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Un cadre quantitatif pour la Ludique
François Maurel
To cite this version:
François Maurel. Un cadre quantitatif pour la Ludique. Autre [cs.OH]. Université Paris-Diderot Paris VII, 2004. Français. �tel-00152476�
HAL Id: tel-00152476
https://tel.archives-ouvertes.fr/tel-00152476
Submitted on 7 Jun 2007
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Simp Proba
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D
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E ∈ G ⊥s
E ∈ G ⊥s
E ∈ G ⊥p
D⊥E
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G = Simp(Proba(G))
D⊥E
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Proba
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Î
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Ñ
Îvu !u Ð Ð Ñ
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EE â H = Simp(G)
â H⊥s = Simp(G⊥p )
a
Î
Î Ñ
Ñ
Îu !u Ð Ð Ñ
Ñ
Ù£8 Þ S Ú ÓDޔÓDÙ SwÚ Ý Ü × Ó2Ó×
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Ü©Ü Ó×
Ñ
S
Î
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G = Proba(H)
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⊥s
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H⊆G
⊥s
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s
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Simp((G1 &G2 )⊥p )⊥s
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D = |D|G = ∩{D0 ∈ G|D0 ⊆ D} = ∩{D0 ∈ Simp(G)|D0 ⊆ D} = |D|Simp(G)
D ∈ |Simp(G)|
D ∈ |Simp(G)|
D = |D|Simp(G) = ∩{D0 ∈ Simp(G)|D0 ⊆ D} = ∩{D0 ∈ G|D0 ⊆ D} = |D|G
D ∈ Simp(|G|)
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:=
:=
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:=
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‚N–
sut"„Ÿ ‹%$3„{’]‚ƒ
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0
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‚N–
sut"„—Ÿ ‹ $„{’]‚ƒ
‹“ 7 † sutt“(t † s
#  Ÿ ’‹ $H„{’]‚=ƒ ƒaHˆ50N‹.Cut
A
σ
(+, ξ, I)
ξ,I
H = {σ, τ }
„c’ ”s+†s
s„
st„ Ÿ “(ƒ ’]¦ “$sŽ%‹ $„c’c‚ƒ ƒ^ˆN‹„c’ ”3s+†^“V‹Œ]s ( % [‹C‡%‹ $u’.ƒas
AH 0
τ
†£“ † st5tsu’kƒ˜ƒaˆ$‹„{’.¡¢†s+(‹Œts
)
ξ`Λ ›
’
Œ
‹
]

