HAL Id: tel-00579490https://tel.archives-ouvertes.fr/tel-00579490v2
Submitted on 12 Apr 2011
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Vérification symbolique de modèles à l’aide de systèmesde 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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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%+$& 9#& +&9.&
1"++%8%.%$* )& !*)9,$%"# ,"--& .( +9%5(#$& @ t1 →r2 t(⊥, r(⊥, i(⊥,⊥)))3
!"! #$%&'%()* +"
!"!"!+ ,-$%.'/0%1& (/$ 23/&/24*) 53/'')**%6%2%0-
! "#$%&'()# *# +! ,-.'/$!"'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→
⋆R s ∧ 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→⋆R att8
78)9(2) : !"!#$#% & '(!)!*+'! ,- #$.% $/0!#$#% '(!#%!*/'! 1! 0!2*!% 344!%%5/'!%
& +32052 1. 0!2*! t1 6
R⋆({t1}) =
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 ∈ X 7
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 %#& !$$%.&/ #1'- 4 ! *"% +/,*$&'("
!"! #$%&'%()* +
!"#$% &'(&)* +)* #,-(.&%'/* !'**%0+)*1 2 !"#&%# -) .) &)#$) .+'* )& $)/"/& 2
(/) .'/34(#"&%'/ -) +" 5'#$) q(t) '6 q )*& (/ ,&"& 3/"+7
8) +"/4"4) -9"#0#) L(A) #).'//( !"# A )*& +9)/*)$0+) -) &'(* +)* &)#$)* .+'*
"..)!&,* !"# A7 :/ )/*)$0+) L -) &)#$)* .+'* )*& #).'//"%**"0+) *% L = L(A)
!'(# ;()+;()* "(&'$"&)* -9"#0#)* 3/%* A7 <'(* !'(='/* -%#) ;() .)& )/*)$0+)
-) &)#$)* )*& ,="+(, !"# +9"(&'$"&) -9"#0#)* A7
,-).(/) "0 !"# Fbin = {t, i,⊥}$ %!&'"()*!&' +,-.#!/-#0 (,-*1*0' A =
(Q,Fbin, Qf ,△) ()2&" 3-* 4 Q = {qt, qi, q⊥}5 Qf = {qt}5 0# △ +,0&'0/1+0 (0 *67
8+0' (0 #*-&'"#"!& 0'# (!&&) 9"7(0''!.' 4
{⊥ → q⊥(⊥), i(q⊥(x), q⊥(y))→ qi(i(x, y)), t(qi(x), qi(y))→ qt(t(x, y))}
%0# 0&'0/1+0 30.# :#*0 *03*)'0&#) 3+.' '"/3+0 9!//0 +0 '.";-� 4
{⊥−→ q⊥, (q⊥, q⊥)
i−→ qi, (qi, qi)
t−→ qt}
<!.' (!&&!&' +,0=0/3+0 (0 *)(.9#"!& -;09 +- *0+-#"!& (0 ()3+-90/0&# →A1 4
t1 = t(i(⊥,⊥), i(⊥,⊥)) →A1 t(i(q⊥(⊥), q⊥(⊥)), i(q⊥(⊥), q⊥(⊥))) →A1
t(qi(i(⊥,⊥)), qi(i(⊥,⊥))) →A1 qt(t(i(⊥,⊥), i(⊥,⊥)))$ >&2&5 +0 #0*/0 t1 0'#
-9903#) 3-* A15 c.?.d.t1 ∈ L(A1)$ @."'A.0 t1 0'# +,.&"A.0 #0*/0 -9903#) 3-* A15
-"&'" L(A1) = {t1}
• 1$2&*34'5)4$ 362$7$)*
:/ &#"/*-(.&)(# -9"#0#)* "*.)/-"/& )*& (/ ;(%/&(!+) T =
(Q,Fbin,F′bin, Qf , δ) '6 > )*& (/ )/*)$0+) 3/% -)* ,&"&*? Fbin )& F ′
bin
+)* )/*)$0+)* 3/%* -)* +)&&#)* )/&#,)* )& -)* +)&&#)* *'#&%)*1 *'/& -)* "+!@"A
0)&*? Qf ⊆ Q )*& -)* )/*)$0+)* 3/"(B? )& δ )*& (/ )/*)$0+) -) #C4+)* -)
&#"-(.&%'/ -) &D!) *(%="/&) E
f(q1(x1), q2(x2))→δ q(u), u ∈ TF ′bin[{x1, x2}] F7G1
q(x)→δ q′(u), u ∈ TF ′bin[x] F7H1
⊥ →δ q(u), u ∈ TF ′bin
F7F1
'6 ⊥ ∈ Fbin, f ∈ Fbin, x, x1, x2 ∈ X ? )& q1, q2, q, q′ ∈ Q7
I&"/& -'//, t (/ &)#$) )/&#,? T !'(#*(%& +) !#,.,-)/& E %+ .'$$)/.) )/
#)$!+"J"/& ;()+;()* 5)(%++)* *)+'/ +)* #C4+)* F7F7 K'(# +9%/*&"/.)? *% (/) 5)(%++)
)*& ,&%;()&,) ⊥ )& +" #C4+) ⊥ →δ q(u) )*& -"/* δ? !(%* ⊥ )*& #)$!+"., !"# q(u)7
8" *(0*&%&(&%'/ !'(#*(%& L(*;(92 +" #".%/)7 M% +" #C4+) f(q1(x1), q2(x2))→δ q(u)
)*& -"/* δ? "+'#* T #)$!+".) (/) '..(##)/.) -9(/ *'(*A"#0#) f(q1(t1), q2(t2))
! "#$%&'() *+ ,(&-.&%)/ 0) 1$ 23(&4.$'&5- 6 7$/) 0) 89:/
!" #$ %$"&$ q(u[x1 ← t1, x2 ← t2])' () *+!,-$ (**-""$.*$ /$ #! 0!"1!2#$ xi
/!.3 t $3% "$& #!*4$ !" ti5 6$ *!#*-# *(.%1.-$ 7-3,-89 *$ ,-$ #! "!*1.$ /$ t
3(1% !%%$1.%$5
6$ %"!.3/-*%$-" T /4:.1 #! "$#!%1(. "4;-#1<"$ 3-10!.%$ $.%"$ /$3 !"2"$3 RT =
{(t, t′) ∈ TFbin × TF ′bin|t→⋆
δ q(t′)' (-" ,-$#,-$3 q ∈ Qf}5
=(1% L ⊆ TFbin -. #!.;!;$ /8!"2"$' .(-3 /4:.133(.3 /(.* #8$.3$&2#$ RT (L) =
{t′ ∈ F ′bin|∃t ∈ L, (t, t′) ∈ RT }5
*+;+ + !"#$%& '&!! ()*!$ +,!-./0"
>(-3 (-0(.3 *!#*-#$" #! *#?%-"$ "4@$A10$ %"!.31%10$ $. -%1#13!.% /$3 %"!.3/-*B
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!#$.*$ /- !"#$%&'#&()*+ $. "4B4*"1%-"$ (-" #$3 3C3%<&$3 :.135
DE(-!77!.1 FGGFH /4.(%$ !" RnT #! *(& (31%1(. /$ RT ' . I(135 J1.31' R⋆
T =⋃
n≥0RnT /4.(%$ #! *#?%-"$ %"!.31%10$ $% "4@$A10$ /$RT 5 DJ2/-##! FGGFH .(-3 "43$.B
%(.3 4;!#$&$.% -. &(C$. /$ *!#*-#$" #! *#?%-"$ %"!.31%10$ /$3 "$#!%1(.3 "4;-#1<"$3
/8!"2"$3 $. -%1#13!.% -. %"!.3/-*%$-" /8!"2"$3 /8+13%(1"$5 K#-3 "4*134&$.%' 4%!.%
/(..4 -. %"!.3/-*%$-" /8!"2"$3 T ' .(-3 ;4.4"(.3 -. .(-0$# %"!.3/-*%$-" /8!"2"$3
1.:.1 H' ! $#4 #$ %"!.3/-*%$-" /8!"2"$3 /8+13%(1"$ %$# ,-$ RH = R⋆T 5
<=)>%1) ;; ,!)- A1 $./0-! /-# "./121# "# $.#3# 4$# 567 ,!)- 0* -1/*8"0&-#01
T = (Q,Fbin,F′bin, Qf , δ) "9:*) 4/1 ; Q = {qt, qi, qr, q⊥}< Fbin = {t, i,⊥}< F ′
bin =
{t, i, r,⊥}< Qf = {qt} #- δ $.#*8# 2$# "# -1/*8)-)!*8 &! # $#8 80)=/*-#8 ;
(⊥,⊥)−−−→ q⊥ (q⊥, q⊥)
(i,i)−−→ qi (q⊥, q⊥)
(i,r)−−→ qr
(qi, qi)(t,t)−−→ qt (qi, qr)
(t,t)−−→ qt (qr, qi)
(t,t)−−→ qt
(qr, qr)(t,t)−−→ qt
>!08 /=!*8 "!*& R⋆T (L(A1)) = R⋆
T ({t1}) = R⋆T ({t(i(⊥,⊥), i(⊥,⊥))}) =
{t(i(⊥,⊥), i(⊥,⊥))< t(r(⊥,⊥), i(⊥,⊥))< t(i(⊥,⊥), r(⊥,⊥))< t(r(⊥,⊥), r(⊥,⊥))}7
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/-*%$-"3 /$ &(%V!"2"$ :.1 "4;-#1$" c.9.d.' 3(1% -.$ "$#!%1(. "4;-#1<"$ R $% -. #!.;!;$
!"! #$%&%'%() *+,$-.&$/0) ,$-%$)1')2& 34,#5 "6
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u = t(i(i(⊥,⊥), i(⊥,⊥)), i(i(⊥,⊥), i(⊥,⊥))) 5@ )5(6 1/6 4&5'/66/(&6 652) !"# /) 1, '/11(1/
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i(x, r(y, z)) → r(x, r(y, z)) !"#$
i(r(x, y), z) → r(r(x, y), z) !"%$
i(⊥,⊥) → r(⊥,⊥) !"&$
!"! #$%&%'%() *+,$-.&$/0) ,$-%$)1')2& 34,#5 "6
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<=< 1>4=
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@( #(+)( ! "# $%&'%( )*+ v (&# '##($-!'>%( 8 2'+#$+ ." #(+)( ! "# ),#% u 2"$& /"(
u→⋆Rreq v= A:"& %$&#:!& *$B.(&&"& %' +3."*#$:! .3#'$%%3( C
u = t(i(i(⊥,⊥), i(⊥,⊥)), i(i(⊥,⊥), i(⊥,⊥)))→r3.6
t(i(r(⊥,⊥), i(⊥,⊥)), i(i(⊥,⊥), i(⊥,⊥)))→r3.6
t(i(r(⊥,⊥), r(⊥,⊥)), i(i(⊥,⊥), i(⊥,⊥)))→r3.6
t(i(r(⊥,⊥), r(⊥,⊥)), i(r(⊥,⊥), i(⊥,⊥)))→r3.6
t(i(r(⊥,⊥), r(⊥,⊥)), i(r(⊥,⊥), r(⊥,⊥)))→r3.5
t(r(r(⊥,⊥), r(⊥,⊥)), i(r(⊥,⊥), r(⊥,⊥)))→r3.4
t(r(r(⊥,⊥), r(⊥,⊥)), r(r(⊥,⊥), r(⊥,⊥))) = v
A:"& +()'+/":!& /"( *( *D()$! !E(&# 2'& "!$/"(5 c.8.d. $% (F$&#( '" ):$!& "!
'"#+( *D()$! .( u 8 v=
A:"& *:!#$!":!& 8 2+3&(!#(+ %(& '"#+(& +,-%(& .( GHI= @' *(%%"%( .( +'*$!( '**:+.(
%' +(&&:"+*( ' '" 2%"& "! (!J'!#5 (# %( 2+$7$%,-( &( 2+:2'-( .( D'"# (! >'& K"&/"E8 "!
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J'$# .:!* "! *D:$F !:!B.3#(+)$!$&#( C
t(x, r(y, z)) → b(x, t(y, z)) !"#$
t(r(x, y), z) → b(t(x, y), z) !"%$
L'+ *:!&3/"(!#5 .( *(##( J'M:!5 &("% "! 2+:*(&&("+ '6'!# "!( .()'!.( .( %'
+(&&:"+*( %E:>#$(!.+'= N"'!. %( 2+:*(&&("+ ' ;!$ '7(* %' +(&&:"+*(5 $% (!7:$( "!(
+(/"0#( .( %$>3+'#$:! /"$ (&# 2+:2'-3( .( >'& (! D'"#=
b(x, t(⊥,⊥)) → t(x, i(⊥,⊥)) !"&$
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! "#$%&'() *+ ,(&-.&%)/ 0) 1$ 23(&4.$'&5- 6 7$/) 0) 89:/
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0!"# ",' .'(("(' 2
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b(x, t(i(y, z), i(u, v))) → t(x, i(i(y, z), i(u, v))) !"##$
b(t(i(x, y), i(u, v)), z) → t(i(i(x, y), i(u, v)), z) !"#%$
3 4% ", )'$ ',56,+$ )7",' +'((' .'(("(' )'/6,)' (6 #'$$!"#.'- 6(!#$ 2
3 $!%+ (78+6+ $"%&6,+ )' .'++' .'(("(' $'#6 !,$!-")*. '+ (' *'+!, &6 9 (6 .'(("('
&!%$%,' 0("$ :6"+' $"%&6,+' ', "+%(%$6,+ )'$ #=;('$ $"%&6,+'$
b(x, t(r(y, z), i(u, v))) → t(x, r(r(y, z), i(u, v))) !"#!$
b(x, t(i(y, z), r(u, v))) → t(x, r(i(y, z), r(u, v))) !"#&$
b(t(r(x, y), i(u, v)), z) → t(r(r(x, y), i(u, v)), z) !"#'$
b(t(i(x, y), r(u, v)), z) → t(r(i(x, y), r(u, v)), z) !"#($
3 $!%+ (78+6+ $"%&6,+ )' .'++' .'(("(' $'#6 "#/!* 0! .6# (6 .'(("(' 6 .:!%$% )'
)!,,'# (' *'+!, 9 ", )'$ ',56,+$ !,$!-")*. ', "+%(%$6,+ )'$ #=;('$ >?<@A !"
>?<BA<
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!1+',%# (7',$'/1(' )' +'#/'$ 6..'$$%1('$ ', .6(."(6,+ (RTAP )⋆({u})< G' 0("$- ,!"$
0!"&!,$ 8;6('/',+ &8#%F'# H"' (' +'#/' "#"'& !,$!-")*. v '$+ 6++'%;,61(' 9 06#+%#
)" +'#/' "#"'& )+&! u 0"%$ H"' u→⋆RTAP v<
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C!"$ "+%(%$!,$ (' /I/' &!.61"(6%#' Fbin 6&'. (6 $'.+%!, 0#8.8)',+' c.9.d ('$
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C!"$ 6&!,$ )!,. ", 6"+!/6+' )76#1#' F,% U 6..'0+6,+ u .!//' $"%+ 2
⊥−→ q⊥
(q⊥, q⊥)i−→ qi (qi, qi)
i−→ qi (qi, qi)
t−→ qt
C!"$ 6&!,$ 8;6('/',+ ", 6"+!/6+' )76#1#' F,% V 6..'0+6,+ v .!//' $"%+ 2
⊥−→ q⊥
(q⊥, q⊥)r−→ qr (qr, qr)
r−→ qr (qr, qr)
t−→ qt
!K qt '$+ ", 8+6+ 6..'0+8<
9);$(<=)/ > 2& 3'$" *#"! ,$! +'*- &4!5!67&! 7 8(8+!*" 9 &4'$"#6'"! +4' 0 ! U
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(#67&!5!-;
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#$ 7!89 !:+-:; %1- 02&2- -,+2 &..1%0- qi, qr, qt, qreq, qrel, qgrant, qm, qrt9 <+2$:2:=161+2;
31- 02&2- -:>+:?1+2 3,661 %1- -$:=&+2- @
• qi @ 7,$- %1- +,1$#- &$A#1--,$- #$ +,1$# 3,$/&+2 &:+-: B$1 31 +,1$# -,+2 !"#9
• qr @ 7,$- %1- +,1$#- &$A#1--,$- #$ +,1$# 3,$/&+2 -,+2 -,:2 !"# -,:2 $#%&#'( )*;
&=13 &$ 6,:+- $+ $#%&#'( )*9 <% C & &$3$+1 ./,.&>&2:,+ &$A#1--,$- #1 31 +,1$#9
• qt @ D1 E12,+ 1-2 -,:2 &$A#1--,$- #1 31 +,1$#; -,:2 %1 E12,+ +1 ",$>1&:2 .&-9
• qreq @ D1 +,1$# 3,$/&+2 1-2 1+ 2/&:+ #1 #16&+#1/ %1 E12,+ .,$/ %$: 6F61 ,$
.,$/ $+ +,1$# &$A#1--,$- #1 31 +,1$#9 G1221 /1B$F21 02&:2 ./,.&>019
• qrel @ D1 E12,+ 1-2 #0.%&30 #1 "&- 1+ H&$2 I .&/2:/ #$ +,1$# 3,$/&+29
• qgrant @ D1 E12,+ 1-2 #0.%&30 #1 H&$2 1+ "&- E$-B$4&$ +,1$# 3,$/&+29
• qm @ D1 E12,+ 1-2 #&+- ,$ &$A#1--,$- 31 +,1$#; %1 E12,+ ",$>1&:2 JG41-2 $+
02&2 &331.20K9
• qrt @ D1 E12,+ 1-2 -,:2 #&+- -,:2 &$A#1--,$- #1 31 +,1$#; c.I.d; :% +4C & .&- #1
3H&+>161+2 3:A#1--$- #$ +,1$# 3,$/&+2 JG41-2 $+ 02&2 &331.20K9
L+ $2:%:-&+2 31- 02&2-; %1- 2/&+-:2:,+- #$ 2/&+-#$321$/ -,+2 %1- -$:=&+21- @
(⊥,⊥)−−−→ q⊥
(q⊥, q⊥)(i,i)−−→ qi (q⊥, q⊥)
(i,r)−−→ qreq (q⊥, q⊥)
(r,r)−−−→ qr
(q⊥, q⊥)(r,t)−−→ qgrant (q⊥, q⊥)
(t,t)−−→ qt (q⊥, q⊥)
(t,i)−−→ qrel
(qi, qi)(i,i)−−→ qi (qr, qi)
(i,i)−−→ qi (qi, qr)
(i,i)−−→ qi
(qr, qr)(i,i)−−→ qr (qreq, qi)
(i,i)−−→ qreq (qi, qreq)
(i,i)−−→ qreq
(qreq, qr)(i,i)−−→ qreq (qr, qreq)
(i,i)−−→ qreq (qr, qr)
(i,r)−−→ qreq
(qr, qi)(i,r)−−→ qreq (qi, qr)
(i,r)−−→ qreq (qr, qr)
(r,r)−−−→ qr
(qr, qi)(r,r)−−−→ qr (qi, qr)
(r,r)−−−→ qr (qreq, qr)
(r,r)−−−→ qreq
! "#$%&'() * +(&,-&%). /) 0$ 12(&3-$'&4, 5 6$.) /) 789.
(qr, qreq)(r,r)−−−→ qreq (qr, qi)
(r,t)−−→ qgrant (qi, qr)
(r,t)−−→ qgrant
(qr, qr)(r,t)−−→ qgrant (qi, qi)
(t,t)−−→ qt (qi, qr)
(t,t)−−→ qt
(qr, qi)(t,t)−−→ qt (qr, qr)
(t,t)−−→ qt (qi, qreq)
(t,t)−−→ qrt
(qreq, qi)(t,t)−−→ qrt (qr, qreq)
(t,t)−−→ qrt (qreq, qr)
(t,t)−−→ qrt
(qi, qgrant)(t,b)−−→ qm (qgrant, qi)
(t,b)−−→ qm (qgrant, qr)(t, b) −→ qm
(qr, qgrant)(t,b)−−→ qm (qt, qi)
(b,b)−−→ qt (qt, qr)
(b,b)−−→ qt
(qr, qt)(b,b)−−→ qt (qi, qt)
(b,b)−−→ qt (qm, qr)
(b,b)−−→ qm
(qm, qi)(b,b)−−→ qm (qr, qm)
(b,b)−−→ qm (qi, qm)
(b,b)−−→ qm
(qt, qreq)(b,b)−−→ qrt (qreq, qt)
(b,b)−−→ qrt (qr, qrt)
(b,b)−−→ qrt
(qrt, qr)(b,b)−−→ qrt (qi, qrt)
(b,b)−−→ qrt (qrt, qi)
(b,b)−−→ qrt
(qi, qrel)(b,t)−−→ qm (qrel, qi)
(b,t)−−→ qm (qr, qrel)
(b,t)−−→ qm
(qrel, qr)(b,t)−−→ qm (qi, qi)
(t,i)−−→ qrel (qi, qr)
(t,r)−−→ qrel
(qr, qi)(t,r)−−→ qrel (qr, qr)
(t,r)−−→ qrel
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&%,0'"!*@5 E(*# /<"(&("# +!, FNT -:%"#%0.-% /% #D0.(-%# "("=&%,0'"!*@ &%- 4*%
FNT ∩ Fbin = ∅ %& +(*, $A!4*% H ∈ FNT ) -:!,'&< /% H %#& *"%5 F(*, 8!',% -!
/'#&'"$&'(" %"&,% /%# #D0.(-%# 8("$&'(""%-# +-*# 8!$'-%) /<#(,0!'#) #%*-# -%# #D0.(-%#
8("$&'(""%-# /% FNT #%,("& <$,'&# %" -%&&,%# 0!7*#$*-%#5 G" 123 8("$&'(""%- Rλ
%#& *" %"#%0.-% /% ,>B-%# /% -! 8(,0% H(t) → α (H H ∈ FNT ) t ∈ T (Fbin,X ) %&
α ∈ T (Fbin∪FNT ,X )5 G" I123 Rλ '"/*'& *"% ,%-!&'(" /% ,<=<$,'&*,% 8("$&'(""%--%
→Rλ#*, /%# &%,0%# /!"# T (Fbin ∪ FNT ) J u→Rλ
v ##:'- %@'#&% *"% +(#'&'(" p /% u
%& *"% ,>B-% H(t)→ α /!"# Rλ &%--%# 4*:'- %@'#&% *"% #*.#&'&*&'(" σ : X → T (Fbin)
!9%$ u|p = H(t)σ %& v = u[ασ]p5
E(*# /<"(&("# <B!-%0%"& +!, →⋆Rλ
-! $-;&*,% &,!"#'&'9% /% -! ,%-!&'(" /% ,<=
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T (Fbin) !$$%##'.-%# +!, -! ,<=<$,'&*,% 8("$&'(""%--% /:*" &%,0% /% T (Fbin ∪ FNT )5
;3<9&'&89 = >R⋆λ? 8.+%/" Rλ ,/ 2458 %" E ⊆ T (Fbin ∪ FNT ) "%0 >,% E ∩
T (Fbin) = ∅? @A%/ %$B0% &% "%'$% 3**% +B0% &% T (Fbin) ; C3'"+' &% E % " &(/."(
C3' R⋆λ(E) %" % " &(D/+ *+)&% ., E
R⋆λ(E) = {β ∈ T (Fbin)|α→
⋆Rλ
β ∧ α ∈ E}.
2)0$(@6). A F/ ! "#$% &% '()(*'+",'% -./*"+.//%0 3 "'.+ &+G('%/*% C3' '3CC.'"
3,H ! "#$% &% '()(*'+",'% *03 +>,% E
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#*, *" %"#%0.-% /% &%,0%# /% Fbin∪FNT ) "(*# !9("# -! +,(+,'<&< -'"<!',% #*'9!"&% J
P H(s1) +H(s2) = H(s1 + s2)
!"! #$%&'()% *) +,-,./0&1/) 234.&0344)5% 627+#%8 9:
!"# H $% &'()*+" ," FNT - .% /0 123$"4 $%" 1"++" 10 %&5*0( 12*% &2(/+" "&1 //"+6"
+ # %*%2& 12*%-
7*$& ++*%& 2++$&10"0 #"11" &10$#1$0" / 0 $% "8"(/+"-
;<)(=5) "> !"#$%!""&'&"$ (& )*+,
+!%$ Rλ -& )*+ ./%01"$ 2
Rλ =
F (x)→ H(G(x))
H(x)→ a(x, x)
G(x)→ 0(⊥,⊥)
G(x)→ 1(⊥,⊥)
.
34&".&'5-& (& $&6'&. 1##&..%5-&. R⋆λ({⊥}) &.$ #1-#/-7 2
8 &" '!(& 9!"#$%!""&- 2
F (⊥)→ H(G(⊥))
→ H(0(⊥,⊥)) + H(1(⊥,⊥))
→ a(0(⊥,⊥), 0(⊥,⊥)) + a(1(⊥,⊥), 1(⊥,⊥))
8 &" '!(& "!":9!"#$%!""&- ;#!''& -& '!(& (& 9!"#$%!""&'&"$ (&. )*+. #-1.:
.%</&.= 2
F (⊥)→ H(G(⊥)
→ H(0(⊥,⊥)) + H(1(⊥,⊥)) + a(G(⊥), G(⊥))
→ a(0(⊥,⊥), 0(⊥,⊥)) + a(1(⊥,⊥), 1(⊥,⊥))
+ a(0(⊥,⊥), 1(⊥,⊥)) + a(1(⊥,⊥), 0(⊥,⊥))
31 (%>76&"#& &.$ (1". -& '!(& "!":9!"#$%!""&-?H(x) → a(x, x) &.$ 1@@-%#15-&
'A'& .% t = G(⊥) #!"$%&"$ /" "!":$&6'%"1- G,
+)(?/@1)% 9 +% /" )*+ &.$ -%"71%6& B1/#C& &$ (6!%$&? "!/. @!/0!". '!"$6&6
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'!(& (& 9!"#$%!""&'&"$ (&. )*+. #-1..%</&.,
#=,.0A.?&034 *1 =/3&3.35) *BC/D0&/?E) C/D3/)%.)4& )4 27+#
9 %& + &"#12*% :-;4 %*$& !*%& (*%106 #*(("%1 %*$& /*$!*%& 10 21"0 +" <=>4 $%
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/ 01 B6"4 "% <CD&-
E"11" &"#12*% (*%10" #*(("%1 +" <=> /"$1 F10" 5*0( +2&6 "% G<CD& &*$& 5*0("
,@$% /0*)+?(" ," # +#$+ ,@ ##"&&2)2+216-
D*21 FNT $% "%&"()+" ," &'()*+"& %*%H1"0(2% $8 1"+& 3$" FNT = {H,Arbiter}-
I@"%&"()+" Fbin "&1 #"+$2 ,6J%2 , %& + &"#12*% :-;4 c.A.d.4 {i, r, t, b,⊥}-
! "#$%&'() * +,-'./)- 0) 12324(&'5() 6784'&788)9-
1.:9)- :282(2)-* !"#$%& '()*& +,'-#*& .& /01 α → β 2&'# 34'& 5,+6&'47&
&+ %+ 8/01 '()*& .& *# -#+7('& 2%76#+4& 9 H(α)→ β #6&5 α, β ∈ T (Fbin,X ):
;#' &<&-=*&> Rreqλ )?+?'?&2 @ =#'47' .&2 '()*&2 .& '?A?5'74%'& BC:DE> BC:FE &4
BC:GE .,++?&2 .#+2 *# 2&547,+ C:H:
H(i(x, r(y, z)))→ r(x, r(y, z)) BD:IE
H(i(r(x, y), z))→ r(r(x, y), z) BD:HE
H(i(⊥,⊥))→ r(⊥,⊥) BD:CE
1.:9)- 0) 4&(459$'&78 Rcirλ * ;,%' $%& *&2 '()*&2 )?+?'?&2 2,7&+4 #==*75#J*&2>
+,%2 #6,+2 J&2,7+ .K%+ &+2&-J*& .& '()*&2 .& 8/01 5,--& 7+.7$%? 57A.&22,%2 9
H(a(x, y))→ a(H(x), y),
H(a(x, y))→ a(x,H(y))|a ∈ Fbin,H ∈ FNT
!&2 '()*&2 +,%2 =&'-&44&+4 .& 57'5%*&' *& +,+A4&'-7+#* H @ 4,%2 *&2 &+.',742
.% 4&'-& .,++?:
1.:9)- 0) ;7&8' 6&<) RFPλ * L& 5#*5%* .& =,7+4 M<& &24 27-%*? =#' *&2 '()*&2
2%76#+4&2 9
Arbiter(x)→ x
Arbiter(x)→ Arbiter(H(x))|Arbiter,H ∈ FNT
!&2 '()*&2 +,%2 =&'-&44&+4 .& )?+?'&' *& +,+A4&'-7+#* H #%4#+4 $%K,+ # J&A
2,7+:
N,%2 =,%6,+2 ,J4&+7' *K&+2&-J*& .& 4&'-&2 #55&227J*&2 &+ 5#*5%*#+4
(Rreqλ )⋆({Arbiter(u)}): O& =*%2> +,%2 =,%6,+2 ?)#*&-&+4 6?'7M&' $%& *& 4&'-& ! "#
$%&'%( )*+ v &24 #44&7)+#J*& @ =#'47' .% 4&'-& ! "# ),#% u =%72 $%& u→⋆Rreqλ
∪RcirλRFPλ
v:
L& 4&'-& ! "# $%&'%( )*+ v &24 #44&7)+#J*& @ =#'47' .% 4&'-& ! "# ),#% u =%72 $%&
u→⋆ v: N,%2 *724,+2 57A.&22%2 *# '?.%547,+ .?4#7**?& 9
Arbiter(u) = Arbiter(t(i(i(⊥,⊥), i(⊥,⊥)), i(i(⊥,⊥), i(⊥,⊥))))→RFPλ
Arbiter(H(t(i(i(⊥,⊥), i(⊥,⊥)), i(i(⊥,⊥), i(⊥,⊥)))))→⋆Rcirλ
Arbiter(t(i(H(i(⊥,⊥)), i(⊥,⊥)), i(i(⊥,⊥), i(⊥,⊥))))→r4.3
Arbiter(t(i((r(⊥,⊥)), i(⊥,⊥)), i(i(⊥,⊥), i(⊥,⊥))))→RFPλ
Arbiter(H(t(i(r(⊥,⊥), i(⊥,⊥)), i(i(⊥,⊥), i(⊥,⊥)))))→⋆
...
