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|yxyx|sqt wpw{t{q|s~ y{~ty tpyx {yx|t yqxsqwxs w|yxsp|sqyx{t
wpsqsqyx {t ypsqtqxstp p~swsqyx {t p~sws ystx ts w |ywypwsqyx txspt qtp
szt w |yxsp|sqyx {t wpsqsqyx {t ypsqt{yqs txt ts rwpwxsqp w|yv~ptx|t { szt sys
tx |yxtpwxs yx qxstp p~swqqs~ y p~txsyx w |yxsp|sqyx {qx{q|wstp yp wxsqtp
qz p~|qqyx {t w ypsqt ts w wqs~ y tppyx wq qtp z~svy{t {t |ywypwsqyx
txspt szt {qx~ptx|t sqqwsqyx tx wpwt y t |vwxwrt {t szt {qx~ptx|t yt
wx {t qztp {t szt {t rpwx{t {qztxqyx tx |y wxs t p~sws {t qtp
szt {t tsqst {qztxqyx
xxt p~txst xt w q|wsqyx wwpq|wsqyx pyzwrpt { uyzs~ yt|sq
{t |t spwwq ts {t p~yqp t |wpw|s~pqsqt {x pyzwrt w p wxwrt
'
zyq wpsqp
{t qxypzwsqyx {q yxqt wwxs wxwrt y {~swqtpyx t pqx|q t {t wpq|wsqyx {
uyzs~ q ptst xt wpq|wsqyx wpsqwxwt wt| xt ypst wpqwqqs~ txspt t pyzwrt tyx w
pyzwrtpqtt wqs sqq~w wqyx ts t z~svy{t {t wpq|wsqyx { pyzwrtp w |ywypwsqyx
wt| t uu yprwxqzt {t |yxtq ts {t ypzwsqyx {t pyzwrtp w {yxx~ qt qtp
szt {t prt oqtx t |tsst ~s{t t |yx|txspt p x yqxs qtx p~|q {xt w q|wsqyx
p~{q|sqyx {t w tpzts~ {t w st { pyzwrt xy twyx {t {~rwrtp xt z~svy{yyrqt
yp spwqstp xt |wt qtx qz ypswxst {w q|wsqyx t w q|wsqyx y w |yxxwqwx|t
t tpst ts qz ypswxst ts y w ypsqt t pqzt {t zwxqpt |yxsqxt
yp w|qqstp w t|spt yx spytpw x rywqpt tx
pw twxs t stpzt qz
ypswxs tx wpsq|qtp p w yrqt ts t prt yt sqq~ {wx |tsst svt
pw ttpw t qxypzwsqyx |yxxwspt p w sv~ypqt {t yqqqs~ yp qtx |yz ptx{pt
t |vw qspt
{~swqtpw t stpzt ts wpqwt q~ w wpq|wsqyx pyzwrpt {
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
wx |t |vw qsptxy pw tyx xt s yyrqt {t {q ~ptxst prt yt t prt yt
yxs {t prt sqqwxs {t txtzt y w{tv
t txtzt y pt p~txstxs
{t |yx|t s y } tpztsstxs {t t pw py|vtp {t w pt p~txswsqyx t
t pqs vzwqx w {t yts q txsyptxs x tsqs |vqtx xt stz ~pwspt |vw{t x
vyzzt txt y |t |yx|t s yxs {wx t pqs vzwqx {t {~xqsqyx qz p~|qt x xt
wt w pswtztxs {t txt qt t yp {t xyspt qzt wxxqtpwqpt ut |yx|t s
yxs rpw{t ts qz p~|q
w sv~ypqt {t yqqqs~ w ~s~ qxspy{qst wp w{tv tx w{tv tt tpzts
w pqt tx |yz st {t qx|tpsqs{t {wx t pwqyxxtztxs p {t |yxxwqwx|t qz p~|qt t
t|stp xyx vwqs~ w sv~ypqt {t yqqqs~ spytpw tx wxxt t o t pqx|q wt {~xqsqyx
{t |tsst sv~ypqt utsst {tpxqpt ts w~t p t xysqyx {t ! ts {t ts
py yt x txtzt {t ztpt tpztsswxs {~wtp t yqxs w p~wqwsqyx {x ~~xtztxs
{yxx~ts yqty |tpswqxt p|t wsv~ypqt {t yqqqs~yx zy{~qtpw{tqxypzwsqyx
qz p~|qt qx|tpswqxt y rpw{tt wp {t {qspqsqyx {t yqqqs~ w {qspqsqyx {t
yqqqs~ sqq~t {~ tx{pw { s t {qxypzwsqyx zy{~q~t yp {t |yxxwqwx|t t tpst
t pqzwxs {t |yxspwqxstyx sqqtpw {t {qspqsqyx {t yqqqs~ yp pt p~txstp {t wqs
ytp~w {qspqsqyx {t yqqqs~ rwpwxsqt ts w{w s~t
xt zwxqpt {t pwqyxxtp wpsqp {t wsv~ypqt {t yqqqs~ ts {sqqtp {t prt yt
y zyxsptpyx |yzztxs yx|sqyxxt t pwqyxxtztxs wt {t prt yt tx t|sqyx
y pw tyx txqst w s yyrqt {t {q ~ptxst prt {wx w t|sqyx s yyrqt q w
~s~ py y~t wp yq pw{t ts rvtssy yq pw{t rvtssy y
tppyx qts yqt {t{~|yz ytp t prtyt tx{t |ws~rypqt pqx|q wt t prt
qz q|wsqt yq pw{t w q yx|sqyxxtxs wt| {t {qspqsqyx {t yqqqs~ ts
t prt |yxyx|sqt wz{wxq qqwx q sqqtxs {t {qspqsqyx {t yqqqs~
