Amplifying lower bounds by means of self- reducibility Eric Allender Michal Koucký Rutgers University Academy of Sciences Czech Republic Czech Republic

2 P  NP  P SPACE  EXP AC 0  ACC 0  TC 0  NC 1  L  ≈ poly-size circuits ≈ poly-size circuits O(log n)-depth poly-size circuits O(log n)-depth poly-size circuits O( 1 )-depth poly-size circuits CC 0 QuestionMOD-q ,  , , MAJ , , MAJ , , MOD-q , , MOD-q

3 Current status Goal: Show SAT  CKT-SIZE( n k ), for all k >1. We have: explicit f  CKT-SIZE( 5 n ) explicit f  CKT-SIZE( 5 n ) lower-bounds Ω( n 1+  d ) lower-bounds Ω( n 1+  d ) formula size Ω( n 3 ), branching programs Ω( n 2 ) formula size Ω( n 3 ), branching programs Ω( n 2 ) Razborov-Rudich: a natural proof of f  CKT-SIZE(n k )  pseudorandom generators  CKT-SIZE(n k’ )

4 Main results Thm: Let f be quickly downward self-reducible and C be a usual circuit class. f is in C -SIZE( n k ) for some k > 1.  f is in C -SIZE( n 1+  ) for any  > 0.

5 Some corollaries: W5-STCONN  TC 0 W5-STCONN  TC 0  W5-STCONN  TC 0 -SIZE( n 1+  ) for any  > 0. W5-STCONN  TC 0 -SIZE( n 1+  ) for any  > 0. MAJ  ACC 0 MAJ  ACC 0  MAJ  ACC 0 -SIZE( n 1+  ) for any  > 0. MAJ  ACC 0 -SIZE( n 1+  ) for any  > 0. W5-STCONN: …  TC 0 =NC 1  ACC 0 =TC 0

6 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 f n using  -gates, fan-in 2 ,  -gates and gates computing f n . 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 f n using  -gates, fan-in 2 ,  -gates and gates computing f n . E.g., W5-STCONN: E.g., W5-STCONN: nnnn fnfnfnfn fnfnfnfn fnfnfnfn fnfnfnfn fnfnfnfn nnnn

7 Thm: W5-STCONN  C-SIZE( n k )  W5-STCONN  C-SIZE( n (k + 1) /2 ). Pf: C’ n CnCnCnCn CnCnCnCn CnCnCnCn CnCnCnCn CnCnCnCn C’ n of size (  n +1)∙O (  n k ) + O( n ) = O( n (k + 1) /2 ) the size of the reduction the size of the reduction

8 Recap: TC 0 =NC 1 TC 0 =NC 1  W5-STCONN  TC 0 -SIZE( n 1+  ) for any  > 0. W5-STCONN  TC 0 -SIZE( n 1+  ) for any  > 0. ACC 0 =TC 0 ACC 0 =TC 0  MAJ  ACC 0 -SIZE( n 1+  ) for any  > 0. MAJ  ACC 0 -SIZE( n 1+  ) for any  > 0. If multiplying n matrices of dim. 2  log n  2  log n over ring ({0,1}, ,  ) is not in NC 1 -SIZE ( n 1+  ) then NC 1  NL. If multiplying n matrices of dim. 2  log n  2  log n over ring ({0,1}, ,  ) is not in NC 1 -SIZE ( n 1+  ) then NC 1  NL. Q: Can such lower bounds be proven?

9 Natural proofs Razborov-Rudich: T n  {h :{0,1} n  {0,1}} is a natural property if T n  {h :{0,1} n  {0,1}} is a natural property if 1) “ f  T n ?” is decidable in time 2 n O(1), and 2) |T n |>2 2 n /2  n. { T n } is a useful property against C if for every function { f n }  { T n }, f  C. { T n } is a useful property against C if for every function { f n }  { T n }, f  C. Thm [RR’95]: If { T n } is a natural and useful property against C- SIZE( m ) then there are no pseudorandom function generators in C-SIZE( m  ).

10 Natural proofs Example: T n = {h :{0,1} n  {0,1}, h does not have circuits of depth log*n and size n 2 consisting of  and MAJ gates} T n = {h :{0,1} n  {0,1}, h does not have circuits of depth log*n and size n 2 consisting of  and MAJ gates} Claim: { T n } is natural and useful against TC 0 -SIZE( n 1.5 ). Q: Is downward self-reducibility natural property? 1) It is sparse. 2) It is not really a property as it relates different input sizes !

11 Q: Can the self-reducibility be applied to SAT? Thm: 1) If f is quickly downward self-reducible to f n  then f  NC. 2) If f is downward self-reducible to f n  by poly-time computation then f  P. 2) If f is downward self-reducible to f n  by poly-time computation then f  P.Pf:a a’ a’ a’ … a’ a’’ a’’ a’’ … a’ a’’ a’’ a’’ … a’ … n c n  c n  2 c n  3 c … < n c/1- 

12 Q: Can the self-reducibility be applied to SAT? Thm (A. Srinivasan 2001): If computing weak approximations to MAX-CLIQUE cannot be done in det. time n 1+  then P  NP. n  - approximating MAX-CLIUQE: n  - approximating MAX-CLIUQE: G  by calculating MAX-CLIQUE exactly on each of the n  pieces we can n  - approximate MAX-CLIQUE of G |maximal clique|/ n  ≤ output value ≤ |maximal clique|

13 Q: Can the self-reducibility be applied to SAT? G  by calculating MAX-CLIQUE exactly on each of the n  pieces we can n  - approximate MAX-CLIQUE of G Thm (J. Håstad 1994): MAX-CLIQUE is reducible in polynomial time to n 1/3 – approximation of MAX-CLIQUE. MAX-CLIQUE  approx. of MAX-CLIQUE  MAX-CLIQUE Thm: Håstad’s reduction of MAX-CLIQUE to n 1/3 – approximation of MAX-CLIQUE must map instances of size n to instances of size n 3/2 unless P=NP. Håstad Srinivasan

14 Open problems Are there downward self-reducible function beyond NC 1 ? Are there downward self-reducible function beyond NC 1 ? Does NP in non-uniform CC 0 [6]  SAT  CC 0 [6]-SIZE( n 2 ) ? Does NP in non-uniform CC 0 [6]  SAT  CC 0 [6]-SIZE( n 2 ) ? What is the size of Håstad’s reduction ? What is the size of Håstad’s reduction ?

15 Thm: Let f have NC 1 circuits of depth d ( n ). f  TC 0 -SIZE( 3 d ( n ) )  NC 1  TC 0. Thm: If multiplying n matrices of dim. 2  log n  2  log n over ring ({0,1}, ,  ) is not in NC 1 -SIZE ( n 1+  ) then NC 1  NL.

16 Q: To which functions can this be applied? Thm: If A and B are complete for C and A is downward self- reducible then so is B. Pf: B ≤ A : b  a |a| ≤ |b| c ba A ≤ B : a’  b’ |b’| ≤ |a’| c ab A ≤ A : a  a’ |a’| ≤ |a|   b  a  a’  b’ |b’| ≤ |b| c ba  c ab |b’| ≤ |b| c ba  c ab