Uniqueness of Optimal Mod 3 Circuits for Parity

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Uniqueness of Optimal Mod 3 Circuits for Parity

• Frederic Green Amitabha Roy
• Clark University Akamai

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Goal Lower bounds on parity for circuits of this
shape
d
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Reduces to Upper bounds bounds on correlation
s
s gt 1/e
d
Hajnal et al. Correlation with parity lt e here
implies
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Correlation Defn 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
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Why?
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Many reasons. Here are two

• and yet we don't know if they can simulate any
• more of ACC (e.g., parity).

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Main concern here m 3, d 2
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Reduction to Exponential Sums
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., d2)
where,
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Generalizations
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Recent History (since ca. 2001)
Here are some things we now know
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Results Known to be Tight
Exhaustive list
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Can We Get Tighter Results?
wherever we can
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So Let's see if we can extend this
to ALL n.
Two key ingredients in Dueñez et al.'s proof

• The optimal polynomials are unique.

• There is a "gap" in the correlation between the
• optimal polynomials and the "first suboptimal"
  ones.

Conjecture (Dueñez et al.) these are true for
all n.
Question Can we even prove this when m3?
Yes!
Our answer, and main result
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Optimal Polynomials
Uniqueness Theorem These are the only ones!
Gap Theorem Anything less is "a lot" less!
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Uniqueness Theorem These are the only ones!
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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.
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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

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Uniqueness Theorem Proof, continued
Wlog, write,
and,
where t2 is a quadratic form, l and r linear
forms in the indicated variables.
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Uniqueness Theorem Proof, continued
Wlog, write,
and,
where t2 is a quadratic form, l and r linear
forms in the indicated variables.
Then, summing over x1 and using (i), (ii), (iii),
obtain
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Uniqueness Theorem Proof, continued
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Uniqueness Theorem Proof, continued
but getting back to what we set out to prove
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Uniqueness Theorem Proof, continued
Not hard to see
Thus, by induction
t2 (l - r)2 and t2 - (l r)2 are both of
optimal form
Underlying graphs must hence have the same shape
but they could be differently labeled
or could they??
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Uniqueness Theorem Proof, concluded
Now,
Supposed to be the "difference" of two "optimal
graphs"
Such a difference consists of "loops" (or single
edges)
BUT l2r2 has too many cross terms to represent
this!
HENCE l r 0, and the polynomials are
identical.
and t is uniquely determined from t2
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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.

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Gap Theorem continued
Hence assume this is optimal, this is not.
Hence, by the uniqueness theorem,
Hence,
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Gap Theorem continued
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Gap Theorem concluded
Now, this
is not so easy to evaluate.
But if we "linearize" l2 and r2 as follows, it
becomes possible
so that,
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Gap Theorem concluded
Now, this
is not so easy to evaluate.
But if we "linearize" l2 and r2 as follows, it
becomes possible
so that,
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Gap Theorem concluded
Now, this
is not so easy to evaluate.
But if we "linearize" l2 and r2 as follows, it
becomes possible
so that,
If l, r have many terms, this dominates, giving a
1/3 factor.
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Conclusions
(well, mostly questions)

• We have proved that optimal quadratic
  polynomials
• are unique for m3, 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 m3 case.

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Danke schön!