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On the degree of symmetric functions on the Boolean cube

On the degree of symmetric functions on the Boolean cube. Joint work with Amir Shpilka. The basic question of complexity. The basic question of complexity. How complex is it (how hard it is to compute f?). The basic question of complexity. How complex is it (how hard it is to compute f?)

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On the degree of symmetric functions on the Boolean cube

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  1. On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka

  2. The basic question of complexity

  3. The basic question of complexity How complex is it (how hard it is to compute f?)

  4. The basic question of complexity How complex is it (how hard it is to compute f?) That depends on the computational model at hand. e.g. Turing machines, Circuits, Decision trees, etc…

  5. Polynomials as computers How complex is it (how hard it is to compute f?) That depends on the computational model at hand. e.g. Turing machines, Circuits, Decision trees, etc… Our model of computation – Polynomials.

  6. Polynomials as computers Our model of computation – Polynomials.

  7. Polynomials as computers Our model of computation – Polynomials.

  8. Tight lower bound Nisan and Szegedy (94) proved assuming f depend on all n variables.

  9. Tight lower bound Nisan and Szegedy (94) proved assuming f depend on all n variables. Can we get stronger lower bounds on more restricted natural classes of functions?

  10. Symmetric Boolean functions

  11. Symmetric Boolean functions Von zurGathen and Roche (97) proved assuming f is non-constant.

  12. Symmetric Boolean functions

  13. Symmetric Boolean functions

  14. Symmetric Boolean functions c .. . 2 1 0 0 1 2 3 4 5 6 . . . n

  15. Symmetric Boolean functions c .. . 2 1 0 0 1 2 3 4 5 6 . . . n

  16. Symmetric Boolean functions What can be said about ? c .. . 2 1 0 0 1 2 3 4 5 6 . . . n

  17. Symmetric functions What can be said about ? For c=1 we got For c=n the function has degree 1.

  18. Symmetric functions What can be said about ? For c=1 we got For c=n the function has degree 1. How does the degree behaves?

  19. Symmetric functions Von zurGathen and Roche noted that

  20. Symmetric functions Von zurGathen and Roche noted that In particular, even for this observation doesn’t exclude the existence of a parabola interpolating on some function.

  21. Relative degree Define

  22. Relative degree Define is monotone decreasing in c.

  23. Relative degree Define is monotone decreasing in c. has a crazybehavior in n.

  24. Relative degree Define is monotone decreasing in c. has a crazybehavior in n.

  25. 6 stages of first-time research Stage 1

  26. 6 stages of first-time research Stage 2

  27. 6 stages of first-time research Stage 3

  28. 6 stages of first-time research Stage 4

  29. 6 stages of first-time research Stage 5

  30. 6 stages of first-time research Stage 6

  31. 6 stages of first-time research Stage 1…

  32. Our main result Main theorem This proves a threshold behavior at c=n.

  33. Our main result Main theorem This proves a threshold behavior at c=n. Yet another theorem

  34. Proof strategy – reducing c Lemma 1. For any n there exist a prime p such that and

  35. Proof strategy – reducing c Lemma 1. For any n there exist a prime p such that and Together with the trivial bound , we already get a threshold behavior

  36. Proof strategy – reducing n Lemma 2. For every c,m,n such that , it holds that Dream version

  37. Proof strategy – reducing n Lemma 2. For every c,m,n such that , it holds that Dream version

  38. Proof strategy – reducing n Lemma 2. For every c,m,n such that , it holds that Dream version

  39. Proof strategy – reducing n Lemma 2. For every c,m,n such that , it holds that

  40. Proof of the main theorem A computer search found that . By Lemma 2 By Lemma 1

  41. Periodicity and degree Low degree Strong periodical structure Dream version

  42. Periodicity and degree Low degree Strong periodical structure Strong periodical structure High degree Dream version

  43. Periodicity and degree Low degree Strong periodical structure Strong periodical structure High degree Hence no function has “to low” degree. Dream version

  44. Periodicity and degree Low degree Strong periodical structure Strong periodical structure High degree Not the same sense of periodical structure…

  45. Low degree implies strong periodical structure Lemma 3. Let with . Let be a prime number. Then for all such that it holds that c .. 1 0 0 1 2 3 . . . d . . . p n

  46. Low degree implies strong periodical structure Lemma 3. Let with . Let be a prime number. Then for all such that it holds that c .. 1 0 0 1 2 3 . . . d . . . pq n

  47. Low degree implies strong periodical structure Lemma 3. Let with . Let be a prime number. Then for all such that it holds that c .. 1 0 0 1 2 3 . . . d . . . pq r n

  48. Strong periodical structure implies high degree Definition. Let and define

  49. Strong periodical structure implies high degree Definition. Let and define

  50. Strong periodical structure implies high degree Definition. Let and define Lemma 4. Let . Then for all If then If then or

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