Amplifying lower bounds
by means of selfreducibility
Eric Allender
Michal Koucký
Rutgers University
Academy of Sciences
Czech Republic
Question
MOD-q
, , MOD-q
, , MAJ
CC0
AC0  ACC0  TC0  NC1  L  P  NP  PSPACE  EXP
, 
≈ poly-size circuits
O(log n)-depth poly-size
circuits
O( 1 )-depth poly-size circuits
2
Current status
Goal: Show SAT  CKT-SIZE( n k ), for all k >1.
We have:

explicit f  CKT-SIZE( 5 n )
1+d

lower-bounds Ω( n

formula size Ω( n ), branching programs Ω( n )
3
)
2
Razborov-Rudich: a natural proof of f  CKT-SIZE(n k )
 pseudorandom generators  CKT-SIZE(n k’ )
3
Main results
Thm: Let f be quickly downward self-reducible and C be a
usual circuit class.
k
f is in C -SIZE( n )
for some k > 1.

f is in C -SIZE( n
1+
)
for any  > 0.
4
Some corollaries:

W5-STCONN  TC0

W5-STCONN 
TC0-SIZE(
 TC0=NC1
n
1+
)
for any  > 0.
MAJ  ACC0


MAJ  ACC0-SIZE( n
W5-STCONN:
 ACC0=TC0
1+
)
for any  > 0.
…
5
Downward self-reducibility

f is quickly downward self-reducible if for some  > 0 there exists a
O(1)-depth and O(n poly-log n)-size circuit family computing fn
using -gates, fan-in 2 , -gates and gates computing fn  .

E.g., W5-STCONN:
fn
n
fn
fn
n
fn
fn
6
Thm:
k
W5-STCONN  C-SIZE( n )
 W5-STCONN  C-SIZE( n
(k + 1) /2
).
Pf:
Cn
C’n
Cn
Cn
Cn
Cn
C’n of size (n +1)∙O(n k ) + O( n ) = O( n (k + 1) /2
)
7
Recap:

TC0=NC1

W5-STCONN 

TC0-SIZE(
)
for any  > 0.
ACC0 =TC0

MAJ  ACC0-SIZE( n

n
1+
1+
)
for any  > 0.
If multiplying n matrices of dim. 2log n  2log n over ring ({0,1},
1+
1
, ) is not in NC -SIZE ( n ) then NC1  NL.
Q: Can such lower bounds be proven?
8
Natural proofs
Razborov-Rudich:

Tn  {h :{0,1}n{0,1}} is a natural property if
1) “ f  Tn ?” is decidable in time 2n
n
n
2
2) |Tn |>2 /2 .

O(1)
, and
{ Tn } is a useful property against C if
for every function { fn }  { Tn }, f  C.
Thm [RR’95]: If { Tn } is a natural and useful property against CSIZE( m ) then there are no pseudorandom function generators

in C-SIZE( m ).
9
Natural proofs
Example:

Tn = {h :{0,1}n{0,1}, h does not have circuits of depth
log*n and size n2 consisting of  and MAJ gates}
Claim: { Tn } is natural and useful against TC0-SIZE( n1.5 ).
Q: Is downward self-reducibility natural property?
1)
It is sparse.
2)
It is not really a property as it relates different input sizes !
10
Q: Can the self-reducibility be applied to SAT?
Thm: 1) If f is quickly downward self-reducible to fn  then f 
NC.
2) If f is downward self-reducible to fn  by poly-time
computation then f  P.
a
Pf:
a’
a’’
a’
a’’
a’
… a’
a’’
…
a’
…
2
nc n c n c n
3c
… < n c/1-
11
Q: Can the self-reducibility be applied to SAT?
Thm (A. Srinivasan 2001): If computing weak approximations to
1+
MAX-CLIQUE cannot be done in det. time n
then P  NP.

n  - approximating MAX-CLIUQE:
G
 pieces we
 by |maximal
calculatingclique|/
MAX-CLIQUE
exactly
on
each
of
the
n

n ≤ output value ≤ |maximal clique|

can n - approximate MAX-CLIQUE of G
12
Q: Can the self-reducibility be applied to SAT?
 by calculating MAX-CLIQUE exactly on each of the n pieces we
can n  - approximate MAX-CLIQUE of G
G
Thm (J. Håstad 1994): MAX-CLIQUE is reducible in polynomial
time to n1/3 – approximation of MAX-CLIQUE.
MAX-CLIQUE  approx. of MAX-CLIQUE  MAX-CLIQUE
Håstad
Srinivasan
Thm: Håstad’s reduction of MAX-CLIQUE to n1/3 – approximation
of MAX-CLIQUE must map instances of size n to instances of
size n3/2 unless P=NP.
13
Open problems

Are there downward self-reducible function beyond NC1?
CC0[6]
 SAT 

Does NP in non-uniform

What is the size of Håstad’s reduction ?
CC0[6]-SIZE(
2
n )?
14
Thm: Let f have NC1 circuits of depth d ( n ).
d(n)
f  TC0-SIZE( 3
)  NC1  TC0.
Thm: If multiplying n matrices of dim. 2log n  2log n over ring
1+
1
({0,1}, , ) is not in NC -SIZE ( n ) then NC1  NL.
15
Q: To which functions can this be applied?
Thm: If A and B are complete for C and A is downward selfreducible then so is B.
Pf:

cba
|b|
B≤A:
ba
|a| ≤
A≤B:
a’  b’
|b’| ≤ |a’| ab
A≤A:
a  a’
|a’| ≤ |a|
c

b  a  a’  b’
|b’| ≤
cba  cab
|b|
16