Recent Results on Polynomial Identity Testing
40
Recent Results on Polynomial Identity Testing Amir Shpilka Technion November 17, 2009 PIT Survey – Oberwolfach 1
Transcript of Recent Results on Polynomial Identity Testing
Re
ce
nt
Re
sults
on
P
oly
no
mia
l Id
en
tity
Te
stin
gP
oly
no
mia
l Id
en
tity
Te
stin
g
Am
ir Sh
pilk
a
Tec
hn
ion
No
ve
mb
er
17,
2009
PIT
Su
rve
y –
Ob
erw
olfa
ch
1
Go
al o
f ta
lk
•Su
rve
y k
no
wn
re
sults
•Exp
lain
pro
of
tec
hn
iqu
es
•G
ive
an
inte
rest
ing
se
t o
f `a
cc
ess
ible
’ o
pe
n
qu
est
ion
s
PIT
Su
rve
y –
Ob
erw
olfa
ch
No
ve
mb
er
17,
2009
2
Talk
ou
tlin
e•
De
fin
itio
n o
f th
e p
rob
lem
•C
on
ne
ctio
n t
o lo
we
r b
ou
nd
s (h
ard
ne
ss)
–K
ab
an
ets
-Im
pla
glia
zzo
–D
vir-S
-Ye
hu
da
yo
ff–
He
intz
-Sc
hn
orr
, A
gra
wa
l–
Ag
raw
al-V
ina
y
•Su
rve
y o
f p
osi
tiv
e r
esu
lts
PIT
Su
rve
y –
Ob
erw
olfa
ch
•Su
rve
y o
f p
osi
tiv
e r
esu
lts
•So
me
pro
ofs
:–
Spa
rse
po
lyn
om
ials
–P
art
ial d
eriv
ativ
es
tec
hn
iqu
e–
De
pth
-3 c
irc
uits
–D
ep
th-4
circ
uits
–R
ea
d-O
nc
e f
orm
ula
s
•C
on
ne
ctio
n t
o p
oly
no
mia
l fa
cto
riza
tio
n
No
ve
mb
er
17,
2009
3
Arith
me
tic
Circ
uits
Fie
ld: F
Va
ria
ble
s:X
1,...,X
n
Ga
tes:
+,
×
Eve
ry g
ate
in t
he
c
irc
uit c
om
pu
tes
a
PIT
Su
rve
y –
Ob
erw
olfa
ch
circ
uit c
om
pu
tes
a
po
lyn
om
ial i
n F
[X1,...,X
n]
Exa
mp
le:
(X1
⋅X
2)
⋅(X
2+
1)
Size
= n
um
be
r o
f g
ate
s
De
pth
= le
ng
th o
f lo
ng
est
inp
ut-
ou
tpu
t p
ath
De
gre
e=
ma
x d
eg
ree
of
inte
rna
l ga
tes
No
ve
mb
er
17,
2009
4
Arith
me
tic
Fo
rmu
las
Sam
e,
exc
ep
t u
nd
erlyin
g g
rap
h is
a t
ree
PIT
Su
rve
y –
Ob
erw
olfa
ch
No
ve
mb
er
17,
2009
5
Bo
un
de
d d
ep
th c
irc
uits
•Σ
Πc
irc
uits:
de
pth
-2 c
irc
uits
with
+a
t th
e t
op
a
nd
×a
t th
e b
ott
om
. Si
ze s
circ
uits
co
mp
ute
s-
spa
rse
po
lyn
om
ials
.
•Σ
ΠΣ
circ
uits:
de
pth
-3 c
irc
uits
with
+a
t th
e t
op
, ×
at
the
mid
dle
an
d +
at
the
bo
tto
m.
PIT
Su
rve
y –
Ob
erw
olfa
ch
×a
t th
e m
idd
le a
nd
+a
t th
e b
ott
om
. C
om
pu
te s
um
s o
f p
rod
uc
ts o
f lin
ea
r fu
nc
tio
ns.
I.e
. a
sp
ars
e p
oly
no
mia
l co
mp
ose
d w
ith
a
line
ar
tra
nsf
orm
atio
n.
•Σ
ΠΣ
Πc
irc
uits:
de
pth
-4 c
irc
uits.
Co
mp
ute
su
ms
of
pro
du
cts
of
spa
rse
po
lyn
om
ials
.
No
ve
mb
er
17,
2009
6
Wh
y A
rith
me
tic
Circ
uits?
•M
ost
na
tura
l mo
de
l fo
r c
om
pu
tin
g p
oly
no
mia
ls
•Fo
r m
an
y p
rob
lem
s (e
.g.
Ma
trix
Mu
ltip
lica
tio
n,
De
t) b
est
alg
orith
m is
an
arith
me
tic
circ
uit
•G
rea
t a
lgo
rith
mic
ac
hie
ve
me
nts
:
PIT
Su
rve
y –
Ob
erw
olfa
ch
•G
rea
t a
lgo
rith
mic
ac
hie
ve
me
nts
:
–Fo
urie
r Tr
an
sfo
rm
–M
atr
ix M
ultip
lica
tio
n
–P
oly
no
mia
l Fa
cto
riza
tio
n
•St
ruc
ture
d m
od
el (
co
mp
are
d t
o B
oo
lea
n
circ
uits)
P v
s.N
P m
ay b
e e
asi
er
No
ve
mb
er
17,
2009
7
Po
lyn
om
ial I
de
ntity
Te
stin
g
Inp
ut:
Arith
me
tic
circ
uit c
om
pu
tin
g f
Pro
ble
m:
Do
es
f≡0
?