rwpwxsqt wrp~ tp |yz wsqqqs~ wt| t y ~pwstp yrqt t prt qz q|wsqt yxs ~s~
p~txs sp t sqq~t tx pwsqt |yxspwqptztxs w prt |yxyx|sqt
y {~|pqpyx |t {q ~ptxs s t {t prt wx {t zyxsptp ypyq t prt qz q|wsqt
rpw{tt yxs t w{w s~t w pt p~txswsqyx {t |yxxwqwx|t t tpst {wx ttt
qxstpqtxxtxs {t |yx|t s ~ywxs {t w&yx |yxsqxt
xxtx t|sqyx xy |yz wptpyx t prt qz q|wsqt rpw{tt w prt t
|ypwzztxs sqq~t {wx t szt {qx~ptx|t yt t prt |yxyx|sqt yqqqs~
utsst |yz wpwqyx zyxsptpw {xt wps t wwxswrt {t prt qz q|wsqt rpw{tt wp pw
yps w prt |yxyx|sqt yqqqs~{wspt wps w {q|s~ qzqswxs w zqt tx pt {t
prt qz q|wsqt t |w| {t qx~ptx|t wpsqp {txsp~t yt B B E
y wyx yqp q|q t yx|sqyxxtztxs ts qxs~ps { pwqyxxtztxs wt {t prt yt
{wxxty sqt {tpt p~txswsqyx {tw |yxxwqwx|t t t|stp twzqqtp wt|t {q ~ptxst
xysqyx {t szt {qx~ptx|t yt spytpw x rywqpt tx wxxt t t| {t prt yt
t |yxxwqwx|t pt p~txs~t ttxs spt qz wpwqst |ts{qpt qz p~|qt qx|tpswqxt y
wrt
t prt yt yxs ~s~ |p~~t yp pt p~txstp {t |yxxwqwx|t ~wxzyqx tt yxs ~s~
txqst tw|y sqq~t yp t |yxspt y {t py|~{~ wz pqwxy t
szt{tprtytyxs wypsqq~|yzztxysq{w py qzwsqyx{tyx|sqyx xz~pqt
ts ttxsqtt y pw tyx t qxstp p~swqqs~ {x szt ts t {~xqp wp w w|qqs~
{t |yz p~vtxqyx wpw x t tps |yz ptx{pt t ptwsqyx {txsp~t ypsqt {~xqt wp t
szt
?(@<@ >?( ?6@ 9 @
xt prt ts xt |yxxwqwx|t q tps wqpt t qtx txspt {t qxypzwsqyx |yxxt ts
{wspt qxypzwsqyx t yx ts {~{qpt y qx~ptp
xprttsxt t ptqyx{t wypzt q
X
tsA
wypZ
tsO
t t pqzt xtptwsqyx txsptt wpqwtX
tsZ
x w ttX
tsA
w p~zqt{twprttsZ
tsO
w|yx|qyx {t wprt x r~x~pwyx tsqzt t w|yxxwqwx|t t pqz~t wp |tsst prt ts wq{t ts t{t pwqyxxtztxs sqq~ ts {yx| {~{|sq|wpyx twt{t {~{qpt{tqxypzwsqyx p twtp
{t ypsqt wpsqp {t wtp {t txsp~t
$;=866@:@6< ;@ @ 9 @
t pwqyxxtztxs wt {t prt p~txst |tpswqx wwxswrt sys {wyp{ q ts w|qt
|yz ptx{pt ts qxstp p~stp t q ts wt xwspt |wp spt vzwqx pwqyxxt ytxs y
ypzt {t prt q ts {tvypwypt wq ptx{pt x wpw qt x wspt wsys {t |ts t
{t pwqyxxtztxs ts w zy{wpqs~ tx t ts q ts yqt {wystp {t prt y {tx txttp
qz tztxs tyx t tyqx {t sqqwstp yp pwqyxxtp wt {t prtq ws xt wt {t |yxxwqwx|t t prt x txtzt {t wqs t {yxx~t x zystp {qx~ptx|t t py|t { pwqyxxtztxs wt {t prt ts t qwxs {wx x ptzqtp stz yx {~stpzqxt t wqs tpsqxtxs yp w wt {t prt txqst yx {~stpzqxt wpzq t prt w q|wt w y t prt q |yxqtxs {t {~|tx|vtp t t|sqtztxs yp xqp yx wrprt w
wpsqt |yx|qyx {t prt yp ystxqp t p~sws qx~p~ wp t wqs x xystpw t tyx
t s t {t prt sqq~t q ts spt x~|twqpt {wrp~rtp w ptwsqyx txspt t prt wwxs {t
pwsqtp qx~ptx|t
wp pw yps w prt yrqt yy~txxtt prt yt p~txstxs qxs~ps {tw pyrpt
qqs~ { pwqyxxtztxsts {t sqqwsqyx {~sqtsst qxrqsqt yt ut tx{wxs tp zqt
txpt ts tw|y |yz t t t |tt {t prt |wqt|yzzt xy t tppyx
yqx
&@:@ @ <:@ @ 9 @
t szt qwxs zy{~qt t |yz ypstztxs {x wsyzyqqst pwqyxxwt w|t x t
spq|yypt x ptzwpt t t prt sqq~t yxs wt qxsqsqt ts zy{~qtxs t szt {t
zwxqpt qxstp p~swt w rpt zyxspt st tt prt txqwrtwtq w{pwqs qtx
Si le feu est ...
et Si ma vitesse est ...
et Si le feu est ...
rouge
rouge
orange
vert
faible
moyenne
faible
élevée
proche
loin
loin
proche
accélère
freine doucement
maintiens ma vitesse
freine fort
Alors, je ...