Arbiter(t(r(r(⊥,⊥), r(⊥,⊥)), r(r(⊥,⊥), r(⊥,⊥))))→RFPλ
t(r(r(⊥,⊥), r(⊥,⊥)), r(r(⊥,⊥), r(⊥,⊥))) = v
N,%2 +,4,+2 =#' RTAPλ *& 8/01 5,-=*&4 .?M+7 &+ 5&44& 2&547,+:
!"! #$%&'()% *) +,-,./0&1/) 234.&0344)5% 627+#%8 "
!"!# $%&'()*%+&,-*' ./ $01( 2/%( 3$01(
!"# $%#&"'!(# )*%('+(*(' ,-+./0+##%1%'2 $+ 3456# +' $+ 456#7 8* /0!/!#%'%!(
9 )!('0+ :"-"( /*# $+ 02;2&0%'"0+ "'%,%#*(' R /+"' <'0+ #%)",2 /*0 "( 3456 Rλ
*1+& "( #+", (!(;'+0)%(*, Fl→r7
9/3:3%0&034 " !"# R $% &' ($) T (Fbin,X ) × T (Fbin,X )* α ∈ T (Fbin)* β ∈
T (Fbin) +# l→ r ∈ R, !"# Rλ -+ .&' #+- /$+
Rλ = {Fl→r(l)→ r} ∪
Fl→r(a(x, y))→ a(Fl→r(x), y),
Fl→r(a(x, y))→ a(x, Fl→r(y))|
a ∈ Fbin
.
0"%("*
α→{l→r} β ⇔ Fl→r(α)→⋆Rλ
β.
+)(;/<1)% 1- 23$# %!#+) /$+ 43%( Fl→r(α)→⋆Rλ
β +(# -3 5-6#$)+ #)3%("#"7+ 4+ -3
)+-3#"!% 4+ )8985)"#$)+ 2!%5#"!%%+--+ :!$) /$+ α→{l→r} β +(# 8#3;-"+ +<35#+=+%# $%+
2!"(,
9/)1=) " 69/)1=) %1..04.&) 6>30/ .?! *;4% 5@;44)A) B88 ,
> α→{l→r} β ⇒ Fl→r(α)→⋆Rλ
β ?
+-!% -3 48@%"#"!%* "- +<"(#+ $%+ :!("#"!% p 4+ α +# $%+ ($;(#"#$#"!% σ : X →
T (Fbin) #+--+( /$+ α|p = lσ +# β = α[rσ]p, !"# α′$% #+)=+ 5!%(#)$"# 5!==+
-+ ($"73%# ? α′ = α[Fl→r(α|p)]p,
A-3")+=+%#* +% $#"-"(3%# (+$-+=+%# -+( )BC-+( 4+ 5")5$-3#"!%* c.D.d. -+( )BC-+(
4+ -3 2!)=+ Fl→r(a(x, y)) → a(Fl→r(x), y) +# Fl→r(a(x, y)) → a(x, Fl→r(y))*
%!$( 37!%(
Fl→r(α)→⋆Rλ
α′. =>7>?
+-!% -3 5!%(#)$5#"!% 4+ α′* "- +<"(#+ $%+ )BC-+ C8%8)8+ 4+ Rλ* c.D.d Fl→r(l)→
r* $%+ ($;(#"#$#"!% σ : X → T (Fbin) +# $%+ :!("#"!% p #+--+( /$+ α′|p =
Fl→r(l)σ, E% 5!%(8/$+%5+* %!$( :!$7!%( 5!%(#)$")+ -+ #+)=+ α′[rσ] /$" +(#
8C3- D β :3) -3 5!%(#)$5#"!% 4+ α′, F!$( 37!%( 4!%5 ?
α′ →Rλβ. =>7@?
0"%("* +% $#"-"(3%# GH,HI +# GH,JI* %!$( :!$7!%( 484$")+ /$+ Fl→r(α)→⋆Rλ
β,
K3) 5!%(8/$+%#* α→{l→r} β ⇒ Fl→r(α)→⋆Rλ
β,
> α→{l→r} β ⇐ Fl→r(α)→⋆Rλ
β ? EL+5#"7+=+%#* :!$)($"7!%( +% "%4$5#"!% ($)
-+ 5M+="% 4+ )8985)"#$)+ /$" %!$( 5!%4$"# D -3 )+7+%4"53#"!%,
A%(#%B ,* '0*(#C!0)*'%!( =/02#+('2+ /*0 ,* DE&F+ =*? $+ ,* GH"0+ >79?B 2'*(' ,+
02#",'*' /0%(&%/*, $+ &+''+ #+&'%!(B +#' $!((2 /*0 ,+ 'F2!0E)+ 9 c.I.d. :"+ /!"0 '!"#
456 RB (!"# /!"1!(# &!(#'0"%0+ "( 3456 Rλ #%)",*(' R +( "'%,%#*(' $+# (!(;
'+0)%(*". Fl→r /!"0 &F*:"+ l→ r ∈ R +' "( (!(;'+0)%(*, G /!"0 ,+ /!%(' G.+7
! "#$%&'() * +,-'./)- 0) 12324(&'5() 6784'&788)9-
:#27(./) ; !"# R $% &' ($) T (Fbin,X )×T (Fbin,X )* E ⊆ T (Fbin)+ !"# l→ r
$%, )-./, 0, R ,# Rl→r /, 1&' #,/( 2$,
Rl→r = {Fl→r(l)→ r} ∪
Fl→r(a(x, y))→ a(Fl→r(x), y),
Fl→r(a(x, y))→ a(x, Fl→r(y))|
a ∈ Fbin
.
!"# RFP /, 1&' #,/ 2$,
RFP = {G(x)→ G(Fl→r(x)) | l→ r ∈ R} ∪ {G(x)→ x}.
!"# Rλ /, 1&' #,/ 2$,
Rλ =⋃
l→r∈R
Rl→r ∪RFP .
3"%("*
R⋆(E) = R⋆λ(E
′)
45,6 E′ = {G(t)|t ∈ E}+
1)/$(<5)- = 7/ 84$# %!#,) 2$, /4 6/9#$), ):;,<"5, ,# #)4%("#"5, 0, &' ,(# ("=$/:,
>4) RFP = {G(x)→ G(Fl→r(x)) | l→ r ∈ R} ∪ {G(x)→ x}+
>()5?) ! @>()5?) -544&84') @A7&( 4B* 0$8- 9C$88)D) EFF +
? R⋆(E) ⊆ R⋆λ(E
′) @ !"# α ∈ E ,# β ∈ R⋆({α})+ ,/!% /4 0:A%"#"!% 0, R⋆* "/
,<"(#, l1 → r1, . . . ln → rn ∈ R ,# t0, . . . , tn #,//,( 2$, α→{l0→r0} t0 →{l1→r1}
. . . →{ln−1→rn−1} tn−1 →{ln→rn} tn = β. ! "#$% Rl0→r0 &!'()*+ )!, -./0*1
+*./, 2++!/03!, ,(4-*54!, 6 )2 (&.(.,*+*./ 78
9/ :4/4&2)*,2/+ -! (&.-!,,3,; /.3, .<+!/./, )! -=!'*/ 0! &414-&*+3&! -*1
0!,,.3, > G(t0) →⋆Rλ
G(t1) . . . →⋆Rλ
G(tn) →G(x)→x tn = β8 ?2& -./,4@3!/+;
β ∈ R⋆λ(E
′)8
A*/,*; /.3, (.3B./, 0403*&! @3!
R⋆({α}) ⊆ R⋆λ({G(α)}). !"#$
C R⋆λ(E
′) ⊆ R⋆(E) > D+30*./, R⋆λ({G(α)})8
C %3((.,./, @3E3/! &F:)! 0! )2 G.&'! G(x)→ G(Fl→r(x)) !,+ 2(()*@34!8 A*/,*;
G(α)→G(x)→G(Fl→r(x)) G(Fl→r(α)))8 H! ()3,; 0E2(&F, )2 (&.(.,*+*./ 7; /.3,
(.3B./, 0403*&! @3!
R⋆({α}) ⊇ R⋆λ({G(α)}). !"%$
C %3((.,./, @3E23-3/! &F:)! 0! )2 G.&'! G(x) → G(Fl→r(x)) !,+ 2(()*@34!8
?2& -./,4@3!/+; 23-3/! &F:)! 0!
⋃
l→r∈RRl→r (!3+ I+&! 2(()*@34! 8
E3/*@3! +!&'! 2++!*/+ 0! T (Fbin) !,+ 0./- .<+!/3 !/ 2(()*@32/+ )2 &F:)!
G(x)→ x8 A*/,*; α ∈ R⋆λ({G(α)})8 HE2(&F, )2 045/*+*./; α ∈ R
⋆({α})8
?.3& -./-)3,*./; (.3& +.3, α ∈ E; 0! JK8LM !+ JK8NM; /.3, .<+!/./, @3!
R⋆(E) = R⋆λ(E
′)8
!"! #$%&'()% *) +,-,./0&1/) 234.&0344)5% 627+#%8 9
!"!# $%&'()*%+&,-*' ./ 0$12( 3/%( $12(
!"# $! #%&'()" *+,&,-%"'%. ")/# !0)"# *+)/0, 1/% #)(' /" 234. ($ %5(#'% /"
6234 &)++%#*)"-!"'7 8! +,&(*+)1/% %#' *)#,% -!"# &%''% #%&'()"7
8! '+!"#9)+:!'()" ;*+,#%"',% *!+ $! <=&>% ;?@ -% $! AB/+% C7D@ %#' -)"",% *!+
$% $%::% D c.E.d *)/+ /" 6234 Rλ #!'(#9!(#!"' /"% '%$$% &)"-('()". ")/# *)/0)"#
&)"#'+/(+% /" 234 R &)++%#*)"-!"' 7
:)(() " !"# $% &'( Rλ )$* T (Fbin ∪ FNT ,X ) × T (Fbin ∪ FNT ,X )+ !"# ,-
'( )$* T (Fbin,X ) × T (Fbin,X ) #-,) .$- /!$* 012.$- *34,- l → r ∈ R5 ", -6")#-
$% )!$) -%)-78,- &'( Rl→r ⊂ Rλ #-, .$-
Rl→r = {Fl→r(l)→ r} ∪
Fl→r(a(x, y))→ a(Fl→r(x), y),
Fl→r(a(x, y))→ a(x, Fl→r(y))|
a ∈ Fbin
.
-# ", -6")#- 942,-7-%# $%- *34,- rFP ∈ RFP #-, .$-
rFP = G(x)→ G(Fl→r(x))
-# G(x)→ x ∈ RFP #-,) .$-
⋃
l→r∈R
Rl→r ∪RFP = Rλ.
:"%)"5
R⋆λ(E
′) = R⋆(E)
2;-0 E ⊆ T (Fbin) -# E′ = {G(t)|t ∈ E}+
;/)1<) 9 6;/)1<) %1..04.&)8 $//!)!%) $% #-, '( 0!%)#*$"# 0!77- <2%) ,-
,-77- =+ -,!% ,- #19!*37- =5 %!$) !8#-%!%) R⋆λ(E
′) = R⋆(E)+
F'!"' -)"", /" 234. ($ %5(#'% ')/G)/+# /" 6234 &)++%#*)"-!"'7 H!(# $! +,I
&(*+)1/% "J%#' *!# #)/0%"' *)##(?$%7 3%B!+-)"# $J%5%:*$% -)"", &(I-%##)/#7
=>)(?5) "9 >!%)"<9*!%) $% &'( .$" #*2;2",,- 0!77- $% ?,#*24-
OnlyOneRequest <-) 9#2#) <$ ':@ 2A2%# $%- )-$,- <-72%<- <- *-))!$*0-
.$" -)# <9BC /*!/249- B$).$DC ,2 *20"%-+ :"%)"5
Rλ =
OnlyOne(t(r(x, y), i(z, t)))→ t(OnlyOne(r(x, y)), i(z, t))
OnlyOne(t(i(x, y), r(z, t)))→ t(i(x, y), OnlyOne(r(z, t)))
OnlyOne(r(r(x, y), i(z, t)))→ r(OnlyOne(r(x, y)), i(z, t))
OnlyOne(r(i(x, y), r(z, t)))→ r(i(x, y), OnlyOne(r(z, t)))
OnlyOne(r(⊥,⊥))→ r(⊥,⊥)
.
!"#$%&'( ) *+,&-.(, /( 01213'%&4'( 5673&%677(8,
! "#$%&'($ )*' +$ ,-. /0& 12"3(&2""$ +$ !* 454$ 4*"&67$8 9*&' "20' )20:2"'
+;<"&7 0"$ )72)7&;(; OnlyOneRequest '07 α ∈ T (Fbin) ($!!$ /0$ )207 (20( p, p′ ∈
Pos(α)= '& α(p) = r, p '= p′= *!27' p ( p′ 20 p′ ( p8 >*"' !* '0&($= "20' *!!2"' 7$:$"&7
? 3$ '0@$( ? !* '$3(&2" !"#$%& '("))&%%*+*#*,- ./01 #&% *!2"1*"!,% +0 3A*)&(7$ B8
>*"' !* '$3(&2" C8C= "20' *!!2"' 3A$73A$7 0"$ 32":$7'&2" &"+&7$3($ $" 32"'&+;7*"(
!$' 'D'(64$' 12"3(&2""$!' 3244$ 0" (D)$ ');3&*! +$' 'D'(64$' 3!*''&/0$'8
!" #$%&' ()*+,-./01,' 2(#$%&'3
!"!# $%&'(&)*+
>*"' 3$(($ '$3(&2"= ;(*"( +2""; 0" ,-. R '07 T (Fbin,X ) × T (Fbin,X ) $( 0"
($74$ t ∈ T (Fbin)= "20' 42"(72"' /0$ 3#$'( )2''&E!$ +$ 3*!30!$7 $%*3($4$"( !$ 454$
$"'$4E!$ +$ ($74$' *33$''&E!$' $" 0(&!&'*"( 0" F,-. +$ !* 1274$ )*7(&30!&67$ c.?.d.
F,-. ;!;4$"(*&7$8
91:7%&%67 ; 3/*, Rλe 0! 45637 Rλe &%, '*, '(8,1& -#-9&!,"*1& :-;"#&9&!, "..&#-
<4563= %&0#&9&!, %* )>"?0& 1@;#& '& Rλe &%, 0!& '&% A/19&% )*B'&%%/0% C
D7 H(a(x, y)) → α "2&) a ∈ FbinE H ∈ FNT E x, y ∈ X E x '= yE α ∈ T (Fbin ∪
FNT ,X ) &, Var(α) = {x, y}
F7 H(⊥)→ α "2&) α ∈ T (Fbin ∪ FNT )
G7 H(a(x,⊥))→ x "2&) x ∈ X E H ∈ FNT &, a ∈ Fbin7
G" 1*&(= !$' GF,-.' +$ !* +;<"&(&2" H= '2"( '0I'*44$"( $%)7$''&:$' )207 !*
');3&<3*(&2" +0 4;3*"&'4$ +$ 7;J;37&(07$ )207 0"$ 76K!$ l→ r ∈ R L
M8 +$ (720:$7 0"$ )2'&(&2" )207 !* 7;J;37&(07$=
H8 +$ :;7&<$7 '& l 32"327+$ *:$3 !$ '20'J($74$ 3207*"(= )0&'
N8 +$ 3*!30!$7 !* '0E'(&(0(&2" 7;'0!(*"( σ= $(
C8 +$ 7$)!*3$7 !$ '20'J($74$ )*7 rσ8
*$13%:3#&%67 /4 $'6&6368( /<='>%&'#?( ='>6'(,3(7& (7 @5A0*
>*"' !* '$3(&2" N8H= "20' *:2"' 42"(7; 3244$"( "20' )20:2"' (7*&($7 !$ ,OP
$" ,-.'8
Q$(($ '$3(&2" 42"(7$ 3244$"( !$ ,OP )$0( 5(7$ 1274*!&'; $" GF,-.' '20' 1274$
+#0" )72E!64$ +$ 3*!30! +#*33$''&E&!&(;8
.2&( FNT 0" $"'$4E!$ +$ 'D4E2!$' "2"J($74&"*0% ($! /0$ FNT =
{H,R,RT, TI, I, Arbiter}8 R#$"'$4E!$ Fbin $'( 3$!0& +;<"& +*"' !* '$3(&2" N8H= c.?.d=
{i, r, t, b,⊥}8
!"! #$%&' ()*+,-./01,' 2(#$%&'3 4
!" #$%&!" '! #()(*#+,-#! ./0123 ./042 !, ./052 '677(!" '87" &8 "!*,+67 /09 "67,
,#87":6#;(!" *6;;! &!" #$%&!" '<=>?@A Rreqλe *6;;! +7'+B-( *+)'!""6-"0
H(i(x, y)) → r(x,R(y)) !"#$
H(i(x, y)) → r(R(x), y) !"%$
R(r(x, y)) → r(x, y) !"&'$
R(⊥)→ ⊥ !"&&$
C6-" +&&-",#67" &8 #()(*#+,-#! '! &8 *67D%-#8,+67 "-+E87,! t0 = H(t(i(⊥,⊥), i(⊥
,⊥)))0 F-*-7! #$%&! ;!7,+677(! *+)'!""-" 7! G!-, H,#! 8GG&+B-(!0 F+7"+3 8-*-7 ,!#;!
'! T (Fbin) 7<!", 8**!""+I&! '! H(t(i(⊥,⊥), i(⊥,⊥)))0
C6-" 8E67" '67* I!"6+7 '! #$%&!" G6-# "G(*+D!# &8 *+#*-&8,+67 '- "J;I6&!H 8E87,
'<8GG&+B-!# &!" #$%&!" .10K23 0 0 0 3 .10LL20 M!" #$%&!" Rcirλe "67, '677(!" *+)'!""6-" !,
"67, '(D7+!" G6-# *N8B-! "J;I6&! '! Fbin "8-: ⊥0
H(i(x, y))→ i(H(x), y) !"&($
H(i(x, y))→ i(x,H(y)) !"&)$
H(r(x, y)) → r(H(x), y) !"&!$
H(r(x, y)) → r(x,H(y)) !"&*$
H(b(x, y))→ b(H(x), y) !"&+$
H(b(x, y))→ b(x,H(y)) !"&,$
H(t(x, y)) → t(H(x), y) !"&#$
H(t(x, y)) → t(x,H(y)) !"&%$
O8+7,!787, 76-" +&&-",#67" &<-,+&+"8,+67 '!" #$%&!" .10L9230 0 0 3 .10LP20 =7 8GG&+)
B-87, &8 #$%&! .10L92 .#!"G0 .10L/22 "-# t03 &! ,!#;! t′1 = t(H(i(⊥,⊥)), i(⊥,⊥))
.#!"G0 t′′1 = t(i(⊥,⊥),H(i(⊥,⊥)))2 !", 6I,!7-0 =7 8GG&+B-87, &8 #$%&! .10K2 !, &8
#$%&! .10LL2 "-# t′1 .#!"G0 t′′123 76-" 6I,!767" &! ,!#;! t
′2 = t(r(⊥,⊥), i(⊥,⊥)) .#!"G0
t′′2 = t(i(⊥,⊥), r(⊥,⊥))20
C6,!Q B-! t′2 !, t′′2 7! *67,+!77!7, 8-*-7 "J;I6&! '! FNT 0 <=>?@A 7! G!-, '67*
G&-" "<8GG&+B-!#0 F+7"+3 76-" +7,#6'-+"67" &! "J;I6&! Arbiter '67, &!" "(;87,+B-!"
"67, '(*#+,!" G8# &!" #$%&!" .10/R2 S .10//20 F-,#!;!7, '+,3 *! "J;I6&! 76-" G!#;!, '!
#()(*#+#! 8-,87, B-! 76-" 8E67" I!"6+70 !" #$%&!" +7'+B-(!" *+)'!""6-" #!;G&+""!7,
&8 "G(*+D*8,+67 '- ?FT G8# -7 =>?@A0
! "#$%&'() * +,-'./)- 0) 12324(&'5() 6784'&788)9-
H(t(x, y)) → b(x,RT (y)) !"#$%
H(t(x, y)) → b(RT (x), y) !"#&%
RT (r(x, y)) → t(x, y) !"##%
H(b(x, y)) → t(x, TI(y)) !"#'%
H(b(x, y)) → t(TI(x), y) !"#!%
TI(t(x, y)) → i(I(x), I(y)) !"#(%
TI(t(x, y))→ r(R(x), I(y)) !"#)%
TI(t(x, y))→ r(I(x), R(y)) !"#*%
I(i(x, y)) → i(x, y) !"#+%
I(⊥) → ⊥ !"#,%
Arbiter(t(x, y)) → t(x, y) !"'$%
Arbiter(t(x, y))→ Arbiter(H(t(x, y))) !"'&%
Arbiter(b(x, y)) → b(x, y) !"'#%
Arbiter(b(x, y)) → Arbiter(H(b(x, y))) !"''%
!"#$%#!#$ &'#(")*+'#( ,% $%+-% ). )*/.$ t0 = Arbiter(t(i(⊥,⊥), i(⊥,⊥)))0
1% t0 %$ %# !22,"3.!#$ ,%( +45,%( 67089:; 6709<:; 670=: %$ 67099:; ,% $%+-% t1 =
Arbiter(t(i(⊥,⊥), r(⊥,⊥))) %($ &!,&.,*0 ># !22,"3.!#$ ,%( +45,%( 670<?: %$ 670<<:
)% t1; ,% $%+-% t2 %($ &!,&.,* !@%& t2 = Arbiter(b(i(⊥,⊥), t(⊥,⊥)))0 A'$%B 3.% )%
t1 6+%(20 t2:; #'.( 2'.@'#( *5!,%-%#$ !22,"3.%+ ,! +45,% 6708?: 6+%(20 708<: %$ 2."(
'/$"%#$ ,% $%+-% t(i(⊥,⊥), r(⊥,⊥)) 6+%(20 b(i(⊥,⊥), t(⊥,⊥)):0 C%( )%.D $%+-%( -%#E
$"'##*( &"E)%((.( +%2+*(%#$%#$ )%.D &'#F5.+!$"'#( @+!"-%#$ !&&%(("/,%(0
A'.( 2'.@'#( '/$%#"+ ,G%#(%-/,% )% $%+-%( !&&%(("/,%( H 2!+$"+ ). $%+-% ! "#
$%#& %# &!,&.,!#$ (Rreqλe )
⋆({Arbiter(u)})0 1% 2,.(; #'.( 2'.@'#( *5!,%-%#$ @*+"F%+
3.% ,% $%+-% ! "# '&()&* $+, v %($ !$$%"5#!/,% H 2!+$"+ ). $%+-% ! "# $%#& u 2."( 3.%
u→⋆Rreq
λev0
>#F#; #'.( #'$'#( 2!+ RTAPλe ,G>IJKL &'-2,%$ )*F#" %# &%$$% (%&$"'#0 A'.(
+%-!+3.'#( 3.% RTAPλe %($ .# )%( /%#&M-!+N( .$","(*( )!#( ,% &M!2"$+% =0
1)/$(:5)- ! -.&* /!**$0#& %& *$1)#&' #" '2,#& H(x)→ α(x) /"' #&* 3#31&+ "$'&*4
5!)* '"$ !+* %&)6 7"* %& 8"'$"0#& x %& 7& & '2,#& 9
: H(⊥)→ α(⊥)4
: H(a(x, y)) → α(a(x, y))4 "8&7 x ∈ X ; H ∈ FNT ; a ∈ Fbin; α(x), α(⊥) &
α(a(x, y)) ∈ T (Fbin ∪ FNT ,X ).
<!)' *$1/#$=&'; +!)* 7!+*$%3'!+* %3*!'1"$* ()& #" '2,#& H(x) → α(x) &* 3#31&+>
"$'&4 <"' 7!+*3()&+ ; #&* '2,#&* %.?@ABC* %& /!$+ =6& D4EF G D4EE /&)8&+ H '&
37'$ &* 7!11& #&* *)$8"+ 9
Arbiter(x)→ x
Arbiter(x)→ Arbiter(H(x))
C! $+!#(O'+-!$"'# %($ *@")%#$% )!#( #'$+% *$.)% )% &!( 6JPQ: -!"( %# 5*#*+!,;
,! $+!#(O'+-!$"'# )% JKL( @%+( >IJKL( %($ $%&M#"3.% 2'.+ $+!"$%+ $'.( ,%( &!(
2'(("/,%( )%( +45,%( )'##*%(0
!"! #$%&' ()*+,-./01,' 2(#$%&'3 4
!"!" #$%&'()$*%+,)& -. #/0' 1.$' 23#/0'
!"# $%&#'()!(# * '(#'+,-'# .' %/0-'# .123456# .' 7!(7!%.8(7' Rl→rcheck9 .' 7!(:
#)%"7);!( .18%,%' .' <8%;8,-'# Rl→rσ 9 .' )'#) .' #",#);)");!( Rl→r
GS ') .' #",#);)");!(
Rl→rσ−apply .' -8 #!"# #'7);!( *=>=>=? @ -8 #!"# #'7);!( *=>=>=*9 ') +!()%'% 7!++'() -'#
");-;#'% $!"% #;+"-'% -8 %&:&7%;)"%' .'# 456# .8(# -8 #!"# #'7);!( *=>=>=A=
B'# $%;(7;$'# .' 7C8D"' '(#'+,-' .' %/0-'# #!() -;#)&# 7;:.'##"# E
?= 5/0-'# .123456# .' 7!(7!%.8(7' Rl→rcheck '#) "( 23456 (!"# $'%+'))8() .'
