2 L1L1L1L1 L2L2L2L2... LiLiLiLi... LjLjLjLj... LdLdLdLd L' 1 L' 2... L' i... L' j... L' d M 1 (x)= M 2 (x)= Fact: linear functions are irreducible polynomial. Corollary: C  0 then M 1, M 2 have the same factors. Corollary: 9 matching i  j(i) s.t. L i ~ L' j(i) PIT algorithm: look for such a matching.

13. What's next: Sketch of lower bound for 2-LDC. Sketch of lower bound for 2-LDC. PIT for depth 3 circuits with top fan-in = 2. PIT for depth 3 circuits with top fan-in = 2. PIT for depth 3 circuits with top fan-in = 3.PIT for depth 3 circuits with top fan-in = 3. General depth 3 circuits (sketch)General depth 3 circuits (sketch) Structural theorem for identically zero depth 3 circuits.Structural theorem for identically zero depth 3 circuits. PIT algorithms for depth 3 circuitsPIT algorithms for depth 3 circuits

14. Preliminaries Claim: wlog linear functions are homogeneous (no constant term). Claim: A: F n  F n invertible linear map, then C(x)  0, C(A ¢ x)  0. Definition: r, rank(C), rank(linear functions in C). Corollary 1: wlog L i 's depend only on x 1,...,x r. Corollary 2: wlog x 1,...,x r appear as linear functions in C.

15. Input: C(x) = c 1 ¢ M 1 + c 2 ¢ M 2 + c 3 ¢ M 3 x 1,...,x r are linear functions in C wlog assume g.c.d.(M 1,M 2,M 3 ) = 1 M 2 (x)= L 2d+1 L 2d+2... L 2d+i... xtxtxtxt... L 3d M 3 (x)= M 1 (x)= Idea: reduction to  (2): C  0 ) C| x s =0  0 ) if x s 2 M 1 then c 2 ¢ M 2 | x s =0 +c 3 ¢ M 3 | x s =0 =0. Lemma: 8 x s 9 d pairs ( i,j(i) ) s.t. L i | x s =0 ~ L j(i) | x s =0 0 L d+1 L d+2... L d+i... L d+j... L 2d L1L1L1L1 L2L2L2L2... LiLiLiLi... xsxsxsxs... LdLdLdLd 0

16. Lemma: i= 1,2 L i 2 M i : L 1 | x s =0 ~ L 2 | x s =0 ) x s 2 span(L 1,L 2 ) Proof: Otherwise L 1 ~ L 2 ) L 1 | M 1,M 2 ) if C  0 then L 1 | M 3 ) L 1 2 g.c.d(M 1,M 2,M 3 )  ? Define E(x) = L 1 (x),...,L 3d (x) Claim: 8 s 9 d pairs (i,j(i)) s.t. x s 2 span(E(x) i,E(x) j(i) ). Corollary: E is a 2-LDC of length 3d. Corollary: 3d=exp(  (r)) ) r=O(log(d)). Thm: If C  0 is  (3) then rank(C) = O(log(d)). PIT Algorithm: brute force. time = exp(log(d) 2 ). Input: C(x) = c 1 ¢ M 1 + c 2 ¢ M 2 + c 3 ¢ M 3 M 1 =  i=1...d L i (x), M 2 =  i=1...d L d+i (x), M 3 =  i=1...d L 2d+i (x) x 1,...,x r are linear functions in C Lemma: 8 x s 9 d pairs ( i,j(i) ) s.t. L i | x s =0 ~ L j(i) | x s =0

17. What's next: Sketch of lower bound for 2-LDC. Sketch of lower bound for 2-LDC. PIT for depth 3 circuits with top fan-in = 2. PIT for depth 3 circuits with top fan-in = 2. PIT for depth 3 circuits with top fan-in = 3. PIT for depth 3 circuits with top fan-in = 3. General depth 3 circuits (sketch)General depth 3 circuits (sketch) Structural theorem for identically zero depth 3 circuits.Structural theorem for identically zero depth 3 circuits. PIT algorithms for depth 3 circuitsPIT algorithms for depth 3 circuits