‚
5
‡
t
‚$– st„oŸ %‹ $„c’c‚ƒ
‹“ 7 † sutt“(t¢† s
# Ÿ ‹ H$„{’]=‚ƒCHƒ^ˆ0.Cut
AH = {σ, τ } σ
(+, n : ξ, I)
n:ξ,I
N‹„c’ ”3s † s
s„ sut"„Ÿ %‹ $„c’c‚ƒ1ƒaˆ$‹„{’H”s †^“‹ŒcsA‹“ 7Q†sut5t"“(t† sAŸ ‹ $„{’]‚ƒ
AH 0
τ
‰‚t’ „{’H”s† s
ASautH0 (n,ξ) ›
‹Œ]‚‡5t
‚N–
sut„FŸ ‹ $„{’]‚ƒ
‹O“ 7Q† st5t"“(t † s
#  Ÿ ’‹ $H„{’]‚=ƒ Hƒaˆ50N‹.Vis
AH = {σ}
σ
(+, π, I)
(+,π,I)
„{’H”s † s
A 0›
’
# ƒ^ˆN‹H„c=’ ”3s H†0s.Vis(zξ ) ‹Œc‚3‡5Ht AH = {σ} ‚N– σ sut„ Ÿ %‹ $3„{’c‚3ƒ zξ ‹“ 7 † sutt“(t † s Ÿ ‹ $„{’]‚ƒ
AH 0 ›
‹Œc‚3‡5t
‚N–
sut„AŸ %‹ $3„{’c‚3ƒ
‹“ 7Q†sut5t"“(t+† s
#  Ÿ ’‹ $H„{’]‚=ƒ ‰H‚0t5.Vis
AH = {σ}
σ
(−, ξ, I)
(−,ξ,I)
’H„{’ ”3s† s
AH 0 ›
H
R SVUW Y[ZYc\^W _q`cb_˜d
.3
‚’ „
“Œƒ ‡ˆuts‹“
† ˆ„ su‡5R
1k’ ƒ’kt„ s )Xt“(‡>c› st
t5’
H0 + H ›
¤ ƒ1‰q‡‚$ ", † sF‰9‹Œ‡ $ ‹Œt
ah.j m
›
&
,l3p^hlm
%
‰9‹‡„{’.‡ †s
žw‚ƒ †ˆ 4 ƒ ’H„“(ƒ^s  < ~'!N|y xzfv € v € ( ƒa‚3ƒ
’.t„Q‚3‡5’c¦“Nst H/ 0ƒ =&k’ t„ ‚‡5’]¦“Ns
s t„“Œƒ >cy €|xOy"!X~a}€N~ }  <(€} ?~
H
R
›
>V ! 5!3 3R 2R
Ÿ sut"„ ]sŠ‰¢‚’.ƒ^„ † s † ˆQ‰i‹Œ‡„
› H = ›
›
‹( ) ’9]s† sutts’kƒ™‰q‡5’.ƒ'$u’‰9‹Œ
† s
st„?‰¢‚3t5’ „c’k¡>‹Œ]‚‡5t‚ƒ1‰q‡‚$ ,"† sw‰9‹‡5$‹ŒtAt“(‡
•
D
R
D
5
t
’
Œ
‹
]

‚
5
‡
t
# t5’ D = Fid ‹Œc‚‡t H Fid &
# Dt’ =suzt„Aξ ”3’kt’ Œ]s‹Œc‚3‡5t
# t’kƒa
H
Vis(zξ )
# t5’
# t5’kƒ^D‚ƒ −ξ = Fid− ‹c‚‡t H Fid
#
H
z
‹Œc‚‡t
# t5’ Dt’ =su(+,
ξ, I).(D1 , . . . , Dn )
# t’kƒa t„A”3’kt’ Œ]s‹Œc‚3‡5t H Vis(+,ξ,I)
# t5’
‹"‰‰9‹Œ‡u‹ „ †O‹ƒ t − ‹Œ]‚‡5t
# t5’kƒ^(−,
ξ, I)
Dξ
H
Cutξ,I &
‚
ƒ
#
H
Fid &
(  ) ’9]s † suttsu’.ƒ‘‰q‡’k'ƒ $’ ‰9‹ D † s R sut„ ƒaˆ5N‹„{’.¡>‹Œc‚‡t csk’]s“ ƒaˆ5N‹„c’k¡ ξ † s[‹T‹Œts
† s
sut"„A”’.t5’ Œcs ¤ ƒC‰a‡‚ $ ,"†sF‰i‹Œ‡ $‹ŒtAt"“(‡
•
›
D
D
5
t
’
Œ
‹
]

‚
5
‡
t
−
# t5’ D“(ƒa=s %‹ Fid
H
Fid− ›
3
$
{
„
]
’
‚
T
ƒ
a
ƒ
ˆ
N‹
O
{
„
’
3
”
s
‹"‰ ‰9‹‡„{’]suƒ^„ %
žq‹Œ]‚‡5t
# ¤ ƒT‡suC‹Œ‡¦“Nscs—ƒ^‚3ƒ 7Q†ˆ(−,
ξ, I)
D
H
H.Vis(−,ξ,I) ›
„Qs‡51’kƒ ’.t5 s† s $"s3„5„ s‡ ,V]s
›
# t5’kƒ^‚ƒ D = Skunk s„ H sut„A“(ƒ^s¡H‚3‡5 s—ƒa‚‡5C‹Œ]s ›
‚N“
0
› H = H0 .Cutπ,I
(−,ξ,I) ›
‚’H„ σ Ÿ ‹ $„{’]‚ƒ ƒaˆ5N‹H„{=’H”sH‹Œ.Vis
‚’H„ K Ÿ %‹ $3„{’c‚3ƒš‰¢‚t’ „c’ ”s™t"“(’ ”‹Œƒ^„ σ › ¤ ƒ
t5t‚ $’cˆs %
H›
‰q‡‚ $ ,"† sw‰9‹Œ‡ $‹ŒtAt“(’H”‹Œƒ^„
K›
‹( )
K = (+, ξ, I)
¤ ƒ™‰q‡‚ $ ,† s $"‚1Ts ‰¢‚N“Œ‡—]5
s $‹Œt ( ‹ )Ž•
1
5
t
’
u
s
"
t
A
„
”
.
