HAL Id: tel-00579490
https://tel.archives-ouvertes.fr/tel-00579490v2
Submitted on 12 Apr 2011
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Vérification symbolique de modèles à l’aide de systèmes
de ré-écriture dédiés
Duy-Tùng Nguyên
To cite this version:
Duy-Tùng Nguyên. Vérification symbolique de modèles à l’aide de systèmes de ré-écriture dédiés. Ordinateur et société [cs.CY]. Université d’Orléans, 2010. Français. �NNT : 2010ORLE2030�. �tel-00579490v2�
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5'-$)06"'-$ ,-! $,7$ ) , )'- σ = {x ← i(⊥, ⊥), y ← ⊥} 89')" &1 :;,"! <*=>. 1&'"$ ? t2σ = t2{x ← i(⊥, ⊥), y ← ⊥} = t1 i i t i y t =2 t =1 = { x , y } i t x !"# O8P Q /+).+ +( #"4#('("('!&
! "#$%&'() *+ ,(&-.&%)/ 0) 1$ 23(&4.$'&5- 6 7$/) 0) 89:/
!" #$%&'&$! p #$() (! '")*" t "%' (! *$' %()N+ ,- %./("!0" 1&2" ε 2.!$'"
3-#3(% 4-('" #$%&'&$!+ ,5"!%"*63" Pos(t) 2" #$%&'&$!% 25(! '")*" t "%' &!2(0'&1"*"!' 2.7!& #-) Pos(f(t1, t2)) = {ε} ∪ {0.p | "' p ∈ Pos(t1)} ∪ {1.p | "' p ∈ Pos(t2)}8
"' Pos(t) = {ε} /(-!2 t ∈ X $( t =⊥+ 9$&' (! $)2)" #-)'&"3 # %() 2"% #$%&'&$!%+ 9$&' p1 "' p2 2"(: *$'% 2" N⋆8 !$(% 2&%$!% /(" p1 # p2 %5&3 ":&%'" ω ∈ N⋆ '"3 /("
p1= p2.ω+
9& p ∈ Pos(t)8 -3$)% t|p %&;!&7" 3"% %$(%<'")*"% 2" t = 3- #$%&'&$! p "' t[s]p 2.!$'"
3" '")*" $6'"!( #-) 3" )"*#3-0"*"!' 2( %$(%<'")*" t|p = 3- #$%&'&$! p #-) 3" '")*" s+
>$(% 2.!$'$!% .;-3"*"!' #-) t(p) 3" %?*6$3" -##-)-&%%-!' 2-!% t = 3- #$%&'&$! p+ @'-!' 2$!!. (! '")*" t ∈ T (Fbin, X )"' A (! "!%"*63" 2" %?*6$3"%8 A-&' PosA(t) =
{p ∈ Pos(t) | t(p) ∈ A}+ B-) 0$!%./("!'8 PosF(t)"%' 35"!%"*63" 2" #$%&'&$!% 2" t8 =
04-0(!" 2$!' (! %?*6$3" A$!0'&$!!"3 -##-)-C'+ ,5"!%"*63" 2" #$%&'&$!% !"#$%&! 25(! '")*" t8 2.!$'. #-) FPos(t)8 "%' 2.7!& 0&<2"%%$(% D FPos(t) = {p ∈ Pos(t) | p.0 /∈ Pos(t)}. ,5"!%"*63" 2" 1-)&-63"% -##-)-&%%-!' (! '")*" t ∈ T (Fbin, X ) "%' 2.!$'.
