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

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The basic question of complexity

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How complex is it (how hard it is to compute f?)

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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…

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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.

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Polynomials as computers Our model of computation – Polynomials.

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Polynomials as computers Our model of computation – Polynomials.

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

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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?

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Symmetric Boolean functions

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Von zur Gathen and Roche (97) proved assuming f is non-constant.

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Symmetric Boolean functions

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0 1 2 3 4 5 6... n 0 1 2....... c

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

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

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

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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?

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Symmetric functions Von zur Gathen and Roche noted that

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

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Relative degree Define

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Relative degree Define is monotone decreasing in c.

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Relative degree Define is monotone decreasing in c. has a crazy behavior in n.

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Relative degree Define is monotone decreasing in c. has a crazy behavior in n.

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6 stages of first-time research Stage 1

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6 stages of first-time research Stage 2

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6 stages of first-time research Stage 3

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6 stages of first-time research Stage 4

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6 stages of first-time research Stage 5

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6 stages of first-time research Stage 6

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6 stages of first-time research Stage 1…

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

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

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

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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

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

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

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

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

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

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Periodicity and degree Low degree Strong periodical structure Dream version

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

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

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

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

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

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

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Strong periodical structure implies high degree Definition. Let and define

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Strong periodical structure implies high degree Definition. Let and define

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

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Proof of Lemma 1 Lemma 1. For any n there exist a prime p such that and

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Proof of Lemma 1 0 1 2... n 0 1 2....... n-1

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Proof of Lemma 1 0 1 2... p... 2p n 0 1 2....... n-1 o(n)

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Proof of Lemma 1 0 1 2... p... 2p n 0 1 2....... n-1 o(n)

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Proof of Lemma 1 0 1 2... p... 2p n 0 1 2....... n-1 o(n) We might as well assume that non-constant

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Proof of Lemma 1 0 1 2... p... 2p n 0 1 2....... n-1 o(n) We might as well assume that non-constant

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Proof of Lemma 1 Define

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Proof of Lemma 1 Define From Lemma 3

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Proof of Lemma 1 Define From Lemma 3 and also

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Proof of Lemma 1 From Lemma 3 Hence

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Proof of Lemma 1 Case 1: g is a non-constant and we are done.

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Proof of Lemma 1 Case 2: g is a constant G Hence, by Lemma 4

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Proof of Lemma 1 Case 2: g is a constant G Hence, by Lemma 4 or is linear.

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Proof of Lemma 1 Case 2: If happens to be linear, apply the proof so far on. Since we are done unless it also happens that is linear. But this means f itself must be linear. Since f is not constant it means f assumes n+1 distinct values – a contradiction.

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Open Questions Main question - Better understand. Improve the lower bounds to non-linear, if possible.

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Thank you!

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