C
f(x 1
,...,x
n)
PIT
Su
rve
y –
Ob
erw
olfa
ch
Ra
nd
om
ize
d a
lgo
rith
m [
Sch
wa
rtz,
Zip
pe
l, D
eM
illo
-Lip
ton
]: e
va
lua
te f
at
a r
an
do
m p
oin
tG
oa
l: d
ete
rmin
istic
alg
orith
m
x 1x 2
x n
No
ve
mb
er
17,
2009
8
Bla
ck B
ox
PIT
≡ ≡≡≡Exp
licit H
ittin
g S
et
Inp
ut:
A B
lac
k-B
ox
circ
uit c
om
pu
tin
g f
.P
rob
lem
:D
oe
s f=
0? P
IT S
urv
ey –
Ob
erw
olfa
ch
Cf(
x 1,...,x
n)
(x1,...,x
n)
Go
al:
de
term
inis
tic
alg
orith
m (
a.k
.a.
Hittin
g S
et)
S,Z,
DM
-L: ∃
sma
ll H
ittin
g S
et
(no
t e
xplic
it)
No
ve
mb
er
17,
2009
9
Mo
tiva
tio
n
•N
atu
ral a
nd
fu
nd
am
en
tal p
rob
lem
•St
ron
g c
on
ne
ctio
n t
o c
irc
uit lo
we
r b
ou
nd
s
•A
lgo
rith
mic
imp
ort
an
ce
:
–P
rim
alit
yte
stin
g [
Ag
raw
al-K
aya
l-Sa
xen
a]
–P
ara
llel a
lgo
rith
ms
for
fin
din
g m
atc
hin
g
PIT
Su
rve
y –
Ob
erw
olfa
ch
–P
ara
llel a
lgo
rith
ms
for
fin
din
g m
atc
hin
g
[Ka
rp-U
pfa
l-W
igd
ers
on
, M
ulm
ule
y-V
azi
ran
i-V
azi
ran
i]
No
ve
mb
er
17,
2009
10
Po
lyn
om
ial I
de
ntity
Te
stin
g
D
efin
itio
n o
f th
e p
rob
lem
•C
on
ne
ctio
n t
o lo
we
r b
ou
nd
s (h
ard
ne
ss)
–K
ab
an
ets
-Im
pla
glia
zzo
–D
vir-S
-Ye
hu
da
yo
ff
–H
ein
tz-S
ch
no
rr, A
gra
wa
l
–A
gra
wa
l-V
ina
y
PIT
Su
rve
y –
Ob
erw
olfa
ch
–A
gra
wa
l-V
ina
y
•Su
rve
y o
f p
osi
tiv
e r
esu
lts
•So
me
pro
ofs
:–
Spa
rse
po
lyn
om
ials
–P
art
ial d
eriv
ativ
es
tec
hn
iqu
e
–D
ep
th-3
circ
uits
–R
ea
d-O
nc
e f
orm
ula
s
•C
on
ne
ctio
n t
o p
oly
no
mia
l fa
cto
riza
tio
nN
ove
mb
er
17,
2009
11
Ha
rdn
ess
: P
IT ≡
low
er
bo
un
ds
[Ka
ba
ne
ts-I
mp
ag
liazz
o]:
•2
Ω(n
)lo
we
r b
ou
nd
fo
r P
erm
an
en
t ⇒
PIT
in n
po
lylo
g(n
)tim
e
•P
IT ∈
P ⇒
sup
er-
po
lyn
om
ial l
ow
er
bo
un
ds:
B
oo
lea
nfo
r N
EX
P, o
r a
rith
me
tic
for
Pe
rma
ne
nt
[Dvir-S
-Ye
hu
da
yo
ff]:
(alm
ost
) sa
me
as
K-I
for
PIT
Su
rve
y –
Ob
erw
olfa
ch
[Dvir-S
-Ye
hu
da
yo
ff]:
(alm
ost
) sa
me
as
K-I
for
bo
un
de
d d
ep
th c
irc
uits
[He
intz
-Sc
hn
orr
,Ag
raw
al]
: P
oly
no
mia
l tim
e B
lac
k-
Bo
x P
IT ⇒
Exp
on
en
tia
l lo
we
r b
ou
nd
s fo
r a
rith
me
tic
circ
uits
Less
on
: d
era
nd
om
izin
gP
IT e
sse
ntia
lly e
qu
iva
len
t to
pro
vin
g lo
we
r b
ou
nd
s fo
r a
rith
me
tic
circ
uit
No
ve
mb
er
17,
2009
12
No
n B
lac
k-B
ox
P.I.T
. ⇔
Low
er
Bo
un
ds
[K-I
]
[Va
lian
t,To
da
,Im
pa
glia
zzo
-Ka
ba
ne
ts-W
igd
ers
on
]:
NE
XP
⊆P
/P
oly
⇒
Pe
rm
is
NE
XP
-co
mp
lete
K-I
: P
erm
ha
s p
oly
siz
e a
rith
. c
irc
uit ⇒
Pe
rm in
NP
PIT
Ide
a:g
ue
ss c
irc
uit f
or
Pe
rm.
ve
rify
co
rre
ctn
ess
u
sin
g s
elf r
ed
uc
ibili
ty a
nd
PIT
.
PIT
Su
rve
y –
Ob
erw
olfa
ch
usi
ng
se
lf r
ed
uc
ibili
ty a
nd
PIT
.
⇒:If
NE
XP
⊆P
/Po
ly a
nd
Pe
rm h
as
po
ly s
ize
c
irc
uits
an
d P
IT in
Pth
en
NE
XP
in N
P⇒
⇐
Oth
er
dire
ctio
n f
ollo
ws
by u
sin
g a
rith
me
tic
ve
rsio
n
of
N-W
ge
ne
rato
r a
nd
Ka
lto
fen
’sfa
cto
riza
tio
n
the
ore
m.