Conclusion
Prémisses
rt zy{~qwxs t |yz ypstztxs {x wsyzyqqst w|t x t spq|yypt
x ptzwpt t {wx |ts t tz t |tpswqxt wpqwt yxs yt ts {wspt xt t yxs w
wp t tz t x t spq|yypt ts yqs tps yqs ypwxrt y yqs pyrt q xt ts w spt txspt
t tps ts t pyrt qt t |vwxrtztxs {t |ytp ts qxswxswx~ x ptwx|vtw qstt ts
s qtztxs xt wpqwt yt w {q ~ptx|t txspt xt qstt wqt ts xt qstt zytxxt
xts w xt wtp spq|st x xt ts w {qpt z v w qstt ts wqt ts tt ts
zytxxt z v yp qsptp |t yx|sqyxxtztxs yq|q t txtzt y |yppt yx{wxs
0
1
vert orange rouge
couleur du feu
proche
loin
15
30 distance du
feu (en m)
vitesse
(en km/h)
faible moyenne
élevée
30
40
50
0
0
1
1
PREMISSES
CONCLUSIONS
freine
fort
freine
doucement
vitesse
accélère
réaction
0
1
20 22
0.3
0.7
maintient
y{~qwsqyx {t {q ~ptxs txtzt y |yppt yx{wxs w |yz ypstztxs {x w
syzyqqst w|t x t spq|yypt
t pqx|q t ts txqst t qwxs w |yx{qsqyx p t p~zqt ts pwqt w|sqyx
p~|yxq~t yp t |yx|qyx {yqs spt pt t|s~t x wqs tyx t {yxx~t {t p~zqt
|tpswqxt prt tpyxs {~|tx|v~t x |tpswqx {trp~ t| t prt vwqsttztxs sqq~t
{wx t szt {qx~ptx|t yt s t wz{wxq rtxyyqp wxxt t w wpsqt |yx|qyx
ts tx wqs xt qxstp ywsqyx txspt t |yx|qyx {t {q ~ptxst prt w|sq~t
wp t tz tqw qstt ts {t z v|tt|q tpw|yxq{~p~t |yzzt wqt wt| x {trp~ ~rw ' ts zytxxt wt| x {trp~ {t ' | rpt wx |t |w yx |yxq{pt t w
qstt ts wqt t zytxxt t prt wt| xt qstt wqt wpyxs {yx| {qz w|s
p w |yx|qyx
E
"866;=;6@ 6 ;<=@ @< 866;=;6@ 8=<=@
ypyx ypzt xt py yqsqyx |tt|q ts wqpt w t {t ypzt {t |yxxwqwx|t
x~rwsqt y yqsqt wp t tz t q yx wpzt t ~|yt yxs ytpst w tzwqxt yx
t pqzt psys t t py yqsqyx yp ttt |tsst py pq~s~ ts wt yxs qz yqt
uts{qpt t t ~|yt xt yxs w ytpst t tttx{ x ts wyp tx p~tx|t {t |yxxwq
wx|t x~rwsqt |wp |tt|q t pqzt xt |yxspwqxst t ~|yt xt yxs w ytpst t tt
tx{ ypyx t pqzt xt |yxspwqxst y xt ptspq|sqyx p t {yzwqxt {xt wtp wyp
yx wpt {t |yxxwqwx|t x~rwsqt
yxs ytpst wyp qt pqzt sstwqs|t vypwqptt ~|yt yxsytpst{t zwxqpt
rwpwxsqt utw xt t tz |vt w {spt ytpst {wspt vtptzwq yx ts p tt yxs
ytpst {wx |tsst wrt vypwqpt utsst qxypzwsqyx ts {yx| rwpwxsqt yqtqwrqs wyp {t
|yxxwqwx|t yqsqt |wp w |yxxwqwx|t ts t |yzzt x t tz t {t |t q ts yqt
|y p x wqs ytp~ wp t tz t
t xt qxypzwsqyx x~rwsqt ts spt qxstp{qst wp {wspt wtp q t qst {t
yp {wx w tzwqxt y ~|yt ts tpz~t yp ~pq~ ztp|pt{q t w zzt zwxqpt xt
|yxxwqwx|t yqsqt xyx ~pq~t wp t |yxxwqwx|t w|stt p t ts ts spt wq{~t
wp {wspt qxypzwsqyx q ts wyqp |yp {wspt vtpt txspt ' ts vtpt
@? 7; 86 @ 8=9 ?6@ 9 @
w zwxqpt {yxs yx |yx|tpw t prt qxtpw p w zwxqpt {wrp~rtp {q ~ptxst prt
}zwrqxyx x szt {t prt |vtp|vwxs {~stpzqxtp q {t tpyxxt ttxs |yyqsptp
txtzt yp x yp p~|q
yqs txtzt {t prt qwxs
q ytps ts {w||yp{ yp |yyqsptpwyp q wps w spwwq txspt v ts v
q wpqt ts {w||yp{ yp |yyqsptp wyp tt wps w spwwq txspt v ts v
q w| ts {w||yp{ yp |yyqsptp wyp q wps w spwwq txspt v ts v
