1. Uniqueness of Optimal Mod 3 Circuits for Parity Frederic Green Amitabha Roy Frederic Green Amitabha Roy Clark University Akamai Clark University Akamai

3. d Goal: Lower bounds on parity for circuits of this shape:

4. Reduces to: Upper bounds bounds on correlation: s Hajnal et al.: Correlation with parity <  here s > 1/  implies d

5. Correlation: Def n : normalized # of agreements-disagreements: In this case: interested in f = the parity function: and g computed by a polynomial mod m of degree d, for odd m: x1x1 x2x2 xnxn...

7. …and yet we don't know if they can simulate any more of ACC (e.g., parity). Many reasons. Here are two:

8. Main concern here: m = 3, d = 2: d < 2

9. Reduction to Exponential Sums where, The correlation can be related to an exponential sum, [Cai, Green & Thierauf 1996], like those that arise in number theory. When m = 3, this reduction is especially simple (e.g., d=2):

10. Generalizations

11. Recent History (since ca. 2001) Here are some things we now know:

12. Results Known to be Tight Exhaustive list:

13. Can We Get Tighter Results? …wherever we can…

15. So: Let's see if we can extend this: Two key ingredients in Dueñez et al.'s proof: The optimal polynomials are unique. Question: Can we even prove this when m=3? Conjecture (Dueñez et al.): these are true for all n. Our answer, and main result: …to ALL n. There is a "gap" in the correlation between the optimal polynomials and the "first suboptimal" ones.

16. Optimal Polynomials Uniqueness Theorem: These are the only ones! Gap Theorem: Anything less is "a lot" less!

17. Uniqueness Theorem: These are the only ones!

18. Uniqueness Theorem: Proof sketch: The proof relies heavily on these identities: Note: (i) and (ii) can be readily generalized to other moduli; but (iii) seems rather mysterious.

19. Uniqueness Theorem Proof, continued: The proof is by induction on n. Consider the (harder) case of n odd. Thus our induction hypothesis is: It is useful to think of the graph underlying t. E.g., for the optimal polynomials:.. 345612 x1x2x1x2 x3x4x3x4 x5x6x5x6 +++

20. Uniqueness Theorem Proof, continued: Wlog, write, and, where t 2 is a quadratic form, l and r linear forms in the indicated variables.

22. Uniqueness Theorem Proof, continued: Then, summing over x 1 and using (i), (ii), (iii), obtain: Wlog, write, and, where t 2 is a quadratic form, l and r linear forms in the indicated variables.

23. Uniqueness Theorem Proof, continued:

24. …but getting back to what we set out to prove…

25. Uniqueness Theorem Proof, continued: Not hard to see: Thus, by induction: t 2 + (l - r) 2 and t 2 - (l + r) 2 are both of optimal form Underlying graphs must hence have the same shape:.. 345612 …but they could be differently labeled… or could they??

26. Uniqueness Theorem Proof, concluded: Now, Supposed to be the "difference" of two "optimal graphs":.. 345612 Such a difference consists of "loops" (or single edges): BUT: l 2 +r 2 has too many cross terms to represent this! HENCE: l = r = 0, and the polynomials are identical. and t is uniquely determined from t 2

27. Gap Theorem: Proof sketch: Again by induction on n. Also, make use of uniqueness. Start at a place we were at before: Easy analysis: if both of these are either optimal or suboptimal, the induction follows through.

28. Gap Theorem continued: Hence assume this is optimal, this is not. Hence, by the uniqueness theorem, Hence,

29. Gap Theorem continued:

30. Gap Theorem concluded: Now, this: is not so easy to evaluate. But if we "linearize" l 2 and r 2 as follows, it becomes possible: so that,

31. Gap Theorem concluded: Now, this: is not so easy to evaluate. But if we "linearize" l 2 and r 2 as follows, it becomes possible: so that,

32. Gap Theorem concluded: Now, this: is not so easy to evaluate. But if we "linearize" l 2 and r 2 as follows, it becomes possible: so that, If l, r have many terms, this dominates, giving a 1/3 factor. Only remaining case: l, r have two terms! Gives the.

33. Conclusions We have proved that optimal quadratic polynomials are unique for m=3, and that there is a gap between suboptimal sums and the optimal ones. We know of no similar exact characterizations for non-trivial circuits Of course, we want to do this for m other than 3. How? Perhaps by finding other properties than uniqueness and gap that will be sufficient to push through an inductive argument? Perhaps by generalizing the mysterious identity (iii)? The problem of tight (or just tighter!) bounds for higher degrees remains a great challenge even for the m=3 case. (well…, mostly questions):

34. Danke schön!