<&%;F'% #; "( )'%+' .' T (Fbin) !" !#$% 8<'7 l= G"8(. -8 7!(7!%.8(7' '()%' t
') l (1'#) $8# $!##;,-'9 (Rl→rcheck)
∗({F εl→r(t)}) = ∅=
>= 5/0-'# .123456# .' 7!(#)%"7);!( .18%,%' .' <8%;8,-'# Rl→rσ '#) "( 23456
(!"# $'%+'))8() .' 7!(#)%";%' "(' -;#)' !%.!((&' .' )'%+'# ;(.'H&# $8% .'#
<8%;8,-'# .' l= I'))' -;#)' '#) %'$%&#'()&' $8% "( )'%+' ') #&+8();D"'+'()
8" 7!"%8()9 (!"# $!"<!(# 7!(#;.&%'% 7'))' -;#)' 7!++' -8 #",#);)");!( %&#"-:
)8() ." $8# !" !#$&" %= 6!;) t′ -' )'%+' %&#"-)8() .' (Rl→rcheck)
∗({F εl→r(t)})=
(Rl→rσ )∗(F l→r
copy(t′)) +/(' @ -1"(;D"' )'%+' t′′ %'$%&#'()8() -8 #",#);)");!( %&:
#"-)8() .' -8 !" !#$&" % '()%' t ') l=
J= 5/0-'# .123456# .' )'#) .' #",#);)");!( Rl→rGS '#) "( 23456 #$&7;F8() -8
<&%;F78);!( .1"(' #",#);)");!( ,;'(:K!%+&'= 2L'7);<'+'()9 #; l (1'#) $8# -;(&8;%'
8-!%# "(' <8%;8,-' x 8$$8%8M) .'"H K!;# 8" +!;(#= 6; )!")'# -'# !77"%%'(7'# .' x
$8%)80'() -' +N+' )'%+' 7!++' <8-'"% 8-!%# -8 #",#);)");!( '#) ,;'(:K!%+&'=
(Rl→rcheck)
∗(F l→rcheck(t
′′)) = {t′′} '#) -8 #",#);)");!( ,;'(:K!%+&'9 ∅ #;(!(=
*= 5/0-'# .123456# .' #",#);)");!( Rl→rσ−apply '#) "( 23456 #$&7;F8() -18$$-;:
78);!( .' -8 #",#);)");!( ,;'(:K!%+&' %&#"-)8() .' t ') l #"% -' )'%+' r= O;(#;9
(Rl→rσ−apply)
∗(F l→rrewrite(t
′′)) = {rσ}=
2( ");-;#8() -'# * '(#'+,-'# .' %/0-'# .123456#9 (!"# #!++'# 78$8,-'# .'
#;+"-'% -8 %&:&7%;)"%' .1"( 456=
!"!"!5 %67),' 89(#$%& 8, :;-:;18/-:, < Rl→rcheck
Rl→rcheck '#) "( 23456 (!"# $'%+'))8() .' <&%;F'% #; "( )'%+' .' T (Fbin) !"'
!#$% 8<'7 l= B18$$-;78);!( .' 7') 23456 '#) %'(<!P& $8% -8 $%&#'(7' ." #P+:
,!-' (!(:)'%+;(8- F εl→r= 2)8() .!((& "( )'%+' t ∈ T (Fbin) 7!(7!%.8() 8<'7 l9
(Rl→rcheck)
∗({F εl→r(t)}) '#) {t
′} !Q
R $!"% )!") p ∈ PosF (l)9 t(p) = t′(p) S
R $!"% )!") p ∈ PosX (l)9 t′(p) = ⊕x,p ∈ Fbin S
R $!"% )!") p ∈ FPos(l)\PosX (l)9 ') $!"% )!") p′ ∈ Pos(t|p)9 t(p.p
′) = t′(p.1.p′)=
B' #P+,!-' ⊕x,p '#) "( +8%D"'"% #;0(;F8() D"' -' )'%+' #!"# 7' +8%D"'"% '#) -8
<8-'"% D"' -8 <8%;8,-' x @ -8 $!#;);!( p .8(# l !,);'() D"8(. -8 7!(7!%.8(7' 8$$8%8M)=
G"8(. -8 7!(7!%.8(7' '()%' t ') l (1'#) $8# $!##;,-'9 (Rl→rcheck)
∗({F εl→r(t)}) = ∅=
!)'T D"'9 @ 7' $!;()9 (!"# (' <&%;F!(# $8# -' 78# !Q .'"H <8%;8,-'# ;.'();D"'#
.!;<'() $8%)80'% -8 +N+' <8-'"%= U!(7 (Rl→rcheck)
∗({F εl→r(t)}) = ∅ #'"-'+'() #; t ') l
#!() #)%"7)"%'--'+'() .;L&%'()#=
! "#$%&'() * +,-'./)- 0) 12324(&'5() 6784'&788)9-
:2;8&'&78 < =Rl→rcheck> !"# l→ r $%& '()*& +& T (Fbin,X )×T (Fbin,X ), -!$. +/%!0
#!%. 12' Rl→rcheck *& 3456 7!%.#'$"# 8 12'#"' +& l 7!99& .$"# :
;, F pl→r(x) → ⊕l|p(x) ∈ R
l→rcheck ." p ∈ PosX (l) &# !< ⊕l|p &.# $% .=9>!*& .1/7"2*
?$& %!$. 7!%."+/'!%. +@A#'& +2%. Fbin
B, F pl→r(a(x, y)) → a(F p.0
l→r(x), Fp.1l→r(y)) ∈ Rl→r
check ." p ∈ Pos(l) \ FPos(l) &#
a = l(p)
C, F pl→r(⊥)→⊥∈ Rl→r
check ." p ∈ FPos \ PosX (l)
?@)/%9) A D!%."+/'!%. *2 '()*& a(b(x,⊥), c(d(⊥, y), z))→ r 7!99& $%& '()*& +&
'/0/7'"#$'&E x, y, z ∈ X &# Var(r) ⊆ {x, y, z}, F!$' +&. '2".!%. +& *".">"*"#/E %!$.
+/%!#!%. 7& #&'9& 12' l, G"%."E
Rl→rcheck =
F εl→r(a(x, y))→ a(F 1
l→r(x), F2l→r(y))
F 1l→r(b(x, y))→ b(F 11
l→r(x), F12l→r(y))
F 11l→r(x)→ ⊕x(x)
F 12l→r(⊥)→⊥
F 2l→r(c(x, y))→ c(F 21
l→r(x), F22l→r(y))
F 21l→r(d(x, y))→ d(F 211
l→r(x), F212l→r(y))
F 211l→r(⊥)→⊥
F 212l→r(x)→ ⊕y(x)
F 22l→r(x)→ ⊕z(x)
.
! "#$ %! &'!()!*"&! %! +,-&!) %! +./.0+1$#+! Rl→rcheck 2+.)!($. 01/%!))#) !)$
%! 3.+14!+ 5#'#( $!+*! t %6((. !)$ %! &7 86+*! )6#971$.! c.:.d l; <( 722&15#7($
&'<=>?@ %! &'!A!*2&! BC )#+ &! $!+*! F εl→r(t) %,)5#! (6#) 6"$!(6() #( 261($ 4A!
%! &'!()!*"&! %! $!+*!) )#00!))!#+) %7() T (Fbin)D )1 0!$ !()!*"&! !)$ 31%!D 7&6+)
t (! 06(06+%! 27) 73!0 l; @1(6(D &'!()!*"&! %! $!+*!) )#00!))!#+) !)$ 3+71*!($ #(
)1(-&!$6( %6$. %# $!+*! t *7+5#. : &7 26)1$16( %!) 37+17"&!) %7() l 73!0 #( )E*"6&!
)2.017& ⊕x 6F x !)$ #( 37+17"&! %! l;
!"!"!" #$%&'( )*+,-#. )' /01(234/2501 )*6373' )' 863567&'( 9 Rl→rσ
Rl→rσ !)$ #( <=>?@ (6#) 2!+*!$$7($ %! 06()$+#1+! #(! &1)$! 6+%6((.! %! $!+*!)
1(%!A.) 27+ %!) 37+17"&!) %! l; G!$$! &1)$! !)$ +!2+.)!($.! 27+ #( $!+*! !$ ).*7(/
$15#!*!($ 7# 06#+7($D (6#) 26#36() 06()1%.+!+ 0!$$! &1)$! 06**! &7 )#")$1$#$16(
+.)#&$7($ %# 27) !" !#$%" &; @61$ t′ &! $!+*! +.)#&$7($ %! (Rl→rcheck)
∗({F εl→r(t)});
(Rl→rσ )∗(F l→r
copy(t′)) *,(! : &'#(15#! $!+*! t′′ +!2+.)!($7($ &7 )#")$1$#$16( +.)#&$7($
%! &7 !" !#$%" & !($+! t !$ l;
H6#) 1($+6%#1)6() &7 (6$16( %'%#'#& $& (%#)%'*&+ 06()$+#1$! : 27+$1+ %'#( $!+*!;
I6#+ )1*2&14!+D &!) 37+17"&!) %! l )6($ +!(6**.!) %7() %!) 37+17"&!) xi 73!0 i .$7($
#( !($1!+;
!"! #$%&' ()*+,-./01,' 2(#$%&'3 4
5*6-0.07- !"# t $% #&'(& )& T (Fbin,X )* !"# x0, . . . , xn +&, -.'"./+&, .00.1
'.",,.%# ).%, t )& 2.$34& 5 )'!"#&* 67.'/'& )& -.'"./+&, -")& 8.'/'& )& -.'"./+& &%
/'&9: )& t; )<%!#< 0.' ⊤X (t); &,# )<=%" 0.' +& #&'(& ,$"-.%# > ⊤(⊕x0(⊥),⊤(⊕x1(⊥
), . . . ,⊤(⊕xn(⊥),⊥⊤) . . .))*
(8,+9), :; !"# t $% #&'(& )& T (Fbin,X ) #&+ ?$& t = a(b(x0,⊥), c(d(⊥, x1), x2))*
@"%,";
⊤X (t) = ⊤(⊕x0(⊥),⊤(⊕x1(⊥),⊤(⊕x2(⊥),⊥⊤))).
A!%,")<'&' +& #&'(& t; ,!"# t′ &,# +& #&'(& a(b(a(⊥,⊥),⊥), c(d(⊥,⊥), a(⊥,⊥)))*
A+."'&(&%#; t =+#'& t′*
!"# !$%&'(# "' )*+,- '!". /�&%%1'% 2& 3!'.%#"(#& 451#$#& 2& 61#(1$4&. ('7
.%1'3(8 2& t9 '!". 16!'. 45('%&'%(!' 2& 28:'(# "' *+,- %&4 ;"&9 45"'(;"& %�&
133&..($4& 2& t′ &.% ⊤(a(⊥,⊥),⊤(⊥,⊤(a(⊥,⊥),⊥⊤)))<
=1 2(>3"4%8 (3( &.% ;"& '!". 6!"4!'. .&"4&0&'% 2&. )*+,-.< =& /!('% 348 &.% 4&.
.?0$!4&. ⊕x< )@&3%(6&0&'%9 t′′3!'%(&'% 2& %&4. .?0$!4&.< !"# 3A1;"& .!".7%�&
s 2& t′′ %&4 ;"& s = ⊕x,p(s′)9 '!". /!"6!'. 45('%&#/#8%&# 2& 41 01'(B#& ."(61'%& C
/!"# 41 ."$.%(%"%(!' σ %&44& ;"& t = t′σ9 σ(x) = s′< D!". /!"6!'. 2!'3 3!'.(28#&#
4&. .?0$!4&. ⊕x 3!00& 2&. 01#;"&"#. /!"# ('(%(14(.&# 41 3!/(& 2& σ(x)<
!"# #8."0&# 41 .(%"1%(!' 3!"#1'%&9 /!"# 2&"E %�&. t ∈ T (Fbin) &% t′ ∈
T (Fbin,X )9 /!"# 4&.;"&4. (4 &E(.%& "'& ."$.%(%"%(!' σ : X → T (F) %&44& ;"& t′σ = t9
F< '!". /!"6!'. 3!'.%#"(#& "' %�& t′′ 2!%8 2&. 01#;"&"#. 1?1'% 4& 4(&' 16&3 σ G
H< I&. 01#;"&"#. .!'% 8J14&0&'% 21'. 451#$#& 2&. 61#(1$4&.⊤X (t′) &% 4&"# /!.(%(!'
21'. ⊤X (t′) &.% 3!''"& /"(.;"& t′ &.% 2!''8<
=& $"% 2& 45)*+,- 2!''8 21'. 41 28:'(%(!' K &.%
F< 2& 68#(:&# ;"5"' %�& t′ &.% 2& 41 L!#0& 2!''8& MtN
H< 2& 01#;"&# 21'. t′ 41 /!.(%(!' 2& 61#(1$4&. 2& t &%
O< 2& 3!/(&# .?0$!4& /1# .?0$!4& 3A1;"& 614&"# ."# 2&. 01#;"&"#. 21'. 451#$#&
2&. 61#(1$4&. t<
5*6-0.07- ; 2Rl→rσ 3 !"# l → r $%& 'B2+& )& T (Fbin,X ) × T (Fbin,X )* !"#
x0, . . . , xn +7&%,&(/+& )& -.'"./+&, .00.'.",,.%# ).%, l &# ?$" 0&$# C#'& +$ )& 2.$34&
5 )'!"#& ).%, l* @"%,"; +7DEFG Rl→rσ−compute &,# )<=%" 3!((& ,$"# >
! F l→rcopy(x)→ Dcopy(RWx0 (RWx1 (. . . RWxn (⊤(F ε
t (x),⊤X (t))) . . .)))
"! RWxi (x)→ RWxi (RWxi (x))
#! RWxi (x)→ NMAxi (x)
$! {NMAxi (a(x, y)) → a(NMAxi (x), NMAxi (y)) | a ∈ Fbin}
%! {NMAxi (⊕xj (x))→ ⊕xj ,(x) | xi &= xj}
! "#$%&'() *+ ,-.'/0). 1) 23435(&'6() 7895'&899):.
! NMAxi (⊥)→⊥
"! NMAxi (⊤(x, y)) → ⊤(NMAxi (x), NMAxi (y))
#! NMAxi (⊥⊤) →⊥⊤
$! RWxi (x) → RWxi,a(x) %&'( a ∈ Fbin
)*! RWxi,a(⊤(x, y)) → ⊤(Rxi,a(x), SWxixi,a(y)) %&'( a ∈ Fbin
))! {SWxx0,a
(⊤(x, y)) → ⊤(Wx,a(x), y)} ∪ {SWxxi,a
(⊤(x, y)) → ⊤(x, SWxxi−1,a
(y)) | i > 0 ∧ i ≤ n∧ x ∈
Var(t)}
)+! Rx,a(g(x, y)) → g(Rx,a(x), y) %&'( g ∈ Fbin
),! Rx,a(g(x, y)) → g(?x(x), Rx,a(y)) %&'( g ∈ Fbin
)-! Rx,a(⊕x(x)) → R⊕x,a (x) %&'( a ∈ Fbin
).! R⊕x,a (a(x, y)) → a(⊕x(x),⊕x(y))
) ! R⊕x,⊥(⊥) →⊥
)"! Wx,a(g(x, y)) → g(Wx,a(x), y) %&'( g ∈ Fbin
)#! Wx,a(g(x, y)) → g(?⊕x (x),Wx,a(y)) %&'( g ∈ Fbin
)$! Wx,a(⊕x(⊥)) → a(⊕x(⊥),⊕x(⊥))
+*! Wx,⊥(⊕x(⊥)) →⊥
+)! Dcopy(⊤(x, y)) → D′copy(⊤(y,D(x))
++! D′copy(⊤(x,⊥)) → x
+,! D(x) → D(D(x))
+-! D(⊥) →⊥
+.! D(x) → D′(x)
+ ! D′(g(x, y)) → g(D′(x), D′(y))
+"! D′(⊥) →⊥
+#! D′(g(x,⊥)) → x
+$! ?⊕x (g(x, y)) → g(?⊕x (x), ?⊕x(y)) %&'( g ∈ Fbin '/ x ∈ Var(t)
,*! ?⊕x (⊥) →⊥
!"# $%&'()!*# +, '-+, $,# './+,# $, '%0%&'(1"', 2'%#,*1%,# $3*# +3 $%4*(1(!* 56
73 './+, 86 ,#1 "*, './+, 2'(*&(23+, !" "*, &!92!#(1(!* $,# !2%'31(!*#6 :,11,
'./+, +3*&, +3 &!2(, +3 )3+,"' !;1,*", 23' &<3=", )3'(3;+, 32'.# 3)!(' 93'="%, =", +,
1,'9, #1!&>% $3*# +3 )3'(3;+, x6 ?*, &!2(, $" 1,'9, %1(=",1% ,#1 %13;+(, @ +3 '3&(*,
$, +A3';', $, )3'(3;+,#6
;<)0%:) => !"# t′ $% #%&'% a(b(a(⊥,⊥),⊥), c(d(⊥, b(⊥,⊥)), c(⊥,⊥))) %# (!"# t $%
#%&'% a(b(x0,⊥), c(d(⊥, x1), x2)).
)* +&%'",&% &,-$% %(# .!/0
Fcopy0t(x)→
Dcopy(RWx0(RWx1(RWx2(⊤(Fεt (x),⊤(⊕x0(⊥),⊤(⊕x1(⊥),⊤(⊕x2(⊥))))))))).
)* &1210&"#3&% +%3# 4#&% &1*$"(1% (3& $% #%&'% Fcopy(t′) %# +3"( $% #%&'%
Dcopy(RWx0(RWx1( RWx2(⊤(Fεt (t
′),⊤(⊕x0(⊥),⊤(⊕x1(⊥),⊤(⊕x2(⊥)))))))))5
6/ 3#"$"(*/# $%( &,-$%( .% Rtcheck .!//1%( .*/( $7%8%'+$% 9:; /!3( !<#%/!/( $%
#%&'% (3"=*/# >
Dcopy(RWx0(RWx1(RWx2( ⊤(t′′,⊤(⊕x0(⊥),⊤(⊕x1(⊥),⊤(⊕x2(⊥),⊥⊤))))))))
!? t′′ %(# $% #%&'% a(b(⊕x0(a(⊥,⊥)),⊥), c(d(⊥,⊕x1(b(⊥,⊥))),⊕x2(c(⊥,⊥))))5
!"! #$%&' ()*+,-./01,' 2(#$%&'3 45
! "#$%& '( )*+,-.& /0102 2304&!0 *!) 5& %! ,3*-& 5& %! 4!%&0" )63,7+& 5!2) %!
4!"-!8%& xi 932,6-322&( :;&,6-4&<&26= 5!2) %& <&-%%&0" ,!)= 02& !**%-,!6-32 5& RWxi
,3""&)*325 > %! ,3*-& 5102 )?<83%&= !-2)- > ,3*-&" &26-#"&<&26 %! 4!%&0" )63,7+&
5!2) xi= 230) !432) 8&)3-2 51!06!26 <3-2) 50 23<8"& 51!**%-,!6-32) /0& %& 23<8"&
5& *3)-6-32) 932,6-322&%%&) 5!2) %! 4!%&0"(
(6,+7), 58 !"#$%&' () !)*+) t′′ #) (,)-)+.() .*/0/#)&!1 '"* () '2+3%() 4%&0!$%&&)(
⊕x2 5 $( 2 6 !*%$' .%'$!$%&' 4%&0!$%&&)(()' 0%**)'.%&! 6"- '2+3%()' c5 ⊥ )! ⊥1 7$&'$
(6 *89() RWx2(x) → RWx2(RWx2(x)) #):*6 ;!*) 6..($<"/) 6" +%$&' #)"- 4%$' .%"*
"&) 0%.$) 0%+.(8!) #) (6 :6()"* '!%0=/) '"* ⊕x2 1
>"$'5 &%"' .%":%&' %3!)&$* () !)*+) t′′′ =
Dcopy(RWx0(RWx1(RWx2(RWx2(RWx2(⊤(t′′,⊤(⊕x0(⊥),⊤(⊕x1(⊥),⊤(⊕x2(⊥
),⊥⊤))))))))))1
! "#$%& @( &A*"-<& /0& %! ,3*-& 5& %! 4!"-!8%& xi &)6 6&"<-2+ 5#) /01-% 21? !
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4!"-!8%& xi( :2 9!-6= )1-% &A-)6& 02 6&% <!"/0&0"= *0-)/0& ,& ,!) &)6 *!) 6"!-6+ *!" %!
"#$%& B= !%3") %& )?<83%& 932,6-322&% NMAxi 2& *&06 *!) F6"& &;!,+ &6 *0-) !0,02
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5& ,3*-& 5&4"!-6 F6"& %!2,+(
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%! "#$%& H( )*+,-.& /0& %! %&,60"& &6 %1+,"-60"& )-<0%6!2+&) )326 3"5322+&) *30" 02
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&) "#$%&) IN > IB &A*%-/0&26 ,3<<&26 230) 4+"-.32) /0& %& )?<83%& )0-4!26 >
%-"& 5!2) %& 6&"<& )63,7+ 5!2) %! 4!"-!8%& x &)6 4"!-<&26 ,&%%& )0**3)+& -26+$"+& 5!2)
%& )?<83%& 932,6-322&% Rx,a( G%0) *"+,-)+<&26= Rx,a &)6 "+,0")-4&<&26 !**&%+ 5!2) %&
63* 50 6&"<& 5& %1!"8"& 5& 4!"-!8%&( O36&P /0& %! "#$%& II( )*+,-.& /0& 230) ,E&",E32)
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!0,02 6&"<& 5& T (Fbin) )&"! *3))-8%& 5& 5+"-4&"( ! %&,60"& 50 )?<83%& 932,6-322&%
)0-4!26 &)6 )*+,-.+& *!" %&) "#$%&) I' &6 I@( :;&,6-4&<&26= %&) )+<!26-/0&) 5& Rx,a
)326 /0& %& )?<83%& )0-4!26 > %-"& 53-6 F6"& a( S#) /0& %& <!"/0&0" 5& 4!"-!8%& x
&)6 <-) 5!2) %! "#$%& I'= %& *!) )0-4!26 &)6 5& 4+"-.&" /0& %& )?<83%& )0-4!26 &)6
&;&,6-4&<&26 a = &6 )- ,1&)6 ,& ,!)= %&) <!"/0&0") )0-4!26) )326 +6!8%-) 5-"&,6&<&26
)0" ,& )?<83%&( & *"3,&))0) 51+,"-60"& 5!2) %1!"8"& 5& 4!"-!8%&) &)6 *"3,E& 5& ,&%0-
! "#$%&'() *+ ,-.'/0). 1) 23435(&'6() 7895'&899):.
!"#$% "$& '(()(* (+), -)' . /0( -)' 1' 2+#-)')# '(% %#3)4! 5(6!"$7! 6+# 1+ #081' 9:;*
)< <3)4'+) %'#2' '(% "#!! c.=.d a(⊥,⊥)>
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'(% 1' 2D2' 63)# 1'( 4+#$+C1'( x1 '% x0>
E% 7<+1'2'<%* 0( -)' 1+ "36$' '( 4+1')#( (%3"B!'( +<( '( 4+#$+C1'( '(%
%'#2$<!'* 1' %'#2' +) %36 ' 1F+#C#' ' 4+#$+C1'( '(% 3C(310%'> G13#(* 1'( #081'( 9H
= AI ' 1+ !7<$%$3< J !"#$4'<% "322'<% !%#)$#' 1' %'#2' 3C(310%' '% "322'<%
#'%3)#<'# = 1F+#C#' ' 4+#$+C1'( 7<+1 $<(%+<"$!>
*+!+!+; 2/<:). 1=>7?2, 1) ').' 1) .6@.'&'6'&89 A Rl→rGS
Rl→rGS '(% )< EKLMN (6!"$7+<% 1+ 4!#$7"+%$3< F)<' ()C(%$%)%$3< C$'<&,3#2!'>
EO'"%$4'2'<%* ($ l <F'(% 6+( 1$<!+$#' +13#( )<' 4+#$+C1' x +66+#+P% ')Q ,3$( +)
23$<(> N$ %3)%'( 1'( 3"")##'<"'( ' x 6+#%+8'<% 1' 2D2' %'#2' "322' 4+1')# +13#(
1+ ()C(%$%)%$3< '(% C$'<&,3#2!'> (Rl→rcheck)
∗(F l→rcheck(t
′′)) = {t′′} '(% 1+ ()C(%$%)%$3<
C$'<&,3#2!'* ∅ ($<3<> Rl→rGS %#+$%' "R+-)' 4+#$+C1' +66+#+$((+<% 61)( F)<' ,3$(
"322' 2'<%$3<<! "$& '(()(>
B3C9&'&89 D ERl→rGS F !"# l → r $%& '()*& +& T (Fbin,X ) × T (Fbin,X ), !"#
x0, . . . , xn *&- ./'"/0*&- +& l +& )/$12& 3 +'!"#&, !"# Indexes *4&%-&50*& +& 1!$6&-
+4"%+&7&- +& ./'"/0*& 1!''&-6!%+/%# 3 */ 585& ./'"/0*& "%"#"/*&, !"# Rl→rGS *4 9:;<
'=-&'.= 3 .='">&' ?$& +&$7 ./'"/0*&- !%# *& 585& #&'5& -$' *4/'0'& +& ./'"/0*&- "%@
-#/%#/%=, A"%-"B Rl→rGS &-# 1!56!-= 6/' +&- '()*&- -$"./%#&- C
! F l→rcheck
(x) → CForin,jn (CForin−1,jn−1 (. . . CFori1,j1 (x) . . .)) "#$% Indexes ={(i1, j1), . . . , (in, jn)}
&! PutMarksi(⊤(x, y)) → ⊤(x, PutMarksi−1(y)) '()* i > 1
+! PutMarks0(⊤(x, y)) → ⊤(⊕=(x), y)
,! CFori,j(x) → Test=i,j (PutMarksi(PutMarksj(x)))
-! Test=i,j (x) → Test=i,j (RRi,j(x))
.! Test=i,j (x) →?x,p(x)
/! {RRi,j(x) → RRi,j,g(x)} "#$% g ∈ Fbin
0! {RRi,j,g(⊤(x, y)) → ⊤(x,RRi−1,j−1,g(y))} "#$% g ∈ Fbin $1 i > 0
2! {RR0,j,g(⊤(x, y)) → ⊤(Rg(x), Rj−1,g(y))} "#$% g ∈ Fbin $1 i > 0
3! {RRi,g(⊤(x, y)) → ⊤(x,Ri−1,g(y))} "#$% g ∈ Fbin $1 i > 0
! {RR0,g(⊤(x, y)) → ⊤(Rg(x), y)} "#$% g ∈ Fbin $1 i > 0
&! {Rg(g(x, y)) → g(Rg(x), y), Rg(g(x, y)) → g(?⊕= (x), Rg(y))} "#$% g ∈ Fbin
+! Rg(⊕=(x)) → Rg,⊕= (x)
,! Rg,⊕= (g(x, y)) → g(⊕=(x),⊕=(y))
-! R⊥,⊕= (⊥) →⊥
*+!+!+* 2/<:). 1=>7?2, 1) .6@.'&'6'&89 A Rl→rσ−apply
Rl→rσ−apply '(% )< EKLMN (6!"$7+<% 1F+661$"+%$3< ' 1+ ()C(%$%)%$3< C$'<&,3#2!'