q tq|w ts {w||yp{ yp |yyqsptpwyp tt wps w spwwq txspt v ts 'v
q wpqt ts wtxsttt t prt ts yxs w|sq~tvypwqpt yqt ysqtxs wp
qxstpt|sqyx {t p~sws {t |vwt prt x w wyp
[7, 9] ∩ U ∩ [8, 9] ∩ [8, 10] = [8, 9]
wt|U
xqtp {t w ypsqt sys txsqtp x ptzwpt t ypxt prt xt w qt wwx {t xt w zy{qtp t p~sws {t ypsqt w |yx|qyx ts |yxq{~p~t |yzztU
xqtp {t ypsqt qtU
ts x ~~ztxs xtspt yp qxstpt|sqyx wx |tsst pt p~txswsqyx yx yqs w prt |yzzt xt |yxspwqxstx ptwx|vtypyx |yxq{pt txtzt {t prt qwxs
y t tz y~ {t txspt pqt spwwqtxs {t v v ts txspt v ts v
t |w{pt {t txspt pqt spwwqtxs wq {t v v
t qxypzwsq|qtx {t txspt pqt spwwqtxs wq {t v v
t qrqt {t txspt pqt spwwqtxs wq {t v v
}zwrqxyx t t prt ts yqtxs {~|tx|v~t x ts {yx| tx p~tx|t {x |w{pt
qxypzwsq|qtx tz y~ wp |tsst txspt pqt t vypwqpt {t spwwq yxs p~txs {yxx~t wp
xqyx {t {q ~ptxs vypwqpt {tspwwq
([9, 12], [14, 17])∪ [17, 18]∪[13, 14]∪∅ = [[9, 12], [13, 18]]
wx |ts t tz tyx yqs t ~~ztxs xtspt {t xqyx ts txtzt q{t utsst pt p~txswsqyx |yxq{pt |vwt prt |yzzt xt qxypzwsqyx ~ztxswqpt
w ptzqpt |wt ptzqtp t tz t {~xqs t ! {qst ! w t|yx{t |wt
{t qzt t tz t t !
utsst {q ~ptx|t txspt t {t pt p~txswsqyx ts yx{wztxswt qtt {q ~ptx|qt t
prt tyx {t |wt w ptzqpt |wt |yppt yx{ xt qz q|wsqyx
A → O
w {t qzt xt |yxyx|sqyxA ∧ O
yp {t {~swq p t y ~pwstp sqq~ yx ts t p~~ptp yq pw{t wrtp y tqpy{ ut{t pt p~txswsqyx yxsxtypxt ~pqtpt tsqx~pqtpt{twptwsqyx
R
txspt txsp~t ts w ypsqt {t w prt qX
tsA
wypZ
tsO
A ∧ O ⊆ R ⊆ A → O
qt |t prt yxs {q ~ptxsttp ypsqt qxstp pstxs {q ~ptzztxs {t t tz t {t wtp |yxyqxstztxs yqt tx |t q |yx|tpxt t prt |yxyx|sqt
yqs t wtp |yppt yx{wxs w pt p~txswsqyx
A ∩ O
{t ptspq|sqyx p x txtzt {t wtp yqt tx |t q |yx|tpxt t prt qz q|w
sqt t| |tsst zy{~qwsqyxyx |yxq{pt t t prt t pqztxs wqxq
¯
A ∪ O
|t q ptqtxs ~qzqxtp t |yxsptt tz t {t w prt A ∩ ¯
O
x w {yx| tx wqs w ptwsqyx qwxst q |yppt yx{ w yxt vw|vp~t {t rpt ts
A ∩ O ⊆ R ⊆ ¯
A ∪ O
U
W
A
1
1
Exemples de la r`egle
O
A
∩ O
Valeurs non mod´elis´ees
par la r`egle
t p~txswsqyx {t wtp {xt prt |yxyx|sqt
U
A
W
1
1
O
Valeurs possibles par
d´efaut ¯
A
∪ O
r`egle A → O
Contre exemples de la
t p~txswsqyx {t wtp {xt prt qz q|wsqtA → O
y tppyx wp w qst t |t {t |wt {t prt t {~|yz ytxs |vw|xt tx y
|ws~rypqt } w{yx| |ws~rypqt pqx|q wt {tprtyt tprt |yxyx|sqt yqqqs~
t prt |yxyx|sqt wxsqrpw{ttt prt qz q|wsqt |tpsqs{t ts t prt qz q|wsqt
rpw{tt yq pw{t w wxs {t {~swqtp t yx|sqyxxtztxs {t |t {q ~ptxs s t
{t prt {yxxyx tt pw t p w xysqyx {t p~q{wsqyx q w |yx{qpt x s t
{qz q|wsqyx wpsq|qtp
$=?;<=86
w yrqt p~q{~t ts x zy{t pqx|q wtztxs sqq~ {wx t {yzwqxt {t w yrqt yt
ut zy{t pt yt p xt wrpt p~q{~t wpsqp {xt stt wrptyx ts txqst {~xqp
w xysqyx {qz q|wsqyx p~q{~t w xysqyx {t xypzt spqwxrwqpt ts wq sp qz ypswxst {wx
|t |yxst st utpswqxt {t prt qz q|wsqt t xy sqqtpyx wp w qst pt ytxs p {t
