of DepthThree and Arithmetic Circuits with General Gates Oded Goldreich Weizmann Institute of Science Based on Joint work with Avi Wigderson Original title On the Size of DepthThree Boolean Circuits for Computing ID: 541981
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Slide1
Boolean Circuits of Depth-Three and Arithmetic Circuits with General Gates
Oded GoldreichWeizmann Institute of Science
Based on Joint work with
Avi
Wigderson
Original title
:
“On
the Size of Depth-Three Boolean Circuits for Computing
Multilinear
Functions”, ECCC TR13-043.Slide2
Constant Depth Boolean Circuits
Parityn requires depth d circuits of size exp(
(n
1/(d-1)
)).Famous frontier: Stronger circuit models.Another frontier: Stronger lower bounds (i.e., exp((n))).
Multi-linear functions : x=(x(1),…,x(t)), x(i)0,1n F(x(1),…,x(t)) = (i_1,…,i_t)T xi_1(1) xi_t(t) associated with tensor T [n]t
Conj (sanity check): For every t>1, there exists a t-linear function that requires depth-three circuits of size exp((tnt/(t+1))). [holds for t=1…]
Think of t=2,… log nSlide3
The Program*
t-linear functions x=(x(1),…,x(t)), |x(i)
|=n
F(x(1),…,x(t)) = (i_1,…,i_t)T x
i_1(1) xi_t(t) Conj (1st sanity check): For every t>1, there exists a t-linear function that requires depth-three circuits of size exp((tnt/(t+1))). [holds for t=1]Goal: For every
t>1, present an explicit t-linear function that requires depth-three circuits of size exp((tnt/(t+1))). [holds for t=1]A 2
nd sanity check: Consider a restricted model of (depth-three) circuits, and prove the L.B. in it.
*) Taking advantage of
Avi’s
absence.Slide4
Arithmetic Circuits with General Gates
Motivation: Depth-three Boolean Circuits for Parityn are obtained by implementing a sqrt(n)
-
way sum of
sqrt(n)-way sums. In general, depth-three BC are obtained via depth-two AC with general ML-gates.Model:
Depth-two (set-)multi-linear circuits with arbitrary (set-)multi-linear gates. Complexity measure (C2) = the (max.) arity of a gate.Recall: We use a fix partition of the variables, and multi-linear means being linear in each variable-block.We get depth-three BC for F of size exponential in C2(F)Depth-three BC obtained this way are restricted in (1) their structure arising from direct composition, and (2) ML gates. Slide5
Arithmetic Circuits with General Gates (cont.)
Model: Unbounded-depth (set-)multi-linear circuitswith arbitrary (set-)multi-linear gates.
Complexity measure (
C
) = max(arity, #gates).PROP:
Every ML function F has a depth-three BC of size exp(O(C(F)).PF: guess & verify.THM: There exist bilinear functions F such that C(F)=sqrt(n) but C2(F)=(n2/3).OBS: For every t-linear F, Ct+1(F) ≤ 2C(F)
.Slide6
Arith. Circuits with General Gates: Results
Model: Unbounded-depth (set-)multi-linear circuitswith arbitrary (set-)multi-linear gates.
Complexity measure (
C
) = max(arity, #gates); C2
for depth-two.THM 1: There exist bilinear functions F such that C(F)=sqrt(n) but C2(F)=(n2/3).THM 2: For every t-linear function F it holds that C(F) ≤ C2(F) = O(tnt/(t+1)).
THM 3: Almost all t-linear functions F satisfy C2(F) ≥ C(F) = (tnt/(t+1)).Open: An explicit function as in Thm 3; for starters
(tn0.51).Slide7
Arith. Circuits with General Gates: Results (cont.)
Model: Unbounded-depth (set-)multi-linear circuitswith arbitrary (set-)multi-linear gates.