#!()1%+<% ' t '% l ()# 1' %'#2' r> G$<($* (Rl→rσ−apply)
∗(F l→rrewrite(t
′′)) = {rσ}>
!"! #$%&' ()*+,-./01,' 2(#$%&'3 45
!"#$% &'( &)( &)( &*( +,( -*
⊕x0
a
RWx2
RWx2
RWx1
RWx0
Dcopy
RWx2
RWx2
RWx1
RWx0
Dcopy
RWx1
RWx0
Dcopy
RWx1
RWx0
Dcopy
RWx2RWx2
RWx2
RWx1
RWx0
Dcopy
RWx2RWx2
RWx2
RWx1
RWx0
Dcopy
c
⊕x2⊕x2
⊥ ⊥
RWx2
RWx1
RWx0
Dcopy
RWx2
RWx1
RWx0
Dcopy
⊕x2
c
⊥ ⊥
c
⊤
⊤
⊤
⊤⊕x1
⊕x0
⊥
⊥
⊥
⊥⊤
a
c
⊥⊤
⊤
⊤
⊤
⊤⊕x1
⊥
⊕x0
⊥
c
⊕x2⊕x2
⊥ ⊥
Rx2,⊥(a)
c
⊥⊤
⊤
⊤
⊤
⊤⊕x1
⊥
⊕x0
⊥
c
⊥
⊥
c
⊥ ⊕x2
⊥
a
c
⊥⊤
⊤
⊤
⊤
⊤⊕x1
⊥
⊕x0
⊥
c
⊥ ⊥
c
⊥ ⊥
Rx2,c(⊕x2)
Wx2,c(⊕x2)
Wx2,⊥(⊕x2)
a
⊥
⊥ ⊥ b
⊥ ⊥
⊕x1
⊥ d ⊕x2
c
⊥ ⊥
b c
RWx2RWx2RWx2
RWx2
RWx1
RWx0
Dcopy
⊤
⊤
⊤
⊤
⊕x2
⊕x1
⊕x0
⊥
⊥
⊥
⊥⊤
c
⊥⊤
⊤
⊤
⊤
⊤⊕x1
⊥
⊕x0
⊥
c
⊕x2
⊥ ⊥
c
⊕x2⊕x2
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a
c
⊥⊤
⊤
⊤
⊤
⊤⊕x1
⊥
⊕x0
⊥
c
⊥ ⊕x2
⊥
c
⊥ ⊕x2
⊥
a
c
⊥⊤
NMAx2(⊤)
⊤
⊤
⊤⊕x1
⊥
⊕x0
⊥
c
⊥ ⊥
c
⊥ ⊥
a
c
⊥⊤
⊤
⊤
⊤
⊤⊕x1
⊥
⊕x0
⊥
c
⊥ ⊥
c
⊥ ⊥
Rx2,⊥(a)
!"#$% .( +( '( /( )
Wx2,⊥(⊕x2 )
!"#$% .'( .)( .*( &,( +*
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!"#$% &'( &)( &+( &*( +,( -*
!"#$% &( -( 0( *( .,( -)
!"# !" # $%&'( )( *+ ,+*(-. /0%123( )+4/ x2
! "#$%&'() !* +,-'./)- 0) 12324(&'5() 6784'&788)9-
:2;8&'&78 < =Rl→rσ−apply> !"# l → r $%& '()*& +& T (Fbin,X ) × T (Fbin,X ), -"%."/
Rl→rσ−apply 0!%#"&%# *&. '()*&. .$"12%#&. 3
! F l→rrewrite(x) → D⊤(RW rew
x0(RW rew
x1(. . . RW rew
xn(⊤(rσl→r, x)) . . .)))
"! RW rewxi
(x) → RW rewxi
(RW rewxi
(x))
#! RW rewxi
(x) → NMAxi (x)
$! RW rewxi
(x) → RW rewxi,a
(x) %&'( a ∈ Fbin
)! RW rewxi,a
(⊤(x, y)) → ⊤(Wxi,a(x), SRxi,a(MSxixi
(?⊕xiy))) %&'( a ∈ Fbin
*! {MSzx0
(⊤(x, y)) → ⊤(⊕z(x), y)}∪{MSzxi
(⊤(x, y)) → ⊤(x,MSzxi−1
(y)) | i > 0∧i ≤ n∧z ∈ Var(r)}
+! {SRx0,a(⊤(x, y)) → ⊤(Rx0,a(x), y)} ∪ {SRxi,a(⊤(x, y)) → ⊤(x, SRxi−1,a(y)) | i > 0 ∧ i ≤ n}
,! D⊤(⊤(x, y)) → D⊤(⊤(x,D(y))
-! D⊤(⊤(x,⊥)) → x
.! D′(⊤(x, y)) → ⊤(D′(x), D′(y))
! D′(⊤(x,⊥⊤)) → x
"! D′(⊤(x,⊥)) → x
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'&#'./%' 0' ,12%'# 0-LVXYZ 0' 3*,3"%$4*!& Rl→rvisit 4'% +"'
Rl→rvisit = {Fl→r(a(x, y))→ a(Fl→r(x), y), Fl→r(a(x, y))→ a(x, Fl→r(y))}
'4 "& '&#'./%' 0' ,12%'# 0-LVXYZ 0' ,'%$&3' Rl→rTV 4'% +"'
Rl→rTV = {Fl→r(x)→ F l→r
rewrite(Fl→rcheck(F
l→rcopy( F
εl→r(x))))}F
&!"# #!..'# 3$)$/%'# 0' #*."%', %$ ,5W53,*4",' 0-"& XYZ8
!"! #$%&' ()*+,-./01,' 2(#$%&'3 44
!"# $%&'$("!)# ("% *%# $+,*%# -./0123 -% 45$4"*'65!) Rl→rvisit !"# $%&'%()
(% ( *% +,&+"-%& -% ! )(%&', .- Fl→r / (!"# -%# % *&!,(# *" (%&'% *! 0 1". *
-%# &23-%# *456789 *% &%-. +% Rl→rTV %#( " % +!'$!#,(,! *% : ! )(%&', .";
F l→rrewrite, F
l→rcheck, F
l→rcopy %( F ε
l→r <+!&%##$! *. ( .=%+ : % #%'>-%# *% &23-%# *0? ,# +,)
*%##"#@A
5 +! #01"% +%B !"# .=! # -% &0#"-(.( $!"& " $.# *% &0)0+&,("&% +!''% #",( C
!"#"$%&%"' ( !"# R $% &' ($) T (Fbin,X ) × T (Fbin,X )* α ∈ T (Fbin)* β ∈
T (Fbin) +%, l→ r ∈ R- ./ 01"(#0 $% 23&' Rλe #0/ 4$0
α→{l→r} β ⇔ Fl→r(α)→∗Rλe
β.
!)*+) , - !)*+) $*..%'.&) -/"%! .01 23'$ 453'')6) 788 50 6!"%# 7/8 0(# ,0
7!%(#)$")0 $% 23&' Rλe 4$" "96/890%#0 /0 6)!70(($( ,0 )8:87)"#$)0-
50 23&' Rλe 0(# 7!%(#)$"# 7!990 /0 ($";+%# <
Rλe = Rl→rvisit ∪R
l→rTV ∪Rl→r
check ∪Rl→rσ ∪Rl→r
GS ∪Rl→rσ−apply
= α →{l→r} β ⇒ Fl→r(α) →∗Rλe
β < 2>07#";090%#* F l→rrewrite(t) →
∗Rl→rσ−apply
rσ ∈
T (Fbin)- 2% 7!%(84$0%70* α[F l→rrewrite(t)]p →∗
Rl→rσ−apply
α[rσ]p ∈ T (Fbin) 0#
6$"(4$0 α[rσ]p = β* β ∈ R∗λe({Fl→r(α)}).
= α →l→r β ⇐ Fl→r(α) →∗Rλe
β < 5+ 6)0$;0 6!$) 70 7+( 0(# 6)!7?0 +$ 7+(
("9"/+")0 #)+"#8 ,+%( /+ 6)0$;0 ,0 /+ 6)!6!("#"!% @ ,+%( /0 (0%( 4$0 /0( 6+( ,0
)8:87)"#$)0 (!%# !),!%%8( 0# ,860%,+%#( ,$ #0)90 "%"#"+/ c.A.d Fl→r(α)-
B"%("* 6!$) )8($98* "/ 01"(#0 $%0 6!("#"!% ,0 α 0# $%0 ($C(#"#$#"!% σ : X →
T (Fbin) #0//0( 4$0 α|p = lσ* β = α[rσ]p 0# Fl→r(α)→∗Rλe
β-
D!$( 6!$;!%( ,8,$")0 4$0 α→l→r β ⇒ Fl→r(α)→∗Rλe
β 4$" 7!%7/$# /+ 6)0$;0-
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0(.(# .++%##,>-%#B 0(. ( 03.-%'% ( -% &0#"-(.( *% -. (&. #D!&'.(,! <+@ *% -. ?3"&%
:AEA
9:;"!<=) ( !"# R $% &' ($) T (Fbin,X )× T (Fbin,X ) 0# E ⊆ T (Fbin)- B"%("*
"/ 01"(#0 $% 23&' Rλe 0# $% (E9C!/0 G ∈ FNT #0/( 4$0 <
R∗(E) = R⋆λe(E
′),
!F E′ = {G(α)|α ∈ E}.
!)*+) > G#+%# ,!%%8 /H23&' Rλe =⋃
l→r∈R(Rl→rvisit ∪ R
l→rTV ∪ Rl→r
check ∪ Rl→rσ ∪
Rl→rGS ∪ Rl→r
σ−apply) ∪ RFP +;07 RFP = {G(a(x, y)) → G(Fl→r(a(x, y))) | l → r ∈
R ∧ a ∈ Fbin} ∪ {G(a(x, y)) → a(x, y) | a ∈ Fbin}* /+ 6)0$;0 0(# #)I( 6)!7?0 A 70//0
,$ #?8!)I90 @-
F. # +%((% #%+(,! B !"# .=! # $&!"=0 1"% #!,( " 789B ,- %;,#(% " 56789
+!&&%#$! *. (A 5 #",(%B / $.&(,& *%# 56789#B !"# $!"=! # -%# +! =%&(,& =%&# *%#
789# +!''% *. # -. #%+(,! :AEA
! "#$%&'() *+ ,-.'/0). 1) 23435(&'6() 7895'&899):.
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!"#$%& $' ()$*%( +! ,-(.$! /( 0123* 4(5* 0123* (! 6#**#!% 6#5 '(* 78,-(*
9:; (% 9/; /( '# <=+5( >?@? A+%5(.(!% /$%& B(6(!/#!%& '# %5#!*"C5.#%$C! /$5(,%( /(
'# <=+5( >?@ 9:; /( 0123 4(5* 123 !D(*% 6#* %C+EC+5* "#$*#:'(& !C+* /(4C!* #'C5*
65C,F/(5 /( .#!$85( $!/$5(,%(?
BC!*$/F5C!* +! ()(.6'( G
;<)0%:) => !"!#$#% & '(!)!*+'! ,- ."!/ '! 01 2 OnlyOneRequest 3(401 2
/$55!%+$#6.#7 !%7 '! %89".#7 :
Rλe =
OnlyOne(t(x, y))→ t(OnlyOne(R(x)), I(y))
OnlyOne(t(x, y))→ t(I(x), OnlyOne(R(y)))
OnlyOne(r(x, y))→ r(OnlyOne(R(x)), I(y))
OnlyOne(r(x, y))→ r(I(x), OnlyOne(R(y)))
OnlyOne(r(⊥,⊥))→ r(⊥,⊥)
R(r(⊥,⊥))→ r(⊥,⊥)
I(i(⊥,⊥))→ i(⊥,⊥)
.
HC+* 65C6C*C!* /#!* ,(%%( *(,%$C! +!( %5#!*"C5.#%$C! /$5(,%( /+ 0123 4(5*
+! (!*(.:'( /( 58='(* /( 0123* I+$ *C!% *+J*#..(!% ()65(**$4(* 6C+5 '# *6FK
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5FKF,5$%+5( +%$'$*#!% Rλ 9c.L.d +!( 58='( 0123 /( '# "C5.( H (t) → α; 6(+% M%5(
*$.+'F 6#5 +! 0123 Rλe ?
@? /( %5C+4(5 +!( 6C*$%$C! 6C+5 '# 5FKF,5$%+5(&
N? /( 4F5$<(5 *$ H (t) ,C!,C5/( #4(, '( *C+*K%(5.( ,C+5#!% (! ,C!*$/F5#!% H
,C..( +! *O.:C'( *6F,$#' (! Fbin& 6+$*
P? /( ,#',+'(5 '# *+:*%$%+%$C! 5F*+'%#!% σ& (%
>? /( 5(.6'#,(5 '( *C+*K%(5.( 6#5 ασ?
Q# %5#!*"C5.#%$C! 9/; /( '# <=+5( >?@ (*% /C!!F( 6#5 '# 65C6C*$%$C! *+$4#!%(?
?(8%8.&'&89 @ 2$97 Rλ 8# 01 2 α ∈ T (Fbin); <' !)9%7! 8# 401 2 Rλe 7!' =8!
{α} ∪⋃
H∈FNT ,β∈Reachables
(R⋆λ({H(β)})) = {α} ∪
⋃
H∈FNT ,β∈Reachables
(Rλe⋆({H(β)})).
?()6A) ! >7.#7 6$##? '(401 2 Rλe @ '. +5!8"! !%7 75A% +5$/B! & /!''! 6! '. +5$+$C
%979$# D;
3! +$9#7 /'? !%7 6! /$#%75895! 8# 401 2 Rλe =89 9*+'?*!#7! '! +5$/!%%8% 6!
5?C?/59785! !# /$#%96?5.#7 H /$**! 8# %E*F$'! %+?/9.' !# Fbin !7 !# 9G#$5.#7 '!%
5AG'!% 6! /95/8'.79$#;
3! 401 2 Rλe !%7 /$#%75897 /$**! '! %89".#7 :
Rλe = RH(t)→α
TV ∪RH(t)→α
check ∪RH(t)→ασ ∪R
H(t)→α
GS ∪RH(t)→α
σ−apply
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,* +($!(1*( ,>"& 7896: ?*(# ,*# 67896:#<
!"!# $%&'(&)*+
G*++* #*-+.!& '()#*&+* "&* #!"# -%$##* ,* #/#+01*# 2!&-+.!&&*%# )%)1*&+$.(*#3
%*# )%)1*&+$.(*# 4 ,(!.+* 567896:#;3 '()#*(?$&+ %$ '".##$&-* ,>*B'(*##.!& ,*# #/#A
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%*# 67896:# *#+ '%"# #.1'%* D"* %*# $"+(*# 896:#3 &!+$11*&+ %$ '(!-),"(* ,*
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67896:# '*"+ $"@1*&+*( %>*I-$-.+) ,* %$ +*-E&.D"* ,* -$-E* !" #$$ *+ %>$%@!A
(.+E1* ,* #$+"($+.!&<
7+$&+ ,!&&) "& 7896: Rλe 3 &!"# $%%!&# %* +($.+*( '$( %>$&$%/#* ,* -$#<
J< H(⊥) → β $?*- β ∈ T (Fbin ∪ FNT ) K G* +/'* *#+ -!&#.,)()* -!11* "&
%&'(%) *+,- . K Rλe→A<
L< H(a(x,⊥))→ x $?*- x ∈ X 3 H ∈ FNT $&, a ∈ Fbin K G* +/'* *#+ -!&#.,)()*
-!11* "& %&'(%) *+,- # K Rλe→B<
M< H(a(x, y))→ α $?*- a ∈ Fbin3 H ∈ FNT 3 x, y ∈ X 3 α ∈ T (Fbin ∪ FNT ,X ) *+
Var(α) = {x, y} K G* +/'* &>*#+ '$# *&-!(* /!/0-1*"23- $" -N+) ,(!.+ -!11*
%*# $"+(*#< O$( -!&#)D"*&+3 &!"# +($.+!&# ,*# +/'*# ,* H(a(x, y)) → α '!"(
#.1'%.P*( %* -N+) ,(!.+ ,* -* +/'* ,* (0@%*<
!"# $%%!&# ,.#+.&@"*( +(!.# +/'*# ," -N+) ,(!.+ ,* %$ (0@%* H(a(x, y))→ α $?*-
a ∈ Fbin3 H ∈ FNT 3 x, y ∈ X 3 α ∈ T (Fbin ∪ FNT ,X ) *+ Var(α) = {x, y}< Q*# ,*"B
'(*1.*(# 67896: +/'* F *+ +/'* FF D". -!&#*(?*&+ %* #/1C!%* a $'(0# $?!.( $''%.D")
%* &!&A+*(1.&$% H #!&+ $''*%)# 4516-37/ !" 8530-< R$%@() %* +/'* +(0# #.1'%* D".
(*1'%.+ C.*& %*# -!&,.+.!&# '!"( %*# +*-E&.D"*# ,>$--)%)($+.!& C$#)*# #"( %>$%@!(.+E1*
,* #$+"($+.!&3 %* 67896: +/'* F $''$($S+ +(0# #!"?*&+3 &!+$11*&+ ,$&# %$ -.(-"%$A
+.!&< Q* 67896: +/'* FF *#+ )@$%*1*&+ .1'!(+$&+* -$( .% &!"# '*(1*+ ,* -!1'!#*(
"&* #".+* ,>$"+$&+ ,>!')($+.!&# D"* &!"# ?!"%!&#< Q* +(!.#.01* 67896: +/'* FFF
D". &!"# '*(1*++*&+ ,* -E$&@*( %* %*++(* -!"($&+* a3 ,* '*(1"+*( ,*"B #!"#A+*(1*#
5-!&-!(,$&+ $?*- x *+ y;3 *+ ,>$T!"+*( "&* %*++(* 4 %>*B+)(.*"( ," +*(1* -!"($&+ *#+
$''*%) 151 9516-37/ !" 8530- <
9!"+ ,>$C!(,3 &!"# -!11*&U!&# '$( %* 67896: +/'* F 5$''*%) 4516-37/ !"
8530- :; %* +/'* %* '%"# #.1'%* ,*# (0@%*# ,* +/'* H(a(x, y)) → α< R$%@() %* +/'*
+(0# #.1'%* D". (*1'%.+ C.*& %*# -!&,.+.!&# '!"( %*# +*-E&.D"*# ,>$--)%)($+.!& C$#)*#
#"( %>$%@!(.+E1* ,* #$+"($+.!&3 *%%*# $''$($.##*&+ +(0# #!"?*&+3 &!+$11*&+ ,$&# %$
-.(-"%$+.!&<
! "#$%&'() *+ ,-.'/0). 1) 23435(&'6() 7895'&899):.
;3<9&'&89 ! !"# $ %&'( ) *
Rλe (+% ,-% !"# $ %&'( ) +- %./%(+ 0(+ 1230(+ 4&45% 67%8 34/69( H (a(x, y)) +.5%
+./+ :.1;(
< H (a(x, y))→ a (x, y)
< H (a(x, y))→ a(G (x), y)
< H (a(x, y))→ a(x,D (y))
4=(6 H,G,D ∈ FNT , a ∈ Fbin*
>./+ ,85.%.5+ '41 Rλe→I *
!"#$%&!&!'() '*"+ ,-.'$++*'+ /! 012304 (56! 77) 8"$ !+( !'9*:! ;66!/- ?.5@
+(1=8 04 :.1;() !( 8"$ '*"+ 6!:&!( ,<!#-9"(!: "'! 9*&6*+$($*' ,! ,!"# ;"(:!+ *6-:;=
($*'+> ?<!+( "'! :%@/! (:%+ $&6*:(;'(! 9;: !//! '*"+ 6!:&!( ,! 9*&6*+!: "'! +"$(!
,<;"(;'( ,<*6-:;($*'+ 8"! '*"+ A*"/*'+>
;3<9&'&89 = !"# $ %&'( ))*
Rλe (+% ,-% !"# $ %&'( )) +- %./%(+ 0(+ 1230(+ 4&45% 67%8 34/69( H (a(x, y)) +.5%
+./+ :.1;(
< H (a(x, y))→ G (D (a(x, y)))
4=(6 H,G,D ∈ FNT , a ∈ Fbin*
>./+ ,85.%.5+ '41 Rλe→II *
1' "($/$+;'( +!"/!&!'( ,!+ :%@/!+ ,! (56! 7 !( 77) '*"+ 6*"A*'+ +$&"/!: /; :%@/!
H (a(x, y))→ a(G (x), D (y)) 9*&&! /!+ +"$A;'(!+ B
C H (a(x, y))→ G′ (D′ (a(x, y)))
C G′ (a(x, y))→ a(G (x), y)
C D′ (a(x, y))→ a(x,D (y))>
D*"+ ;//*'+ ,-.'$: 9$=,!++*"+ /!+ :%@/!+ 6:$&$($A!+ 012304 (56! 777 8"$ '*"+
6!:&!((!'( ,! 9E;'@!: /; /!((:! 9*":;'(!) ,! 6!:&"(!: ,!"# +*"+=(!:&!+) !( ,<;F*"(!:
"'! /!((:! G /<!#(-:$!": ," (!:&! 9*":;'(>
;3<9&'&89 >? !"# $ %&'( )))*
Rλe (+% ,-% !"# $ %&'( ))) +A-0 (B-+%( /5( 1230( %(00( C/(
< H (a(x, y))→ b(x, y)
< H (a(x, y))→ a(y, x)
< H (a(x, y))→ a(⊥, a(x, y))
4=(6 H ∈ FNT et a, b ∈ Fbin*
>./+ ,85.%.5+ '41 Rλe→III *
,%35&<5$'&89 16 @AB )9 2C7@2,
?!((! +!9($*' &*'(:! 9*&&!'( /! 3HI 6!"( J(:! K*:&;/$+- !' 012304+ +*"+
K*:&! ,<"' 6:*L/%&! ,! 9;/9"/ ,<;99!++$L$/$(->
M!+ :%@/!+ ,! :-=-9:$(":!NO>PQR !( NO>PPR ,*''-!+ !' +!9($*' O>S 3HI !' 12304
+*'( ,-FG -/-&!'(;$:!+ G ,:*$(!>
!"! #$%&'( ) *+,-./ 0&#$%&'(1 23
!"# $%&# '(# !&$)(# *!# "%" +'+,("$!-)(#. "%&# *)+%"# /0!1%)/ &"( '-#$( /(#
"%"2$(),-"!&3 FNT "+*(##!-)(#4 5&-# *6!7&( H ∈ FNT 8! (9(*$&() &"( %:+)!$-%"
+'+,("$!-)(4 5!) (3(,:'(. '(# );<'(# /( )+2+*)-$&)( =>4?@ ($ =>4A@ /%""+(# /!"# '!
#(*$-%" >4B #%"$ $)!"#C%),+(# *%,,( '(# );<'(# /( DEFGDH Rreqλe →∪
-"/-7&+(# *-2
/(##%
H(i(x, y)) → H4.8(H′4.8(i(x, y))) !"#!$
H4.8(i(x, y)) → r(x, y) !"#%$
H′4.8(i(x, y)) → i(x,R(y)) !"#&$
H(i(x, y)) → H4.9(H′4.9(i(x, y))) !"#'$
H4.9(i(x, y)) → r(x, y) !"#($
H′4.9(i(x, y)) → i(R(x), y) !"#)$
H-,-'!-)(,("$. "%&# !8%"# '(# );<'(# /( *-)*&'!$-%" Rcirλe →∪ = Rcir
λe 4 I(# );<'(#
-"/-7&+(# *-2/(##%&# )(,:'-##("$ '! #:+*-J*!$-%" /& GK5 :!) &" DEFGDH4
H(t(x, y)) → H4.20(H′
4.20(t(x, y))) !"!*$
H4.20(t(x, y)) → b(x, y) !"!+$
H′4.20(t(x, y)) → t(x,RT (y)) !"!,$
H(t(x, y)) → H4.21(H′
4.21(t(x, y))) !"!#$
H4.21(t(x, y)) → b(x, y) !"!!$
H′4.21(t(x, y)) → t(RT (x), y) !"!%$
RT (r(x, y)) → t(x, y) !"!&$
H(b(x, y)) → H4.23(H′
4.23(b(x, y))) !"!'$
H4.23(b(x, y)) → t(x, y) !"!($
H′4.23(b(x, y)) → b(x, TI(y)) !"!)$
!"%*$
H(b(x, y)) → H4.24(H′
4.24(b(x, y))) !"%+$
H4.24(b(x, y)) → t(x, y) !"%,$
H′4.24(b(x, y)) → b(TI(x), y) !"%#$
H(t(x, y)) → H4.25(H′
4.25(t(x, y))) !"%!$
H(t(x, y)) → H′′′4.25(H′′
4.25(t(x, y))) !"%%$
H′′′4.25(t(x, y)) → i(x, y) !"%&$
H′4.25(t(x, y)) → t(I(x), y) !"%'$
H′′4.25(t(x, y)) → t(x, I(y)) !"%($
H(t(x, y)) → H4.26(H′
4.26(t(x, y))) !"%)$
H(t(x, y)) → H′′′4.26(H′′
4.26(t(x, y))) !"&*$
H′′′4.26(t(x, y)) → r(x, y) !"&+$
H′4.26(t(x, y)) → t(R(x), y) !"&,$
H′′4.26(t(x, y)) → t(x, I(y)) !"&#$
!"&!$
! "#$%&'() *+ ,-.'/0). 1) 23435(&'6() 7895'&899):.
H(t(x, y)) → H4.27(H′
4.27(t(x, y))) !"#$%
H(t(x, y)) → H′′′4.27(H′′
4.27(t(x, y))) !"##%
H′′′4.27(t(x, y)) → r(x, y) !"#&%
H′4.27(t(x, y)) → t(I(x), y) !"#'%
H′′4.27(t(x, y)) → t(x,R(y)) !"#(%
I(i(x, y)) → i(x, y) !"&)%
I(⊥) → ⊥ !"&*%
Arbiter(t(x, y)) → t(x, y) !"&+%
Arbiter(t(x, y)) → Arbiter(H(t(x, y))) !"&,%
Arbiter(b(x, y)) → b(x, y) !"&!%
Arbiter(b(x, y)) → Arbiter(H(b(x, y))) !"&$%
!"# $!%!$# &'( RTAPλe →∪ )* +,-.+/ 0!1&)*% 234$5 *$ 0*%%* #*0%5!$6 !"#
&!"7!$# !8%*$5( )9*$#*18)* 2* %*(1*# '00*##58)*# : &'(%5( 2" %*(1* ! "# $%#& *$
0')0")'$% (Rreqλe →∪
)⋆({Arbiter(u)})6 ;* &)"#< $!"# &!"7!$# 3=')*1*$% 73(54*( >"*
)* %*(1* ! "# '&()&* $+, v *#% '%%*5=$'8)* : &'(%5( 2" %*(1* ! "# $%#& u &"5# >"*
u→⋆Rreq
λe →∪
v6
!"!# $%&'()*%+&,-*' ./ 01$23( 4/%( 201$23(
?0%"*))*1*$%< 2*# +,-.+/#< &(3#*$%3# &'( )*# 234$5%5!$# @< A *% BC #!$% #"DE
5#'11*$% *F&(*##57*# &!"( )' #&30540'%5!$ 2" 130'$5#1* 2* (3E30(5%"(* &!"( "$*
(G=)* ,-.+/ 'H'$% 0I%3 ='"0J* H (a(x, y))→ α ∈ Rλe K
B6 2* %(!"7*( "$* &!#5%5!$ &!"( )' (3E30(5%"(*<
L6 2* 73(54*( #5 H (a(x, y)) 0!$0!(2* '7*0 )* #!"#E%*(1* 0!"('$%< &"5#
M6 2* 0')0")*( )' #"8#%5%"%5!$ (3#")%'$% σ< *%
N6 2* (*1&)'0*( )* #!"#E%*(1* &'( ασ '7*0 α O;*# #"8#%5%"%5!$# 2*# 7'(5'8)*#
&*"7*$% P%(* 1")%5&)53*#Q6
,$ 0!$#3>"*$0*< $!"# '7!$# )* (3#")%'% )5#%3 05E2*##!"#6 !"# (*1'(>"!$# >"*
)*# #H#%G1*# 2* (3E30(5%"(* &(515%57*# )5#%3# 05E2*##"# #!$% )5$3'5(*# 2(!5%*# *% #!$%
0'&'8)*# 2* #51")*( $951&!(%* >"*) #H#%G1* ,-.+/ )5$3'5(* 2(!5%* !" 1P1* $!$
)5$3'5(* 2(!5%*6
R' %('$#S!(1'%5!$ O*Q 2* )' 4="(* N6B *#% 2!$$3* &'( )' &(!&!#5%5!$ #"57'$%*6
;(8%8.&'&89 * -!$ Rλe )+ ./01- & α ∈ T (Fbin)2 3# &4$* & %&* 1./01-* Rλe→I 5
Rλe→II 5 Rλe→III 5 Rλe→A & Rλe→B 6&+ 7'&8 Rλe→∪9 &#* ()&
{α}∪⋃
H∈FNT ,β∈Reachables
(R⋆λe({H(β)})) = {α}∪
⋃
H∈FNT ,β∈Reachables
(Rλe⋆→∪({H(β)})).