qz q|wsqyx p~q{~t |yzzt xy t tppyx tx t|sqyx y |yzztx&yx wp {~xqp
xt wrpt p~q{~t ts w sxypzt wx {t yyqp {~xqp t qz q|wsqyx p~q{~t
xt wrpt p~q{~t ts t {~xqp wpsqp { w{p ts
(L, ≤, >, J)
y (L, ≤)
ts x txtzt wpsqttztxs yp{yxx~ x sptqq y~{wxs x yzzts ~rw1
>
ts w xypzt spqwxrwqpt tztxs ts w ''' y sxypzt wy|q~t
(L, ≤)
xt sxypzt>
{~xqt p[0, 1] × [0, 1]
{wx[0, 1]
ts xt yx|sqyx {~xqt wqxq tt ts |yzzswsqt x>y = y>x
tt ts wy|qwsqt (x>y)>z = x>(y>z)
tt ts zyxysyxt a>b ≤ c>b
qa ≤ c
tsb ≤ d
1
ts ~~ztxs xtspt 1>x = x
0
ts ~~ztxs wypwxs 0>x = 0
J
ts xt qz q|wsqyx p~q{~t y {~xqpyx tx {~swq t py pq~s~ {t qz q|wsqyx
a>
min
b = min{a, b}
a>
prod
b = a.b
a>
luka
b = max{0, a + b − 1}
t zqxqzzt py{qs ts wsxypzt {t wqtqs yxs t sxypzt t |ypwxst x
ts xystp t w sxypzt { zqxqzz
>
min
ts xt ypxt ~pqtpt {t syst t sxypzt
t qswxst xt qz q|wsqyx p~q{~t {t
>
ts xt y ~pwsqyx qxwqptJ
p[0, 1]
{~xqt wp J(α, γ) = sup{δ ∈ [0, 1]|α>δ ≤ γ}
ut q ts wq ~|pqpt ∀α, β, γ ∈ [0, 1]
α>β
qβ ≤ J(α, γ)
t pqx|q wt py pq~s~ {t qz q|wsqyx p~q{~tzqt tx ~q{tx|t wp ypq ypq ''
yxs {~|pqst {wx wxxt t o
$ @ =:=;<=@
t prt qz q|wsqt {yxs w qrxq|wsqyx w ~s~ {yxx~t tx ttxs spt t |yzzt
xt w q|wsqyx {qpt|st {t sv~ypqt {t w{tv w{tv p t pwqyxxtztxs w py|v~ tyx
w{tv|vwt rpwqx {t |yxxwqwx|t ts spt |yzzt xt ptspq|sqyxp xtxtzt {t y
qt x sv~ypqt {t yqqqs~ yq pw{t yx pt p~txst t |yxxwqwx|t
wp {t {qspqsqyx {t yqqqs~
π
w py yqsqyxX ∈ A
ts wyp spt qxstp p~s~t wqxq ∀u ∈ U, π
X
(u) ≤ µ
A
(u)
y
µ
A
(u)
ts w yx|sqyx {w wpstxwx|t w ytxtzt yA
xt yx|sqyx {w wp stxwx|tµ
A
(u)
{~xqs yp sys yqxs {t xqtp t {trp~ {w wpstxwx|t {t
u
txtzt yA
qx~rwqs~ pyqtxs { wqs t qxypzwsqyxX
tsA
xt |s w xt xytt qxypzwsqyx ptsptqrxt tx|ypt txtzt {t wtp yqt {tX
t prt qz q|wsqt ttxs spt t |yzzt xt t stxqyx {t w yrqt |wqt tx
yrqt |wqtt zy{ yxtx |yppt yx{
A ∧ (A → O) |= O
y
|=
pt p~txst qx~ptx|t yrqt∧
ts w sxypzt wy|q~t w ptwsqyx |yxq{~p~t x yrqt ytyx wt zy{ yxtx r~x~pwq~ py y~ wp w{tv tx w{tv wpsqxuywqpt
ut q qrxqt t yp x wqs
A
0
yx ts |w wt {t {~{qpt xt wtpO
0
{t qz q|wsqyxA → O
yp xt txsp~tA
0
w wtp {t ypsqtO
0
|yppt yx{wxst ts {yxx~t wp µ
0
O
(w) = sup
u∈U
>(µ
A
0
(u), µ
A
(u) → µ
O
(w))
utsst |yzqxwqyx pyt|sqyx ts |yxxt y t xyz {t |yz yqsqyx
sup −>
y t ts ytxs xys~t◦
t y ~pwstp {t |yxyx|sqyx ts {qz q|wsqyx xt ttxs w spt |vyqq qx{~ tx{wzztxs x {t wspt t |vyq {t>
{~stpzqxt {yx| y ~pwstp {qz q|wsqyxwrp~rwsqyx {t |t prt t wqs {t zwxqpt |yxyx|sqt |wp |tt|q pt p~txstxs xt yq
qqs~ xyxrwpwxsqt {twtp |yxq{~p~t |yzzt yqt wp xtprt ttxs sptqxstp{qst
wp {wspt prt ut q ts wq ~|pqpt
O
0
= A
0
◦
n
\
i=1
A
i
→ O
i
t {t s t {t prt qz q|wsqt |yppt yx{txs w {t qxstp p~swsqyx yrqt {xt
prt {t s t
A → O
yqsA
ts w yO
ts pwqt yqsO
ts pwqt |vwt yq tA
ts pwqt|ts{qpt tA → O
ts syswtztxs pwq qO
ts w zyqx wqpwq tA
ut{t qxstp p~swsqyx yxs ~qwtxstypyxptrwp{twswt{t ~pqs~ {tqz q|wsqyx
