Amplifying lower Amplifying lower bounds by means bounds by means of self-reducibilityof self-reducibility

Eric Allender Eric Allender MichalMichal KouckýKoucký

Rutgers University Rutgers University Academy of SciencesAcademy of Sciences

Czech RepublicCzech Republic

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P P NP NP PPSPACESPACE EXP EXPACAC00 ACC ACC00 TC TC00 NCNC1 1 L L

≈ ≈ poly-size circuitspoly-size circuits

O(log O(log nn)-depth poly-size )-depth poly-size circuitscircuitsO( 1 )-depth poly-size circuitsO( 1 )-depth poly-size circuits

CCCC00

QuestionQuestionMOD-qMOD-q

, ,

, , , MAJ, MAJ , , , MOD-q, MOD-q

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Current statusCurrent status

Goal: Goal: Show SAT Show SAT CKT-SIZE( CKT-SIZE( n n k k ), for all ), for all kk >1.>1.

We have:We have:

explicit explicit ff CKT-SIZE( 5 CKT-SIZE( 5 nn ))

lower-bounds lower-bounds ΩΩ( ( n n 1+1+d d ))

formula size formula size ΩΩ( ( n n 33 ), branching programs ), branching programs ΩΩ( ( n n 22 ) )

Razborov-Rudich:Razborov-Rudich: a a natural natural proof of proof of ff CKT- CKT-SIZE(SIZE(n n k k ) ) pseudorandom generators pseudorandom generators CKT-SIZE(CKT-SIZE(n n k’ k’ ))

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Main resultsMain results

Thm: Thm: Let Let ff be be quickly downward self-quickly downward self-reduciblereducible and and CC be a be a usualusual circuit class. circuit class.

ff is in is in CC -SIZE( -SIZE( n n kk ) ) for some for some kk > 1.> 1.

ff is in is in CC -SIZE( -SIZE( n n 1+1+ ) ) for any for any > 0.> 0.

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Some corollaries:Some corollaries: W5-STCONN W5-STCONN TC TC00

W5-STCONN W5-STCONN TC TC00-SIZE( -SIZE( n n 1+1+ ) ) for any for any > 0.> 0.

MAJ MAJ ACC ACC00

MAJ MAJ ACC ACC00-SIZE( -SIZE( n n 1+1+ ) ) for any for any > 0.> 0.

W5-STCONN:W5-STCONN: … …

TCTC00=NC=NC

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ACCACC00=TC=TC00

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Downward self-reducibilityDownward self-reducibility

ff is is quickly downward self-reduciblequickly downward self-reducible if for some if for some > 0 > 0 there exists a O(1)-depth and O(there exists a O(1)-depth and O(nn poly-log poly-log nn)-size )-size circuit family computing circuit family computing ffnn using using -gates, fan-in 2 -gates, fan-in 2 , , --gates and gates computing gates and gates computing ff

n n . .

E.g.,E.g., W5-STCONN: W5-STCONN:

nn

ffnn ffnn ffnn ffnn

ffnn

nn

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Thm:Thm: W5-STCONN W5-STCONN CC-SIZE( -SIZE( n n kk ) )

W5-STCONN W5-STCONN CC-SIZE( -SIZE( n n ((k k + 1) /2+ 1) /2 ) .) .

Pf:Pf:

C’C’nn

CCnn CCnn CCnn CCnn

CCnn

C’C’n n of size (of size (nn +1)∙O+1)∙O((n n kk ) + O( ) + O( n n ) = ) = O( O( n n ((k k + 1) /2+ 1) /2 ) )

the size of the reductionthe size of the reduction

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Recap:Recap: TCTC00=NC=NC11

W5-STCONN W5-STCONN TC TC00-SIZE( -SIZE( n n 1+1+ ) ) for any for any > 0.> 0.

ACCACC0 0 =TC=TC00

MAJ MAJ ACC ACC00-SIZE( -SIZE( n n 1+1+ ) ) for any for any > 0.> 0.

If multiplying If multiplying nn matrices of dim. 2 matrices of dim. 2log log nn 2 2log log nn over ring ({0,1}, over ring ({0,1}, , , ) is not in NC) is not in NC11-SIZE ( -SIZE ( n n 1+1+ ) ) then NCthen NC1 1 NL. NL.

Q: Can such lower bounds be proven?Q: Can such lower bounds be proven?

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Natural proofsNatural proofs

Razborov-Rudich:Razborov-Rudich:

TTnn { {h h :{0,1}:{0,1}nn{0,1}} is a {0,1}} is a natural property natural property if if

1) “ 1) “ ff T Tn n ?” is decidable in time 2?” is decidable in time 2nnO(1)O(1), and, and

2) |T2) |Tn n |>2|>222n n /2/2

nn..

{ T{ Tnn } is a } is a useful property against useful property against C C if if

for every function { for every function { ffnn } } { T { Tn n }, }, ff CC..

Thm [RR’95]: Thm [RR’95]: If { TIf { Tnn } is a natural and useful } is a natural and useful

property against property against CC-SIZE( -SIZE( mm ) then there are no ) then there are no pseudorandom function generators in pseudorandom function generators in CC-SIZE( -SIZE( m m ).).

