Linear Systems With Composite
Moduli
Arkadev Chattopadhyay
(University of Toronto)
Joint with:
Avi Wigderson
The Problem.
Question: What can we say about the
boolean solution set of such systems?
Outline of Talk.
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Motivation.
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Natural problem.
Circuits with MOD Gates .
Surprising power of composite moduli.
Our Result.
Some Circuit Consequences.
High Level Argument.
Circuits With MOD Gates.
Theorem (Razborov’87, Smolensky’87). Addition
of MODp gates to bounded-depth circuits,
does not help to compute function MODq , if
(p,q)=1 and p is a prime power.
Nagging Question: Is ‘and p is a prime power’
essential?
Smolensky’s Conjecture.
Conjecture: MODq needs exponential size
circuits of constant depth having
AND/OR/MODm gates if (m,q)=1.
Not known even for m=6.
Barrier: Prove any non-trivial lower
bounds for AND/OR/MOD6.
The Weakness of Primes.
MODp Gates
Fermat’s Gift for prime p:
Conclusion: AND cannot be computed by constant-depth
circuits having only MODp gates (in any size).
The Power of Composites.
MODm
MODm
MODm
MODm
C
Fact: Every function can be computed by depth-two
circuits having only MODm gates in exponential size,
when m is a product of two distinct primes.
Power of Polynomials Modulo
Composites.
Defn: Let P(x) reperesent f over Zm, w.r.t A:
Def: The MODm -degree of f is the degree of
minimal degree P representing f, w.r.t. A.
Fact: The MODm -degree of OR is (n).
Power of Composite Moduli.
Theorem(Barrington-Beigel-Rudich’92): MODm-degree
of OR is O(n1/t) if m has t distinct prime factors, i.e.
for m=6 it is
.
Theorem(Green’95, BBR’92): MODm -degree of MODq is
(n).
Theorem(Hansen’06): Let m,q be co-prime. MODmdegree of MODq is O(n1/t) if m has t distinct prime
factors, as long as m satisfies certain condition, i.e.
MOD35 – degree of PARITY is
.
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.
Linear Systems: Our Result.
A i µ Zm
Theorem: The boolean solution set,
pseudorandom to the MODq function.
(independent of t)
, looks
Circuit Consequence.
Corollary: Exponential size needed by MAJ ± AND ± MODm
to compute MODq, if m=p1p2 and m,q co-prime.
(Solves Beigel-Maciel’97 for such m).
Remark: Obtaining exponential lower bounds on size of
MAJ ± MODm ± AND is wide open.
Proof Strategy.
Gradual generalization leading to result.
 Singleton Accepting Sets.
Exponential sums
 Low rank systems.
of Bourgain.
 Low rigid rank
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Deal with high rigid rank separately.
(Extend Grigoriev-Razborov).
Singleton Accepting Set.
Set of
Boolean solns
Assume Ai={0}
A linear form
Fourier Expansion
Finishing Off For Singleton
Accepting Set.
Exponential sum
reduction
(Goldman, Green)
Non-Singleton Accepting Sets.
Union Bound:
+
+
j · (m-1)t singleton systems
Low Rank Systems.
Shouldn’t High Rank be Easy?
Tempting Intuition from linear algebra: If L has
high rank, then the size of the solution set BL
should be a small fraction of the universe,
and hence correlation w.r.t MODq is small.
Caveat: Our universe is only the boolean cube!
Example:
rank is n.
BL ´ {0,1}n
Sparse Linear Systems.
Observation: For each i, there exists a polynomial
Pi over Zm of degree at most k, such that
Polynomial Systems With
Singleton Accepting Set.
Degree · k
Relevant Sum for Correlation:
Bourgain’s breakthrough:
Low Rigid Systems.
We can combine low rank and sparsity into rigidity:
rank=r
(k,r)-sparse
Strategy:
k-sparse
Rank With Respect To
Individual Prime Factors.
Chinese
Remaindering
Low Rigidity Over Prime Fields
is Enough.
Otherwise: High Rigid Rank.
Theorem: If L does not admit a partition into
L1 [ L2 such that L1 (and L2) has k-rigid
rank over Z (resp. Z ) at most r. Then,
Extends ideas of Grigoriev-Razborov for
arithmetic circuits.
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!