→
| swtw x wqsA → O
ts w q ts ttztxs qA
ts pwqt tsO
ts wA → O
O
¬O
A
pwqt w¬A
pwqt pwqt wt {t ~pqs~ {tA → O
ut tx{wxs |t {t s t {qxstp p~swsqyx zxtxs {t s t {qz q|wsqyx t qz q|wsqyx p~q{~t yp tttA → O
ts syswtztxs pwqt q ts ttztxs qO
ts w zyqx wqpwqt tA
x w tt |t prt utsst qxstp p~swsqyx qrxqt t t qzwrt {tA
wp wptwsqyxR
yxs {wxO
|t q ts ~|pqpt A ◦ R ⊆ O
t qz q|wsqyx {t wypzt¬A∨ O
y¬
ts wx~rwsqyx ts∨
w {qyx|sqyxyt ut prt yxs w t~ prt |tpsqs{t utsst qxstp p~swsqyx qrxqt tx wqs t t ~~ztxs vyp{t
A
y {wxO
w wpsqtxxtxs wptwsqyx|t qt xyst R ⊆ ¯
A ∪ O
tzwpyx tx yrqt ytw ypzwsqyx¬A ∨ O
qxstp pst {q ~ptzztxs {t w yrqt |wqt m ! ! !! t prt qz q|wsqt rpw{tt ttxs spt |yz pqt |yzzt X
tsA
Z
tsO
ut prt |yppt yx{txs w ptwsqyxA ◦ R ⊆ O
ypt◦
ts w |yz yqsqyxsup
∧
y∧
ts ytxs t zqxqzz w ptwsqyx ts wyp t xystp
sup
u∈U
(µ
A
(u) ∧ π
Z|X
(w, u) ≤ µ
O
(w))
π
Z|X
(w, u)
ts xt {qspqsqyx {t yqqqs~ |yx{qsqyxxtt t t pqzt w ptwsqyx {t {~ tx{wx|ttxsptX
tsZ
tt tpzts {t{~stpzqxtp t yqxs x~~xtztxsw ∈ W
ts yqt w|vwxs t ~~xtztxsu ∈ U
ts yqt w {trp~π
X
(u)
} t qst {q ~ptxs s t {t prt qz q|wsqt rpw{tt tyx qz q|wsqyx sqq~t {wx~wsqyx y tx ~s{qtpyx spyq
}z q|wsqyx p~q{~t {t {t
a → o =
1
qa ≤ b
o
qxyxU
0
α
1
A
0
1
O
O’
α
W
uyx{qsqyx {t w prt ypsqt {t w prt
}x~ptx|t wt| xt prt qz q|wsqt rpw{tt qz q|wsqyx {t {t ts xt txsp~t spq|st }z q|wsqyx p~q{~t {t yrtx
a → o =
min(1, o/a)
qa 6= 0
1
ysvtpqtU
0
α
1
A
0
1
O
O’
α
W
uyx{qsqyx {t w prt ypsqt {t w prt
}x~ptx|t wt| xt prt qz q|wsqt rpw{tt qz q|wsqyx {t yrtx ts xt txsp~t
spq|st
t| yrtxw sxypzt ts t py{qs
}z q|wsqyx xyx p~q{~t {t t|vtpwqxt
a → o =
1
qa ≤ o
0
qxyxx ts xystp t qz q|wsqyx {t t|vtpwqxt ts w ptwsqyx xyxyt w zyqx ~
|qt ~pqwxs w ptwsqyx qtt ts ~rwt w xyw {t wspt qz q|wsqyx w
rpt zyxspt qx~ptx|t {xt txsp~t spq|st wpsqp {xt prt qz q|wsqt rpw{tt zy{~q~t wp qz q|wsqyx {t t|vtpwqxt
U
0
α
1
A
0
1
O
O’
α
W
uyx{qsqyx {t w prt ypsqt {t w prt
}x~ptx|t wt| xt prt qz q|wsqt rpw{tt qz q|wsqyx {t t|vtpwqxt ts xt
x |vyqqpw xt qz q|wsqyx {t t|vtpwqxt q yx yvwqst ystxqp x p~sws qz p~|q
zwq xyx y x yppw wq |vyqqp qz q|wsqyx {t yrtx yp w ypzt spw ~yq{wt
yp {t {~swq p t prt qz q|wsqt rpw{ttyx ts t p~~ptp pw{t
y yq pw{t ut prt yxs wpsq|qptztxs w{~wst yp {t py|t
{qxstp ywsqyx m ! ! t prt |tpsqs{t ttxs qxstp p~stp wqxq
X
tsA
q ts |tpswqx tZ
yqsO
ut s t {t prt |yppt yx{ w pt p~txswsqyxR ⊆ ¯
A ∪ O
ypt xqyx ts tmax
ts w x~rwsqyx {t
A
1 − A
yx ysqtxs w |yxspwqxst qwxst q ptqtxs sqqtp qz q|wsqyx {t ttxtqtxta → o = max(1 − a, o)
π
Z|X
(w, u) ≤ max(1 − µ
A
(u), µ
O
(v))
U
0
α
1
A
0
1
O
O’
α
1 − α
W
uyx{qsqyx {t w prt ypsqt {t w prt
}x~ptx|t wt| xt prt qz q|wsqt |tpsqs{t qz q|wsqyx {t ttxtqtxt
ut s t {t prt tpzts {t py wrtp {t qx|tpsqs{t } xts w t|w|t yp wqpt {t
qxstp ywsqyx
$ @ 8686<=@
uyzztxywyx p~|~{tzztxstprt|yxyx|sqtpt p~txstxs{t{yxx~t|ts