Complexity measure (
C
) = max(arity, #gates); C2
for depth-two.An approach (a candidate): The 3-linear function assoc. with tensor T=(i,j,k): |i-(n/2)|+|j-(n/2)|+|k-(n/2)|≤n/2. Open: An explicit function as in Thm 3; for starters (tn0.51).Note: A restricted notion of (“structured”) rigidity suffices.Open:
Show that Toeplitz matrix w. rigidity n1.51 for rank n0.51.PROP: The complexity of the above 3-linear function is lower bounded by the maximum complexity of all bilinear functions associated w. Toeplitz
matrices.THM: If matrix M has rigidity m3 for rank m, then the corresponding bilinear function has complexity
(m
)
.Slide8
Comments on the proofs
Model: Multi-linear circuits with arbitrary multi-linear gates. Complexity measure (C) = max(arity, #gates
)
;
C2 for depth-two.THM 1: There exist bilinear
functions F such that C(F)=sqrt(n) but C2(F)=(n2/3).THM 2: For every t-linear function F it holds that C(F) ≤ C2(F) = O(tnt/(t+1)).THM 3: Almost all t-linear functions F satisfy C2
(F) ≥ C(F) = (tnt/(t+1)).THM 4: If matrix M has rigidity m3 for rank m, then the corresponding bilinear function has complexity (m).
PF: Covering by m cubes of side m.
PF:
A counting argument.
PF idea:
s=
sqrt
(n)
,
f(
x,y
)=g(x,L
1
(y),…,
L
s
(y
)
)
.
PF idea:
The
m
linear function yield a rank
m
matrix, whereas the
m
quadratic forms (in variables) cover
m
3
entries.Slide9
Add’l comments on the proof of THM 1
Model: Multi-linear circuits with arbitrary multi-linear gates. Complexity measure (C) = max(arity
, #gates
)
; C2 for depth-two.THM 1: There exist
bilinear functions F such that C(F)=sqrt(n) but C2(F)=(n2/3).PF: For s=sqrt(n), let f(x,y)=g(x,L1(y),…,Ls(y)), where g is generic (over n+s bits), each Li computes the sum of s
variables in y.A generic depth-two ML circuit of complexity m computes f asB(F1(x),…,Fm(x),G1(y),…,Gm(y)) + i[m]Bi(x,y)
where the Bi’s are quadratic and each function has arity
m
.
Hitting
y
with a random restriction that leaves one variable alive in each block, we get
B(F
1
(x),…,
F
m
(x),
G’
1
(y’),…,
G’
m
(y’))
+
i
[
m]
B’
i
(
x,y
’)
w
here each
B’
I
(and
G’
I
) depends on
O(m/s)
variables.
Hence, the description length is
O(m
3
/s)
; cf. to
ns=n
2
/s
.
Slide10
Structured Rigidity
THM 4’: If matrix M has (m,m,m)-structured rigidity for rank m, then the corresponding bilinear function has complexity
(m
)
.PF idea: The proof of Thm 4 goes through w.o
. any change.DEF: Matrix M has (m1,m2,m3)-structured rigidity for rank r if matrix R of rank r the non-zeros of M-R cannot be covered by m1 (gen.) m2-by-m3 rectangles.
Rigidity m1m2m3 implies (m1,m2,m3) structured rigidity for the same rank, but not vice versa.THM 5: There exist matrices of (m,m,m)-structured rigidity for rank m
that do not have rigidity 3mn for rank 0 (let alone for rank m).For every
m
[n
0.51
,n
0.66
]
.
PF:
Consider a random matrix with
3mn
one-entries.Slide11
Summary
t-linear functions x=(x(1),…,x(t)), |x(i)|=n
F
(x(1),…,x(t)) = (i_1,…,i_t)T xi_1
(1) xi_t(t) Conj (1st sanity check): For every t>1, there exists a t-linear function that requires depth-three circuits of size exp((tnt/(t+1))). [holds for t=1]
Long-term/dream goal: For every t>1, present an explicit t-linear function that requires depth-three circuits of size exp((tnt/(t+1))). [
holds for t=1]2nd sanity check:
Prove L.B. for a restricted model of (depth-three) circuits;
specifically,
Arithm.Ckts
with
general
gates:
Current goal
:
Show that
e
xplicit
t
-linear functions
F
satisfy
C
2
(F) ≥ C(F) =
(
t
n
t
/(t+1)
)
or
so (i.e. super-
sqrt
).Slide12
END
Slides available athttp://www.wisdom.weizmann.ac.il/~oded/T/kk.pptxPaper available athttp://www.wisdom.weizmann.ac.il/~oded/p_kk.html