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Natural proofsNatural proofs

Example:Example:

TTnn = { = {h h :{0,1}:{0,1}nn{0,1}, {0,1}, hh does not have circuits of does not have circuits of

depth log*depth log*nn and size and size nn22 consisting of consisting of and MAJ and MAJ gates}gates}

Claim: Claim: { T{ Tnn } is natural and useful against TC } is natural and useful against TC00-SIZE( -SIZE( nn1.51.5

).).

Q: Q: Is downward self-reducibility natural property?Is downward self-reducibility natural property?

1)1) It is sparse.It is sparse.

2)2) It is not really a property as it relates different It is not really a property as it relates different input sizes !input sizes !

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Q: Q: Can the self-reducibility be applied to SAT?Can the self-reducibility be applied to SAT?

Thm: Thm: 1) If 1) If ff is quickly downward self-reducible to is quickly downward self-reducible to ffn n

then then f f NC. NC.

2) If 2) If ff is downward self-reducible to is downward self-reducible to ffn n

by poly-time by poly-time computation then computation then f f P. P.

Pf:Pf:aa

a’ a’ a’ a’ a’ a’ …… a’ a’

a’’ a’’ a’’ a’’ a’’ a’’ …… a’a’

……

nncc n n cc nn22cc nn33cc … < … < n n c/1-c/1-

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Q: Q: Can the self-reducibility be applied to SAT?Can the self-reducibility be applied to SAT?

Thm (A. Srinivasan 2001): Thm (A. Srinivasan 2001): If computing weak If computing weak approximations to MAX-CLIQUE cannot be done in approximations to MAX-CLIQUE cannot be done in det. time det. time n n 1+1+

then P then P NP. NP.

nn - approximating MAX-CLIUQE: - approximating MAX-CLIUQE:GG

by calculating MAX-CLIQUE exactly on each of the by calculating MAX-CLIQUE exactly on each of the nn pieces we can pieces we can nn - approximate MAX-CLIQUE of - approximate MAX-CLIQUE of GG

|maximal clique|/ |maximal clique|/ nn ≤ output value ≤ |maximal clique| ≤ output value ≤ |maximal clique|

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Q: Q: Can the self-reducibility be applied to SAT?Can the self-reducibility be applied to SAT?

GG

by calculating MAX-CLIQUE exactly on each of the by calculating MAX-CLIQUE exactly on each of the nn pieces we can pieces we can nn - approximate MAX-CLIQUE of - approximate MAX-CLIQUE of GG

Thm (J. Håstad 1994):Thm (J. Håstad 1994): MAX-CLIQUE is reducible in MAX-CLIQUE is reducible in polynomial time to polynomial time to nn1/31/3 – approximation of MAX- – approximation of MAX-CLIQUE.CLIQUE.

MAX-CLIQUE MAX-CLIQUE approx. of MAX-CLIQUE approx. of MAX-CLIQUE MAX-MAX-CLIQUE CLIQUE

Thm:Thm: Håstad’s reduction of MAX-CLIQUE to Håstad’s reduction of MAX-CLIQUE to nn1/31/3 – – approximation of MAX-CLIQUE must map approximation of MAX-CLIQUE must map instances of size instances of size nn to instances of size to instances of size nn3/2 3/2 unless unless P=NP.P=NP.

HåstadHåstad SrinivasSrinivasanan

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Open problemsOpen problems

Are there downward self-reducible function Are there downward self-reducible function beyond NCbeyond NC11? ?

Does NP in non-uniform CCDoes NP in non-uniform CC00[6] [6] SAT SAT CC CC00[6]-[6]-SIZE( SIZE( n n 22 ) ?) ?

What is the size of Håstad’s reduction ?What is the size of Håstad’s reduction ?

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Thm: Thm: Let Let ff have NC have NC11 circuits of depth circuits of depth d d ( ( n n ). ).

ff TC TC00-SIZE( 3-SIZE( 3d d ( ( n n )) ) ) NC NC1 1 TC TC00..

Thm: Thm: If multiplying If multiplying nn matrices of dim. 2 matrices of dim. 2log log nn 2 2log log nn over ring ({0,1}, over ring ({0,1}, , , ) is not in NC) is not in NC11-SIZE ( -SIZE ( n n 1+1+ ) ) then NCthen NC1 1 NL. NL.

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Q: Q: To which functions can this be To which functions can this be applied?applied?

Thm: Thm: If If AA and and BB are complete for are complete for CC and and AA is downward is downward self-reducible then so is self-reducible then so is BB..

Pf: Pf: BB ≤ ≤ AA : : bb aa ||aa| ≤ || ≤ |bb||ccbaba

AA ≤ ≤ BB : : a’a’ b’b’ ||b’b’| ≤ || ≤ |a’a’||ccabab

AA ≤ ≤ AA : : aa a’a’ ||a’a’| ≤ || ≤ |aa||

bb aa aa’’ b’b’

||b’b’| ≤ || ≤ |bb||ccba ba ccabab