Amplifying lower bounds by means of self-reducibility

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

MOD-q , , MAJ , , MOD-q ,  Question ≈ poly-size circuits O(log n)-depth poly-size circuits O( 1 )-depth poly-size circuits CC0 AC0 ACC0 TC0 NC1  L  P  NP  PSPACE EXP

Current status Goal: Show SAT  CKT-SIZE( n k ), for all k>1. We have: • explicit f  CKT-SIZE( 5 n) • lower-bounds Ω( n 1+d ) • 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’ )

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.

 TC0=NC1 Some corollaries: • W5-STCONN  TC0  W5-STCONN  TC0-SIZE( n 1+) for any  > 0. • MAJ  ACC0  MAJ  ACC0-SIZE( n 1+) for any  > 0. W5-STCONN: …  ACC0=TC0

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 fn fn n

Thm:W5-STCONN  C-SIZE( n k)  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) the size of the reduction

Recap: • TC0=NC1  W5-STCONN  TC0-SIZE( n 1+) for any  > 0. • ACC0 =TC0  MAJ  ACC0-SIZE( n 1+) for any  > 0. • If multiplying n matrices of dim. 2log n 2log nover ring ({0,1}, , ) is not in NC1-SIZE ( n 1+) then NC1  NL. Q: Can such lower bounds be proven?

Natural proofs Razborov-Rudich: • Tn {h :{0,1}n{0,1}} is a natural property if 1) “ f  Tn ?” is decidable in time 2nO(1), and 2) |Tn |>22n /2 n. • { 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 C-SIZE( m) then there are no pseudorandom function generators in C-SIZE( m ).

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? • It is sparse. • It is not really a property as it relates different input sizes !

a a’ a’ a’ … a’ a’’ a’’ a’’ … a’ … 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. Pf: nc n cn2cn3c … < n c/1-

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: 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|

 by calculating MAX-CLIQUE exactly on each of the n pieces we can n - approximate MAX-CLIQUE of G Q: Can the self-reducibility be applied to SAT? 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 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. Srinivasan Håstad

Open problems • Are there downward self-reducible function beyond NC1? • Does NP in non-uniform CC0[6]  SAT  CC0[6]-SIZE( n 2) ? • What is the size of Håstad’s reduction ?

Thm: Let f have NC1 circuits of depth d ( n ). f TC0-SIZE( 3d ( n ))  NC1  TC0. Thm: If multiplying n matrices of dim. 2log n 2log nover ring ({0,1}, , ) is not in NC1-SIZE ( n 1+) then NC1  NL.

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|cba A ≤ B : a’ b’|b’| ≤ |a’|cab A ≤ A : a a’|a’| ≤ |a| b a a’ b’ |b’| ≤ |b|cba cab

Amplifying lower bounds by means of self-reducibility