No
ve
mb
er
17,
2009
13
Bla
ck-B
ox
P.I.T
. ⇒
Low
er
Bo
un
ds
[He
intz
-Sc
hn
orr
,Ag
raw
al]
: B
lac
k-B
ox
P.I.
T fo
r si
ze s
circ
uits
in t
ime
po
ly(s
) (i
.e.
po
ly(s
)si
ze
hittin
g s
et)
imp
lies
exp
on
en
tia
l lo
we
r b
ou
nd
s fo
r a
rith
me
tic
circ
uits:
Giv
en
H=
pi,
fin
d n
on
-ze
ro lo
g(|
H|
)+1-v
aria
te
po
lyn
om
ial f
suc
h t
ha
t f(
p)=
0 f
or
all
i.
PIT
Su
rve
y –
Ob
erw
olfa
ch
i
po
lyn
om
ial f
suc
h t
ha
t f(
pi)=
0 f
or
all
i.
⇒f
do
es
no
t h
ave
siz
e s
circ
uits
Giv
es
low
er
bo
un
ds
for
fin
PS
PA
CE
Co
nje
ctu
re[A
gra
wa
l]:
H=
(y
1,…
, y
n)
: y
i=y
kim
od
r, k,r
< s
20
is a
hittin
g
set
for
size
sc
irc
uits
No
ve
mb
er
17,
2009
14
Imp
ort
an
ce
of
ΣΠ
ΣΠ
circ
uits
[Ag
raw
al-V
ina
y,R
az]
: Exp
on
en
tia
l lo
we
r b
ou
nd
s fo
r
ΣΠ
ΣΠ
circ
uits
imp
ly e
xpo
ne
ntia
l lo
we
r b
ou
nd
s fo
r g
en
era
l circ
uits.
Pro
of:
1. D
ep
th r
ed
uc
tio
n a
-la
P=
NC
2[V
alia
nt-
Skyu
m-
Be
rko
witz-
Ra
cko
ff]
2.
Bre
ak t
he
circ
uit in
th
e m
idd
le
an
d in
terp
ola
te e
ac
h p
art
usi
ng
ΣΠ
circ
uits.
PIT
Su
rve
y –
Ob
erw
olfa
ch
an
d in
terp
ola
te e
ac
h p
art
usi
ng
ΣΠ
circ
uits.
Co
r[A
gra
wa
l-V
ina
y]:
Po
lyn
om
ial t
ime
PIT
of
ΣΠ
ΣΠ
circ
uits
giv
es
qu
asi
-po
lyn
om
ial t
ime
PIT
fo
r g
en
era
l c
ircu
its.
Pro
of:
By [
He
intz
-Sc
hn
orr
,Ag
raw
al] p
oly
no
mia
l tim
e
PIT
⇒e
xpo
ne
ntia
l lo
we
r b
ou
nd
s fo
r Σ
ΠΣ
Πc
ircu
its.
[A
gra
wa
l-V
ina
y]
⇒e
xpo
ne
ntia
l lo
we
r b
ou
nd
s fo
r g
en
era
l circ
uits.
No
w u
se [
K-I
].
No
ve
mb
er
17,
2009
15
Po
lyn
om
ial I
de
ntity
Te
stin
g
D
efin
itio
n o
f th
e p
rob
lem
C
on
ne
ctio
n t
o lo
we
r b
ou
nd
s (h
ard
ne
ss)
K
ab
an
ets
-Im
pla
glia
zzo
D
vir-S
-Ye
hu
da
yo
ff
H
ein
tz-S
ch
no
rr, A
gra
wa
l
A
gra
wa
l-V
ina
y
PIT
Su
rve
y –
Ob
erw
olfa
ch
A
gra
wa
l-V
ina
y
•Su
rve
y o
f p
osi
tiv
e r
esu
lts
•So
me
pro
ofs
:–
Spa
rse
po
lyn
om
ials
–P
art
ial d
eriv
ativ
es
tec
hn
iqu
e
–D
ep
th-3
circ
uits
–R
ea
d-O
nc
e f
orm
ula
s
•C
on
ne
ctio
n t
o p
oly
no
mia
l fa
cto
riza
tio
nN
ove
mb
er
17,
2009
16
Ra
nd
om
ize
d a
lgo
rith
ms
for
PIT
Sch
wa
rtz,
Zip
pe
l,De
Mill
o-L
ipto
n:
Ev
alu
ate
Ca
t a
ra
nd
om
inp
ut
Giv
es
err
or-
ran
do
mn
ess
tra
de
off
Ch
en
-Ka
o: tr
ad
e t
ime
fo
r e
rro
r o
ve
r R
:
πi=
±p
i,1…
±p
i,r ,
fo
r d
iffe
ren
t p
rim
es,
ra
nd
om
sig
ns
The
n C
≡0
iff
C(π
1,π
2,…
,πn)
= 0
PIT
Su
rve
y –
Ob
erw
olfa
ch
The
n C
≡0
iff
C(π
1,π
2,…
,πn)
= 0
Tru
nc
atin
g a
fte
r t
dig
its
giv
es
err
or
O(1
/t)
Intu
itio
n: ra
nd
om
co
nju
ga
te w
on
’t v
an
ish
mo
d 2
-t
For
mu
ltili
ne
ar
po
lyn
om
ials
, C
-Ku
se n
ran
do
m b
its
for
1/p
oly
err
or,
S-Z
-DM
-Lu
se n
log
(n)
bits
for
err
or .
Lew
in-V
ad
ha
n: g
en
era
lize
d C
-Kto
fin
ite
fie
lds:
irr
ed
uc
ible
po
lyn
om
ials
↔p
rime
s,
po
we
r se
ries
↔sq
ua
re r
oo
ts. Tr
un
ca
tio
n m
od
xt .
