amplifying lower bounds by means of self-reducibility
DESCRIPTION
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 - PowerPoint PPT PresentationTRANSCRIPT
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