’
5
t
’
Œ

c

s
‹
c

‚
‡
t
# t5’kƒ^
H
H.Vis(+,ξ,I)
# t’
‹"‰ ‰9‹Œ‡‹ „ †O‹Œƒ t − ‹Œc‚3‡5t
# t’kƒa(−,
ξ, I)
Dξ
H
H.Cutξ,I &
‚
ƒ
#
H
H.Fid &
(  ) K = (+, n : ξ, I)
# t5t5’’kƒ^asut"ž„At‚”’H’.„ t5’ ŒcsŸ %‹ ‹$3c„{‚’]‡‚t ƒH‰‚t’ „{H.Vis
¤ ƒ ‰q‡‚ $ ,† s ‰9‹Œ
’H”s† s (+,n:ξ,I)
‡ $‹Œtwt“(’H”‹ƒa„
# [‹ $"‚3ƒa„c’kƒ^“V‹σ„c’c‚ƒ † s
ASautH0 (n,ξ) ›
σ›
’ t5’
t“Œ’ „ ‹Œc‚‡t
›
(−, ξ, I)
σ
H
H.Cutn:ξ,I &
’k’ t5’kƒ^‚ƒ
H
H.Fid ›
›
$( *) K = z ›
¤ ƒ ‹
H
H.z ›
( † ) K = zξ ›
¤ ƒ™‰q‡‚ $ ,† s $"‚1Ts ‰¢‚N“Œ‡—]5
s $‹Œt
•
(+, ξ, I)
5
t
’
u
s
"
t
A
„
”
.
’
5
t
’
Œ

c

s
‹
c

‚
‡
t
# ξ
H
H.Vis(zξ ) &
# t5’kƒ^t‚’ ƒ
# t’kƒaD‚ƒ −ξ = Fid− ‹Œ]‚‡5t H H.Fid &
#
H
H.z
&
R32*_ ! ukN# O #
(s)
! "O3QŒ#V K = zn:ξ ›
¤ ƒC‰a‡‚ $ ,"†s $‚1TsŠ‰‚N“(‡ ]s5$‹Œt
•
(+, n : ξ, I)
# tt’’kƒaξ‚3ƒasut"ž „—t‚”3’ ’.„ t5’ csŸ ‹%‹Œ$3]„{‚’c‡5‚3t ƒ H‰‚t5’H„{H.Vis
¤ ƒ+‰a‡‚ $ ,"†s ‰9‹Œ‡$‹ŒtFt"“(’ ”‹Œƒ^„
’ ”3s†s (zn:ξ ) &
#  ‹ $"‚ƒ^„c’kƒ^“V‹σ„c’c‚ƒ
ASautH0 (n,ξ) ›
† s
D−
σ›
5
t
’
Œ
‹
c

3
‚
5
‡
# t5’.ƒaD‚ƒ − = Fid− t H H.Fid &
#
H
H.z
&
› H = H0 .Vis(+,π,I) ›
‚’H„ σ Ÿ ‹$„{’]‚ƒ ( ‰¢‚t’ „c’ ” s*)‹Œt5t‚$’cˆs % H ›
( ‹ ) ’ sut„‘t"“(’ ”3’]s † Ÿ ‹“  ‚’kƒ t (“ ƒas ‹%$3„{’]‚ƒ ƒaˆ5N‹„c’ ”s
σ
H.Vis(−,ξ.i,J) ›
( )
($)
¤ ˜
ƒ ‡suC‹Œ‡¦“Ns]s—ƒa‚ƒ37 † ˆ„ su‡51’kƒ’kt5 s†s $s„„Qs‡ ,cs
›
’ σ sut"„ t"“(’ ”3’]s† s Fid− ‹Œc‚‡t H H.Vis − ›
)
’ ƒ Ÿ st„t“Œ’ ”3’cs †XŸ ‹%“ $“(ƒas‹$„{’]‚ƒ‘ƒaˆ5N‹„c’ ”(Fid
sža‹Œ]‚‡5t
σ
H
(−, ξ.i, J)
ž ‹Œ]‚‡5t
H
st„ “(ƒ^s—¡H‚‡5 s ƒa‚3‡5C‹Œcs
›
_q`cb !"$#%&'( )+*,$*'-/.01*32!" -4$!" $'5'-!"1'-/67
2 8$!"*9:<; 1+2'=*!">&?'=*,@@A**=B0DC(EFGFH2*,IA*,J4 *,2*K4L!"I4 M12'N*!"