No
ve
mb
er
17,
2009
17
Ra
nd
om
ize
d a
lgo
rith
ms
for
PIT
Ag
raw
al-B
isw
as:
Ob
serv
e:
C ≡
0iff
C(y
,yD,y
D2,…
, y
Dn)
≡0
Pro
ble
m:
de
gre
e t
oo
larg
e
A-B
giv
e a
“sm
all”
se
t o
f p
oly
no
mia
ls
f i(y)
s.t
. C
≡0
iff
∀iC
(y,y
D,y
D2,…
, y
Dn)
≡0
mo
df i(
y)
−Si
mila
r id
ea
use
d in
prim
alit
yte
st o
f A
-K-S
PIT
Su
rve
y –
Ob
erw
olfa
ch
−Si
mila
r id
ea
use
d in
prim
alit
yte
st o
f A
-K-S
−U
ses
less
ra
nd
om
bits
tha
nS-
Z-D
M-L
−N
on
bla
ck-b
ox
Ag
raw
al’s
co
nje
ctu
re:
(y
1,…
, y
n)
: y
i=y
kim
od
r,k
,r<
s20
is a
hittin
g s
et
for
size
sc
irc
uits
No
ve
mb
er
17,
2009
18
De
term
inis
tic
alg
orith
ms
for
PIT
•Σ
Πc
ircu
its
(a.k
.a.,
spa
rse
po
lys)
[B
en
Or-
Tiw
ari,
G
rigo
riev
-Ka
rpin
ski,
Kliv
an
s-Sp
ielm
an
,…]
–B
lac
k-B
ox
in p
oly
no
mia
l tim
e
•N
on
-co
mm
uta
tiv
e f
orm
ula
s [R
az-
S]
–N
on
-Bla
ck-B
ox
in p
oly
no
mia
l tim
e
•Σ
ΠΣ
(k)
circ
uits
[Dv
ir-S,
Ka
ya
l-Sa
xen
a,A
rvin
d-
Mu
kh
op
ad
hya
y,K
arn
in-S
,Sa
xen
a-S
esh
ad
ri,K
aya
l-Sa
raf]
PIT
Su
rve
y –
Ob
erw
olfa
ch
Mu
kh
op
ad
hya
y,K
arn
in-S
,Sa
xen
a-S
esh
ad
ri,K
aya
l-Sa
raf]
–
Bla
ck-B
ox
in q
ua
si-p
oly
no
mia
l tim
e*
–N
on
-Bla
ck-B
ox
in t
ime
nO
(k)
•Su
m o
f k R
ea
d-o
nc
e f
orm
ula
s [S
-Vo
lko
vic
h]
–B
lac
k-B
ox
in n
O(lo
g(n
) +
k)
–N
on
-Bla
ck-B
ox
in t
ime
nO
(k)
•M
ultili
ne
ar
ΣΠ
ΣΠ
(k):
[Ka
rnin
-Mu
kh
op
ad
hya
y-S
-Vo
lko
vic
h]
–B
lac
k-B
ox
in q
ua
si-p
oly
no
mia
l tim
e
No
ve
mb
er
17,
2009
19
Wh
y s
tud
y r
est
ric
ted
mo
de
ls
•[A
gra
wa
l-V
ina
y]
PIT
fo
r Σ
ΠΣ
Πc
irc
uits
imp
lies
PIT
fo
r g
en
era
l de
pth
.
•G
ain
ing
insi
gh
t to
mo
re g
en
era
l qu
est
ion
s:
Intu
itiv
ely
:lo
we
r b
ou
nd
s im
ply
PIT
Mu
ltili
ne
ar
form
ula
s: s
up
er
po
lyn
om
ial b
ou
nd
s
PIT
Su
rve
y –
Ob
erw
olfa
ch
Mu
ltili
ne
ar
form
ula
s: s
up
er
po
lyn
om
ial b
ou
nd
s [R
az]
bu
t n
o P
IT a
lgo
rith
ms
No
t e
ve
n f
or
De
pth
-3 m
ultili
ne
ar
form
ula
s!
Sum
of
RO
Fs,
de
pth
-3,4
mu
ltili
ne
ar
form
ula
s re
laxa
tio
ns
of
the
mo
re g
en
era
l pro
ble
m
•In
tere
stin
g r
esu
lts:
Str
uc
tura
l th
eo
rem
s fo
r Σ
ΠΣ
(k)
an
d Σ
ΠΣ
Π(k
) c
irc
uits.
No
ve
mb
er
17,
2009
20
Po
lyn
om
ial I
de
ntity
Te
stin
g
D
efin
itio
n o
f th
e p
rob
lem
C
on
ne
ctio
n t
o lo
we
r b
ou
nd
s (h
ard
ne
ss)
K
ab
an
ets
-Im
pla
glia
zzo
A
gra
wa
l
D
vir-S
-Ye
hu
da
yo
ff
A
gra
wa
l-V
ina
y
PIT
Su
rve
y –
Ob
erw
olfa
ch
A
gra
wa
l-V
ina
y
Su
rve
y o
f p
osi
tiv
e r
esu
lts
•So
me
pro
ofs
:–
Spa
rse
po
lyn
om
ials
–P
art
ial d
eriv
ativ
es
tec
hn
iqu
e
–D
ep
th-3
circ
uits
–R
ea
d-O
nc
e f
orm
ula
s
•C
on
ne
ctio
n t
o p
oly
no
mia
l fa
cto
riza
tio
nN
ove
mb
er
17,
2009
21
Pro
ofs
–ta
ilore
d f
or
the
mo
de
l
Pro
ofs
usu
ally
use
`w
ea
kn
ess
’ in
he
ren
t in
mo
de
l•
De
pth
2: fe
w m
on
om
ials
. Su
bst
itu
tin
g y
aito
xiw
e
ca
n c
on
tro
l `c
olla
pse
s’ o
f d
iffe
ren
t m
on
om
ials
.