! ! "#$%&'()#$ *&( (+(,-.&( /#$0,)#$$&1( %&'( 234( 56
7'&8%& 9 :7'&8%& (800)$0,& :;#)' 0/! *<$( 1=<$$&>& ?@@ !"#! $%##& '()
*+,-*.) Rλe→I / Rλe→II / Rλe→III / Rλe→A (! Rλe→B/ '" 01(23( ()! !14) 01%56(
7 5(''( $( '" 01%0%)8!8%# 9:
;( 0%8#! 5'& ()! $( 5%#)!1281( 2# *+,-*. <28 8=0'&=(#!( '( 01%5())2) $( 1&>
&518!21(:
;( *+,-*. ()! 5%#)!128! 5%==( '( )283"#! ?
Rλe→∪ = RH (a(x,y))→α
TV ∪RH (a(x,y))→α
check ∪ RH (a(x,y))→ασ ∪R
H (a(x,y))→α
σ−apply
!"# $%&'$!(# "( )*)+,-) ,!"% ./-&'%)% -& ,%!,!#'$'!( ,%./.0)($)1
A>&.B1& 6C +!"#! $%##&( 2#( 14@'( +,-*. $( '" A%1=( H(a(x, y)) −→
b(a(x,⊥), a(⊥, y)): +''( ()! )8=2'&( 0"1 '() 14@'() *+,-*.) 5%==( )28! ?
B H (a(x, y))→ H”(H ′(a(x, y)))
B H ′ (a(x, y))→ b(x, y)
B H” (b(x, y))→ b(G (x), D (y))
C%21 D(x)→ a(⊥, x)/ #%2) "3%#) 2# (#)(=D'( $( 14@'() )283"#!() ?
B D (⊥)→ a(⊥,⊥)
B D(c(x, y))→ A(D′(c(x, y))) ∀c ∈ FbinB A (c(x, y))→ a(x, y)
B D′ (c(x, y))→ c(⊥, c(x, y))
B D′ (⊥)→ a(⊥,⊥)
C%21 G(x)→ a(x,⊥)/ #%2) A"8)%#) 2# "00(' $( E ?
B G (⊥)→ a(⊥,⊥)
B G(c(x, y))→M(D(c(x, y))) ∀c ∈ FbinB M (a(x, y))→ a(y, x)
2($"'$'3)+)($4 (!"# !5$)(!(# -) %.#"-$&$ ,-"# 6.(.%&-4 .$&($ -& /!+5'(&'#!( 0)#
$%&(#7!%+&$'!( 8,%.#)($.)# ,&% -)# 9:/;)# 8/< )$ 8)< 0) -& =6"%) >1? ! "#$%&'($ )*+
,&%%-.% / 0-.# (-.( 12345! '-.% "-.6-'% )-'%(#.*#& ,&% 412345% $7.*689&'(% "-.#
9:8))&%%*;*9*($ %.# T (Fbin)< 1' #$%.=$! (-.% 345 "&.( >(#& %*=.9$ "8# 9&% 412345%<
!"#$%&' ( !"# R $% &' ($) T (Fbin,X ) × T (Fbin,X ) *# α ∈ T (Fbin)+
,"%("- ". */"(#* $% '01&' Rλe→∪ #*. 2$*- 3!%("45)!%( Reachables = {α} ∪⋃
H∈FNT ,β∈Reachables(Rλe
⋆→∪({H(β)}))- R
⋆(α) = Reachables+
)$'*+' , *.!% .*( 6)!6!("#"!%( 7 *# .8 #95!):;* <+
! "#$%&'()#$ *&( (+(,-.&( /#$0,)#$$&1( %&'( 234(
?8'% )&((& %&)(*-'! '-(#& 8=;*(*-' &%( ,& )@&#)@&# .'& )-'6&#%*-' *',*#&)(& &'
)-'%*,$#8'( 9&% %A%(B=&% C-')(*-''&9% )-==& .' (A"& %"$)*89 ,&% %A%(B=&% )98%%*7.&%<
199& &%( #&"#$%&'($& "8# 9&% DB)@&% "-*'(*99$&% ,& 98 EF.#& G<H<
! "#$%&'() *+ ,-.'/0). 1) 23435(&'6() 7895'&899):.
!"# Rλ $% &'( )$* T (Fbin∪FNT )+ ,!$) -!%)#*$")!%) $% %!$./0$ '( Rconv+
'!$# 1203!*14 52/%)/635/ 1/ )763!5/) T (Fbin) 8 90*#"* 1/ FNT /# Fbin /)# 1:%!#:
90* Fconv/)# 1:;%" -"<1/))!$) =
> ?!$* -@0A$/ )763!5/ 3"%0"*/ a ∈ Fbin B7 -!69*") ⊥C4 0D!$#/* a ∈ Fconv
> ?!$* -@0A$/ )763!5/ $%0"*/ H ∈ Fbin4 0D!$#/* h ∈ Fconv
#/5 A$/ @ /)# 3"%0"*/
/# h $∈ FbinE%)$"#/4Rconv /)# -!%)#*$"# 90* 1/) *FG5/) 1/Rλ /% */6950H0%# #!$) 5/) )763!5/)
$%0"*/) H(...) 90* 1/) )763!5/) 3"%0"*/) h(...,⊥) !I ⊥ .0 */695"* 5/ )!$)<#/*6/ 1*!"#
1/ h %!% $#"5"):+
;<)0%:) != !"#$%&'!"# () *+,- %) (.)/)01() 234 56$#76) OnlyOne ∈ FNT )#8
8'9%6$8 : onlyone ∈ Fconv; 9$"#$ () +,- Rconv !<8)"6 )#8 =!00) () #6$>9"8 ?
Rconv =
onlyone(t(r(x, y), i(z, t)),⊥)→ t(onlyone(r(x, y),⊥), i(z, t))
onlyone(t(i(x, y), r(z, t)),⊥)→ t(i(x, y), onlyone(r(z, t),⊥))
onlyone(r(r(x, y), i(z, t)),⊥)→ r(onlyone(r(x, y),⊥), i(z, t))
onlyone(r(i(x, y), r(z, t)),⊥)→ r(i(x, y), onlyone(r(z, t),⊥))
onlyone(r(⊥,⊥),⊥)→ r(⊥,⊥)
.
J05@/$*/$)/6/%#4 5/ 9*!35F6/ 1:)!*60") ."/%# 1$ 6!1/ 1/ K!%-#"!%%/6/%#+
L0%) 5/ -0) 1/ 52/M/695/ NO4 5/ 6!1/ 1/ K!%-#"!%%/6/%# 1/ -/ '( /)# )"6"50"*/
0$ 6!1/ 1$ &'( + P/9/%10%#4 10%) 5/ -0) 1/ 52/M/695/ QN4 %!$) 0.!%) !3#/%$ 5/)
*:)$5#0#) 1"R:*/%#)+
E% *:)$6:4 %!$) 0.!%) 9*!$.: A$/ 9!$* #!$# '( 4 "5 /M")#/ $% &'( -!**/)9!%<
10%#+ ?!$* 50 *:-"9*!A$/4 )" $% &'( )0#")K0"# 50 -!%1"#"!% 1/ 50 )/-#"!% 4 "5 /M")#/
$% '( -!**/)9!%10%#+ L20$#*/ 90*#4 )" -/ &'( -!%#"/%# A$/ 1/) *FG5/) */)9/-#/%#
5/ 6!1/ 1/ K!%-#"!%%/6/%# 1/ '( 4 %!$) 9!$.!%) #*0%)K!6/* ./*) $% '( 90* $%/
60%"F*/ "%1"*/-#/+
!"#$ %& '(#)!*+, -
!"# $% $&!'()*%+ ",-# !.,"# '*,',#/ 0%# #1#)23%# 4% */5/$*()-*% 6,"$)(,""%0#7
8,-# 3,")*,"# 9-% ",)*% 3,420% ! 0! '-(##!"$% 4:%;'*%##(," 4%# #1#)23%# 4% */5
/$*()-*% %) 9-:(0 %#) <(%" !4!')/ = 0:/)-4% 4% '*,'*(/)/# 4% #>*%)/ 4% %) 4% '*,'*(/)/#
4% 0,?(9-% )%3',*%00% 4% #1#)23%# '*/#%")/% %" 4/)!(0 = 0! '!*)(% @@@7
8,-# !.,"# 3(# %" /.(4%"$% -"% #,-# $0!##% 4% #1#)23%# 6,"$)(,""%0#+ 0%# /0/3%"5
)!(*%#+ '*/#%*.!") 0! '-(##!"$% 4:%;'*%##(," 4%# #1#)23%# 6,"$)(,""%0# %) 4%# )%$&5
"(9-%# 4:!$$/0/*!)(," 4%# $!0$-0# !<,-)(##!") = -" ,-)(0 4% ./*(A$!)(," #13<,0(9-%
%B$!$%7 8,-# %;'0(9-,"# $,33%") (0# #,") /.!0-/# !- $&!'()*% C7
% '0-#+ ",-# !.,"# 3(# %" /.(4%"$% -"% #,-# $0!##% 4% #1#)23%# 6,"$)(,""%0#
/0/3%")!(*%#+ 0%# /0/3%")!(*%# = 4*,()%+ '*/#%*.!") 0! '-(##!"$% 4:%;'*%##(," 4%# #1#5
)23%# 6,"$)(,""%0# %) 4%# )%$&"(9-%# 4:!$$/0/*!)(," 4%# $!0$-0#7
!"# 0! '!*)(% %;'/*(3%")!0% D%" $&!'()*% E F+ ",-# !00,"# $,3'!*%* ",)*% ,-)(0
<!#/ #-* ",)*% 6,*3!0(#3% #,() !.%$ 4%# ,-)(0# 4% */5/$*()-*% )%0# 9-% G(3<-H %)
I!-4%+ #,() !.%$ 4%# ,-)(0# 4% ./*(A$!)(," )%0# 9-% JK@8+ 8-JIL+ JIMNG+ OJ +
%)$ !A" 4% 3,")*%* ",# '%*6,*3!"$%# $,3'/)()(.%#7
!"#$%&' (
!"#$"%&'( )*+ +,+%-.*+
/#/.*(%"&0*+
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".//0123
!" #$%&'()*+, -./01$21)(&3 4$155(62, ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 78
!9 :;)(+(51)(&3 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 8"
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.%# #6#&71%# 2% ,-8-$,'&*,% -.-1%"&!',%# 9:;<=># (* =:;<=>#?@ A%# !*&,%# 9.%#
$.!##'5*%# %& .%# ;<=>#? 2('4%"& +!##%, +!, *"% &,!"#B(,1!&'(" #'1+.% 9+,(+(#-% !*
$0!+'&,% +,-$-2%"&? !4!"& .3-4!.*!&'(" +!, "(# !./(,'&01%#@
>('& *" :;<=> Rλe 96 $(1+,'# *" =:;<=>? 5*' '"2*'& *"% ,%.!&'(" 2% ,-8
-$,'&*,% →Rλe#*, 2%# &%,1%# E ⊆ T (Fbin)) .3(CD%$&'B %#& 2% $!.$*.%, .! $.E&*,%
&,!"#'&'4% -$,'&% R⋆λe(E)@ ("$ $%& :;<=> Rλe "% $("&'%"& 5*% 2%# ,7/.%# 2% .!
B(,1%
F@ H(a(x, y)) → α !4%$ a ∈ Fbin) H ∈ FNT ) x, y ∈ X ) x $= y) α ∈ T (Fbin ∪
FNT ,X ) %& Var(α) = {x, y}
G@ H(⊥)→ β !4%$ β ∈ T (Fbin ∪ FNT )
H@ H(a(x,⊥))→ x !4%$ x ∈ X ) H ∈ FNT %& a ∈ Fbin@
I!'"&%"!"& "(*# 2(""("# 5*%.5*%# !./(,'&01%# 5*' %J+.'5*%"&
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H(a(x, y)) → α) .3!./(,'&01% F 4! B!',% !++%. M .3!./(,'&01% G +(*, &,!'&%, "3'18
+(,&% 5*%. α ∈ T (Fbin ∪ FNT ,X )@ A% 2%,"'%, &,!'&% .% $!# (N '. %J'#&% .%# ,7/.%# 2%
+('"& KJ% H(a(x, y)) → a(x, y),H(a(x, y)) → H(H ′(a(x, y))) ∈ Rλe @ >' $3%#& C'%"
$% $!#) .3!./(,'&01% H #%,! !++%.-@
#$%&'()*+, " O!.$*. 2% HRλe(s)
9'/4&3-()(&3 : s ⊆ T (Fbin)) H ∈ FNT %& Rλe *" %"#%1C.% 2% ,7/.%# 23:;<=>#
! res← ∅
"! 5( ⊥ ∈ s %& H(⊥)→ β ∈ Rλe 1$&'5
#! ,%# +← β
$! ;3 5(
%! <&2' a(u, v) ∈ s %& H(a(x, y))→ α ∈ Rλe =1(',
&! ,%# +← αRλe(u, v)
'! ;3 <&2'
(! <&2' a(u,⊥) ∈ s %& H(a(x,⊥))→ x ∈ Rλe =1(',
)! ,%# +← u
*! ;3 <&2'
! <&2' a(u, v) ∈ s %& H(a(x, y)) → a(x, y),H(a(x, y)) → H(H ′(a(x, y))) ∈ Rλe
=1(',
"! ,%# +← FPH′
Rλe({a(u, v)})
#! ;3 <&2'
$! ',)&2'3,' res
! "#$%&'() *+ ,-$./$'&01 2)3 343'56)3 7.76)1'$&()3
8.90(&'#6) : !"#$" %& αRλe(u, v)
;(7<012&'&01 = u, v ∈ T (Fbin)' α ∈ T (Fbin ∪ FNT ,X ) &( Rλe $) &)*&+,"& %&
-./"&* %012345*
! 3& 6α = ⊥7 $.0(3
"! ()'0/(1)( ⊥
#! 3&101 3& 6α = x7 $.0(3
$! ()'0/(1)( u
%! 3&101 3& 6α = y7 $.0(3
&! ()'0/(1)( v
'! 3&101 3& 6α = H(α′) !8&# H ∈ FNT 7 $.0(3
(! ()'0/(1)( HRλe(α′Rλe
(u, v))
)! 3&101 3& 6α = a(α1(x, y), α2(x, y)) !8&# a ∈ Fbin7 $.0(3
*! ()'0/(1)( a(α1Rλe(u, v), α2Rλe
(u, v))
! >1 3&
9!)* "0!"/:-;(<+& =' "! ):(!(;:) αRλe(u, v) 8;*;(& "& #>(? %-:;( %& H(a(x, y))→
α ∈ Rλe &( *$,*(;($& -&*@&#(;8&+&)( "&* 8!-;!,"&* x, y @!- u &( v %!)* αA 9& @"$*'
*; $) *B+,:"& ):)C(&-+;)!" %& FNT &*( -&)#:)(-? @&)%!)( "0&D@":-!(;:)' HRλe&*(
!@@&"?A &"! /!-!)(;( "& +?#!);*+& %0?8!"$!(;:) &) +:%& E:)#(;:))&"A & @-:#&**$*
&*( %?#-;( %!)* "0!"/:-;(<+& FA G& %&-);&- (-!;(& "& #!* :HRλe )0&*( @!* ";)?!;-& %-:;(&A
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?@)6%.) :A !"# $%&'(&)*$& +& ,-.$&( +& ,/0/1,"#2,& Rreqλe 345678 345978 345:;7 &#
345::7 <"'(" =2& $&( ,-.$&( +& 1",12$<#"!' Rcirλe > 345:?78 555 345:97 +& '!#,& /#2+& +&
1<( @AB +<'( $< (&1#"!' 45?5 A2 +/*2#8 '!2( C!2$!'( 1<$12$&, 2' D<( +& ,/0/1,"#2,&
E!'1#"!''&$ &' 2#"$"(<'# $%<$.!,"#F)& : >
HR
req
λe ∪Rcirλe
→∪
(u) = H(t(i(i(⊥,⊥), i(⊥,⊥)), i(i(⊥,⊥), i(⊥,⊥))))
G' <DD$"=2<'# $&( ,-.$&( +& 1",12$<#"!' Rcirλe 8 '!2( #,!2C!'( +&( #&,)&( 1"0+&((!2( >
t(H(i(i(⊥,⊥), i(⊥,⊥))), i(i(⊥,⊥), i(⊥,⊥))) + t(i(i(⊥,⊥), i(⊥,⊥)),H(i(i(⊥,⊥), i(⊥,⊥))))
G# D2"( $&( (!2( #&,)&( <H<'# H I $< ,<1"'& 1!'#"'2&'# I J#,& (")D$"K/( &' 2#"$"(<'#
$& Rcirλe >
t((i(H(i(⊥,⊥)), i(⊥,⊥)), i(i(⊥,⊥), i(⊥,⊥))) + t((i(i(⊥,⊥),H(i(⊥,⊥))), i(i(⊥,⊥), i(⊥,⊥))) +t(i(i(⊥,⊥), i(⊥,⊥)), i(H(i(⊥,⊥)), i(⊥,⊥))) + t(i(i(⊥,⊥), i(⊥,⊥)), i(i(⊥,⊥),H(i(⊥,⊥)))).
A 1& )!)&'#8 $&( ,-.$&( Rcirλe +&C"&''&'# <DD$"1<*$&(8 '!2( <$!,( <##&".'!'( 2' &'0
(&)*$& +& #&,)&( 1$!( 1"0+&((!2( >
t(i(r(⊥,⊥), i(⊥,⊥)), i(i(⊥,⊥), i(⊥,⊥))) + t(i(i(⊥,⊥), r(⊥,⊥)), i(i(⊥,⊥), i(⊥,⊥))) +t(i(i(⊥,⊥), i(⊥,⊥)), i(r(⊥,⊥), i(⊥,⊥))) + t(i(i(⊥,⊥), i(⊥,⊥)), i(i(⊥,⊥), r(⊥,⊥)))
AD,-( $< 1<'!'"(<#"!'8 HRreq
λe∪Rcir
λe(u) =
t(i(r(⊥,⊥), i(⊥,⊥)) + i(i(⊥,⊥), r(⊥,⊥)), i(i(⊥,⊥), i(⊥,⊥))) +t(i(i(⊥,⊥), i(⊥,⊥)), i(r(⊥,⊥), i(⊥,⊥)) + i(i(⊥,⊥), r(⊥,⊥)))
!"! #$%&'()*+, -./01$21)(&3 4$155(62, 78
#$%&'()*+, 9 !"#$" %& FPHRλe
(s)
:'/4&3-()(&3 ; s ⊆ T (Fbin)' H ∈ FNT &( Rλe $) &)*&+,"& %& -./"&* %012345*
! res1 ← ∅ 6
"! res2 ← s 6
#! )13)62, res1 $= res2 <1(',
$! res1 ← res2 6
%! res2+← HRλe(res2)
&! =3 )1362,
'! ',)&2'3,' res2
57+!)(89$&+&)(' %!)* #&( &:&+;"&' )<$* ;<$=<)* #<)*8%7-&- "& *>+,<"& H
#<++& $) +!-9$&$- 8)%89$!)( ? 9$&" &)%-<8( )<(-& *>*(.+& %& -7@7#-8($-& &*( &)
(-!8) %& (-!8(&-A B$(-&+&)( %8(' #C!9$& (&-+& u %& "0&)*&+,"& %& (&-+&* %<))7 s
*;7#8D!)( $) 12345' H(u) *;7#8D& 9$& "& (&-+& u %<8( E(-& +8* ? F<$-A G!- #<)@
*79$&)(' )<$* !;;"89$<)* Rλe *$- H(u) %!)* "& ,$( %08+;"7+&)(&- "! +8*&@?@F<$-
!$(!)( 9$& ;<**8,"&A
B$(-&+&)( %8(' "&* !"/<-8(C+&* H &( I !;;"89$&)( "& *>+,<"& H *$- #C!9$& (&-+&
%& "0&)*&+,"& %& (&-+&* #"<* &( !;;"89$&)( "& *>*(.+& %& -7@7#-8($-& !$(!)( 9$&
;<**8,"&A
J" K!$( *!=<8- 7/!"&+&)( 9$& "! *7+!)(89$& %$ *>+,<"& K<)#(8<))&" H &*(' %!)*
$) #&-(!8) *&)*' %7D)8& /-L#& ? "012345 Rλe ' ;<$- $) &)*&+,"& %& (&-+&*A J" &*(
8)(7-&**!)( %& #!"#$"&- (<$(&* "&* #<)D/$-!(8<)* ;<**8,"&* &) !;;"89$!)( H *$- s ⊆
T (Fbin) $) )<+,-& )<)@,<-)7 %& K<8* &( !;;"89$&- Rλe !$(!)( %& K<8* 9$& ;<**8,"&'
c.?.d.A FPRλe(s)A 58 $) ;<8)( D:& &*( <,(&)$' "0&)*&+,"& %& (&-+&* %!)* T (Fbin)
!##&**8,"&* &*( #!"#$"7A
M0!"/<-8(C+& H #!"#$"& "0&)*&+,"& %& #<)D/$-!(8<)* !##&**8,"&* %& "0&)*&+,"& %&
(&-+&* %& T (Fbin) !;-.* !=<8- !;;"89$7 H ? "! ;<*8(8<) "! ;"$* C!$(&A B$(-&+&)(
%8(' 8" +<)(-& "! -<,$*(&**& %& #&( !"/<-8(C+&A
28)!"&+&)(' ;<$- #!"#$"&- "0&)*&+,"& %&* !##&**8,"&* N#<++& +&)(8<))7 %!)* "&
#C!;8(-& OP' )<$* !=<)* ,&*<8) %& %7D)8- "& #!"#$" %$ ;<8)( D:&A 0&*( "& -Q"& %&
"0!"/<-8(C+& RA
1) K!8(' "0!"/<-8(C+& H' "0!"/<-8(C+& I &( "0!"/<-8(C+& R ;-<;<*7* #8@%&**$*
<)( ;-<D(7 "&* (&#C)89$&* %0!##7"7-!(8<) %& "! =7-8D#!(8<) *>+,<"89$& !" #$$
ST->!)( HUVW' T->!)( HUUIX (&""&* 9$& Y
Z >1?$,12 -.23(4()/ !" #$$ %& "0&)*&+,"& %& (&-+&* %& T (Fbin) Y 3&#C@
)89$&+&)( %8(' )<$* )0!##&;(<)* 9$0$) $)89$& -&;-7*&)(!)( ;<$- $) (&-+& %&
T (Fbin)A &"! ;&-+&( "&* %<))7&* #<%7&* %&* +<%."&* (-.* #<+;!#(&*A
Z @13&3(51)(&3 %& "0&)*&+,"& %& (&-+&* %& T (Fbin) Y M&* !"/<-8(C+&* ;-<D(&)(
,8&) "&* ;-<;-87(7* "8)7!8-&* %&* (&-+&*' %7F? ;!-"7 ? "! *&#(8<) RAH &( ? "! *&#(8<)
OAHAH' ;<$- 9$& "& #<%!/& %&* +<%."&* *<8&)( ;"$* #<+;!#(&*A
! "#$%&'() *+ ,-$./$'&01 2)3 343'56)3 7.76)1'$&()3
"$8#) !" #$$ ! "#$%&' ($)(&) *' )+$),-./0#1' 2 '0 )+$),-./0#1' 3 *-/0 40.'
1/5 '6 ($(#' 7-&. 89/0'. *' .':;$/.' &6 .85&)0$0 *8<= ($)(&)8>
?+'@'17)' 5&/9$60 9$ /))&50.'. (-11'60 )' 7-/60 A@' '50 ($)(&)8 7-&. 6-0.' 80&*'
*' ($5 BCD>
9:)6%.) ;; %&"'& ()''*+ (,+ ,'+,-.!,+ (, /01!,+ Rλecir ∪ Rλe
req2 (, !3,4,-5!,
678 ')9+ ":)'+ ;"!;9!* u1 = HRreq
λe∪Rcir
λe(u) = {u11, u12, u13, u14} ":,;
u11 = t(i( (⊥,⊥), i(⊥,⊥)), i(i(⊥,⊥), i(⊥,⊥))), u12 = t(i(i(⊥,⊥), (⊥,⊥)), i(i(⊥,⊥), i(⊥,⊥))),u13 = t(i(i(⊥,⊥), i(⊥,⊥)), i( (⊥,⊥), i(⊥,⊥))), u14 = t(i(i(⊥,⊥), i(⊥,⊥)), i(i(⊥,⊥), (⊥,⊥))).
<, (,94=0-, 5"+ (, /*>*;/=&9/, ,+& (, /,>?"=/, ,4";&,-,'& !3,4,-5!, 5/*;*(,'& ":,;
!3,'+,-.!, (, &,/-,+ (, u18 c. .d ')9+ ;"!;9!)'+
u2 = HRreq
λe∪Rcir
λe(u1) = HR
req
λe∪Rcir
λe(u11 + u12 + u13 + u14) =
H(t(i( (⊥,⊥), i(⊥,⊥)), i(i(⊥,⊥), i(⊥,⊥)))) + H(t(i(i(⊥,⊥), (⊥,⊥)), i(i(⊥,⊥), i(⊥,⊥)))) +H(t(i(i(⊥,⊥), i(⊥,⊥)), i( (⊥,⊥), i(⊥,⊥)))) + H(t(i(i(⊥,⊥), i(⊥,⊥)), i(i(⊥,⊥), (⊥,⊥)))).
