1. Linear Systems With Composite Moduli Arkadev Chattopadhyay (University of Toronto) Joint with: Avi Wigderson TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAA A A A

2. The Problem. Question: What can we say about the boolean solution set of such systems?

3. Outline of Talk. Motivation. Natural problem. Circuits with MOD Gates. Surprising power of composite moduli. Our Result. Some Circuit Consequences. High Level Argument.

4. Bounded-Depth Circuits. Theorem (Ajtai, Furst, Saxe and Sipser, Yao, Hastad). Circuits of constant depth, comprising AND/OR/NOT gates, need exponential size to compute PARITY, i.e. MOD 2.

5. Circuits With MOD Gates. Theorem (Razborov’87, Smolensky’87). Addition of MOD p gates to bounded-depth circuits, does not help to compute function MOD q, if (p,q)=1 and p is a prime power. Fermat’s Gift for prime p: Nagging Question: Is ‘and p is a prime power’ essential?

6. Circuits With MOD Gates. Theorem (Razborov’87, Smolensky’87). Addition of MOD p gates to bounded-depth circuits, does not help to compute function MOD q, if (p,q)=1 and p is a prime power. Nagging Question: Is ‘and p is a prime power’ essential?

7. Smolensky’s Conjecture. Conjecture: MOD q needs exponential size circuits of constant depth having AND/OR/MOD m gates if (m,q)=1. Not known even for m=6. Barrier: Prove any non-trivial lower bounds for AND/OR/MOD 6.

8. The Weakness of Primes. MOD p Gates Conclusion: AND cannot be computed by constant-depth circuits having only MOD p gates (in any size). Fermat’s Gift for prime p:

9. The Power of Composites. MOD m C Fact: Every function can be computed by depth-two circuits having only MOD m gates in exponential size, when m is a product of two distinct primes.

10. Status Of Our Ignorance. Barrier 1. No non-trivial lower bounds are known for AC 0 ± MOD 6. Barrier 2(Beigel-Maciel’97): No non-trivial lower bounds were known for depth-3 circuits of the form OR ± AND ± MOD 6. Theorem (Beigel-Maciel’97): Exponential size is required by OR ± AND ± MOD 6 ckts to compute MOD q, if (m,q)=1.

11. Strange Problem With Generalized MOD Gates. Barrier: Depth-2 circuits of type MOD 6 A ± MOD 6, even though exponential lower bounds exist for MOD 6 ± MOD 6 (Barrington-Straubing- Therien). Barrier Before This Work: Depth-3 circuits of the form OR ± AND ± MOD 6. Question: What’s special about these MOD m gates?

12. Power of Polynomials Modulo Composites. Defn: Let P (x) reperesent f over Z m, w.r.t A : Def: The MOD m -degree of f is the degree of minimal degree P representing f, w.r.t. A. Fact: The MOD m -degree of OR is  (n).

13. Power of Composite Moduli. Theorem(Barrington-Beigel-Rudich’92): MOD m -degree of OR is O(n 1/t ) if m has t distinct prime factors, i.e. for m=6 it is. Theorem(Green’95, BBR’92): MOD m -degree of MOD q is  (n). Theorem(Hansen’06): Let m,q be co-prime. MOD m - degree of MOD q is O(n 1/t ) if m has t distinct prime factors, as long as m satisfies certain condition, i.e. MOD 35 – degree of PARITY is.

14. Can Many Polynomials Help? Defn: P represents f if: Question: What is the relationship of t and deg( P )? Observation: n linear polynomials can represent AND and NOR functions.

15. Linear Systems: Our Result. Aiµ ZmAiµ Zm Theorem: The boolean solution set,, looks pseudorandom to the MOD q function. (independent of t )

16. Circuit Consequence. Corollary: Exponential size needed by MAJ ± AND ± MOD m to compute MOD q, if m=p 1 p 2 and m,q co-prime. (Solves Beigel-Maciel’97 for such m). Remark: Obtaining exponential lower bounds on size of MAJ ± MOD m ± AND is wide open.

17. Proof Strategy. Gradual generalization leading to result. Singleton Accepting Sets. Low rank systems. Low rigid rank Deal with high rigid rank separately. Exponential sums (Extend Grigoriev-Razborov). of Bourgain.

18. Singleton Accepting Set. Assume A i ={0} Set of Boolean solns A linear form Fourier Expansion

19. Finishing Off For Singleton Accepting Set. Exponential sum reduction (Goldman, Green)

20. Non-Singleton Accepting Sets. + j · ( m -1) t singleton systems + Union Bound:

21. Low Rank Systems.

22. Shouldn’t High Rank be Easy? Tempting Intuition from linear algebra: If L has high rank, then the size of the solution set B L should be a small fraction of the universe, and hence correlation w.r.t MOD q is small. Caveat: Our universe is only the boolean cube! Example: rank is n. B L ´ {0,1} n

23. Sparse Linear Systems. Observation: For each i, there exists a polynomial P i over Z m of degree at most k, such that

24. Polynomial Systems With Singleton Accepting Set. Degree · k Relevant Sum for Correlation: Bourgain’s breakthrough:

25. Low Rigid Systems. We can combine low rank and sparsity into rigidity: rank= r k -sparse ( k, r )-sparse Strategy:

26. Rank With Respect To Individual Prime Factors. Chinese Remaindering

27. Low Rigidity Over Prime Fields is Enough.

28. Theorem: Let m = p 1 p 2. Let L = L 1 [ L 2 be any system of generalized equations over Z m. If L 1 (and L 2 ) has k -rigid rank over Z (resp Z ) at most r 1 (and r 2 ) then, Using estimates of exponential sums by Bourgain.

29. Otherwise: High Rigid Rank. Theorem: If L does not admit a partition into L 1 [ L 2 such that L 1 (and L 2 ) has k -rigid rank over Z (resp. Z ) at most r. Then, Extends ideas of Grigoriev-Razborov for arithmetic circuits.

30. Combining the Two, We Are Done. Question: What about m=30? Answer: Recently, in joint work with Lovett, we deal with arbitrary m. THANK YOU!

31. Combining the Two, We Are Done. Open Question: Handle m=30. THANK YOU!