•N
on
Co
mm
uta
tiv
e f
orm
ula
s: P
oly
no
mia
l ha
s fe
w
line
arly
ind
ep
en
de
nt
pa
rtia
l de
riva
tiv
es
[Nis
an
].
Ke
ep
tra
ck o
f a
ba
sis
for
de
riva
tiv
es
to d
o P
IT.
PIT
Su
rve
y –
Ob
erw
olfa
ch
Ke
ep
tra
ck o
f a
ba
sis
for
de
riva
tiv
es
to d
o P
IT.
•Σ
ΠΣ
(k):
se
ttin
g a
lin
ea
r fu
nc
tio
n t
o z
ero
re
du
ce
s to
p f
an
-in
. If k
=2
the
n m
ultip
lica
tio
n g
ate
s m
ust
be
th
e s
am
e. C
alls
fo
r in
du
ctio
n.
•M
ultili
ne
ar
ΣΠ
ΣΠ
(k):
in s
om
e s
en
se `
co
mb
ina
tio
n’
of
spa
rse
po
lyn
om
ials
an
d m
ultiln
ea
rΣ
ΠΣ
(k).
•R
ea
d-O
nc
e-F
orm
ula
s: s
ub
fo
rmu
las
of
roo
t c
on
tain
½
of
va
riab
les.
No
ve
mb
er
17,
2009
22
De
pth
2 (
ΣΠ
)c
irc
uits
f(x 1
,…,x
n)
= M
1+
…+
Mm
sum
of
md
eg
ree
dm
on
om
ials
Ide
a: re
pla
ce
xib
y y
aiso
th
at
all
mo
no
mia
ls m
ap
u
niq
ue
ly, in
terp
ola
te r
esu
ltin
g p
oly
no
mia
l.
Pro
ble
m: a
i–s
ne
ed
to
gro
w f
ast
(g
ive
s h
igh
de
gre
e)
[Kliv
an
s-Sp
eilm
an
]: f
or
larg
e p
rime
p,k ≤
p s
et
ai=
ki-1
mo
d p
. Ev
alu
ate
at
np
+1
diffe
ren
t y-s
.
PIT
Su
rve
y –
Ob
erw
olfa
ch
i
mo
d p
. Ev
alu
ate
at
np
+1
diffe
ren
t y-s
.
x 1e
1⋅x
2e
2⋅…
⋅x n
en
ma
pp
ed
to
y^
(e1+
e2k+
…+
enk
n-1
) =
yE(k
)
mm
on
om
ials
de
fin
e m
po
lyn
om
ials
E1(k
), …
,Em
(k).
Th
ey a
re m
ap
pe
d 1
-1 if
kis
no
t ro
ot
of
an
y E
i-Ej.
Ho
lds
for
a la
rge
fra
ctio
n o
f th
e k
’s.
Be
tte
r c
on
stru
ctio
ns
are
kn
ow
n
No
ve
mb
er
17,
2009
23
No
n c
om
mu
tative
fo
rmu
las
Spe
cia
l ca
se:
set-
mu
ltili
ne
ar
de
pth
-3 c
irc
uits
X =
X1⊔
X2⊔
…⊔
Xd
, X
i=x
i,1,…
,xi,n
Mu
ltip
lica
tio
n g
ate
: M
i=
Li,1
(X1)
⋅…
⋅L i
,d(X
d)
C =
M1
+ …
+ M
s
Ma
in o
bse
rva
tio
n:
dim
en
sio
n o
f p
art
ial d
eriva
tive
s o
f C
ac
co
rdin
g t
o X
,…,X
(an
yk)
is a
t m
ost
s
PIT
Su
rve
y –
Ob
erw
olfa
ch
Ma
in o
bse
rva
tio
n:
dim
en
sio
n o
f p
art
ial d
eriva
tive
s o
f C
ac
co
rdin
g t
o X
1,…
,Xk
(an
yk)
is a
t m
ost
s
(sp
an
ne
d b
y L
i,k+
1(X
k+
1)⋅
…⋅L
i,d(X
d)
i=1…
s)
Alg
orith
m [
Ra
z-S]
: c
om
pu
te a
ba
sis
for
all
de
riva
tive
sa
cc
ord
ing
to
X1,…
,Xk
sta
rtin
g f
rom
k=
1 t
ok=
d. C
≡0 if
at
the
en
d a
ll b
asi
s e
lem
en
ts
are
0
Sam
e id
ea
als
o in
th
e g
en
era
l ca
seN
ove
mb
er
17,
2009
24
De
pth
-3 c
irc
uits
(ΣΠ
Σ(k
) c
irc
uits)
L =
Σt=
1...n
at⋅x t
+ a
0,
Mi=
Πj=
1...d
iL i,j,
C =
Σi=
1...k
Mi
De
fin
itio
n:
C s
imp
leif n
o li
ne
ar
fun
ctio
n a
pp
ea
rs in
all
the
Mi-s
C m
inim
ali
f n
o s
ub
set
of
mu
lt. g
ate
s su
ms
to z
ero
Ma
in t
oo
l [D
vir
S]:
If C
≡0
sim
ple
an
d m
inim
al
PIT
Su
rve
y –
Ob
erw
olfa
ch
Ma
in t
oo
l [D
vir
S]:
If C
≡0
sim
ple
an
d m
inim
al
the
n d
im(s
pa
n(L
i,j))
≤ R
an
k(k
,d)
= (
log
(d))
k*
Less
on
: If C
≡0
the
n it
is v
ery
str
uc
ture
d
No
n B
lac
k-B
ox
Alg
orith
m:
fin
d p
art
itio
n t
o s
ub
-c
irc
uits
of
low
dim
en
sio
n (
aft
er
rem
ova
l of
g.c
.d.)
an
d b
rute
fo
rce
ve
rify
th
at
the
y v
an
ish
.
