zeiaeyigd zxez
zerixk izla zeira
`ean
zpekn :dipyd dxcbdd z` xikfp .zety iedifl bpixeih zepekne ,zeivwpet aeyigl bpixeih zepekn dk cr epi`x
dpekn ozipda .F
= {qacc , qrej } dxevdn `id dly miiteqd miavnd zveawy jka zcgein zety iedifl bpixeih
qacc avna zxver M mdilry milind lk sqe`k L (M ) dty mixicbn ,M efky
j` ,hlt yi dpeknly al eniy)
.(epnn minlrzn ep`
.L z`
zlawn M -y mixne` ,L (M ) = L-y jk dpekn M -e idylk dty `id L m` •
.L z`
m`y cera ,xevrz `l llk
w ∈L
w∈
/L
M -y
drixkn M -y mixne` f` ,hlw lkl zxver M
jkl sqepae
L (M ) = L
m`
•
w∈
/ L xear f` ,L z` zlawn M m`y jka `ed zexcbdd izy oia lcadd
qrej avna xevrz dpeknd w ∈
/ L lkly ghaen f` L z` drixkn M
.(qacc -a xevrz dpeknd
w ∈ L dlin xear) zixhniq `l dxeva zxcben dty ly dlaw ik al miyl aeyg
okzii
m` ik miyxec ep` mixwnd ipya)
dlin xeare ,xevrl miaiig
.(miaiig `l
:zety ly zewlgn zexcben el` zexcbd lr qqazda
R ={L| L z` drixknd M
RE ={L| L z` zlawnd M
.(Σ
= {0, 1}
dpekn zniiw}
•
dpekn zniiw}
•
lr xaecn ik cinz migipn ep` ;zetyd a"` dfi` lrn xicbdl yi - zipkh dxrd)
oeqkl iperihe dxitq ilewiy
dxitq ilewiy
.
lr qqean didi eply oey`xd oerihd .
RE-a `l s`e R-a opi`y zety zeniiw ik ze`xl zrk xearp
.dn xac ly meiw i`/meiw zgkeda dnver ax ilk md dxitq ilewiy illk ote`a
dipn za z`xwp miirahd mixtqnd oiaeSdpia lre r"gg dn`zd yiy dveaw ,zeveawd zxezn xekfk
ik ze`xl lw .
Σ∗ =
n
∗
n≥0 Σ oky ,dipn za `id Σ -y miiwzn ,Σ iteq a"` lkl
zepekn ly dipn oa xtqn wx miiwy raep df oerihn .iteq `ed zeiteq zeveaw ly dipn oa cegi`e ,ziteq `id (n
∗
.Σ ly xai` xnelk ,ziteq zfexgn zervn`a ccewl ozip dpekn lk ik epi`x oky ,bpixeih
weica jxe`n
Σ
lrn milind lk sqe`)
Σn
lke
|w1 | = |w2 | m` e` |w1 | < |w2 | m` w1 < w2 :itxbewiqwl xcq zervn`a `id Σ∗ -a milind z` xtqnl zg` jxc
∗
.w1 , w2 , w3 , . . .-k Σ -a milind z` xtnqp .(ze`-ze` mieeynyk ,epiidc) libx ipelin xcqa w2 iptl d`a w1 la`
ik gipp .zety ly dipn oa `l xtqn miiw ik xehpw ly oeqkl`d ly dfl dnec oeriha ze`xdl ozip ,ipy cvn
∗
`id ,wi ∈ Σ milindn zg` lkl ,epiidc .L = {wi |wi ∈
/ Li } dty dpape Σ lrn zetyd lk ly dipn `id L1 , L2 . . .
L-y
o`kne ,wi dlina zelcap od oky
L 6= Li
gxkda
i
lkly o`kn .Li -a z`vnp `l `id m` wxe m`
L-a
z`vnp
.zety ly dipn oa `l xtqn yiy o`kn .dxizq - zetyd ly dipna dpi`
zepekn ici lr zelawzny zety ly dipn oa xtqn wx yiy o`kn .weica zg` dty zlawn bpixeih zpekn lk ,zrk
.bpixeih zpekn ici lr zelawzn opi`y zety zeniiw okle ,illk ote`a zety ly dipn oa `l xtqn yi j` ,bpixeih
1
wtzqdl illk ote`a ozip .xzei minkegn mipeqkl bivp qxewd jynday oeeikn ,zxekfzk wxe j` o`k bved oeqkld
∗
∗
lre Σ ly dwfgd zveaw `ed xnelk ,Σ ly zeveawd zz lk sqe` weica `ed Σ lrn zetyd lk sqe`y dxin`a
∗
.Σ znvern dlecb eznver ,zeveawd zxezn xehpw htyn it
L ∈
R ziteqpi` dty lkl ik gikedl yi
htynn ok lre ,dipn za `id
L
:mipgand cg`a ritedy ,dxitq ilewiy ly sqep yeniyl zrk xearp
0
0
znver :epi`x xaky dnl ddf oerihd .L ∈
/
-y jk L ⊆ L dty-zz zniiw
RE
dty-zz zniiw okle ,dipn za dppi` ,L ly zetyd zz lk sqe` `idy ,dly dwfgd zveaw znvery raep xehpw
.bpixeih zpekn s` ici lr zlawzn dpi`y
zxwean dvxd
RE-a od zeniieqn zetyy dgkedl dwfg dhiy bivpe ziaeig dyibl zrk xearp
:d`ad dtya opeazp dnbec xeza .