@5/0+ !3*!=-='"&=)' (,+ &,/-,+ /,()'("'&+ ,& !" ;"')'=+"&=)'8 ')9+ ;"!;9!)'+
;)--, +9=&
t(i( (⊥,⊥), (⊥,⊥)), i(i(⊥,⊥), i(⊥,⊥))) + t(i(i(⊥,⊥), i(⊥,⊥)), i( (⊥,⊥), (⊥,⊥))) +t(i( (⊥,⊥), i(⊥,⊥)), i( (⊥,⊥), i(⊥,⊥))) + t(i( (⊥,⊥), i(⊥,⊥)), i(i(⊥,⊥), (⊥,⊥))) +t(i(i(⊥,⊥), (⊥,⊥)), i( (⊥,⊥), i(⊥,⊥))) + t(i(i(⊥,⊥), (⊥,⊥)), i(i(⊥,⊥), (⊥,⊥))) +t( ( (⊥,⊥), i(⊥,⊥)), i(i(⊥,⊥), i(⊥,⊥))) + t( (i(⊥,⊥), (⊥,⊥)), i(i(⊥,⊥), i(⊥,⊥))) +t(i(i(⊥,⊥), i(⊥,⊥)), ( (⊥,⊥), i(⊥,⊥))) + t(i(i(⊥,⊥), i(⊥,⊥)), (i(⊥,⊥), (⊥,⊥)))
@5/0+ !" ;"')'=+"&=)'8 ')9+ 5)9:)'+ ).&,'=/ 9', /,5/*+,'&"&=)' 5!9+ ;)-5";& ;=>
(,++)9+ A
t(i(i(⊥,⊥), i(⊥,⊥)), ( (⊥,⊥), i(⊥,⊥)) + (i(⊥,⊥), (⊥,⊥)) + i( (⊥,⊥), (⊥,⊥))) +
t(i( (⊥,⊥), (⊥,⊥)) + ( (⊥,⊥), i(⊥,⊥)) + (i(⊥,⊥), (⊥,⊥)), i(i(⊥,⊥), i(⊥,⊥))) +
t(i( (⊥,⊥), i(⊥,⊥)), i( (⊥,⊥), i(⊥,⊥)) + i(i(⊥,⊥), (⊥,⊥))) +
t(i(i(⊥,⊥), (⊥,⊥)), i( (⊥,⊥), i(⊥,⊥)) + i(i(⊥,⊥), (⊥,⊥)))
<,+ 5"+ (, /*>*;/=&9/, +9=:"'&+ ?)';&=)'',/)'& (, !" -B-, -"'=0/,2 C! ?"9& ')&,/
D9, ("'+ ;, ;"+>! u, v $∈ HRreq
λe∪Rcir
λe(u) -"=+ u, v ∈ FPRreq
λe∪Rcir
λe(u)2
E60&/0/9'1'60F 6-&5 9-G-65 %&' )' 0'.1' A6$) '50 %&$5/:5/1/)$/.' $& 0'.1' /6/0/$)
5$&; )' (#$6,'1'60 *'5 5G1H-)'5 *'5 5-&5:$.H.'5 *' )$ .$(/6' 80/%&'08' 7$. )' 5G1H-)'
t> I+$&0.' 7$.0F )'5 -78.$0/-65 *' (/.(&)$0/-6 7-&. *'5 5-&5:$.H.'5 *' )$ .$(/6' ;-60
)' 141' (-17-.0'1'60>
J6' %&'50/-6 89/*'60' '50 %+&>=! 5)++=.!, (3";;*!*/,/ !, ;"!;9! ,' *:=&"'& !" /*5*&=>
&=)' (,+ )5*/"&=)'+ 5)9/ !" ;=/;9!"&=)' ,' (,+;,'("'& 9', +,9!, ?)=+ !3,'(/)=& ='>
&*/,++* ,& +"&9/,/ !, '),9( ;)9/"'& "9&"'& D9, 5)++=.!,>
"'5 0'(#6/%&'5 6-&5 7'.1'00'60 *' )&00'. (-60.' )+'@7)-5/-6 (-1H/6$0-/.' 7-&.
)'5 1-*K)'5 5G180./%&'5> L$)#'&.'&5'1'60F )$ (-17$(/08 *'5 *-668'5 6-&5 (-6*&/0
= &6 $&0.' 7.-H)K1' %&/ '50 )+'M'0 *'5 *-668'5 /60'.18*/$/.'5> ?'5 7-/605 A@'5 )-:
($&@ N?ODP N-& )$ 50.$08,/' *' 5$0&.$0/-6 Q"/$.*- 3RRSTP 7.85'608'5 *$65 )$ 5'(0/-6
5&/9$60' 5-60 &6' H-66' 5-)&0/-6 7-&. (' 7.-H)K1'>
!"! #$%&'&()%&*+ ,-
./0*1&%2'3 4 !"#$" %& FPHRλe
(s) !'&# %&( )*+"&( %& ,-./0 12&
5167*+8&%&*+ 9 s ∈ T (Fbin)3 H,H⋆ ∈ FNT &0 Rλe $/ &/(&45"& %& )*+"&(
%6789:;(
! res1 ← ∅ <
"! res2 ← s <
#! %)+%:;3 res1 $= res2 <)&13
$! res1 ← res2 <
%! res2+← H⋆Rλe
(res2)
&! res2+← HRλe(res2)
'! =+ %)+:;3
(! 13%*;1+31 res2
!" #$%&'&()%&*+
&00& (�.-/ 0)!.0& "6-,0.4.(!0.-/ %&( !"+-).0=4&( %6>'!"$!0.-/ &/ $0.".(!/0 %&(
0&#=/.?$&( %6!##>">)!0.-/ %&( #!"#$"( !5-$0.((!/0 @ $/ -$0." %& '>).1#!0.-/ (A45-"B
.?$& &C#!#&D
E-$) ?$&"?$&( 0A,&( %& )*+"&( >">4&/0!.)&( @ %)-.0&3 /-$( !""-/( 4-/0)&) "! #!,!#B
.0> %6!##>">)!0.-/ %&( #!"#$"(D F'.%&44&/03 "6&C#!#.0> %& #&00& -,0.4.(!0.-/ %>,&/%
%$ /-45)& %& )*+"&( %& )>B>#).0$)& !,,".#!5"&(3 %& "6.4,-)0!/#& %& #&( )*+"&( !./(.
?$& %$ /.'&!$ %& (A4>0).& %$ 4-%*"& %-//>D
!"!# $%&'(&)*+, &%-. '& /)0&)12*- 3- /&)(0&)*+,
G-0)& 5$0 &(0 %6!##>">)&) "& #!"#$" &/ >'.0!/0 "! )>,>0.0.-/ %&( -,>)!0.-/( ,-$)
"! #.)#$"!0.-/ &/ %&(#&/%!/0 $/& (&$"& H-.( @ "6&/%)-.0 ./0>)&((> &0 (!0$)&) "& /-&$%
#-$)!/0 0!/0 ?$& ,-((.5"&D I&( ,-./0( 12&( "-#!$2 JI8EK J-$ "! (0)!0>+.& %& (!0$)!B
0.-/K (-/0 $/& 5-//& (-"$0.-/D
I! (0)!0>+.& %& (!0$)!0.-/ J!,,&">& >+!"&4&/0 I&( !"#$% &"'(% )!*+,'- )& K
&(0 ,)-,-(>& %!/( L .!)%- MNNOPD 7""& "$00& #-/0)& "6&Q&0 %& "! 0!.""& %&( %-//>&(
./0&)4>%.!.)&(D 7Q�.'&4&/03 #&0 !"+-).0=4& &(0 0)*( &C#!#& ,-$) %&( (0)$#0$)&(
0)*( #-4,!#0&( @ "! RSS 0&"( ?$& TSS L .!)%- MNNOP3 SSS L -$')&$) MNNMP3 ;SS
L -$')&$) MNNUP3 V;SS L9=.&))ABT.&+ MNNWP3 L9=.&))ABT.&+ MNNXPD
9-$0 %6!5-)%3 /-$( ./0)-%$.(-/( $/ /-$'&!$ (A45-"& /-/B0&)4./!" H⋆,-$)
#=!?$& (A45-"& /-/B0&)4./!" HD &( /-$'&!$2 (A45-"&( ,&$'&/0 Y0)& $0.".(>( %!/(
%&( )*+"&( )>B>#).0$)& #-44& $/ (A45-"& /-/B0&)4./!" /-)4!"D Z"( (-/0 ./0&),)>0>(
#-44& Hn,-$) %&( &/0.&)( /-/ />+!0.H( nD
>6=+&%&*+ -- ./+#%0!/1+$"!# 2(% 34&.35% 2( $67( 89
5!"$ Rλe→I ,# (#%(1:;( 2( /<=;(% +6+#$ ,# *>$? =+,*@( A," (%$ 2( ;+ 0!/1(
H (a(x, y)) !B H ∈ FNT ($ a ∈ Fbin9 5(;!# 2(% *+% 2( /<=;(% 2, *>$? 2/!"$- #!,%
+C!#% ;(% #!,C(;;(% /<=;(% %,"C+#$(% D
! "#$%&'() *+ ,-$./$'&01 2)3 343'56)3 7.76)1'$&()3
Rλe→I R′λe→I
H (a(x, y))→ a(x, y) H⋆ (a(x, y))→ a(x, y)
H (a(x, y))→ a(G (x), y) H⋆ (a(x, y))→ a(G⋆ (x), y)
H (a(x, y))→ a(x,D (y)) H⋆ (a(x, y))→ a(x,D⋆ (y))
H (a(x, y))→ a(G (x), y)
H (a(x, y))→ a(x,D (y)) H⋆ (a(x, y))→ a(G⋆ (x), D⋆ (y))
H (a(x, y))→ a(x, y)
H (a(x, y))→ a(x,D (y)) H⋆ (a(x, y))→ a(x,D⋆ (y))
H (a(x, y))→ a(x, y)
H (a(x, y))→ a(G (x), y) H⋆ (a(x, y))→ a(G⋆ (x), y)
H (a(x, y))→ a(x, y)
H (a(x, y))→ a(G (x), y)
H (a(x, y))→ a(x,D (y)) H⋆ (a(x, y))→ a(G⋆ (x), D⋆ (y))
!"# H, G, D ∈ FNT $ H⋆, G⋆, D⋆ ∈ FNT "% a ∈ Fbin
! "#$% &'%() *$( !# %)#&+"'),#%-'& .(+ /012/3 .( %45( 5($% 6%)( #55!-*$7(
+75#)7,(&% 5'$) !8(&+(,9!( .( ):;!(+ H (a(x, y)) → α (% !8(&+(,9!( .( ):;!(+
H (b(x, y)) → β '< α, β ∈ T (Fbin ∪ FNT ,X ) +8-! &8(=-+%( 5#+ .( ):;!( .( %45(
%(!!( *$( H (a(x, y))→ b(x, y) '$ H (b(x, y))→ a(x, y)>
8791&'&01 :! !"#$%&!'"()&# *+$ ,-. ,/$ *+ (01+ 223
/&)( Rλe→II 4# +#$+'56+ *+ !786+$ "0"#( 4# 9:(; 8"49<+ =4) +$( *+ 6" %&!'+
H (a(x, y)) &> H ∈ FNT +( a ∈ Fbin3 /+6&# 6+$ 9"$ *+ !786+$ *4 9:(; *!&)(? #&4$
"@&#$ *+ #&4@+66+$ !786+$ 9&''+ 6+$ $4)@"#(+$ A
Rλe→II R′λe→II
H (a(x, y))→ G(D(a(x, y)))
G (a(x, y))→ a(x, y) H⋆ (a(x, y))→ D⋆ (a(x, y))
H (a(x, y))→ G(D(a(x, y)))
D (a(x, y))→ a(x, y) H⋆ (a(x, y))→ G⋆ (a(x, y))
H (a(x, y))→ G(D(a(x, y)))
G (b(x, y))→ b(x, y) H⋆ (a(x, y))→ G⋆ (c(x, y))
G (a(x, y))→ c(x, y)
G (c(x, y))→ α
D (a(x, y))→ b(x, y) H⋆ (a(x, y))→ D⋆ (b(x, y))
D (c(x, y))→ c(x, y)
D (b(x, y))→ β
"@+9 H, G, D ∈ FNT ? H⋆, G⋆, D⋆ ∈ FNT +( a ∈ Fbin? α, β ∈ T (Fbin ∪ FNT ,X )?
Var(α) = {x, y} +( Var(β) = {x, y}3
!"#$%&'()")$% *&%+ $,(- .,(/,$- 0,$-&*!1)1 2) -3"4,2) H⋆0,"") ($ "#15
'()(1 &$*&'(#$% 6 '()2 )$*1,&% $,%1) -3-%7") *) 1!5!01&%(1) )-% )$ %1#&$ *) 2#$0)1
($ .,&$% 89): ;#1 0,$-!'()$%+ 2)- 17<2)- %1#$-=,1"#%&,$ .)(/)$% $,(- #&*)1 6 2#$0)1
!"! #$%&'&()%&*+ ,-
./0*1&%2'3 !"#$" %& αRλe(u, v) !'&# %&( )*+"&( %& ,-./0 12&
4156*+7&%&*+ 8 u, v ∈ T (Fbin)3 α ∈ T (Fbin ∪ FNT ,X ) &0 Rλe $/ &/(&45"& %&
)*+"&( %6789:;(
! (& <α = ⊥= )/*1(
"! 13%*91+31 ⊥
#! (&+*+ (& <α = x= )/*1(
$! 13%*91+31 u
%! (&+*+ (& <α = y= )/*1(
&! 13%*91+31 v
'! (&+*+ (& <α = H(α′) !'&# H ∈ FNT = )/*1(
(! 13%*91+31 HRλe(α′
Rλe(u, v))
)! (&+*+ (& <α = H⋆(α′) !'&# H⋆ ∈ FNT = )/*1(
*! 13%*91+31 FPHRλe
(α′Rλe
(u, v))
! (&+*+ (& <α = a(α1(x, y), α2(x, y)) !'&# a ∈ Fbin= )/*1(
"! 13%*91+31 a(α1Rλe(u, v), α2Rλe
(u, v))
#! :+ (&
"&( #!"#$"( %& >8? %!/( "&( &/%)-.0( ./0@)&((!/0( ,-$) -,0.4.(&) "& 0&4,( %& #!"A
#$"B C@(-)4!.(3 /-$( &((!D-/( %6./0@+)&) #&( 0)!/(E-)4!0.-/( %!/( /-( !"+-).0F4&(
%6@'!"$!0.-/ #"!((.G$&(B
>6!"+-).0F4& H &(0 "! /-$'&""& '&)(.-/ %& "6!"+-).0F4& I <"6!"+-).0F4& %$ ,-./0
12&= G$. #-4,0& "&( /-$'&!$2 (D45-"&( /-/A0&)4./!$2 H⋆B C!/( "! 5-$#"& %& #&
/-$'&" !"+-).0F4&3 "&( )*+"&( %& "6-,@)!0&$) %& ,-./0 12& (-/0 !,,".G$@&( &/ !,,&"!/0
"! E-/#0.-/ H⋆Rλe
(s) <"6!"+-).0F4& J !'&# %&( (D45-"&( /-/A0&)4./!$2 H⋆=B
>6!"+-).0F4& K &(0 @+!"&4&/0 ($5(0.0$@ ,!) "6!"+-).0F4& LB 7/ )&+!)%!/0 "& /-$A
'&!$ (D45-"& /-/A0&)4./!" H⋆3 #&0 !"+-).0F4& !,,&""& "6!"+-).0F4& H <"6!"+-).0F4&
%$ ,-./0 12&=B
M/0$.0.'&4&/03 (&"-/ "! ,)-,-(.0.-/ L &0 "! #-/(0)$#0.-/ %&( !"+-).0F4&( ".(0@& !$
%&((-$( N
O >.+/&( L %& "6!"+-).0F4& H
O >.+/&( P &0 JQ %& "6!"+-).0F4& L
-/ ,&$0 #-/#"$)& G$& "6@'!"$!0.-/ ,!) "6!"+-).0F4& H &0 ,!) "6!"+-).0F4& I %-//& "&
4R4& )@($"0!0B
7/ E!.03 &/ "!/S!/0 "& #!"#$" H⋆Rλe
(res2)<>.+/& L %& "6!"+-).0F4& H= !'!/0 "& #!"#$"
HRλe(res2)<>.+/& T %& "6!"+-).0F4& H= /-$( ,-$'-/( !00&./%)& "& ,-./0 12& ,"$( '.0&
G$& "6!"+-).0F4& IB
41*$*(&%&*+ !"# Rλe $% &'() *# s ∈ T (Fbin)+ ,"- *."/#* 0*/ Rλe→I ⊆ Rλe
!$ Rλe→II ⊆ Rλe 1*23-"//4%# 0*/ 5!%0"#"!%/ 0* -4 067%"#"!% 88 !$ 0* -4 067%"#"!%
89 4"%/"
(Rλe ∪ (R′λe→I ∪R
′λe→II))
⋆(E′) = R⋆λe(E
′).
! "#$%&'() *+ ,-$./$'&01 2)3 343'56)3 7.76)1'$&()3
!"# E ⊆ T (Fbin) "$ E′ = {H(t)|t ∈ E}%
8()/-) 9 & R⋆λe(E
′) ⊆ (Rλe ∪ (R′λe→I
∪ R′λe→II
))⋆(E′)' #(")$ *!+,"-$ )"./-
Rλe ⊆ (Rλe ∪ (R′λe→I
∪R′λe→II
))
& (Rλe ∪ (R′λe→I
∪R′λe→II
))⋆(E′) ⊆ R⋆λe(E
′). 0/12)1+!/-) "- $2 +$ -$ .( - .3)"
," # ) 4
& (Rλe ∪R′λe→I
)⋆(E′) ⊆ R⋆λe(E
′) 4
& R′λe
⋆H⋆ (a(x,y))→a(x,y)(E
′) ⊆ Rλe⋆H (a(x,y))→a(x,y)(E
′)
& R′λe
⋆H⋆ (a(x,y))→a(G⋆ (x),y)(E
′) ⊆ Rλe⋆H (a(x,y))→a(G (x),y)(E
′) # 2
Hn (a(x, y)) →⋆H (a(x,y))→a(G (x),y) a(Gn (x), y) !"# n )15) 66"-$
72 -,%
& 8+6+. +2"6"-$ 9/12 ." 2")$"%
& (Rλe ∪R′λe→II
)⋆(E′) ⊆ R⋆λe(E
′) 4
& 8+6+. +2"6"-$ 9/12 .") 1$2") # )%
!"# $%%!&# '%%"#()*) +*# &!",*$"- $%.!)'(/0*# 1$) &!()* 2("3* 3* +$# 4567
:;)6%.) <= 8/+$ Rcirλe ,*:-+ , -) .(";"69." 92*#*,"-$% <") 2=7.") ," #+2#1. $+/-
>?%@AB' %%% >?%@CB ," Rcirλe *$ -$ ,") *.*6"-$ +2") D ,2/+$" $39" E >Rcir
λe →IB 9"1!"-$
F$2" /9$+6+)*") 9/12 ." G .#1. ," 9/+-$ :;" #/66" . H I/- )1+! -$" 4
H⋆(i(x, y)) → i(H⋆(x), H⋆(y)) !"#$
H⋆(r(x, y)) → r(H⋆(x), H⋆(y)) !"%$
H⋆(b(x, y)) → b(H⋆(x), H⋆(y)) !"&$
H⋆(t(x, y)) → t(H⋆(x), H⋆(y)) !"'$
J- 99.+K1 -$ .( .7/2+$L6" ," 9/+-$ :;" D . #+2#1. $+/- 9/12 .") 2=7.") ," #+2#1. M
$+/- /9$+6+)*") R′cirλe →I
>N%@B' >N%AB' >N%OB' >N%?B FPHRcirλe
∪Rreq
λe∪R′cir
λe →I
(u) ,/--" 92=)
K1".K1") # .#1.)% < )36*$2+" ,") )/1)M$"26") -/1) 9"26"$ ," 2"M92"-,2" .") 2*)1.$ $)
,") 9/+-$) :;") ./# 1; ,*PD # .#1.*) "- # #L"% Q+- ."6"-$' -/1) $$"+7-/-) ." 9/+-$
:;" 7./R . FPHRcirλe
∪Rreq
λe∪R′cir
λe →I
(u) = t(a(a(⊥,⊥), a(⊥,⊥)), a(a(⊥,⊥), a(⊥,⊥))) /S
a ∈ {i, r}%
8% 9$"( &!(*) :"* %*# $);)*# #!&( 3*# #()"+(")*# ()<# +!01$+(*# .)=+* $" ($;%*$"
3>"&'+'(2 *( ? %$ +$&!&'#$('!&7
!"# )*0$):"!&# $"##' :"* :"*%:"*# +$%+"%# 1!") 3*# )<.%*# 3$&# Rλe→I *(
Rλe→II #!&( '&"('%*# +$) '%# #!&( 32@? +$%+"%2#7 6!")#"',!&# %*# &!",*$"- $%.!)'(/0*#
1!") )23"')* +*# '&+!&,2&'*&(# 3$&# %$ #*+('!& #"',$&(*7
!"! #$%&'&()%&*+ ,
-./*0&%1'2 3 !"#$" %& FPHRλe
(s) !'&# %& ()$'&""&* +,-"&* %& .)/(0 12&
4056*+7&%&*+ 8 s ∈ T (Fbin)3 H,H⋆ ∈ FNT &0 Rλe $( &(*&45"& %& +,-"&*
%6789:;*
! res1 ← ∅ <
"! res2 ← s <
#! %)+%9:2 res1 '= res2 ;)&02
$! res1 ← res2 <
%! res2+← H⋆R′λe→I
∪R′λe→II
(res2)
&! res2+← HRλe ,R′
λe→I,R′
λe→II,Rλe→I ,Rλe→II
(res2)
'! <+ %)+9:2
(! 02%*:0+20 res2
!"!" #$%&'()*+, -./01$21)(&3 &4)(+(5/,
)44& %=>? 4&(0/)((=3 ()$* .+).)*)(* %& ()$'&!$2 !"-)+/0@4&* .)$+ ='/0&+
"&* +&A#!"#$"* /($0/"&* .)$+ %&* +,-"&* %!(* Rλe→I ∪Rλe→II B
C6!"-)+/0@4& D &*0 "! ()$'&""& '&+*/)( %& "6!"-)+/0@4& E F"6!"-)+/0@4& %$ .)/(0
12&G H$/ #)4.0& "6 /(0+)%$#0/)( !$0)4!0/H$& %$ ()$'&!$ 9:;B I!(* "! 5)$#"& %&
"6!"-)+/0@4&3 %&* +,-"&* !'&# "6).=+!0&$+ %& .)/(0 12& *)(0 !.."/H$=&* &( !..&"!(0
$(& J)(#0/)( H⋆R′λe→I
∪R′λe→II
(s) &0 &( !..&"!(0 "6!"-)+/0@4& K HRλe(s) "/4/0= .)$+
"! +,-"& =0!(0 %!(* Rλe→I )$ Rλe→II C6!"-)+/0@4& L %)((& $(& J!M)( %6!.."/H$&+
HRλe(s) *!(* $0/"/*&+ !$#$(& +,-"& %!(* Rλe→I &0 Rλe→II .)$+ "! .+&4/,+& =0!.&
%& "! 0+!(*J)+4!0/)(B
C6!.."/#!0/)( %&* ()$'&!$2 !"-)+/0@4&* %)((& &2!#0&4&(0 "& 4N4& #!"#$" #)44&
"& #!"#$" %& *O*0,4&* $0/"/*!(0 "&* +,-"&* ).0/4/*=&* %& "! *�/)( .+=#=%&(0& .)$+ "&
4N4& .+)5",4& %!(* $( 0&4.* %& #!"#$" H$!*/A/%&(0/H$&B
P(0$/0/'&4&(03 *&")( "! .+).)*/0/)( D &0 "! #)(*0+$#0/)( %&* !"-)+/0@4&* "/*0=&
#/A%&**)$* Q
R C/-(&* S &0 D %& "6!"-)+/0@4& D
R C/-(&* S &0 D %& "6!"-)+/0@4& L
()$* .)$')(* #)(#"$+& H$& "6='!"$!0/)( .!+ "6!"-)+/0@4& D &0 .!+ "6!"-)+/0@4& T
%)((& "& 4N4& +=*$"0!0 %& "6!"-)+/0@4& EB
U)$* !')(* %)(# "! .+).)*/0/)( D3 $( +=*$"0!0 ."$* J)+0 H$& "! .+).)*/0/)( SB
40*$*(&%&*+ 3 !"# Rλe $% &'() *# s ∈ T (Fbin)+ ,"- *."/#* 0*/ Rλe→I ⊆ Rλe
!$ Rλe→II ⊆ Rλe 1*23-"//4%# 0*/ 5!%0"#"!%/ 0* -4 067%"#"!% 88 !$ 0* -4 067%"#"!%
89 4"%/"
(Rλe ∪ (R′
λe→I ∪R′
λe→II) \ (Rλe→I ∪Rλe→II))⋆(E′) = R⋆
λe(E′).
4:*5 E ⊆ T (Fbin) *# E′ = {H(t)|t ∈ E}+
! "#$%&'() *+ ,-$./$'&01 2)3 343'56)3 7.76)1'$&()3
8.90(&'#6) !"#$" %& HRλe ,R′
λe→I,R′
λe→II,Rλe→I ,Rλe→II
(s) !'&# %& ()$'&""&* +,-
."&* %& /)0(1 23&
:(7;012&'&01 <
s ∈ T (Fbin)4 H ∈ FNT
Rλe ,R′
λe→I,R′
λe→II,Rλe→I ,Rλe→II *)(1 %&* &(*&56"&* %& +,."&* 789:;*
! res← ∅
"! 3& ⊥ ∈ s &1
H(⊥)→ β ∈ Rλe \(Rλe→I ∪Rλe→II) ∪(R′
λe→I∪R′
λe→II) $.0(3
#! +&* +← β
$! =1 3&
%! %0/( a(u, v) ∈ s &1
H(a(x, y))→ α ∈ Rλe \(Rλe→I ∪Rλe→II) ∪(R′
λe→I∪R′
λe→II) >$&()
&! +&* +← αRλe∪(R′I∪R′
II)(u, v)
'! =1 %0/(
(! %0/( a(u,⊥) ∈ s &1
H(a(x,⊥))→ x ∈ Rλe \(Rλe→I ∪Rλe→II) ∪(R′
λe→I∪R′
λe→II) >$&()
)! +&* +← u
*! =1 %0/(
! %0/( a(u, v) ∈ s &1 H(a(x, y)) → a(x, y),H(a(x, y)) → H(H ′(a(x, y))) ∈ Rλe
>$&()
"! +&* +← FPH′
Rλe({a(u, v)})
#! =1 %0/(
$! ()'0/(1)( res
:()/-) ?@ !"#$%# &$''()'*# +# ,- ."/./&(*(/) 01
(Rλe ∪ (R′
λe→I ∪R′
λe→II) \ (Rλe→I ∪Rλe→II))⋆(E′) = R⋆
λe(E′).
-%#' E ⊆ T (Fbin) #* E′ = {H(t)|t ∈ E}2
34-."5& ,- ."/./&(*(/) 6 )/$& /7*#)/)&
(Rλe ∪ (R′
λe→I ∪R′
λe→II))⋆(E′) = R⋆
λe(E′).