Imp
rove
d nO(k)a
lgo
rith
m b
y [
Ka
ya
l-Sa
xen
a].
No
ve
mb
er
17,
2009
25
Bla
ck-B
ox
PIT
fo
r Σ
ΠΣ
(k)
Bla
ck-B
ox
alg
orith
m [
Ka
rnin
-S]:
re
stric
t C
to a
low
d
im s
ub
spa
ce
su
ch
th
at
the
dim
en
sio
ns
of
an
ysu
b-c
irc
uit is
no
t re
du
ce
d b
y t
oo
mu
ch
.
Ide
a: su
ch
ma
p p
rese
rve
s st
ruc
ture
of
C
Cla
im: C
|V
≡0
iff
C ≡
0
Ca
n f
ind
po
ly s
et
of
V-s
of
dim
en
sio
nR
an
k(k
,d)
PIT
Su
rve
y –
Ob
erw
olfa
ch
Ca
n f
ind
po
ly s
et
of
V-s
of
dim
en
sio
nR
an
k(k
,d)
Giv
es:
po
ly(n
)⋅d
O(R
an
k(k
,d))
tim
e a
lgo
rith
m[S
axe
na
-Se
sha
dri]:
fin
iteF
, R
an
k(k
,d)
< k
3lo
g(d
)
[Ka
ya
l-Sa
raf]
: o
ve
r Q
,RR
an
k(k
,d)
< k
k
Imp
rove
[D
vir-S
] a
nd
[K
arn
in-S
] (p
lug
an
d p
lay)
To s
ee
th
e p
roo
fs c
om
e t
o t
he
PIT
se
ssio
n!
No
ve
mb
er
17,
2009
26
Bla
ck-B
ox
PIT
fo
r m
ultili
ne
ar
ΣΠ
ΣΠ
(k)
[Ka
rnin
-Mu
kh
op
ad
hya
y-S
-Vo
lko
vic
h]
C =
Σi=
1...k
Mis.
t.M
i=
Pi1
⋅…⋅P
id, P
ijis
siz
es
mu
ltili
ne
ar
ΣΠ
circ
uit. P
i1,…
,Pid
va
ria
ble
dis
join
t
Ob
serv
e:
in e
ac
h M
i, a
t m
ost
po
lylo
g(n
)P
ij-s
ha
ve
m
ore
th
an
n/p
oly
log
(n)
va
ria
ble
s.
⇒M
i=
Ai⋅B
i, A
i=
qu
asi
-po
ly s
pa
rse
an
d
PIT
Su
rve
y –
Ob
erw
olfa
ch
⇒M
i=
Ai⋅B
i, A
i=
qu
asi
-po
ly s
pa
rse
an
d
Bi=
pro
du
ct
of
spa
rse
Pijo
n n
/po
lylo
g(n
) va
rs
Cla
im:
∃p
oly
log
(n)
va
rsth
at
aft
er
de
ria
va
tin
go
r su
bst
itu
tin
g z
ero
es
to t
he
m,
C’
= Σ
i=1...k
B’ i
≠0
Pro
of:
ea
ch
op
era
tio
n r
ed
uc
es
by h
alf t
he
n
um
be
r o
f m
on
om
ials
of
som
e A
i
No
ve
mb
er
17,
2009
27
Bla
ck-B
ox
PIT
fo
r m
ultili
ne
ar
ΣΠ
ΣΠ
(k)
C’
= Σ
i=1...k
B’ i
≠0
s.t.
Bi=
pro
du
ct
of
spa
rse
, va
ria
ble
dis
join
t, P
ij–s
on
n/p
oly
log
(n)
va
rs
Cla
im: C
’c
on
tain
s n
on
-ze
ro m
ultili
ne
ar
ΣΠ
Σ(k
)c
irc
uit
Pro
of:
ra
nd
om
ly f
ix a
ll va
rsa
pp
ea
rin
g w
ith
x1…
PIT
Su
rve
y –
Ob
erw
olfa
ch
1
Ca
n d
era
nd
om
ize
usi
ng
PIT
fo
r sp
ars
e p
oly
s.
Co
nc
lusi
on
: n
ee
d a
bla
ck-b
ox
wa
y o
f `i
sola
tin
g’
po
lylo
g(n
) va
ria
ble
s w
hile
ap
ply
ing
a s
pa
rse
-PIT
fo
r th
e r
em
ain
ing
va
rs.
[S-V
olk
ovic
h]:
ge
ne
rato
r fo
r re
ad
-on
ce
-fo
rmu
las
ha
vin
g t
his
pro
pe
rty.
No
ve
mb
er
17,
2009
28
Re
ad
-On
ce
fo
rmu
las
(RO
Fs)
A f
orm
ula
wh
ere
eve
ry v
aria
ble
lab
els
at
mo
st
on
e le
af.
