Slide 1
Part III Theory of Parallel
Computation
Uniform Computation via
Nonuniform Computation
LECTURE 3-1
Polynomial-size Circuits
edges.-out its ofnumber theisfanout theand inputs)(or
edges-in its ofnumber theis its gate, logicaleach For
.log called are operationsBoolean by labeled nodes The
AND).(or n conjunctio and
OR),(or n disjunctio negation, assuch ,operationsBoolean by labeled are
nodesother and 1 and 0 or values esby variabl labeled are edge-in
no with nodes osedigraph wh acyclican is A
fanin
ical gates
rcuitBoolean ci
snsd0
便利貼
what is circuits? a circuit is a logic gate. Electronic circuit is a real magine. circuit can compute a boolean function.
. variables theaboveonly appear gatesnegation thesuch that
gates ofnumber most twiceat anddepth same with theone equivalent
an toed transformbecan circuit Boolean every law,Morgan de Using
Size = # of gates
THEOREM
circuits. size-polynomial has in set Every P
).()( such that
polynomial a exists thereif has languageA
).2 when )()(( )()(
is of The
.1)( where
functionsBoolean of sequence aby determined becan *}1,0{
).()( . computing and most at fanin of gatesth circuit wi of size
minimum theis of )( the, formulaBoolean aFor
,,,
,
,
2
npnCS
puts-size circpolynomialA
t==CnCS=CnCS
Aityze complexcircuit-si
Axx
A
fCfCft
ffComplexity circuit cf
A
AnAAntAt
An
An
t
=
=
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Simulate Transition function by circuits
PROOF
SIMULATE DTM COMPUTATION BY A CIRCUIT
P/poly
.by denoted is circuits size polynomial having languages of class The P/poly
. Thus, .)()2)((most at size hascircuit resultant The
2.fanin of gates 2)(most at by simulated becan gateeach So,
network. in the nodes 22)(most at are therebecause 12)(
most at isit ,restrictednot is gateeach offanin theAlthough .
function Boolean computing )(most at size ofcircuit aconsider
y,sufficienc see To obvious. isnecessity the,)()( Since
.)()(such that polynomial a exists there
,
P/polyAnpn+np
n+np
n ++npn ++np
χ
np
nCSnCS
npnCSpP/polyA
n,A
AA
,n
PROPOSITION
PROOF
Sparse and poly
sparse. is Then
}.,…,{for ,…, = (n) define , language aFor
.every for |)(||)(||such that polynomial a exists there, which,of
each for strings, tonumbers natural from functions ofset thedenote Let
.)(
such that polynomial a exists thereif is languageA
2121
Spolyh
wwwSwwwhS
xxp|xhph
poly
np||A
psparseA
s
knks
n
=
=
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Characterization of P/poly
.)(, ,|| with for
such that language a and function a is There (c)
.such that set sparse a exists There (b)
circuit. size-polynomial has Language (a)
.equivalent are statements following The
BnhxAxnxx
PBpolyh
PAS
A
S
(b)(c)(a)(b)
PROOF
P/poly and NP
?/ and between iprelationshany say Can we
|)(|,
such that and exist there/
],|))[(|||,(
such that polynomial a and exist there
polyPNP
BxhxAx
polyhPBpolyPA
ByxxpyyAx
pPBNPA
(b)=>(a)
circuits. size-polynomial has then
circuits, size-polynomial has and If
A
BPA
B
LEMMA 1
circuits. size-polynomial hasset sparseEvery
LEMMA 2
(a)=>(c)
.}|)(|, {
and Clearly, . accepts )( decoding from obtainedcircuit the
and )(that condition thesatisfying s’, all ofset thebe Let
.function aobtain we
,)( string a into computingcircuit size-polynomial theEncoding
,
Bxhx|xA=
PBx|x|h
|x|y=hyxB
polyh
nh
An
(c)=>(b)
time.polynomialin accepts oracle with algorithm This
reject; else
accept then if
for-end
;0 then 10 if
;1 then 01 if
begin do )( to1for
;
;input
algorithm. following heConsider t .)()(such that polynomial a be Let
1. is )( of symbolth the01
0. is )( of symbolth the10
:set sparse a Define
AS
Bx#w
wS
wS
|x|pi
w
x
np|n|hp
nhiS
nhiS
S
in
in
in
in
P/poly and NP
?/ and between iprelationshany say Can we
|)(|,
such that and exist there/
],|))[(|||,(
such that polynomial a and exist there
polyPNP
BxhxAx
polyhPBpolyPA
ByxxpyyAx
pPBNPA
.,,,such that and different
havemay we,|||| and with ,For
],|))[(|||,(
such that polynomial a and exist there
221121
212121
ByxByxyy
nxxxxAxx
ByxxpyyAx
pPBNPA
==
/ Possibly, polyPNP
BxhxAx
polyhPBpolyPA
|)(|,
such that and exist there/
.,,,such that same thehave
tohave we,|||| and with ,For
21
212121
ByxByxy
nxxxxAxx
==
2
2
1
1
|)(|,
such that and exist there/
],|))[(|||,(
such that polynomial a and exist there
BxhxAx
polyhPBpolyPA
ByxxpyyAx
pPBNPA
NPpolyP /
. of role play thecannot Thus,
].,|))[(|||,exist may re the
].,|))[(||||),(|(
].,|))[(||||),(|(
./ Suppose
12
2
2
2
BB
Byxxpyy
ByxxpyxhyAx
ByxxpyxhyAx
polyPA
=
=
2
2
1
1
|)(|,
such that and exist there/
],|))[(|||,(
such that polynomial a and exist there
BxhxAx
polyhPBpolyPA
ByxxpyyAx
pPBNPA
NPpolyP /
.sets recursive-non contains / Therefore,
.computable benot may Moreover,
polyP
h
./
.