-%#' E ⊆ T (Fbin) #* E′ = {H(t)|t ∈ E}2
34-$*"# .-"*8 -$ 9/9#)* +/)):8 )/$& ./$%/)& ."/$%#" ;$4$)# "5<,# #) R′
λe→I
#&* -..,('-7,# &#$,#9#)* &( ,- "5<,# '/""#&./)+-)*# #) Rλe→I #&* -..,('-7,#2 3# .,$&
,# ":&$,*-* +4$) .-& +# ":=:'"(*$"# +# ,- "5<,# #) R′
λe→I#&* (+#)*(;$# > '#,$( +# ,- "5<,#
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λe→II-%#' Rλe→II 2 @#,- )/$& .#"9#*
+4/7*#)(" ,# ":&$,*-* +4$) .-& +# ":=:'"(*$"# &-)& $*(,(&#" ,- "5<,# #) (Rλe→I∪Rλe→II)2
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N"!&'
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O H (a(x, y))→ a(⊥, a(x, y))
)K'0 H ∈ FNT et a, b ∈ FbinJ
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u(n(x, y), n(z, t)) → n(n(x, y), n(z, t))
u(p(x, y), z) → p(p(x, y), z)
u(x, p(y, z)) → p(x, p(y, z))
H! +%/"#2!%3/$(!" )& +,- 41%# &" 5+,- G 5+,- ;755'&97%.-%, RPPλ G
# ./*0'# *)()1)'# 2 ./*0'# +3+(% 0' 45%) *+647' H(u(...))8
H(u(p(x, y), z)) → p(p(x, y), z) H(u(x, p(y, z))) → p(x, p(y, z))
H(u(n(x, y), n(z, t))) → n(n(x, y), n(z, t))
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# ./*0'# ,' !"$(% ->'
Percolate(x) → x
Percolate(x) → Percolate(H(x))
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RPPλe G
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H(u(x, y))→ n(Neg(x), Neg(y)) Neg(n(x, y)) → n(x, y)H(u(x, y))→ p(Pos(x), y) Pos(p(x, y))→ p(x, y)H(u(x, y))→ p(x, Pos(y))
# !"#$% )$ *+(*,#-.+/' 01##$% %/'. +)$'.+2,$% 3 *$##$% )$ 45 67
H(u(x, y)) → u(H(x), y)H(u(x, y)) → u(x,H(y))
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H⋆(a(x, y))→ a(H⋆(x),H⋆(y))!
$<= RPP5 *./ >$<= RPP
λ %0 *./ ?>$<= RPPλe ).&&%*;./2(/0 @.1&/+**%/0 -%
'A'% &3*1-0(0 E1 ⊆ T (Fbin) B ;(&0+& 2% -C%/*%',-% +/+0+(- E0 7
E1 = RPP ⋆(E0) = R
PPλ
⋆(Percolate(E0)) = R
PPλe
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O! ",10'(78'&,1 -971 456 8/$00&:7) P 456 RLEPP
c(p(x, y), p(z, t)) → c(c(x, y), p(z, t)) c(p(x, y), p(z, t)) → c(p(x, y), c(z, t))c(p(x, y), n(z, t)) → c(c(x, y), n(z, t)) c(n(x, y), p(z, t)) → c(n(x, y), c(z, t))
J! 4($10;,(<$'&,1 -7 456 =)(0 71 3456 P 3456 6755'&37%.-%, RLEPλ P
# &'()*% (!#!+!*%
H(c(p(x, y), p(z, t))) → c(c(x, y), p(z, t)) H(c(p(x, y), p(z, t))) → c(p(x, y), c(z, t))H(c(p(x, y), n(z, t))) → c(c(x, y), n(z, t)) H(c(n(x, y), p(z, t))) → c(n(x, y), c(z, t))
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H(c(x, y)) → c(H(x), y)H(c(x, y)) → c(x,H(y))
# &'()*% * :0$#/ "<*
Election(x) → x
Election(x) → Election(H(x))
!"! #$%&'() *+,$+()-(./) 01
!"# !" # $%&'%( )*)+, -+ ./%'0+ 1% c
2! 2+*.)3$+4*/5$. %6 7289 :(+) 6. ;7289 3 ;7289 .4&&%(54+1)+,
RLEPλe 3
# !"#$% "&'&(&$%
H(c(x, y)) → c(Elec(x), Neg(y)) H(c(x, y)) → c(Elec(x), Pos(y))H(c(x, y)) → c(Pos(x), Elec(y)) H(c(x, y)) → c(Neg(x), Elec(y))Elec(p(x, y)) → c(x, y) Neg(n(x, y)) → n(x, y)Pos(p(x, y)) → p(x, y) Elec(⊥) → ⊥Pos(⊥) → ⊥ Neg(⊥) → ⊥
# !"#$% )$ *+(*,#-.+/' 01##$% %/'. +)$'.+2,$% 3 *$##$% ), 45 67
H(c(x, y)) → c(H(x), y)H(c(x, y)) → c(x,H(y))
# !"#$% )$ 8/+'. 9:$ 0;/,% .(-),+%/'% )$ *$##$ ), 45 6 $' .(-+.-'. )$% *-%
8/%%+<#$% )$% %=></#$% -,?)$%%/,% )$ #- (-*+'$7
Election(c(x, y)) → c(x, y) Election(c(x, y)) → Election(H(c(x, y)))Election(b(x, y)) → b(x, y) Election(b(x, y)) → Election(H(b(x, y)))
"! <=/545)*/5$. 3 6%( &789%( 1% .0&.-9),04+ (4+, 45,0'0(:%( .4''% 9) (-0,% 3
H⋆(a(x, y))→ a(H⋆(x),H⋆(y))!
6% $;<RLEP= (4+ >$;<RLEP
λ %, (4+ ?>$;<RLEPλe .4&&%(54+1)+, @4-&+0((%+,
9% 'A'% &:(-9,), E2 B 5)&,0& 1% 9C%+(%'D9% 0+0,0)9 E”1 3
E2 = RLEP ⋆(E”
1) = RLEPλ
⋆(Election(E”
1)) = RLEPλe
⋆(Election(E”
1)).
8>)6'/*/) (?=>+54(./*6? @ 1' ,.+#+%-'. '/.($ /,.+# 0)&.-+##& -, *@-8+.($ A7B '/,%
%/>>$% *-8-<#$% )$ .(-+.$( #$ 8(/./*/#$ C1D -=-'. 24008(/*$%%,% $' 138, 699 %$*/'?
)$%E
E 5)&,0& 1- &:(-9,), 1% 9C%+(%'D9% E2= +4-( (4''%( 0+,:&%((:( 5)& 9C%+(%'D9% 1%
5&45&0:,:( )*)+, -+ ./%'0+ .4+,%+)+, -+0F-%'%+, c 1% 9) &).0+% B 9) @%-099% GH40& 9)
I8-&% !"J! H40& .@! 1%( &789%( 1C0+K)&0)+.% %+ ?>$;< 1)+( 9% ./)50,&% (-0K)+,!
!"#$%&'( )* +,-./%0#&%,1 $#' 23456
!" #$%&'() *& +&,-. +/0 1+2'3&/0-'4%.,.546
!"# $%&&% #%$&'(") "(*# +,-#%"&("# .*%/.*%# $(0!1%# 2#('& 0% &3+% $/!##'.*%
#('& 0% &3+% 4'-,!,$4'.*%5 +(*, #'6*/%, *" ,-#%!* 0% 7%&,' 8(,"- -/-6%"&!',%9 :(*#
6("&,("# -1!/%6%"& $(66%"& "(*# +(*;("# $!/$*/%, 0%# #*$$%##%*,# 0<*" 6!,.*!1%
0(""- !'"#' .*% /<%"#%68/% 0% 6!,.*!1%# !$$%##'8/%# = +!,&', 0<*" 6!,.*!1% '"'&'!/
0(""- +!, "(# #3#&>6%#9
?(*& 0<!8(,0) "(*# 0%;("# -;(.*%, /% $(0!1% 0<*" %"&'%, +(*, #'6*/%, *" "(68,%
0% @%&(" 0!"# *"% +/!$% 0<*" ,-#%!* 0% 7%&,'9
5(7#'89(0 ! "#$%& '#!( )*( +,-(& .! /%$)$(* )*( %*01*( %*)( 2/* 0 3./0 450.&
s(⊥, 0) 3./0 1& s(⊥, s(⊥, 0)) 3./0 2& *% #$!($ '* (/$%*6 7* 8*%%* 1#!$90*& .! 3*/%
($1/)*0 /! *!(*1:)* $!;!$ '<*!%$*0(6 =*3*!'#!%& !./( 305"50.!( /%$)$(*0 '*( %*01*(
'#!( Fbin 2/$ 8.!%$*!% /! (*/) (>1:.)* '* 8.!(%#!%* ⊥ #?*8 '*( (>1:.)*( :$!#$0*(
%*)( 2/* zero, one, two, three...6
A" 0-&!'/) /%# %"&'%,# 0, 1, 2... #("& ,%+,-#%"&-# +!, zero(⊥,⊥), one(⊥,⊥)) %&$9
7(*, +/*# 0% #'6+/'$'&-) "(*# #*++(#("# .*% &(*# /%# $(0!1%# %" AB?CD +%*;%"&
0-#(,6!'# !;(', +/*#'%*,# $("#&!"&%#9 ("$ *"% ,>1/% TestChange(zero(x, y)) →
une(x, y) +%*& E&,% ,%+,-#%"&-% +!, TestChange(0)→ 19
F'"#') 0<*"% +!,&) +(*, 6(0'G%, /% "(68,% 0% @%&("# 0!"# *"% +/!$%) "(*# *&'/H
'#("# 0%*I ,>1/%# +,'6'&';%# Minus, P lus /'#&-%# $'H0%##(*# JK 7(*, 0%# ,-#%!*I 0% 7%&,' 0#%10 J
Plus(0) → 1Minus(1) → 0
K 7(*, 0%# ,-#%!*I 0% 7%&,' 1,1 0#%10 J
Plus(1) → 2...Minus(2) → 1...
<!*&,% +!,&) +(*, #%*/%6%"& &%#&%, *" %"&'%,) "(*# *&'/'#("# #(*;%"& 0%*I ,>1/%#
0% Test /'#&-%# $'H0%##(*# JK 7(*, 0%# ,-#%!*I 0% 7%&,' 0#%10 J
Test0(0) → 0Test1(1) → 1
K 7(*, 0%# ,-#%!*I 0% 7%&,' 1,1 0#%10 J
Test2(2) → 2Test3(3) → 3
:(*# !//("# +,-#%"&%, &,('# $(0!1%# 0<AB?CD +(*, /%# ,-#%!*I 0% 7%&,' 8(,"-#
-/-6%"&!',%# R =< S, T, Pre, Post,m0 >9 L<'0-% +,'"$'+!/% %#&
M9 0% $(0%, *" 6!,.*!1% %" *" &%,6% 0!"# T (Fbin)9 F* 0-8*&) /% 6!,.*!1% '"'&'!/
m0 %#& ,%+,-#%"&- +!, *" &%,6% '"'&'!/ tm0
N9 %"#*'&% 0% $("#&,*',% *" %"#%68/% 0% ,>1/%# Rtλe +(*, $4!.*% &,!"#'&'(" t ∈ T 9
F'"#') *" O,!"$4' 0%# &,!"#'&'("# 0* 6!,.*!1% m = *" !*&,% m′%#& 6(0-/'#-
+!, /! ,-H-$,'&*,% 0* tm !* tm′ %" *&'/'#!"& 0%# ,>1/%# Rtλe 9
!"! #$%&'() *& +&,-. +/0 1+2'3&/0-'4%.,.546 78
92:5-.,;<& 7 !"#$" %& '!()$!*&+ !##&++,-"&+ ReachRTλe(tm0)
+-$354*.,.54 = ./ 0&('& ,/,0,!" tm0 ∈ T (Fbin)1
RTλe $/ &/+&'-"& %& (2*"&+ %345678+ 0&" )$& R
Tλe =
⋃
ti∈T
Rtiλe
! res1 ← ∅ 9
"! res2 ← tm0 9
#! ,'4,>(& res1 &= res2 ?'.-&
$! res1 ← res2 9
%! @5(- α ∈ res2 ?'.-&
&! res2+← (RTλe)
⋆({Transti(α)|ti ∈ T})
'! A4 @5(-
(! A4 ,'4>(&
)! -&,5(-4&- res2
4/:/1 /;$+ #!"#$";/+ "3&/+&'-"& %& '!()$!*&+ !##&++,-"&+ Reach[R,m0] <!(
45678 #;''& %!/+ "3!"*;(,0='& >?
@;$+ #;''&/A;/+ <!( $/ #;%!*& /!BC D ;%!*& EF &0 &/+$,0& %&$G !$0(&+
D ;%!*&+ H &0 IF &/ <!(0!*&!/0 %&+ %;//J&+ ,/0&('J%,!,(&+ &0 &/ <(;:0!/0 "& #!"K
#$" -!+J +$( "& #!#=& %& ',&$G &/ ',&$G?
!"#$% &
;/+,%J(;/+ S1 "3&/+&'-"& %& @ <"!#&+ %3$/ (J+&!$ %& L&0(,? L;$( <"$+ %& ",+,-,",0J
&0 +!/+ <&(0& %& *J/J(!",0J1 /;$+ +$<<;+;/+ )$3&""&+ +;/0 /;''J&+ p0, p1, ..., pN−1?
B$A4.,.54 CD !"#$% &'
(!)* +, -./%#+ "% 0%*-) 1!-,. .2.3%,*#)-% < S, T, Pre, Post,m0 >'
!,/*-+)/!,/ +, *%-3% tm0 "#,/ T (Fbin) 4!33% +,% 2)/*% "%/ 52#4%/ %* 2%+-/
3#-6+#$%/ %, 4!,/%-7#,* 2% ,!3 "% 48#6+% 52#4%
tm0 = p0(v0, p1(v1, ...pN−1(vN−1,⊥)))
*%2 6+% 5!+- *!+* pi ∈ S, i = 0..N − 1 9
: vi = 1 /) pi ∈ m0
: vi = 0 /),!,'
!,/*-+)/!,/ +, %,/%312% "% -;$2%/ "% -.<.4-)*+-% Rtλe 5!+- 48#6+% *-#,/)*)!,
t ∈ T ' =!+/ 4!,/)".-!,/ 6+#*-% 4#/ "% p 9: p &∈ Pre(t) %* p &∈ Post(t) 9
Transt(p(x, y)) → p(x, Transt(y)) !"#$
: p &∈ Pre(t) %* p ∈ Post(t) 9
Transt(p(x, y)) → p(Plus(x), T ranst(y)) !"%$
: p ∈ Pre(t) %* p &∈ Post(t) 9
Transt(p(x, y)) → p(Minus(x), T ranst(y)) !"&$
! "#$%&'() *+ ,-./0&1$'&-2 %$( 34567
p ∈ Pre(t) !" p ∈ Post(t) # $%&' ()*+,%-' .&! /! 01*.&12! !'" '&3)*+!&*! 4
5)*%6
Transt(p(x, y)) → p(Test1(x), T ranst(y)) !"#$
Transt(p(x, y)) → p(Test2(x), T ranst(y)) !"%$
... !"!$
$%&' 1(%-' )21/!0!-" 7!'%+- &-! *82/! 3%&* /1 "!*0+-1+'%- 9! /1 /+'"! 9! 3/1:!' #
Transt(⊥)→ ⊥6 ;+-'+< /! :=1-2!0!-" 9>)"1" 9! m0 4 m1 !- &"+/+'1-" /1 "*1-'+"+%-
t !'" 0%9)/+') 31*
tm1 ∈ (Rtλe)
⋆({Transt(tm0)})
!"# $%%!&# '%%"#()*) +* +!,$-* .$) "& */*0.%*1
38)9%0) :; ?%912! @6 A%+" &- *)'!1& 9! B!"*+ 7%*-) 9& B*%7/80! 4 C 3=+/%'%3=!'6
A%+" S &- !-'!07/! 9! @C 3/1:!' #
S = {Idle0,WaitLeft0, HasLeft0,WaitRight0, HasRight0, Fork0,
Idle1,WaitLeft1, HasLeft1,WaitRight1, HasRight1, Fork1}
A%+" m0 = {Idle0, Fork0, Idle1, Fork1}D- "!*0! tm0 !'" 9%-: *!3*)'!-") :%00! '&+" EF%+* /1 ,2&*! G6H61I #
idle0(1, waitleft0(0, hasleft0(0, waitright0(0, hasright0(0, fork0(1,
idle1(1, waitleft1(0, hasleft1(0, waitright1(0, hasright1(0, fork1(1,⊥))))))))))
A%+" &-! "*1-'+"+%- GoEat0 "!//! .&! Pre(GoEat0) = {Idle0} !" Post(GoEat0) =
{WaitLeft0,WaitRight0} J" &-! 1&"*! GoLeft0 "!//! .&! Pre(GoLeft0) = {Fork1,WaitLeft0}
!" Post(GoLeft0) = {HasLeft0}
J//!' (%-" K"*! 3*%9&+"!' 31* /!' *82/!' 9! *)L):*+"&*! G6@ 4 G6H :+L9!''&'6 ;/%*'
m1 = m0 − Pre(GoEat0) + Post(GoEat0) '!*1 :1/:&/) 31* &- 31' 9! *)L):*+"&*! #
tm1 = idle0(0,waitleft0(1, hasleft0(0,waitright0(1, hasright0(0, fork0(1,
idle1(1, waitleft1(0, hasleft1(0, waitright1(0, hasright1(0, fork1(1,⊥))))))))))
;/%*' m2 = m1 −Pre(GoLeft0) +Post(GoLeft0) '!*1 :1/:&/) 31* &- 31' 9! *)L):*+"&*! #
tm2 = idle0(0, waitleft0(0,hasleft0(1,waitright0(1, hasright0(0, fork0(1,
idle1(1, waitleft1(0, hasleft1(0, waitright1(0, hasright1(0, fork1(0,⊥))))))))))
6/1<0'$'1 )8%/(&9)2'$<8 ; J- &"+/+'1-" -%"*! %&"+/ E9)"1+//) 1& :=13+"*! MI< :!
:%912! !'" :1317/! 9! "*1+"!* /! 3*%7/80! 1N1-" M 3=+/%'%3=!' !- 2.666, 321 '!:%-9!'6
!"! #$%&'() *& +&,-. +/0 1+2'3&/0-'4%.,.546 78
+5.4,% 95-,% *( 35*':& *& -$%&'( *& +&,-. 8 !"#$%& '()!$% * )+,)%-+!./
+5.4,% 9'.;2&% *( 35*':& *& -$%&'( *& +&,-. 8 0+12 +32%.-+,2 41% )%55%
25.1)51.% 6% 6+,,7%2 ,8%25 #(2 )+"#()5%& $8(-(,5(9% 6%2 :;<= 2 ,8%25 6+,) #(2
15!$!27/
!"#$ %& '&(#)* +),' %&-,"%%". *),# +'/#"*$)*# %" 0)1&2" #,(3&*$. 0)1&2" 45
!"#$% &
6, 1/7,$. *),# #,++)#)*# -," $),$"# %"# +%&0"# 1&*# S #)*$ *)88/"# 1" p0 9pN−1 0)88" 1&*# %" )1&2" :5 ;),$ 1!&7)'1. *),# 1(#0,$)*# 1"# '<2%"# +'(8($(3"#
1, 0)1&2" 4 Postp "$ Prep +),' ,*" +%&0" +5 Postp *),# +"'8"$ 1!&=),$"' ,* ="$)*1&*# %& +%&0" p -,&*1 Prep +"'8"$ 1" '"$('"' ,* ="$)*# 1" %& +%&0" p5 >"# 1",?+",3"*$ /2&%"8"*$ 1/+%&0"' 9 %& +%&0" #,(3&*$" #( %& +%&0" 0),'&*$" *!"#$ +&# p5
Postp(p(x, y)) → p(Plus(x), y)Prep(p(x, y)) → p(Minus(x), y)
<$=4.,.54 8> >+6(9% ?/
+!5 1, .72%(1 6% @%5.! 3+.,7 7$7"%,5(!.% < S, T, Pre, Post,m0 >/
>+,25.1!2+,2 1, 5%."% tm0 6(,2 T (Fbin) )+""% 1,% $!25% 6%2 #$()%2 %5 $%1.2
"(.41(9%2 %, )+,2%.-(,5 $% ,+" 6% )A(41% #$()%
tm0 = p0(v0, p1(v1, ...pN−1(vN−1,⊥)))
5%$ 41% #+1. 5+15 pi ∈ S, i = 0..N − 1 B
C vi = 1 2! pi ∈ m0
C vi = 0 2!,+,/
>+,25.1!2+,2 1, %,2%"3$% 6% .D9$%2 6% .7E7).!51.% #+1. )A(41% 5.(,2!5!+, t ∈ T
%, )+""%,F(,5 #(. 1,% .D9$% 6% )+"#+2!5!+, #.!,)!#($% 6% $( '+."%
Transt(x)→ Post0(Pre0(x))
(-%) Pre0 1,% .D9$% 6% )+"#+2!5!+, 6% 2+1.)% Pre(t) %5 Post0 )%$$% 6% 6%25!,(5!+,
Post(t)/
C >A(41% p ∈ Pre(t) %25 !,5%.#.757% #(.
Prei(x)→ Prep(Prei+1(x)), 0 ≤ i <‖ Pre(t) ‖ %5 Pre‖Pre(t)‖(x)→ x/
C >A(41% p ∈ Post(t) %25 !,5%.#.757% #(.
Posti(x)→ Postp(Posti+1(x)), 0 ≤ i <‖ Post(t) ‖ %5 Post‖Post(t)‖(x)→ x
G%2 .D9$%2 #.!"!5!-%2 Postp %5 Prep #+1. 1,% #$()% #/ Postp ,+12 #%."%5 68(E
H+15%. 1, H%5+, 6(,2 $( #$()% p 41(,6 Prep #%."%5 6% .%5!.%. 1, H%5+, 6% $( #$()% p/
I%2 6%1J #%1-%,5 79($%"%,5 2(15%. * $( #$()% 21!-(,5% 2! $( #$()% ()51%$$% ,8%25 #(2
p/
C >A(41% #$()% p B Prep(p(x, y))→ p(Minus(x), y)/
C @+1. 5+12 $%2 p′ &= p B Prep(p′(x, y))→ p′(x, Prep(y))/
C >A(41% #$()% p B Postp(p(x, y))→ p(Plus(x), y)/
! "#$%&'() *+ ,-./0&1$'&-2 %$( 34567
!"# $!"% &'% p′ �= p ( Postp(p′(x, y))→ p′(x, Postp(y)))
*+,%+- &' ./0,1'2',$ 345$0$ 3' m0 6 m1 "$+&+%0,$ &0 $#0,%+$+!, t '%$ 2!35&+%5 70#
tm1 ∈ (Rtλe)
⋆({Transt(tm0)})
! "#$% &'%() *$( %'$% (+% ,-&-)- (% #..!/*$- 0 .#)%/) 1( !# )#2/&(3 4'$+ #!!'&+
/!!$+%)() 2( 2'1#,( .#) $& (5(6.!(3
!"#$%" &' 8!301' 9) :!+$ &' #5%'0" 3' '$#+ 3' &4';'27&' 9<) :!+$ S ", ',%'2=&'
3' >9 7&0.'% (
S = {Idle0,WaitLeft0, HasLeft0,WaitRight0, HasRight0, Fork0,
Idle1,WaitLeft1, HasLeft1,WaitRight1, HasRight1, Fork1}
:!+$ m0 = {Idle0, Fork0, Idle1, Fork1} ?, $'#2' tm0 '%$ 3!,. #'7#5%',$5 .!22'
&' %"+@0,$ AB!+# &0 C1"#' D)E)0F (
idle0(1, waitleft0(0, hasleft0(0, waitright0(0, hasright0(0, fork0(1,
idle1(1, waitleft1(0, hasleft1(0, waitright1(0, hasright1(0, fork1(1,⊥))))))))))
:!+$ ",' $#0,%+$+!, GoEat0 $'&&' G"' Pre(GoEat0) = {Idle0} '$ Post(GoEat0) ={WaitLeft0,WaitRight0} *&!#% m1 = m0 − Pre(GoEat0) + Post(GoEat0) %'#0 .0&."&5 70# ",
70% 3' #5H5.#+$"#' 0@'. ",' #I1&' 3' .!27!%+$+!, 3' %!"#.' Pre(GoEat0)- c.6.d. J&&'@0 K$#' 7#!3"+$' 70# &'% #I1&'% 3' #5H5.#+$"#' +,3+G"5'% .+H3'%%!"%)
RPre(GoEat0)λe =
8
<
:
GoEat0Pre0(x)→ PreIdle0 (x)PreIdle0 (p(x, y)) → p(x, PreIdle0 (y))∀p ∈ S, p %= idle0PreIdle0 (idle0(x, y))→ idle0(Minus(x), y)
9
=
;
.
'$ .'&&' 3' 3'%$+,0$+!, Post(GoEat0)- c.6.d.
RPost(GoEat0)λe =
8
>
>
>
>
<
>
>
>
>
:
GoEat0Post0(x)→ PostWaitLeft0 (PostWaitRight0 (x))PostWaitLeft0 (p(x, y)) → p(x, PostWaitLeft0 (y)) ∀p ∈ S, p %= waitleft0PostWaitLeft0 (waitleft0(x, y)) → waitleft0(Plus(x), y)PostWaitRight0 (p(x, y))→ p(x, PostWaitRight0 (y)) ∀p %= waitright0PostWaitRight0 (waitright0(x, y))→ waitright0(Plus(x), y)
9
>
>
>
>
=
>
>
>
>
;
.
*+,%+- ,!"% !=$',!,%
tm1 = idle0(0,waitleft0(1, hasleft0(0,waitright0(1, hasright0(0, fork0(1,
idle1(1, waitleft1(0, hasleft1(0, waitright1(0, hasright1(0, fork1(1,⊥))))))))))
J$ ",' 0"$#' GoLeft0 $'&&' G"' Pre(GoLeft0) = {Fork1,WaitLeft0} '$ Post(GoLeft0) ={HasLeft0} *&!#% m2 = m1 − Pre(GoLeft0) + Post(GoLeft0) %'#0 .0&."&5 70# ", 70% 3' #5H
5.#+$"#' 0@'. ",' #I1&' 3' .!27!%+$+!, 3' %!"#.' Pre(GoLeft0)- c.6.d. J&&' @0 K$#'7#!3"+$' 70# &'% #I1&'% #5H5.#+$"#' +,3+G"5'% .+H3'%%!"%)
RPre(GoLeft0)λe =
GoLeft0Pre0(x) → PreFork1 (PreWaitLeft0 (x))et ainsi de suite.
ff
.
'$ .'&&' 3' 3'%$+,0$+!, Post(GoLeft0)- c.6.d.
RPost(GoLeft0)λe =
GoLeft0Post0(x)→ PostHasLeft0 (x)et ainsi de suite.
ff
.
!"! #$%&'() *& +&,-. +/0 1+2'3&/0-'4%.,.546 78
idle
wait lef t
hasleft
wai t r ight
hasright
fork
1
1
1
1
1
1
1
0
0
0
0
1
idle
wait lef t
hasleft
wai t r ight
hasright
fork
0
0
0
0
0
0
1
0
0
0
0
1
p
p p
0 0 0 1
p
p p
? ? ? ?
p
p
p p
1 0 0 0
p
p p
0 1 1 0
p
p
b) Codage 3
a) Codages 1 et 2
...
...
Bin = 1011
Bin = 0110
Bin = 0101
Bin = 0000
!"# !" # $%&'() &*+, -./)'+ &) 0)1-2 3 4 5627%/%56)/ ), T (Fbin)
!"#$%&%"'( ")*+ ),'%")"+
tm2 = idle0(0, waitleft0(0,hasleft0(1,waitright0(1, hasright0(0, fork0(1,
idle1(1, waitleft1(0, hasleft1(0, waitright1(0, hasright1(0, fork1(0,⊥))))))))))
#$%(2,',% &)9$-.:&4,'() ; -" *'!$!+#"' ")'.% )*'!$ /01'#!$$1 #* 23#4!'.% 56( 2%
2)0#7% %+' 2#4#,$% 0% '.#!'%. $% 4.),$8&% #9#"' : 43!$)+)43%+ %" 63, 632 +%2)"0%+;
+5.4,% <5-,% *( 35*'=& *& -$%&'( *& +&,-. " <)*+ ),+%.=)"+ >*% $% 4#.'#7%
0%+ .87$%+ 4)*. $%+ )41.#'!)"+ 4.!&!'!=%+ 0%+ 4$#2%+ %+' ,!%" ).7#"!+1 0#"+ 2% 2)0#7%;
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-, /)".3, 4 ,% 56 789*",:, ", /;.+!, (-./, ,'% !%9-9'#, ()!< "#%,<29*,< !*, !*9+!,
()'9%9)* =!*, >,!9--,? ".*' -8.<$<,6 @)!' "#A*9'')*' Bin(p)|k /)22, -, /;9B<, 1 -.