Pre
pro
ce
sse
d R
OF:
ca
n r
ep
lac
e
ea
ch
xiw
ith
Ti(x i
)
Sum
of
RO
Fs:
PIT
Su
rve
y –
Ob
erw
olfa
ch
Sum
of
RO
Fs:
F =
F1
+ F
2+
…
+ F
k
ea
ch
Fiis
a (
P-)
RO
F
Re
sult:
Bla
ck-B
ox
PIT
fo
r su
m o
f k
(P)R
OFs
in t
ime
nO
(lo
g(n
) +
k)
No
ve
mb
er
17,
2009
29
X3
Ge
ne
rato
r fo
r R
OFs
A=
α
1,
α2,
α3,
…,
αn ⊆F
Let
ui(x)
be
su
ch
th
at
ui(a
j) =
δ(i
,j)
De
f: F
or
eve
ry i
∈[n
] a
nd
k ≥
1:
Gi k(y
,z) ≜
ui(y
1)·
z 1+
ui(y
2)·
z 2+
… +
ui(y
k)·z
k
G(y
,z) ≜
(G1
, G
2, …
,G
n)
PIT
Su
rve
y –
Ob
erw
olfa
ch
Gk(y
,z) ≜
(G1k,
G2k
, …
,G
nk)
Cru
cia
l Pro
pe
rty:
Gk|
(yk
= α
m)
=G
k-1
+ z
k·ē
m
Gk(y
,z)
en
ab
les
iso
latio
n o
f a
ny k
’≤k
va
ria
ble
s
In a
dd
itio
n, G
k(y
,z)
is g
en
era
tor
for
2k
spa
rse
po
lyn
om
ials
(n
ee
de
d f
or
ΣΠ
ΣΠ
PIT
)
No
ve
mb
er
17,
2009
30
PIT
fo
r R
OFs
The
ore
m: L
et
Pb
e a
no
n-z
ero
RO
P t
he
n
P(G
log
(n)+
1)
≠0
. M
ore
ov
er,
if P
is a
no
n-
co
nst
an
t p
oly
no
mia
l th
en
so
is
P(G
log
(n)+
1)
Pro
of
ide
a: in
du
ctio
n o
n s
tru
ctu
re o
f fo
rmu
la.
PIT
Su
rve
y –
Ob
erw
olfa
ch
Pro
of
ide
a: in
du
ctio
n o
n s
tru
ctu
re o
f fo
rmu
la.
If t
he
to
p g
ate
is ×
the
n b
y in
du
ctio
n w
e
are
ok. If t
op
ga
te is
+, th
en
on
e s
on
ha
s fe
w v
aria
ble
s.
Ca
n k
ee
p a
va
ria
ble
th
at
be
lon
gs
to s
ma
ll so
n ‘
aliv
e’.
No
ve
mb
er
17,
2009
31
Sum
of
RO
Fs
F =
F1
+ F
2+
…
+ F
k
Ide
a: P
IT f
or
RO
Fs g
ive
s a
just
ifyin
g s
et
for
an
y k
R
OFs
of
size
nO
(lo
g n
)
Just
ifyin
g s
et:
co
nta
ins
at
lea
st o
ne
inp
ut
(a1,…
,an)
suc
h t
ha
t if F
id
ep
en
ds
on
xm
the
n
PIT
Su
rve
y –
Ob
erw
olfa
ch
(a1,…
,an)
suc
h t
ha
t if F
id
ep
en
ds
on
xm
the
n
F i(a
1,…
,am
-1,x
m,a
m+
1,…
,an)
de
pe
nd
s o
n x
m.
By c
ha
ng
ing
xi←
x i+
aia
ssu
me
th
at
all
the
Fi-s
a
re 0
-ju
stifyie
d.
I.e.
ass
ign
ing
ze
ros
to a
ll va
ria
ble
s b
ut
x ike
ep
s d
ep
en
de
nc
e o
n x
i
No
ve
mb
er
17,
2009
32
Ha
rdn
ess
of
rep
rese
nta
tio
n
Ha
rdn
ess
of
rep
rese
nta
tio
n:
no
su
m o
f k <
n/3
0-ju
stifie
d R
OFs
ca
n c
om
pu
te x
1·x
2·…
·xn
Pro
of
Ide
a:
By in
du
ctio
n o
n k
. B
y t
akin
g p
art
ial
de
riva
tive
s a
nd
ma
kin
g s
ub
stitu
tio
ns,
ca
n
rem
ove
so
me
of
the
RO
Fsb
ut
pre
serv
e t
he
st
ruc
ture
s o
f F
an
d x
1·x
2·…
·xn.
PIT
Su
rve
y –
Ob
erw
olfa
ch
stru
ctu
res
of
Fa
nd
x1·x
2·…
·xn.
The
ore
m:
Let
Fb
e a
su
m o
f k
0-ju
stifie
dR
OFs
. Le
t A
be
a s
et
of
all
ve
cto
rs in
0,1
no
f H
am
min
g w
eig
ht
≤ k
.Th
en
F ≡
0 ⇔
F|A
≡
0.
Ide
a: Fo
r n
≤ k
cle
ar.
Fo
r la
rge
n,
set
x i=
0.
Ind
uc
tio
n im
plie
s x i
| F
. H
en
ce
x1·x
2·…
·xn
| F
⇒
⇐
No
ve
mb
er
17,
2009
33
Po
lyn
om
ial I
de
ntity
Te
stin
g
D
efin
itio
n o
f th
e p
rob
lem
C
on
ne
ctio
n t
o lo
we
r b
ou
nd
s (h
ard
ne
ss)
K
ab
an
ets
-Im
pla
glia
zzo
D
vir-S
-Ye
hu
da
yo
ff
H
ein
tz-S
ch
no
rr, A
gra
wa
l
A
gra
wa
l-V
ina
y
PIT
Su
rve
y –
Ob
erw
olfa
ch
A
gra
wa
l-V
ina
y
Su
rve
y o
f p
osi
tiv
e r
esu
lts
So
me
pro
ofs
:
Spa
rse
po
lyn
om
ials
P
art
ial d
eriv
ativ
es
tec
hn
iqu
e
D
ep
th-3
circ
uits
R
ea
d-O
nc
e f
orm
ula
s
•C
on
ne
ctio
n t
o p
oly
no
mia
l fa
cto
riza
tio
nN
ove
mb
er
17,
2009
34
PIT
an
d F
ac
torin
g
fis
composed
if f
(X)
= g
(X|
S)⋅
h(X
|T)
wh
ere
Sa
nd
T
are
dis
join
t
[S-V
olk
ovic
h]:
PIT
is e
qu
iva
len
t to
fa
cto
rin
g t
o
decomposable
fac
tors
.