{qpt {t |y t {t wtp |yxyqxstztxs yqt x t ts qyx sqqt x y ~pwstp
{t |yxyx|sqyx txspt txsp~t
A
ts w ypsqtO
{t w prt w prt qrxqt yx ts wyqp qzswx~ztxs t wqsX
tsA
tsZ
tsO
| ~wsqyx yp xt wpqwtX
w {qspqsqyx {t yqqqs~ rwpwxsqt wy|q~tx ∈ A
ts yqt ts stt t ∀u ∈ U, δ
X
(u) ≥ µ
A
(u)
utsst qxypzwsqyx xt pt p~txst xt ptspq|sqyx {t wtp yqtqx~rwqs~ qrxqt
t pqtx xtz |vt {t xytt qxypzwsqyx {~stx{pt t wtp rwpwxsqt yqt {wspt
wtp {t
U
yp {qxypzwsqyx p w sv~ypqt {t yqqqs~ t t|stp t p~~ptpw wxxt t ot prt |yxyx|sqt yxs wrp~r~t {t zwxqpt {qyx|sqt qt tt pt p~txstxs {t
wtp rwpwxsqt yqttp wtp xtspt ts {yx| txtzt q{t q ts x ~~ztxs xtspt
yp xqyx w |yx|qyx
O
0
{x txtzt {t prt |yxyx|sqt{A
i
∧ O
i
, i = 1, . . . , n}
~swxs {yxx~ xt txsp~t ytA
0
ts {yx| {~xqt wp O
0
= A
0
◦
n
[
i=1
A
i
∧ O
i
w ypsqt {xt prt ts rwpwxsq q qtp prt yxs w|sq~t wyp syst tp |yx|qyx
yxs rwpwxsqt y ~pwstp {t {qyx|sqyx r~x~pwtztxs sqq~ ts t
max
ut s t {wrp~rwsqyx ~qws xt w||zwsqyx {qxypzwsqyx yx w{t prt yx w wyqp {qxypzwsqyxts t xyzpt {t |y t {t wtp {txsp~t ypsqt qzswx~ztxs yqt tpw qz ypswxs
t prt |yxyx|sqt pt p~txstxs {yx| {t {yxx~t ytp~t ts xyx w {t |yxspwqxst
|yxxt wp |yx~txs |t prt xt ttxs t pqztp xq qz yqqqs~xq qxstp{q|sqyx xt
wtp xyx ytp~t xts w qz yqttt xts sys qz tztxs w rwpwxsqt yqt
uyxq{pyx xt txsp~t p~|qt
u
0
q yx wµ
A
(u
0
) = α
wt|α ∈ [0, 1]
t wtp {wxO
yxs syyp yqtzwq tt xt yxs rwpwxsqt w {trp~
α
t {trp~ {t yqqqs~ rwpwxsqδ(u, w)
t pqzt t yqxs t ~~xtztxsv
tsw
yxs yqt |yxyqxstztxs
x {trp~ {t yqqqs~ rwpwxsq tx ypsqt { szt
δ(u, w) = 1
qrxqt t ypX = u
w wtpw
ts syswtztxs yqt ypZ
x ptwx|vt
δ(u, w) = 0
qrxqt t qX = u
wyp w|xt prt xt ts rwpwxsqp tw
ts yqt y qz yqt ypZ
q w prt xt w qt wyx ts {wx t |w yδ(u, w) = 0
t wtp yxs {yx| syst yqtzwq tt xt yxs w rwpwxsqt x {trp~ {w wpstxwx|t0
O
0
xqxstp{qs pqtx w wtp xts w rwpwxsqt yqt zwq tt ptst ypswxs yqt txtzt yO
0
ystx |yppt yx{ {yx| wypxt qx~pqtpt {t wtp yqt yp w wtpZ
yp p~ztpt p~sws ystx tx ypsqt {yqs qxstp p~stp {t w zwxqpt qwxst
µ
0
O
(z) = 0
yxxt ts pqtx{qpt p w wtpz
ttxts wrwpwxsqt zwq tt ts yqt qxt wspt prt ts ~txsttztxs w rwpwxsqpµ
0
O
(z) ∈]0, 1[
w yqqqs~ {tz
ts rwpwxsqt x |tpswqx xqtwµ
0
O
(z) = 1
w wtpz
ts syswtztxs yqtwp w qstxy wyx {~swqtp t {t s t {t prt |yxyx|sqt t prt yqqqs~
ts t prt wxsqrpw{tt yq pw{t m ! ! w ptzqpt wpsqt {t qx~wsqyx ts ~|pqpt
µ
A
(u) ∧ µ
O
(w) ≤ µ
R
(u, w)
yqs wt| {t yqqqs~ rwpwxsqt µ
X,Z
(u, w) ≥ µ
A
(u) ∧ µ
O
(w)
't prt |yxyx|sqt yqqqs~ yq pw{t
q
X
tsA
wypZ
tsO
ttxs zwqxstxwxs ~|pqpt X
tsA
wyp qts yqttZ
yqsO
y ~pwstp|yppt yx{ q|q xt |yxyx|sqyx w ypsqt {t w prt ts {yx| xt {qspqsqyx {t
yqqqs~ rwpwxsqt
δ
Z|X
= µ
A
∧ µ
O
q yx |vyqqs yp y ~pwstp {t |yxyx|sqyx t
min
yx ptspyt t prt {t wz{wxqU
0
α
1
A
0
1
O
α
V
O’
uyx{qsqyx {t w prt ypsqt {t w prt
}x~ptx|t wt| xt prt |yxyx|sqt yqqqs~ ts xt txsp~t spq|st
m
! !!