.L
={hM i |fM (x) = x-y
jk
x
miiw}
.zeaygn ody divwpetl "zay zcewp" yiy zepeknd lk zty idef
M
z` uixdl :qb gek qqean yetig heyt dpi`y zay zcewp zniiw m`d wecal jxc dxexa `l , hM i ozpida
Σ∗ zelinl epbvdy dipna ynzyp .x mb `ed hltd m`d wecale x ixyt` hlw lk lr
:`ad mzixebl`d z` lawpe
i = 1, 2, . . .
.y
= M (wi )
.law ,y
`id epzxhn lk ik epl rixtn epi` df xac .mlerl miizqi `l mzixebl`d ,zay zcewp oi`
.xevrl `l "epl xzen"
mzixebl`d z` rval `ed oexztd .x lkl zxver
M -y
L-l
lkl .1
z` ayg (`)
= wi
m` (a)
M -l m`y al miyp
∈ RE-y ze`xdl
zekiiy opi`y milin lr okle ,L
epl ghaen `l oky ,jk mb caer `l mzixebl`d lfnd rexl
zxwean dvxda
:zilnxet z`f x`zp .
j = 1, 2, . . .
i = 1, 2, . . . , j
.law ,wi hltd lawzd m` .wi lr micrv
ipy
M
z` mivixn ep` diipyd divxhi`a .w1 hlwd lr cg` crv
M
j
jyna
.micrv
k
zkynp
wi
lr
M
zvixe ,fM
.lawp ok lre ,zegtl micrv
lkl (`)
z` uxd .
i
z` mivixn ep` dpey`xd divxhi`a ,xnelk
.d`ld oke
-d divxhi`a f`
M
lkl .1
w1 , w2
mihlwd ipy lr micrv
(wi ) = wi -y jk wi dlin zniiw f` hM i ∈ L m`
k jynl z`f dyrpe wi lr M z` uixp j = max {i, k}
dizeklyde dxivrd ziira
:d`ad dtyd zervn`a zx`ezn dxivrd ziira
m` zlawne
x
lr
M
z` dvixn heyt
.HP ={(hM i , x) | x lr dzvixa zxver M }
HP z` zlawnd dpekn - HP ∈ RE xexiaa) HP ∈/ R ik mi`xn d`vxda
.aeyigl mipzip izla md mixg` mixac ik gikedl ozip ef d`vezn cvik zrk d`xp .(dniizqd
x
lr zxver `l
M
m`e ,zxver `idy cr
x
lr dvx
M -y
micrvd xtqn `ed
f (hM i , x)-y
jk
.f
z` miaygn
(hM i , x)
M
zvix
f divwpet xicbp
(hM i , x) = ε f`
f m`
f (hM i , x)
ozpida :`ad ote`a drxkdl zpzip dziid dxivrd ziira ,aeyigl zpzip dziid
eplaiw ,drxkdl zpzip dpi` dxivrd ziiray oeeikn
.migec zxg`
.f
(hM i , x) 6= ε
m` milawne
.dxizq
2
f` ,x lr zxver dpi`
M m`y df oaena "zlwlewn" `id f -y gipp - dl`yd i`pz z` hrn zrk jaqpe dad
f -l mexbl zlbeqn dpi`e zizedn dpi` f ly efd "dira"dy `id df dxwna eply ohad zyegz .f (hM i , x) = 2000
.f zervn`a dxivrd ziira z` xeztl ozip ik ze`xdl mivex oiicr ep`y o`kn .aeyigl zpzipl jetdl
m` dn la`
.lawl ozip okle
e` ,micrv 2000 jez
x
lr zxver
.divleniqd meza dxvr
M
x lr
M -y
m` lawpe
zxver
M
ik xexa
f (hM i , x) 6= 2000
m`
:ziwlg caer eply oexztd
e` :miipyn cg`y epl reci .lelrza ynzyp df dxwna ?f
x lr M
(hM i , x) = 2000
M -y
ly micrv 2000 ly divleniq rvap okl .x lr llk zxver dpi`
.eply iepiyd xg`l mb aeyigl zpzip izla dxzep
3
f -y
d`xn df .dgcp ,zxg`