⇐: f
≡0 if
ff+
y⋅z
ha
s tw
o d
ec
om
po
sab
le f
ac
tors
.
PIT
Su
rve
y –
Ob
erw
olfa
ch
⇐: f
≡0 if
ff+
y⋅z
ha
s tw
o d
ec
om
po
sab
le f
ac
tors
.
⇒: C
laim
: If w
e h
ave
a (
BB
or
NB
B)
PIT
fo
r a
ll c
irc
uits
of
the
fo
rm C
1+
C2⋅C
3,
wh
ere
Ci∈
M
the
n g
ive
n (
BB
or
NB
B)
C ∈
Mw
e c
an
de
term
inis
tic
ally
ou
tpu
t (B
B o
r N
BB
) a
ll d
ec
om
po
sab
le f
ac
tors
of
C.
No
ve
mb
er
17,
2009
35
De
co
mp
osa
ble
fa
cto
rin
g u
sin
g P
IT
Cla
im: If w
e h
av
e a
(B
B o
r N
BB
) P
IT f
or
all
circ
uits
of
the
fo
rm C
1+
C2⋅C
3, w
he
reC
i∈
Mth
en
giv
en
(B
B o
r N
BB
) C
∈M
we
ca
n d
ete
rmin
istic
ally
ou
tpu
t (B
B o
r
NB
B)
all
de
co
mp
osa
ble
fa
cto
rs o
f C
.
Ide
a: U
sin
g P
IT f
ind
a ju
stifyin
g a
ssig
nm
en
t a
for
PIT
Su
rve
y –
Ob
erw
olfa
ch
Ide
a: U
sin
g P
IT f
ind
a ju
stifyin
g a
ssig
nm
en
t a
for
C. Se
t x n
= a
na
nd
fa
cto
r (r
ec
urs
ive
ly).
Ass
um
e S
1,…
, S k
is t
he
pa
rtitio
n o
f [n
-1].
For
eve
ry S
ic
he
ck w
ea
the
r C
(a)⋅
C ≡
C(X
Si←
aSi)⋅
C(X
[n]\
Si ←
a[n
]\Si)
If y
es,
ad
d S
ito
th
e p
art
itio
n. A
t th
e e
nd
pu
t a
ll th
e r
em
ain
ing
va
rs in
a n
ew
se
t.N
ove
mb
er
17,
2009
36
PIT
an
d f
ac
torin
g
•D
ete
rmin
istic
de
co
mp
osa
ble
fa
cto
rin
g is
e
qu
iva
len
t to
low
er
bo
un
ds:
–D
ete
rmin
istic
fa
cto
rin
g im
plie
s N
EX
Pd
oe
s n
ot
ha
ve
sm
all
arith
me
tic
circ
uits
–Lo
we
r b
ou
nd
s im
ply
De
term
inis
tic
de
co
mp
osa
ble
fa
cto
rin
g
PIT
Su
rve
y –
Ob
erw
olfa
ch
fac
torin
g
•P
IT ≡
fac
torin
g f
or
mu
ltili
ne
ar
po
lyn
om
ials
•D
ete
rmin
istic
de
co
mp
osa
ble
fa
cto
rin
g f
or
de
pth
-2,
ΣΠ
Σ(k
), s
um
of
rea
d-o
nc
e…
•O
pe
n p
rob
lem
: is
PIT
eq
uiv
ale
nt
to g
en
era
l fa
cto
riza
tio
n?
No
ve
mb
er
17,
2009
37
Sum
ma
ry o
f ta
lk
D
efin
itio
n o
f th
e p
rob
lem
C
on
ne
ctio
n t
o lo
we
r b
ou
nd
s (h
ard
ne
ss)
K
ab
an
ets
-Im
pla
glia
zzo
D
vir-S
-Ye
hu
da
yo
ff
H
ein
tz-S
ch
no
rr, A
gra
wa
l
A
gra
wa
l-V
ina
y
PIT
Su
rve
y –
Ob
erw
olfa
ch
A
gra
wa
l-V
ina
y
Su
rve
y o
f p
osi
tiv
e r
esu
lts
So
me
pro
ofs
:
Spa
rse
po
lyn
om
ials
P
art
ial d
eriv
ativ
es
tec
hn
iqu
e
D
ep
th-3
circ
uits
R
ea
d-O
nc
e f
orm
ula
s
C
on
ne
ctio
n t
o p
oly
no
mia
l fa
cto
riza
tio
nN
ove
mb
er
17,
2009
38
Som
e `
ac
ce
ssib
le’
op
en
pro
ble
ms
1.
Giv
e a
Bla
ck-B
ox
PIT
alg
orith
m f
or
no
n-
co
mm
uta
tive
fo
rmu
las
2.
Solv
e P
IT f
or
de
pth
-3 c
irc
uits
3.
Solv
e P
IT f
or
mu
ltili
ne
ar
de
pth
-3 c
irc
uits
4.
Bla
ck-B
ox
PIT
fo
r se
t-m
ultili
ne
ar
de
pth
-3
PIT
Su
rve
y –
Ob
erw
olfa
ch
4.
Bla
ck-B
ox
PIT
fo
r se
t-m
ultili
ne
ar
de
pth
-3
circ
uits
5.
Po
lyn
om
ial t
ime
PIT
fo
r (s
um
of)
RO
Fs
6.
P.I.
T. f
or
de
pth
-4 w
ith
re
stric
ted
fa
n-in
7.
P.I.
T. f
or
rea
d-k
fo
rmu
las
(ca
n d
o it
fo
r k=
2)
8.
Is P
IT e
qu
iva
len
t to
ge
ne
ral f
ac
toriza
tio
n?
No
ve
mb
er
17,
2009
39
Tha
nk Y
ou
!
No
ve
mb
er
17,
2009
40
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