4$!A*,'=*,@ @A**=B0-OPRQSG72!"-4$!" EGT4 U*2V'=*!":/WX'(!":? G+)728*,ZY\[^]`_NabAc"dHefP
>V ! 5!3 3R 2'
RTSVUXW>Y[ZY]\^WL_a`cb d Cn ,%! h knqj ^h lgaph ?n m
rŸ x?~ 1'cy
†Ÿ “(ƒ & k’ t"„Q‚‡’c¦“Ns
st„‘“(ƒ^s™‰q‡ˆ
cH
H
’kƒa†£“ $„c’ ”sTsƒ^„
›
rX5
s $‹Œt† s
st„A„{‡‹Œ’ „ ˆ suƒ t"“(’ „ s
cH.Vis(+,n:ξ.i,J )
›
(‹
=
=
=
=
=
=
=
c
cH.z
cH.Fid
cH.Vis(η,ξ,I)
cH.Vis(zξ )
cH.Vis(Fid− )
cH.Cutπ,I
7$
&
‰q‡ˆ
( ˆ”3suƒ^„5“$suk]su
‡‚ƒ ’c¦“Ns
7$
&
cH .z
c0H
cH .(η, ξ, I)
cH .zξ
cH .Fid−
cH
)
‡‚ƒ ’c¦“Ns”3’c†s
st„Aˆ5N‹Œ %
‚$“
s„ [‹Ž‰a‡ˆ 7 $& ‡‚3ƒ ’c¦“Ns 0 sut"„—ˆN‹cs
† sηŸ ‹%$„c’c‚ƒCƒaˆN‹O+
„{’ ”3s ( ‰q−‡‚ ‰q‡s*)‘‹ O’.‘‹Œ]sˆ3”sƒac„H“Nsu.cs
›
ž9‚ƒ † ‚3’ „‡s† ˆ 4ƒ ’k‡Acs ‰¢‚’.ƒ^„Qs“Œ‡
œ ‚N“Œ‡—]5
s $‹Œt
‚N–
[‹ ‰q‡ˆ
%
7 $ ‡‚ƒ
&
’]¦“Ns
cH
‰q‡5’H”ˆ"s
•
n:ξ
cH.Vis(+,n:ξ.i,J )
suƒ^„ ”3’c† s*) † ˆ84 ƒ ’cs
cH.Vis(+,n:ξ.i,J ) = cH .(+, m : ξ.i, J)
‚N–
m
st„¢cs—ƒa‚3TŒ‡s †Ÿ ‹%$3„{’c‚3ƒ t† s[‹¡H‚‡Ts
sTsžN‰‚N“(‡ cs5$‹Œt
Vis(−,ξ,K)
†O‹Œƒ t
cH
‰q‡’ ”ˆ † s
ž9‚ƒ † ‚3’ „‡s† ˆ 4 ƒ ’k‡ ]sŠ‰¢‚’.ƒ^„Qs“Œ‡
cH.Vis(zn:ξ.i )
n:ξ
SautH (n, ξ.i) ›
•
cH.Vis(zn:ξ.i ) = cH .(zm:ξ.i )
‚N–
m
st„¢cs—ƒa‚3TŒ‡s †Ÿ ‹%$3„{’c‚3ƒ t† s[‹¡H‚‡Ts
\ ¢ \ Y]Z3Yc\^WL_a`cb d ‘n', ! h ŽmVl qpHg9jah n m%
Vis(−,ξ,K)
†O‹Œƒ t
cH
‰q‡’ ”ˆ † s
r Ÿ ‚N“ Œk’9†Ÿ “(ƒ & ’kt"„Q‚‡’c¦“Ns†Ÿ “Œƒ ‡ˆuts‹“ t"“(‡ “(ƒas ‹Œts
st„F“ŒƒasŠ‰q‡ˆ
β
Tsƒ^„ ”’]† s )t"“(‡
β›
7$
&
SautH (n, ξ.i) ›
‡‚ƒ ’]¦ “$s
( ˆ3”suƒ^„“Nsu.cs 7
RTSVUXW>Y[ZY]\^WL_a`cb d Agap ! h lh g9j
r ‹TzxO cy{€ a|y xz
†Ÿ “Œƒ ‡ˆuts‹“
† s‘‹ts
sut"„ cs † sutts’kƒ †s (‹Œts
¡H‚‡Tˆ+†s
β
Ÿ “(ƒ ’]‚ƒ † sut ‚N“ Œk’.t† suJRK
t & ’.t„Q‚3‡5’c¦“Nst t“(‡—R