#-) Var(t)+ B3(% A$)*"33"*"!'8 Var(t) = {t|p | p ∈ PosX(t)}+
;<)=%1) > '$(#$ )"##*+ Fbin = {t, i, ⊥} &$ X = {x, y} ,"--& )(#+ ./&0&-1.& 23
4"#+%)*!"#+ .& $&!-& t2 = t(x, i(⊥, y)) (5&, .( !(,%#& )& t2 &+$ t3 6."!+7 ./&#+&-8.&
)& )& 1"+%$%"#+ !"#$%&! FPos(t2) &+$ {10} &$ 19%+:9& Var(t2) = {x, y}7 ./&#+&-8.&
)& 5(!%(8.&+ PosX(t2) &+$ {0, 11} 3
! EF9 R "%' (! "!%"*63" 2" !;<.&+ )& !*=*,!%$9!& (l, r) ∈ T (Fbin, X ) ×
T (Fbin, X )8 .;-3"*"!' 2.!$'. #-) l → r $G Var(r) ⊆ Var(l) "' l '∈ X + ," EF9
R &!2(&' (!" )"3-'&$! 2" ).<.0)&'()" →R %() 2"% '")*"% 2$!' 3- 03H'()" ).I":&1" "'
')-!%&'&1" "%' .0)&'" →⋆
R+ B3(% #).0&%.*"!'8 !$(% 2&%$!% /(" t →R t′ %5&3 ":&%'" (!"
#$%&'&$! p 2" Pos(t)8 (!" )J;3" l → r ∈ R "' (!" %(6%'&'('&$! σ : X (→ T (Fbin)'"33"
/(" t|p = lσ "' t′= t[rσ]p+
;<)=%1) ? >"%$ Fbin = {t, i, r, ⊥}7 X = {x, y} &$ 9# $&!-& ,."+ t1 =
t(i(⊥, ⊥), i(⊥, ⊥))3 4"#+%)*!"#+ 9#& !;<.& r1 = i(⊥, ⊥) → r(⊥, ⊥)7 (."!+ #"9+ (5"#+
? 1"++%8%.%$*+ )& !*)9,$%"# ,"--& .&+ +9%5(#$&+ @ t1→r1 t(r(⊥, ⊥), i(⊥, ⊥))
t1→r1 t(i(⊥, ⊥), r(⊥, ⊥))
4"#+%)*!"#+ 9#& !;<.& r2 = t(x, i(⊥, y) → t(y, r(⊥, x)7 (."!+ %. &0%+$& 9#& +&9.&
!"! #$%&'%()* +" !"!"!+ ,-$%.'/0%1& (/$ 23/&/24*)
! "#$%&'()# *# +! ,-.'/$!"'0& 1!. +2!&!+34# *2!$$#44'5'+'"- #4" 5!4-# 4). +2#&4#67 5+# *# R7*#4$#&*!&"4 *2)& #&4#65+# *# "#.6#48
2#&4#65+# *# R7*#4$#&*!&"4 9!11#+- -:!+#6#&" +! $+;").# .-<#=',# #" ".!&7 4'"',# 0) 1+)4 4'61+# +2#&4#65+# *#4 -"!"4 !$$#44'5+#4> *2)& #&4#65+# *# "#.6#4 E ⊆ T (Fbin) #4" R⋆(E) = {s ∈ T (Fbin) | t →⋆Rs ∧ t ∈ E}8
?). )& #&4#65+# *# "#.6#4 *# Fbin@ &0)4 !,0&4 +#4 1.01.'-"-4 +'&-!'.#4 4)',!&"#4 A
B a(s, s1) + a(s, s2) = a(s, s1+ s2)
B a(s1, s) + a(s2, s) = a(s1+ s2, s)
!,#$ a )& 43650+# *# Fbin8
?0'#&" t )& "#.6# *!&4 T (Fbin) #" s )& #&4#65+# *# "#.6#4 $+048 C0)4 *'.0&4