.
2
2
polyPPHP/polyNP
P/poly
PHP/polyNP
pp
k
p
k
p
=
=
THEOREM
LEMMA 1
LEMMA 2
.,|’ and ,|’, then ,)( if)(
;,|’or ,|’, then ,)( if (b)
;,1 and ,0 (a)
,most at length of QBF-any For
:conditions following thesatisfies Then .)(Set
B].)(,SAT )[|| ,(
such that and exists thereThus,
./SAT So, .in -complete is which ,SATConsider
10
10
k
BwFBwFBwFF’xF= c
BwFBwFBwFF’xF=
BwBw
nF
wnw=h
nhFFnFF
polyh PB
polyP
xx
xx
k
k
k
p
k
p
m
==
==
PROOF OF LEMMA 1
. means This
)].,()()())[(||,))((||,(
Thus,
(c).apply can then we,’)( (b)apply can then we,’)( if
, variables with each For . variables1most at with
allfor trueisit that Assume true.isit (a),by then constant, a is if fact,In
.,SAT tobelongs length of QBF- athen
,conditions above thesatisfies ify that inductivel showcan we,Conversely
2
p
k
k
kk
SAT
BwFcbanFFnpwwSATF
FxFFxF
nFnF
F
BwFnF
w
==
−
PROOF OF LEMMA 2
trouble.a bringst replacemennotation theoflegality theidea, for this
However, proof. finish the would weone, into oracles sparse twoencoding
By oracles. sparse only two containing machine aobtain to and
)( g witnessinmachines twocombineThen .by replace
tois idea obtained-yimmediatelAn . that advantage by taking
set sparse somefor prove want to weand , have we
Now, .set sparse afor that so,/ ,hypothesis
induction By the .for Assume .consider ,1For
. since trivialisit ,1For .on induction by prove We
‘
1
1
1
S
B
SSB
Sp
k
p
k
Bp
k
pp
k
PB
P/polyAP/polyNP
P/polyNP
S’PAPBNPA
SPBpolyP
BNPAAk>
NPk=kP/poly
=
−
−
. ))((,)(,)(,
and ,|| with any for )(|)(,| where
, )(,
.))((,)(,|)(|,
,|| with for Hence, .|)(|, ,|| with for
such that and exist there,/ since Moreover,
fact. apply thecan weThus,
.imply and that Note true.be wouldoracle
sparse afor t replacemen asuch shown that becan it fact, By this
.)(, ,|| with for
such that and exists thereset sparse afor
:fact following theusingby trouble thisaround go wefollowing, In the
npgnhxnx
nxxnpnhx
Rnx
RnpgnhxBxhxAx
nxxRxgxBxnxx
PRpolygpolyPNPB
NPAPBNPA
BnhxAxnxx
polyhNPBSNPA
SSB
S
=
. togoes as zero togoes )8(2most at size
ofcircuit a hasfunction Boolean a ofy probabilit the, variables of functions
Boolean 2 are thereSince .2 << )O(2= ))22(2( ,)8(2For .))22(2(by bounded is most at size and variables of circuits ofnumber theSo, . size and variables of circuits ))22(2(most at have weTherefore, choices. 22most at hasit Thus, choices). (2constant aor choices), 2( literal a choices), most (at gate previous a becan nodes previousEach nodes. previous on two actshat operator t ORor ANDan with assigned is gateeach that Note . size and variables with circuits ofnumber count thefirst We ).2( size of circuits requires variables offunction Boolean every Almost nnn 224/22 2 2 + nn/ n n+s+sn/s= n+s+s sns nn+s+ n+s+n s sn /n n n nsn s s n THEOREM PROOF Open ).log()(such that language a Find nnnCSA A =