cs ‡uˆ ts‹“ β
R›
RTSVUXW>Y[ZY]\^WL_a`& dm% m kh j gap{l g £g?jqn$
s“ † suttsu’.ƒ t
s„
† s ‹tsut†^“V‹csutAt‚ƒ^„FxON| x ?
> xz ? ~
D
¤ ƒ¥ƒ^‚N„Qs "$ s3„5„ st’ „“V‹„{’]‚ƒ
AKK9MK
N2 2 ! % ! > 2 32 2 !
DFEAE
'
(
E
D ⊥ E›
P
QM P M
t’
JD, EK = z ›
R32 ( ! ukN# O #
! "O3QŒ#V (+, 0 : ξ.0.1, {42})
(+, 1 : ξ.0.1, {42})
(−, ξ.0, {1; 2})
(−, ξ.0, {1; 2})
(+, ξ, {0; 2})
(+, ξ, {0; 2})
(−, ξ.0, {1})
(−, ξ.0, {1})
D1 =
D2 =
(+, ξ, {0; 1})
(+, ξ, {0; 1})
(+, ξ2 , ∅)
(+, ξ1 , ∅)
(−, ξ.0.1, {42})
(−, ξ.0.1, {42})
(+, 0 : ξ.0, {1}) (+, 0 : ξ.0, {1; 2})
P2 8
%
E=
C 2 32 > "$3 5 % !
(−, ξ, {0; 1})
T
(−, ξ, {0; 2})
"
3 " 2 D1 , E
%!
Cutξ,{0;1}
Cutξ,{0;1} .Cut0:ξ.0,{1}
Cutξ,{0;1} .Cut0:ξ.0,{1} .Cutξ,{0;2}
Cutξ,{0;1} .Cut0:ξ.0,{1} .Cutξ,{0;2} .Cut0:ξ.0,{1;2}
Cutξ,{0;1} .Cut0:ξ.0,{1} .Cutξ,{0;2} .Cut0:ξ.0,{1;2} .Cut0:ξ.0.1,{42}
Cutξ,{0;1} .Cut0:ξ.0,{1} .Cutξ,{0;2} .Cut0:ξ.0,{1;2} .Cut0:ξ.0.1,{42} .Vis(+,ξ1 ,∅)
5$2 C 3% !! 2
P2 8
%
C 2 32 > "$3 5 % !
JD1 , EK = (+, ξ1 , ∅)
T
"
3 " 2 D2 , E
%!
Cutξ,{0;1}
Cutξ,{0;1} .Cut0:ξ.0,{1}
Cutξ,{0;1} .Cut0:ξ.0,{1} .Cutξ,{0;2}
Cutξ,{0;1} .Cut0:ξ.0,{1} .Cutξ,{0;2} .Cut0:ξ.0,{1;2}
Cutξ,{0;1} .Cut0:ξ.0,{1} .Cutξ,{0;2} .Cut0:ξ.0,{1;2} .Cut1:ξ.0.1,{42}
Cutξ,{0;1} .Cut0:ξ.0,{1} .Cutξ,{0;2} .Cut0:ξ.0,{1;2} .Cut1:ξ.0.1,{42} .Vis(+,ξ2 ,∅)
5$2 C 3% !! 2
! 52 32 I
3d- 5 % ! =! " L *2 3 ! > 2\5 > 5 > 32 8
JD2 , EK = (+, ξ2 , ∅)
2 I 2 A > 2 > 2r5 > 5 > 32 8 % C 2 2 3" 2 A ! 2 5 &> ! - % 2 ) > 2 +PZ % ! 2 I 2$A > 273d- B> % ! 32 ! % ! 3" 2 A ! A62
% C 2 z 2 H! % ! 3 ! > ! % A > % ! | % ! #2 5% ! 3" 2 > 2\32 2 !