()# t ∈ s 44'@ #& $0&4'*-.!&" +#4 1.01.'-"-4 +'&-!'.#4 1.-$-*#&"#4@ s 1#)" D".# -$.'" 1!. $#""# 6!&'E.# A s = t1+ . . . + tn !,#$ ti∈ T (F ) #" '+ #='4"# j "#+ ()# tj = t8 F&
1.!"'()#@ )&# "#++# ".!&4G0.6!"'0& 4'61+# #4" !11#+-# +! $!&0&'4!"'0&8 C0)4 *-&0"0&4 ∅ )& #&4#65+# ,'*#8
C0)4 10),0&4 05"#&'. +2#&4#65+# *# "#.6#4 !$$#44'5+#4 H 1!."'. *2)& "#.6# '&'"'!+ init #& $!+$)+!&" R⋆({init})8 I# 1+)4@ &0)4 10),0&4 -:!+#6#&" ,-.'/#. 4' )& "#.6#
att #4" !""#':&!5+# H 1!."'. *) "#.6# init 1)'4 ()# init →⋆ Ratt8 78)9(2) : !"!#$#% & '(!)!*+'!,- #$.% $/0!#$#% '(!#%!*/'! 1! 0!2*!% 344!%%5/'!% & +32052 1. 0!2*! t1 6 R⋆({t 1}) = t(i(⊥, ⊥), i(⊥, ⊥)), t(r(⊥, ⊥), i(⊥, ⊥)), t(i(⊥, ⊥), r(⊥, ⊥)), t(r(⊥, ⊥), r(⊥, ⊥)) . 7$.% +$."$#% 84252! 893'!*!#0 6 R⋆({t1}) =
t(i(⊥, ⊥), i(⊥, ⊥)) + t(r(⊥, ⊥), i(⊥, ⊥)) + t(i(⊥, ⊥), r(⊥, ⊥)) + t(r(⊥, ⊥), r(⊥, ⊥)) = t(r(⊥, ⊥), i(⊥, ⊥) + r(⊥, ⊥)) + t(i(⊥, ⊥), i(⊥, ⊥) + r(⊥, ⊥))
= t(t(i(⊥, ⊥) + r(⊥, ⊥), i(⊥, ⊥) + r(⊥, ⊥))
:# ;#- #$.% 3"$#% +2$."8 <.! '!% 0!2*!% t(r(⊥, ⊥), i(⊥, ⊥)), t(i(⊥, ⊥), r(⊥, ⊥)) !0 t(r(⊥, ⊥), r(⊥, ⊥)) %$#0 300!59#3/'!% +32 28=84250.2! & +32052 1! t1>
J0). .-4)6#.@ &0)4 .#".0),0&4 #& .-7-$.'").# +2-()',!+#&$# *) *$1!'=4?!4@5#9 10). +#4 434"E6#4 /&'48 I!&4 +! 4#$"'0&K8L8L@ &0)4 *0&&0&4 )&# '++)4".!"'0& *# $#""# "#$%&'()# 10). +! ,-.'/$!"'0& *) J.0"0$0+# *2M.5'".!:# M.50.#4$#&"8
!"#$%&'( )* +'%,-%$(. /( 0# 12'%3-#&%4, 5 6#.( /( 789.
!"!# $%&'()*+',- . /0*'12 12 *3+,4*+25 10*&6&25
!"# $%&&% #%$&'(") "(*# '"&+(,*'#("# -%# .+/-'0'"!'+%# ,%# !*&(0!&%# ,1!+2+% %& ,%# &+!"#,*$&%*+# ,1!+2+%# .(*+ ,%# &%+0%# 3"'# $(0.(#/# .!+ ,%# #402(-%# 5("$6 &'(""%-# 2'"!'+%# %& -! $("#&!"&% ⊥7 8"#*'&%) "(*# 0("&+("# $(00%"& "(*# .(*9("# $!-$*-%+ -! $-:&*+% +/;%<'9% &+!"#'&'9% %" *&'-'#!"& ,%# &+!"#,*$&%*+# ,1!+2+%#7 )*:* *: ;<&4=#&( />#'6'(. (& 7'#,./<-&(<' />#'6'(.
• ;<&4=#&( />#'6'(.
=" !*&(0!&% ,1!+2+%# 3"' #*+ Fbin %#& *" >*!,+*.-% A = (Q, Fbin, Qf, △) (?