18. Input: C(x) = c 1 ¢ M 1 + c 2 ¢ M 2 + c 3 ¢ M 3 +... + c k ¢ M k Def: C is simple if g.c.d.(M 1,...,M k )=1 Def: sim(C) = C/g.c.d.(C) Def: C is minimal if no sub-circuit is zero. Thm: C  0 is simple and minimal, r = rank(C), d = deg(C). Then 9 2-LDC E: F a  F b s.t. a = r/ 2 k 2 log(d) k-3  b = kd Corollary: rank(C) · O ( log(d) k-2 ) Proof: induction on k. Assume x 1,...,x r 2 C. Consider C| x i = 0  0. Top fan-in is k-1. Done? simple? minimal? rank?

19. M 2 (x)= L 3,1 L 3,2... L 3,i... xtxtxtxt... L 3,d M 3 (x)= M 1 (x)= 0 L 2,1 L 2,2... L 2,i... L 2,j... L 2,d L 1,1 L 1,2... L 1,i... xsxsxsxs... L 1,d L k,1 L k,2... L k,i... L k,j... L k,d M k (x)= IsIs... Claim: 8 x s 9 I s s.t. (C I s )| x s =0  0 and minimal Cor: 9 I,r' ¸ r/2 k s.t. (wlog) 8 1 · s · r' (C I )| x s =0  0 and minimal.

20. Cor: 9 I,r' ¸ r/2 k s.t. 8 1 · s · r' (C I )| x s =0  0 and minimal. Optimistic: done? Problematic: what's the rank of (C I )| x s =0 ? Optimistic: lemma: rank(C I ) ¸ r' ¸ r/2 k Problematic: (C I )| x s =0 not simple Optimistic: consider sim((C I )| x s =0 ) (removing g.c.d.) Problematic: what happens to the rank? Optimistic: eh... Lemma: 9 x i s.t. rank(sim((C I )| x s =0 )) ¸ rank(C I )/2 k log(d) Proof: … End of proof: induction on (C I )| x i =0 (from Lemma).

21. What's next: Sketch of lower bound for 2-LDC. Sketch of lower bound for 2-LDC. PIT for depth 3 circuits with top fan-in = 2. PIT for depth 3 circuits with top fan-in = 2. PIT for depth 3 circuits with top fan-in = 3. PIT for depth 3 circuits with top fan-in = 3. General depth 3 circuits (sketch) General depth 3 circuits (sketch) Structural theorem for identically zero depth 3 circuits.Structural theorem for identically zero depth 3 circuits. PIT algorithms for depth 3 circuitsPIT algorithms for depth 3 circuits

22. Structural theorem: C  0 is  (k) then: 9 partition I 1 t I 2 t... t I m = [k] s.t. C Ij  0 minimal (C = C I 1 + C I 2 +... + C I m ) rank(sim(C Ij )) · O(log(d) |I j |-2 ) PIT algorithm: For each I ½ [k] check whether rank(sim(C I )) · O(log(d) |I|-2 ) if yes then brute force check if C I  0 if 9 partition as in theorem then C  0 Running time: exp(log(d) k-1 ).

23. The Multilinear Case If C is multilinear then rank(C)=d. But we proved that if C=0 is simple and minimal then rank(C) · polylog(d) We get that d · polylog(d) Can only hold for finitely many values ! Conclusion: d · O(1) rank(C) · dk · O(1) Polynomial time algorithm

24. Open problems PIT algorithms for stronger models:PIT algorithms for stronger models: –Depth 3 circuits –Bounded depth Tightness of our results:Tightness of our results: –Conjecture: If C  0 is  (k) simple, minimal then rank(C) = poly(k). –[KS06] Not true for finite fields! Example in of a circuit with top fanin=3 and rank ~ log(d) Example in of a circuit with top fanin=3 and rank ~ log(d) ( Multilinear ) (Multilinear)