"
%
$# O # ! "O3QŒ#Vk— !ŒŒk"
!Qu[! Q 3R 2a`
(+, 0 : ξ.0.1, {42})
(−, ξ.0, {1; 2})
(+, ξ, {0; 2})
z
(−, ξ 0 .0, {1})
(−, ξ 0 .0, {2})
(+, ξ 0 , {0})
(−, ξ.0, {1})
P2 2
D01 =
A I A > 2 3d-/8
C 2 3 " 2 % !
(+, ξ, {0; 1})
> %
D01 , E
Cutξ,{0;1}
Cutξ,{0;1} .Cut0:ξ.0,{1}
Cutξ,{0;1} .Cut0:ξ.0,{1} .Vis(+,ξ 0 ,{0})
Cutξ,{0;1} .Cut0:ξ.0,{1} .Vis(+,ξ 0 ,{0}) .Vis(−,ξ 0 .0,{1})
Cutξ,{0;1} .Cut0:ξ.0,{1} .Vis(+,ξ 0 ,{0}) .Vis(−,ξ 0 .0,{1}) .Cutξ,{0;2}
Cutξ,{0;1} .Cut0:ξ.0,{1} .Vis(+,ξ 0 ,{0}) .Vis(−,ξ 0 .0,{1}) .Cutξ,{0;2} .Cut0:ξ.0,{1;2}
Cutξ,{0;1} .Cut0:ξ.0,{1} .Vis(+,ξ 0 ,{0}) .Vis(−,ξ 0 .0,{1}) .Cutξ,{0;2} .Cut0:ξ.0,{1;2} .Cut0:ξ.0.1,{42}
Cutξ,{0;1} .Cut0:ξ.0,{1} .Vis(+,ξ 0 ,{0}) .Vis(−,ξ 0 .0,{1}) .Cutξ,{0;2} .Cut0:ξ.0,{1;2} .Cut0:ξ.0.1,{42} .
Vis(+,ξ1 ,∅)
2
52 C
Cutξ,{0;1}
Cutξ,{0;1} .Cut0:ξ.0,{1}
Cutξ,{0;1} .Cut0:ξ.0,{1} .Vis(+,ξ 0 ,{0})
Cutξ,{0;1} .Cut0:ξ.0,{1} .Vis(+,ξ 0 ,{0}) .Vis(−,ξ 0 .0,{2})
Cutξ,{0;1} .Cut0:ξ.0,{1} .Vis(+,ξ 0 ,{0}) .Vis(−,ξ 0 .0,{2}) .z
3% !! 2
e6f
k ml ck
(&
&
*
!#*
(&
,
0&
&
)&
&
&
, (&)*
&
!#*&
&
(−, ξ 0 .0, {1})
(−, ξ 0 .0, {2})
(+, ξ 0 , {0})
3
(&
$
V +&
j
7
@;T;>7Nq 7
> > 3
3
-,
(&
3 V
, (*
!#*
@rqdj9;@<lnjEq
3 >
&
.&
/!
0!
!#*
&21
, +*
&
z
p4li^j?7Nk‘; q
@ 7
5 ]{3 O5 5
3 > $5
5 3
>&> > 3 V
Z 6] 5 &> > > 5 !#"%$'&
(+, ξ1 , ∅)
JD1 , EK =
7
!
%
'&8&
!#*
> > 5 *A 8 > &
)!
"
/!
3!
, (&)*
!#*
!9*:
&
&
*(!
/"
'&
!#*
3
*8A
(&465
B>
/, &
3!
2 > 3
"
&
'"
/&
-7
1
(&
R32 ! ukN# O #
8A 3 > 5
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