Q %#& *" %"#%02-% ,1/&!&# @*"!'+%#A) Qf ⊆ Q %#& *" %"#%02-% ,1/&!&# 3"!*<)
%& △ %#& *" %"#%02-% ,% +BC-%# ,% &+!"#'&'(" ,% &4.% #*'9!"&% D f (q1(x1), q2(x2)) → q(f (x1, x2)),
(? f ∈ Fbin, q, q1, q2 ∈ Q, x1, x2 ∈ X7
E1!*&(0!&% ,1!+2+%# #*+ Fbin 5("$&'(""% #*+ ,%# &%+0%# $-(# ,% Fbin7 =" !*6
&(0!&% ,1!+2+%# $(00%"$% !*< 5%*'--%# %& 0("&% .!# F .!# ,% 2!# %" G!*&) !#6 #($'!"& ,*+!"& -% &+!H%& '",*$&'9%0%"& *" /&!& !9%$ $G!>*% #(*#6&%+0%7 I- 5!*& "(&%+ >*1'- "14 ! .!# ,1/&!& '"'&'!- ,!"# *" !*&(0!&% ,1!+2+%# 3"') 0!'#) .(*+ *" $("#&!"& #402(-% ⊥) *"% +BC-% ,% &+!"#'&'(" %#& ,% -! 5(+0% ⊥ → q(⊥)7 J!+ $("#/>*%"&) -%# +BC-%# ,% &+!"#'&'(" .(*+ $% #402(-% $("#&!"& .%*9%"& K&+% $("#',/+/%# $(00% -%# !"#$% &'&(&)#$%7 L' -%# #(*#6&%+0%# ,'+%$&# u1, u2
,% t = f(u1, u2) #("& /&'>*%&/# !9%$ -%# /&!&# q1, q2) !-(+# -% &%+0% t #%+! /&'6
>*%&/ .!+ >*%->*%# /&!&# q !9%$ f(q1(x1), q2(x2)) → q(f (x1, x2)) ∈ △7 M(*#
,/3"'##("# 5(+0%--%0%"& -! +%-!&'(" ,% ,/.-!$%0%"& .!+ *" !*&(0!&% ,1!+2+%# 3"'7
L('& A = (Q, Fbin, Qf, △) *" !*&(0!&% ,1!+2+%# 3"' #*+ Fbin7 E! +%-!&'(" ,%
,/.-!$%0%"& →A %#& ,/3"'% .!+ D L('& t, t′ ∈ T (Fbin∪ Q),
t →A t′⇐⇒
∃C ∈ C(Fbin∪ Q), ∃u1, u2 ∈ T (Fbin)),
∃f (q1(x1), q2(x2)) → q(f (x1, x2)) ∈ △,
t = C[f (q1(u1), q2(u2))],
t′ = C[q(f (u1, u2))].
.
→⋆
A %#& -! $-:&*+% &+!"#'&'9% %& +/;%<'9% ,% →A7
=" &%+0% $-(# t ,!"# T (Fbin) %#& !$$%.&/ .!+ *" !*&(0!&% ,1!+2+%# 3"' A =
(Q, Fbin, Qf, △) #'
t →⋆A q(t)
.(*+ >*%->*%# /&!&# q ,!"# Qf 7 M(&%N >*% $%&&% ,/3"'&'(" $(++%#.(", F -!
"(&'(" ,1!*&(0!&% ,1!+2+%# 3"' "("6,/&%+0'"'#&% $!+ $% 0(,B-% ,1!*&(0!&% ,1!+2+%# 3"' .%+0%& N/+() *"% (* .-*# ,% +BC-%# ,% &+!"#'&'(" !9%$ -% 0K0% $:&/ C!*$G%7 8" $("#/>*%"$%) $1%#& .(##'2-% ,1!9('+ .-*# ,1*"% +/,*$&'(" F .!+&'+ ,* 0K0% &%+0% $-(#7 8&) *" &%+0% $-(# t %#& !$$%.&/ #1'- 4 ! *"% +/,*$&'("
