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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

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

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

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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.

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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?

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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?

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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.

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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:

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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.

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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.

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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?

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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).

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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.

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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.

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

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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.

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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.

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

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

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

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Low Rank Systems.

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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

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

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

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

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Rank With Respect To Individual Prime Factors. Chinese Remaindering

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Low Rigidity Over Prime Fields is Enough.

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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.

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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.

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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!

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

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