uts t {tprt sp tsqq~ ts ystx wp |yz ~ztxswpqs~ wp pw yps w spyq wspt
s t {t prt x t tst prt qz q|wsqt rpw{tt ts t prt |yxyx|sqt yqqqs~
wrqtxs syst t {t p t xyw {t w ypsqt t prt qz q|wsqt |tpsqs{t wrqtxs p
t yps {t w ypsqt wp |yz ~ztxswpqs~ yx ts {yx| |yxspqpt {t prt |yxyx|sqt
wrqwxs p t yps {t w ypsqt
y xt {~swqtpyx w tp yx|sqyxxtztxs t |yppt yx{txs w |yxspwqxst qwxst
∀(u, w) ∈ U × W, max(π
X,Z
(u, w), 1 − µ
A
(u)) ≤ µ
O
(w)
ut prt ttxs qxstp p~stp {t w zwxqpt qwxst
X
ts py|vt { xyw {tA
t wtp wsyp { xyw {tO
yxs yqt ypZ
x xystpw t t wtp wsyp { xyw {tO
{yqtxs wx{ zzt spt {wx t yps {tO
yp wyqp xt qxtx|t wrpt ' zyxspt qx~ptx|t wpsqp {xt prt |yxyx|sqt wxsqrpw{tt t xyz {t prtwxsqrpw{tt qtxs { wqs tt yxs x |yz ypstztxs qxtpt |tq {t prt qz q|wsqt
rpw{tt t|vtpwqxt t ptzqpt p~sp~|qtxs t yps ts t t|yx{t ~wprqtxs t xyw
U
0
α
1
A
0
1
O
O’
α
V
1 − α
uyx{qsqyx {t w prt ypsqt {t w prt
'}x~ptx|t wt| xt prt |yxyx|sqt wxsqrpw{tt ts xt txsp~t spq|st
tp qxstp p~swsqyx xts w wq~t {x yqxs {t t ~zwxsqt
$?: @6 ?@ @ ( ?<==;<=86
y wyx t t prt yt ttxs spt {~|yz y~t tx wspt s t {t prt
rvtssy ts w
yq ts w
''
q yxs {t y|ws~rypqt {t prt |yxyx|sqt
ts qz q|wsqt
q w prt pt p~txst
xt |yxspwqxst xt yqrwsqyxxt ptspq|sqyx prt
q w prt pt p~txst w |tpsqs{t y w s q|wqs~ prt xt ~ysqyx rpw{tt y w py qzqs~ prt {t t tz t{t |tqts yqt{t{yxx~t prt t |tsst zwxqpt yx ts |vyqqp t s t {t prt w{w s~ |t yx ts zy{~qtp t prt qz q|wsqt tztxs spt t prt t w{~wst {wx t |w y yx sqqt {t
|yxxwqwx|t t tpst q t pqztxs y ypzt {t |yxspwqxst ypt w ptwsqyx qwxs t
txsp~t w ypsqt ts rpw{tt ts |yxsqxt t prt qz q|wsqt rpw{tt yxs t prt t
w{w s~t
p |tsst ~s{t {x yqxs {t t yrqt xy wyx zwqxstxwxs p~txstp x |yz w
pwsq {t prt qz q|wsqt rpw{tt ts {t prt |yxyx|sqt yqqqs~ {x yqxs {t t
pwsqt BB E EB B EE E E
y |yz wptpyx q|q t szt {t prt qz q|wsqt rpw{tt ts {t prt |yxyx|sqt
yqqqs~ wx {t {~stpzqxtp tp wwxswrt ts qx|yx~xqtxs tyx |tpswqx |pqspt
wrp~rwsqyx {t p~sws tx ypsqt |w w|qs~ {qxstp ywsqyx qxtx|t {t w ypzt {t txtzt y w||zwsqyx {t |yxxwqwx|t {wx xt wt {t prt |yv~ptx|t z~|wxqzt {qx~ptx|t % 9 ;<=86
uyzzt xy wyx {~ t q~ p~|~{tzztxst prt qz q|wsqt wrprtxs |yxyx|
sqtztxstst prt |yxyx|sqt{qyx|sqtztxs wrpt qsptwrp~rwsqyx{t |t {t
s t {t prt yp t szt {t prt qwxs q
