Siyao Guo Ilan Komargodski New York University Weizman Institute of Science Circuit directed acyclic graph Gates labeled by and operations Fanout 2 ID: 480133
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Slide1
Negation-Limited Formulas
Siyao
Guo Ilan Komargodski
New York University
Weizman
Institute of ScienceSlide2
Circuit
: directed acyclic graph. Gates labeled by
,
and
operations. Fan-out 2.Formula: a circuit with fan-out 1.Size = # of gates.
Boolean Circuits and Formulas
output wire
input wires
Slide3
Monotone computation: no
gates.
“
The effect of negation gates on circuit size remains to a large extent
a mystery
”, Stasys Jukna
Monotone vs. Non Monotone
Computation
GeneralMonotone
[ILMR01]
Circuit size lower bound
[Tal14]
[GP14]
Formula
size lower bound
General
Monotone
Circuit size lower bound
Formula
size lower boundSlide4
Bound the # of Negations
Circuits
[
Markov’58,Fischer’75,BNT98]
:
s
ize:
, negations:
size:
, negations:
Formulas
[Nechiporuk’62,Morizumi’09]:size:
, negations: size:
, negations:
# of negations above is tight.
is the input size of the functions.
The size is
not known to be tight. Slide5
Bound the # of Negations
is the input size of the functions.
# of
negations in a
formula
# of
negations in a
circuit
[Markov’58,Fischer’75]
[Nechiporuk’62,Morizumi’09]Slide6
????
????
Bound the # of Negations
# of
negations in a
formula
# of
negations
i
n a
circuit
[Markov’58,Fischer’75]
[Nechiporuk’62,Morizumi’09]
is the input size of the functions.
Slide7
??
Extending Monotone Results
# of
negations in a
circuit
[AM05,Ros15] (size lower bounds),
[BCOST15] (learning),
[
G
MOR15] (crypto)
[Markov’58,Fischer’75]
????
# of
negations in a
formula
[Nechiporuk’62,Morizumi’09]
is the input size of the functions.
IDEA:
Decompose
the
-negation
circuit into
monotone
parts and apply the known results on them (in a smart way).
What about negation-limited formulas?Slide8
??
What about Negation-Limited Formulas?
# of
negations in a
circuit
[AM05,Ros15] (size lower bounds),
[BCOST15] (learning),
[GMOR15] (crypto)
[Markov’58,Fischer’75]
????
# of
negations in a
formula
[Nechiporuk’62,Morizumi’09]
is the input size of the functions.
TrivialSlide9
??
What about
Negation-Limited Formulas?
# of
negations in a
circuit
[AM05,Ros15] (size lower bounds),
[BCOST15] (learning),
[GMOR15] (crypto)
[Markov’58,Fischer’75]
??
# of
negations in a
formula
[Nechiporuk’62,Morizumi’09]
is the input size of the functions.
This Work
IDEA:
Decompose
the
-negation
formula
into
monotone
parts and apply the known results on them (in a smart way).
Slide10
Our Decomposition Theorem
Every
formula
of size
and
negations can be rewritten as a formula of size
of the form
,
where
is a
read-once
formula
every
is a monotone formula.
Monotone
Pushing NOTs to the top
inputs
Slide11
Application 1 – Formulas to Circuits
Circuit
s
ize:
n
egations:
Known
Formulasize:
negations:
Circuitsize: negations:
Formula
size:
negations:
Is this the best we can do?
TrivialSlide12
Application 1 – Formulas to Circuits
Theorem 1:
Formula, size:
, negations:
, depth:
circuit, size:
, negations:
,
depth
Result
for negation-limited circuits
non-trivial result for
negation-limited
formulas
Circuit
size:
n
egations:
Formula
s
ize:
n
egations:
This WorkSlide13
Average-Case Lower Bound Extension
Theorem [Ros15]:
For
any
>0,
there exists an explicit function f
which is ½ + o(1) hard for
NC1 circuits with
negation gates under uniform distribution.
Corollary:
For
any >0, there exists an explicit function f which is ½ + o(1) hard for polynomial size formulas with
negation gates under uniform distribution.
Slide14
Proof of Theorem 1
Theorem 1:
Formula, size:
, negations:
, depth:
circuit, size:
, negations:
, depth
Decomposition Theorem
is a formula with
inputs
View
as a circuit with
inputs
is a circuit with
negations
Apply [BNT98] theoremSlide15
Application 2 – Shrinkage of Formulas
Definition:
is
where each
input is fixed
w.p
to a uniformly random
bit.
What happens to
; the size of
??Applications to PRGs, lower bounds, Fourier results, #SAT algorithms, etc.Definition:
is the largest constant s.t. formula
F
Trivial:
Slide16
Application 2 – Shrinkage of Formulas
- shrinkage exponent of formulas
[Tal14]
.
- shrinkage
exponent of
monotone formulas (conjecture
=3.27
).
-
shrinkage exponent of formulas with negations.
# of
negations
Conj.
[PZ93]
[
Subb’61
,...,Tal’14]
Slide17
Application 2 – Shrinkage of Formulas
Definition:
is the largest constant
s.t.
formula
F
that
contains
negations
Theorem 2:
Let
be a formula with
negations
Corollary:
If
, then
Slide18
Proof of Theorem 2
Theorem 2:
Let
be a formula with
negations
Decomposition Theorem
Each
is monotone
shrinks at rate
There are
’s
Restrict each
with
Slide19
Proof of Decomposition
Statement:
size: S <= 2s,
T
<= max{5t-2,1
}
s
ize: s, negations: t
…
Pushing NOTs to the top
Proof:
Induction on t.
Base case: t=0.
- Induction hypothesis:
holds for < t
(t>=1).
Slide20
Induction Step (NOT Root)
Statement:
…
Pushing NOTs to the top
Proof:
s’<=s,
t’= t-1
…
size: S <= 2s,
T = T’
<=
max{5(t-1)-2,1}
s
ize: s, negations: t
Decompose F’ by IH
size: S <= 2s,
T
<= max{5t-2,1
}Slide21
Induction Step (AND/OR Root)
Statement:
…
Pushing NOTs to the top
Proof:
/
s
1
+s
2
=s, t
1
+t
2
= t
Decompose
F
1
, F
2
by IH
…
/
…
S
1
+S
2
<= 2s, T
1
+T
2
<= 5t-4
size: S <= 2s,
T
<= max{5t-2,1
}
s
ize: s, negations: t
Require: t
1
,t
2
<tSlide22
Induction Step (Overall)
…
Pushing NOTs to the top
Proof:
s’+s
’’=s, t’= t
…
1
s
ize: s, negations: t
Decompose F’
(
previous slides
)
…
s’’+ s’’ + 2s’<= 2s, T = T’+2 <= 5t-2
Statement:
F’
:
t’ =t
, each
subformula
of F’ has
< t
negations
…
F’’ is monotone
0
size: S <= 2s,
T
<= max{5t-2,1
}
T’ <= 5t-4Slide23
Summary
Efficient decomposition theorem for negation-limited formulas
E
xtending results to
negation limited
formulas
from:
Negation limited
circuits
(size lower bounds, learning, crypto, etc). Generic, exp. improvement.Monotone
formulas (shrinkage)Thanks!
??
[Nechiporuk’62,Morizumi’09]
This Work
# of
negationsSlide24
Circuits to Formulas
-negation formula
-negation circuit
Average-case
lower
bound for
PRF
Learning
PRF
Learning
b