()'9%9)* k ", Bin(p)6 C.< ,:,2(-,& .0,/ Bin(p) = 01..0& *)!' .0)*' Bin(p)|0 = 0&
Bin(p)|1 = 1 ,% Bin(p)|(h− 1) = 06 C.< /)*'#+!,*%& -. >,!9--, 0 ,'% <#',<0#, ()!< -.
(-./, p0 =Bin(0) = 00..0?&666 +!.*" -. >,!9--, N − 1 =.0,/ Bin(N − 1) = 11..1? ,'%
<#',<0#, ()!< -. (-./, pN−16 @)!' *8.0)*' ")*/ (-!' $,')9* ", /)*',<0,< -, *)2 ",
/;.+!, (-./,& /, +!9 <#"!9% -. %.9--, "! /)".3,6
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(!)* +, -./%#+ "% 0%*-) 1!-,. .2.3%,*#)-% < S, T, Pre, Post,m0 >' !,/*-+4
)/!,/ +, *%-3% t "#,/ T (Fbin) "% 5#+*%+- h = log2(N) *%2 6+% 7!+- 85#6+% i 9 /)
marquage = pi: #2!-/ i ∈ PosF (marquage): #;%8 marquage ∈ {0, 1} %* 0 ≤ i < N '
5#6+% *-#,/)*)!, t ∈ T "!)* <*-% 7-!"+)*% 7#- 2%/ -=$2%/ "% -.4.8-)*+-% 8!34
3%,>#,* 7#- +,% -=$2% 7-),8)7#2% 8!33% "#,/ 2% 8!"#$% ? 9
Transt(x)→ Post0(Pre0(x))
*%22% 6+% Pre0 7!+- 2@%,/%312% "% /!+-8% Pre(t) %* Post0 7!+- 2@%,/%312% "% "%/*)4
,#*)!, Post(t)'
A 5#6+% p ∈ Pre(t) %/* ),*%-7-.*.% 7#-
Prei(x)→ PrehBin(p)(Prei+1(x)), 0 ≤ i <‖ Pre(t) ‖'
A 5#6+% p ∈ Post(t) %/* ),*%-7-.*.% 7#-
Posti(x)→ PosthBin(p)(Posti+1(x)), 0 ≤ i <‖ Post(t) ‖'
B%/ -=$2%/ "% 8)-8+2#*)!, /!,* $.,.-.%/ %* #772)6+.%/ C 7#-*)- "% 2# -#8),% 9
A 5#6+% Preh: 7!+- 0 < k ≤ h
A PrekBin(p)(p(x, y)→ p(Prek−1
Bin(p)(x), y) /) Bin(p)|k = 0
A PrekBin(p)(p(x, y)→ p(x, Prek−1
Bin(p)(y)) /),!,'
D,E,: 6+#," ,!+/ #--);!,/ C 2# 7!/)*)!, "% 2# 72#8% ),")6+.% 9
Pre0(p(x, y))→ p(Minus(x), y)
Pre1(p(x, y))→ p(x,Minus(y))
A 5#6+% Posth: 7!+- 0 < k ≤ h
A PostkBin(p)(p(x, y)→ p(Postk−1
Bin(p)(x), y) /) Bin(p)|k = 0
A PostkBin(p)(p(x, y)→ p(x, Postk−1
Bin(p)(y)) /),!,'
D,E,: 2%/ !7.-#*)!,/ 7!+- #F!+*%- !+ -%*)-%- +, F%*!, "@+,% 72#8% 6+#," ,!+/
#--);!,/ C 2# 7!/)*)!, ),")6+.% /!,* 9
Post0(p(x, y))→ p(Plus(x), y)
Post1(p(x, y))→ p(x, P lus(y))
G),/): 2% 85#,$%3%,* "@.*#* "% m0 C m1 +*)2)/#,* 2# *-#,/)*)!, t %/* 3!".2)/. 7#-
tm1 ∈ (Rtλe)
⋆({Transt(tm0)})
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9)&:;2& " !"#$% &' (!)* +% ,-.%#/ "% 0%*,) "% +1%2%34+% 56' (!)* S /7 %7.%38+%"% 95 4+#:%. ;
S = {Idle0,WaitLeft0, HasLeft0,WaitRight0, HasRight0, Fork0,
Idle1,WaitLeft1, HasLeft1,WaitRight1, HasRight1, Fork1}
<!/. #=!7. 8%.!)7 "1/7 #,8,% "% >#/*%/, h = 4 4!/, :!"%, +1%7.%38+% S? [email protected]. 7!/.
/*)+).!7. 95 4,%3)A,%. B%/)++%. "% :%* #,8,% C/) .!7* ,-.%,=-%. 4!/, +%. 95 4+#:%. "%
S? "!7: +%. 6 "%,7)A,%. 7% .!7* 4#. /*)+).-.' (!)* m0 = {Idle0, Fork0, Idle1, Fork1}'
D7 *%,3% tm0 %.* "!7: ,%4,-.%7*- :!33% %7 E$/,% F'G'8'
(!)* /7% *,#7.)*)!7 GoEat0 *%++% C/% Pre(GoEat0) = {Idle0} %* Post(GoEat0) ={WaitLeft0,WaitRight0} H+!,. m1 = m0 − Pre(GoEat0) + Post(GoEat0) .%,# :#+:/+- 4#, /7
4#. "% ,-I-:,)*/,% #=%: /7% ,A$+% "% :!34!.)*)!7 "% .!/,:% Pre(GoEat0)? [email protected]. J++%=# K*,% 4,!"/)*% 4#, +%. ,A$+%. "% ,-I-:,)*/,% )7")C/-%. :)I"%..!/. ;
RλePre(GoEat0) =
8
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:
GoEat0Pre0(x)→ Pre0000(x)Pre0000(p(x, y)) → p(Pre000(x), y)Pre000(p(x, y))→ p(Pre00(x), y)Pre00(p(x, y))→ p(Pre0(x), y)Pre0(p(x, y)) → p(Minus(x), y)
9
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%* :%++%. "% "%.*)7#*)!7 Post(GoEat0)? [email protected].
RλePost(GoEat0) =
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:
GoEat0Post0(x) → Post0001(Post0011(x))
Post0001(p(x, y)) → p(Post001(x), y)Post001(p(x, y)) → p(Post01(x), y)Post01(p(x, y))→ p(Post1(x), y)
Post0011(p(x, y)) → p(Post011(x), y)Post011(p(x, y)) → p(Post11(x), y)Post11(p(x, y))→ p(x, Post1(y))
9
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0)! 0)!1 " 3&!16
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#51!3) >
H(sys(x, y))→ SyncII+1(sys(x, y)) !"%$
H(sys(x, y)) → Sync0N−1(sys(x, y)) !"&$
! "#$%&'() *+ ,-./0&1$'&-2 %$( 34567
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SyncII+1(sys(x, y))→ sys(SyncII+1(x), y) !"#$%
SyncII+1(sys(x, y))→ sys(x, SyncII+1(y)) !"##%
6")# &!"# .$&7!&# .(# +,-.(# *( #8&/9+!&)#$0)!& : /9$;"( &)%($" *" /!*$-( 5
SyncII+1(sys(x, y)) → sys(SyncI(x), SyncI+1(y)) !"#&%
Sync0N−1(sys(x, y)) → sys(Sync0(x), SyncN−1(y)) !"#'%
SyncI(sys(x, y)) → sys(x, SyncI(y)) !"#(%
SyncI+1(sys(x, y)) → sys(SyncI+1(x), y) !"#)%
Sync0(sys(x, y)) → sys(Sync0(x), y) !"#!%
SyncN−1(sys(x, y))→ sys(x, SyncN−1(y)) !"#*%
<)&$.(=(&0> $" &)%($" *?"& #!"#3+@#($" *" /!*$-( 5
Sync0(local(x, y))→ α0 !"#+%
SyncN−1(local(x, y))→ αN−1 !"#,%
SyncI(local(x, y)) → βI !"&$%
SyncI+1(local(x, y)) → βI+1 !"&#%
!A α, β #!&0 *(# /!=1!#)0)!&# *( ;"(.;"(# +,-.(# 1+)=)0)%(# Postp $&* Prep 1!"+
;"(.;"(# 1.$/(#B
38)9%0) :; !"#$%&' (') *+$")",*')- ./ 01+23 4/ )"4) !5)'14 161/2 7 ,$18')
{Idle,WaitLeft,HasLeft,WaitRight,HasRight, Fork} 19'8 )'4$'&'/2 : ;'2"/)(1/) $') ,$18') Idle '2 Fork ')2 8"(5 8+<('))"4) =
tlocal = local(idle(1, waitleft(0, hasleft(0, waitright(0, hasright(0, fork(1,⊥))))))
>+/)+3 4/ 8"(1?' (' /+9'14 @ ,"4! : ,*+$")",*') ')2 4/' 8"&,")+2+"/ (' ('4A )"4)<
!5)'14A +('/2+B4') ')2 8"(5 8"&&' )4+2 =
tsys1 = sys(tlocal, tlocal)
>+/)+3 4/ 8"(1?' /+9'14 : ,"4! C ,*+$")",*') ')2 8"&,")5 (' $1 &D&' &1/+%!' =
tsys2 = sys(tsys1 , tsys1 )
C%)*(==(&0> $1+,# .( 1+!'.,=( *(# 69).!#!19(#> .( /!*$-( 9)@+$+/9);"( (& /!=3
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=(&0 "& 1(" *)I@+(&0 *(# $"0+(# /$+ ). *!)0 #8&/9+!&)#(+ 0!"# .(# #!"# +@#($"D $%(/
"&( 1.$/( 1$+0$-@(B
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+$11!+0 $"D $"0+(# $11+!/9(#B
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7+78&)210'&$ 9 .'"02& .& 2:;& <= 10)$0 &++&$ $")2 '&8;+1/7&$ ;1' +1 ',-+& $#0(1)2&
Sync⋆II+1(sys(x, y))→ sys(Sync⋆II+1(x), Sync⋆II+1(y))
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Tλe =
⋃
ti∈T
Rtiλe8 )2&" (9&/,( v ,(" 7$( v = R
tλe(Transt(u)) !:(# t ∈ T ; !&+/& "2<,!, =%<#<.(+,
.$ ,(%-( v (/, .<>+& #*--( /$&:!+, ?
R−1λe (v) = u
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#!"#$"< /$% Rλe D =!%,&% .( init #*--( /$&:!+, ?
deadlock = R⋆λe(init) \ R
−1λe (R
⋆λe(init))
;4)3=) "> ?;4)3=) (3,,-$,/) @
deadlock = R⋆λe(init)\R
−1λe (R
⋆λe(init))
E deadlock ⊆ R⋆λe(init)\R
−1λe (R
⋆λe(init)) ? F+,$&,&:(-(+, /("*+ deadlock ⊆
R⋆λe(init) (, deadlock ∩R
−1λe (R
⋆λe(init)) = ∅
E R⋆λe(init)\R
−1λe (R
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!##(//&0"( (, &" +2(/, =!/ .!+/ R−1λe (R
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t0local = local(idle(0,waitleft(1, hasleft(0, waitright(0,hasright(1, fork(0,⊥))))))
t1local = local(idle(0,waitleft(1, hasleft(0, waitright(0,hasright(1, fork(0,⊥))))))
A* -9&*(* $)/&*%01"' 5
t0local = local(idle(0, waitleft(0,hasleft(1,waitright(1, hasright(0, fork(0,⊥))))))
t1local = local(idle(0, waitleft(0,hasleft(1,waitright(1, hasright(0, fork(0,⊥))))))
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Φ ::= TRUE | ⊥ | a(Φ1,Φ2) | Φ1 ∧ Φ2 | Φ1 ∨ Φ2 | α
(>'. a ∈ Fbin '* α ∈ T (Fbin,X )@
B* )7/#&) -)# CDE# +*,##/(")# 6!"' 48'/A)' "%) &)**) 6'!6'/8&8;
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#&1 -$& (RInv)⋆({t}) %7&,# 2/, ;")& )/%, $%& )&, 0!'(&, ."<)&,,!$, =
5* Invall /;&. %7"(2!'#& -$&1 #&'(& t
RInvall ={
x→ x}
.
>* Inv⊥ /;&. t = ⊥
RInv⊥ ={
⊥ → ⊥}
.
?* InvaInvl,Invr /;&. t = a(tl, tr)@ a ∈ Fbin@ Invl(tl) &# Invr(tr) ,!%# ;'/"&,
a(x, y)→ a(x, y)
/;&. Pos(tl) = 00,Pos(tr) = 01 &#
(RInvl)⋆(tl) ∩ (RInvr)⋆(tr) (= ∅
A* (Inv1 ∨ Inv2)(t) /;&. Inv1(t) !$ Inv2(t) &,# ;'/"&, =
(RInv1∨Inv2)⋆(t) = (RInv1)⋆(t) ∪ (RInv2)⋆(t)
B* (Inv1 ∧ Inv2)(t) /;&. Inv1(t) &# Inv2(t) ,!%# ;'/"&, =
(RInv1∧Inv2)⋆(t) = (RInv1)⋆(t) ∩ (RInv2)⋆(t)
C* InvαRInvα =
{
α→ α, α ∈ T (Fbin,X )}
.
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9:GH &# IJK H@ !"# OneToken 1& 2'3)"./# )34%" .!((& 1& ,$";/%# 2!$' $% #&'(&
α ∈ T (Fbin) =
OneToken(α) =
⊥ si ∀p, p′Pos(α),
α(p) = α(p′) = t⇒ p = p′
∅ sinon.
L/' .!%,3-$&%#@
TAP satisfait φ⇔ ∀α ∈ R⋆({u}), OneToken = ⊥
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TAP satisfait φ⇔ ∀α ∈ R⋆({u}), OneToken (= ∅
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F conflict -( "1!- ,$- $&D*&"- -1&" onlyt 1! *! %1(&- !& $-" conflict)
F not -( "1!- ,$- $&D*&"- -1&" not)
F onlyt -(&1&/
G9$&-$%:.$ '$ #H=.$- '$ @AB ROneToken$-" .(-"2$- ,(>'$--!- 5
a(⊥,⊥) → not(⊥,⊥) with a ∈ {i, r, b} !"#$
t(⊥,⊥) → onlyt(⊥,⊥) !"%$
a(conflict(x, y), z) → conflict(conflict(x, y), z) !"&$
a(x, conflict(y, z)) → conflict(x, conflict(y, z)) !"'$
a(onlyt(x, y), onlyt(z, w)) → conflict(onlyt(x, y), onlyt(z, w)) !"($
t(onlyt(x, y), onlyt(z, w)) → conflict(onlyt(x, y), onlyt(z, w)) !")$
a(not(x, y), not(z, w)) → not(not(x, y), not(z, w)) !"!$
t(not(x, y), not(z, w)) → onlyt(not(x, y), not(z, w)) !"*$
a(not(x, y), onlyt(z, w)) → onlyt(not(x, y), onlyt(z, w)) !"+$
a(onlyt(x, y), not(z, w)) → onlyt(onlyt(x, y), not(z, w)) !"#,$
t(not(x, y), onlyt(z, w)) → conflict(not(x, y), onlyt(z, w)) !"##$
t(onlyt(x, y), not(z, w)) → conflict(onlyt(x, y), not(z, w)) !"#%$
I1!# +2#(3$# 8!$ .$ TAP satisfait φ) ,$.* -!J" '$ "$-"$# -9(. $6(-"$ !&$ ,1&3=>
!#*"(1& 2"(8!$"2 onlyt E .* #*,(&$ 5
TAP satisfait φ⇔
(InvOneToken) ∧ InvonlytInvall,Invall⇔
(ROneToken)⋆({u}) ∩ (InvonlytInvall,Invall)⋆({u}) (= ∅
1K InvonlytInvall,Invall$-" .9$&-$%:.$ '$ #H=.$- 41!# 3."#$# !&$ ,1&3=!#*"(1& 2"(8!$"2
onlyt E .* #*,(&$ 5
onlyt(x, y)→ onlyt(x, y)
*+$, Pos(tl) = 00,Pos(tr) = 01 $" (RInvall)⋆(tl) ∩ (R
Invall)⋆(tr) (= ∅/
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)12%"& )/%, 3/ )12%"#"!% 45 ,6"3 &7",#& $% 89 (!)13",/%# $%& "%:/'"/%.& #&3 -$&
(RInvλe )
⋆({Inv(t)}) %6&,# ;/, :")& )/%, $%& )&, 0!'(&, ."<)&,,!$, =
4* Invall /:&. %6"(;!'#& -$&3 #&'(& t
RInvallλe =
{
Inv(a(x, y))→ a(x, y) ∀a ∈ FbinInv(⊥)→ ⊥
}
.
>* Inv⊥ /:&. t = ⊥
RInv⊥λe =
{
Inv(⊥)→ ⊥}
.
?* InvaInvl,Invr /:&. t = a(tl, tr)@ a ∈ Fbin@ Invl(tl) &# Invr(tr) ,!%# :'/"&,*
RInvaInvl,Invrλe =
{
Inv(a(x, y))→ a(Invl(x), Invr(y))}
.
A* Inv1 ∨ Inv2 /:&. Inv1(t) !$ Inv2(t) &,# :'/"&,*
RInv1∨Inv2λe =
Inv(a(x, y))→ Inv1(a(x, y)) ∀a ∈ FbinInv(a(x, y))→ Inv2(a(x, y)),
Inv(⊥)→ Inv1(⊥),
Inv(⊥)→ Inv2(⊥)
.
B* Inv1 ∧ Inv2 /:&. Inv1(t) &# Inv2(t) ,!%# :'/"&,*
RInv1∧Inv2λe =
{
Inv(a(x, y))→ Inv1(Inv2(a(x, y))) ∀a ∈ Fbin,
Inv(⊥)→ Inv1(Inv2(⊥))
}
.
C* Invα +!$, /:!%, D&,!"% )& 36EF89 Rα→βcheck )!%%1 )/%, 3/ ,&.#"!% A*> -$" %!$,
;&'(&# )& :1'"2&' ," $% #&'(& t)& T (Fbin) .!%.!')& /:&. α*
:# 1*)&; #$+" 6$+5$#" 57')8!' ") 34!#"!(<3! ! (*+5*)" 7&*&" Bad = ¬φ =
¬Inv(R⋆(init)) "$)& &$+=$+'" 1*+,- ># 7&*& ¬φ 5'*) !"& 2$(6&7 2$((! +# !,!(63!-
:0;$-/-2$ 6< !"# t $% #&'(& )& T (Fbin)* G%& "%:/'"/%.& %1H/#":& ¬Inv(t) &,#
:'/"& ,&$3&(&%# ," ¬Inv &,# $%& )&, 0!'(&, ."<)&,,!$, =
4* ¬Invall = ∅
>* ¬Inv⊥ /:&. t = a(t1, t2)
?* ¬InvaInvl,InvrI /:&. t = b(tl, tr)@ b (= a | b ∈ Fbin@ !$
I /:&. t = a(tl, tr) Invl(tl) ∨ Invr(tr) &,# 0/$7*
A* ¬(Inv1 ∨ Inv2) /:&. ¬Inv1(t) ∧ ¬Inv2(t) &,# :'/"&*
B* ¬(Inv1 ∧ Inv2) /:&. ¬Inv1(t) ∨ ¬Inv2(t) &,# :'/"&*
!" #$%&'()* +, -.)'/0%('12 &%) 34567
! ¬Invα "#$% Rα→βcheck &'( t $&) #*+$
! "#$%& $' ()$*%( +!( #+%,( *-'+%$-! .-+, /#'/+'(, ¬Inv (! +%$'$*#!% '( /#'/+' 0(*
1%#%* #//(**$2'(* R⋆3 ''( (*% .,1*(!%1( 0#!* '# /-!4(/%+,( *+$5#!%( 6
#128*0(9)* : ,-.")*/0 +1*0#"(*"0%$!
¬Inv(α) = R⋆(init)\Inv(α)
"#$% α '0 )$(2$ %3/&!
7(5(!-!* 8 !-%,( 1%+0( 0( /#* 9:;3
3;*<&=* :! 4156789 &'*#"0) R¬φλe &:-%*;$ 3" :(/:(*-)- ¬φ!
No t(r(x, y)) → r(No t(x), No t(y)) !"#$%
No t(b(x, y)) → b(No t(x), No t(y)) !"#&%
No t(i(x, y)) → i(No t(x), No t(y)) !"#'%
No t(⊥) → ⊥ !"#(%
Two t(r(x, y)) → r(Two t(x), y) !"#!%
Two t(r(x, y)) → r(x, Two t(y)) !"#)%
Two t(b(x, y)) → b(Two t(x), y) !"#*%
Two t(b(x, y)) → b(x, Two t(y)) !"+,%
Two t(i(x, y)) → i(Two t(x), y) !"+#%
Two t(i(x, y)) → i(x, Two t(y)) !"++%
Two t(t(x, y)) → i(x, Two t(y)) !"+$%
9-2"0)*<'$2$0) +*)= 3$ &>2?/3$ No@t ∈ FNT $&) *0)(/+'*) :/'( #-(*;$( <'1'0
)$(2$ 0$ %/0)*$0) :"& 3$ &>2?/3$ t! 5A$%)*#$2$0)= *3 01> " "'%'0$ (B.3$ +$ 3" C/(2$
No@t(t(x, y))→ β! 50 %/0&-<'$0%$= :/'( '0 )$(2$ α ∈ T (Fbin) :/'( 3$<'$3 *3 $D*&)$
p ∈ Pos(α) )$3 <'$ α(p) = t= (R¬φλe )
∗({No@t(α)}) = ∅!
4$ &>2?/3$ Two@t ∈ FNT $&) *0)(/+'*) :/'( #-(*;$( <'$ 3$ &>2?/3$ t "::"("E)
+"0& '0 )$(2$ "' 2/*0& '0$ C/*&! F*0&*= :/'( '0 )$(2$ α ∈ T (Fbin)=
G (R¬φλe )
⋆({Two@t(Two@t(α))}) = ∅ &*.0*;$ <'$ 3$ &>2?/3$ t "::"("E) +"0& α
'0$ &$'3$ C/*& /' H"2"*& I
G (R¬φλe )
⋆({Two@t(Two@t(α))}) (= ∅ &*.0*;$ <'$ 3$ &>2?/3$ t "::"("E) +"0& α
"' 2/*0& +$'D C/*&!
J"( %/0&-<'$0)=
3$ 7FJ &")*&C"*) φ ⇔(R¬φ
λe )⋆((RTAPλe )⋆({Two t(Two t(t′0))})) ∧ (R¬φ
λe )⋆((RTAPλe )⋆({No t(t′0)})) = ∅
<'"0+ t′0 = Arbiter(t0) $) t0 $&) 3" %/0;.'(")*/0 *0*)*"3$ +' 7FJ!
!"! #$%&'() *+%,,)((-.-&-/0 1234 &)( -$5%4-%$/( 678
!"# $%&#'()!(# "(' *")%' #!+"),!( '( -*+-"+*() R⋆λe({InvOnlyT (α)}) -,./'##!"# 0
InvOnlyT (a(x, y)) → a(InvOnlyT (x), InvNoT (y)) !"#$%
InvOnlyT (a(x, y)) → a(InvNoT (x), InvOnlyT (y)) !"#&%
InvOnlyT (t(x, y)) → t(InvNoT (x), InvNoT (y)) !"#$%
InvNoT (a(x, y)) → a(InvNoT (x), InvNoT (y)) !"#!%
InvNoT (⊥) → ⊥ with a ∈ {i, r, b} !"#&%
!"#$%&%" '()!*+,'-%&#( . ! "#$%$&'!# !(#)* ("#$% + ! ,-#'$% ,'!& %* ./'0$#)*
123 !("& &(44*& .'0'5%*& ,* 6-)$7*) 8"* φ *&# 6)'$ 0(") %* 9:; #)'$#'!# 24000)(<
.*&&*")& *! 2, 933 &*.(!,*&=
!"!# $%&'(%&) *+,-'&) ./0+/&)1&23)
/010202 3!*+45&%+6- 78+-9&*+&-%" 7# ::
!"#$%&'!"# () *'!+!,!() %) -)',!(.+) )" #),+$!" /01010 2!3# "!3# $"+&')##!"#
3"$43)5)"+ .36 *'!*'$&+&# %) E1 43$ ,!"+$)"")"+ #)3()5)"+ %)# &+$43)++)# n7 p )+
⊥ 89!$' (. :;3') /0<=0
E1 = ((RPP )
⋆(E0) ∩ T ({n, p,⊥})).
>" )"#)5?() %) '@;()# %ABCDEF (RFilternp⊥λe )⋆({Inv(α)}) ,!'')#*!"%."+ )#+
%&,'$+ ,$G%)##!3# H
Inv(⊥) → ⊥ !"#'%
Inv(p(x, y))→ p(Inv(x), Inv(y)) !"()%
Inv(n(x, y)) → n(Inv(x), Inv(y)) !"(*%
C$".()5)"+7 "!3# ,!"#$%&'!"# (. *'!*'$&+& +)(() 43) #. '.,$") )#+ &+$43)+&) *.'
p H
E′1 = (E1 ∩ {p(t1, t2)|t1, t2 ∈ T}).
I$"#$7 (RPλe)
⋆({P (α)}) .J), P (p(x, y))→ p(x, y)0
B" ,!",(3#$!"7 E′1 #= ∅ #$;"$:) 43A.3 5!$"# 3") %)# K)3$(()# *!##@%) 3") J.()3'
p L E′1 = ∅ #$;"$:) 43) +!3+)# ()# K)3$(()# "A!"+ 43A3") J.()3' n0
!"#$%&%" '()!*+,'-%&#( 2; ! "#$%$&'!# !(#)* ("#$% +,-#'$%%- '" ./'0$#)* 123 $%
*&# 0(&&$5%* ,* #)'$#*) %* 0)(#(.(%* ,* ;*).(%'#* '>'!# 25000)(.*&&"& *! 4($!& ,?"!*
,*4$* ,* &*.(!,*=
! "#$%&'() *+ ,-(&./$'&01 %$( 23456
*+7+ +7 ,-(&./$'&01 89&1:$(&$1'; 8< %(0'0/0=) 892=)/'&01 >(?0();/)1'
!"#$%&'!"# () *'!+!,!() %-.(),+$!" /'0!')#,)"+ )" #),+$!" 123242
56"# (6 %)78$9:) &+6*); "!7# )##6<)'!"# %) %)#,)"%') )" #&(),+$!""6"+ () ,6"=
%$%6+ >&+$?7)+& *6' p@ #$ *!##$0()2
/ *6'+$' %7 '(+6+ %) E2; "!7# "!7# $"+&')##!"# 678 *'!*'$&+&# 6<6"+ 7" ,A):$"
?7$ ,!"+$)"+ 7"$?7):)"+ c %) (6 '6,$") B (6 C)7$(()2
D" +)( .EFGH (RPathcλe )⋆({Inv(α)}) )#+ %&,'$+ ,$=%)##!7# I
O(a(x, y)) → a(x, y) avec a ∈ {n, p} !"#$%
Inv(⊥) → ⊥ !"##%
O(⊥) → ⊥ !"#&%
Inv(c(x, y)) → c(O(x), Inv(y)) !"#'%
Inv(c(x, y)) → c(Inv(x), O(y)) !"#(%
J-)"#):0() E′2 )#+ $((7#+'& )" KL7') 12M2
N!7# :!"+'!"# &L6():)"+ ,!::)"+ "!7# *!7O!"# :!%&($#)' ,) .EFGH )" FGH I
N!7# *!7O!"# %&"!:0')' () "!:0') %) C)7$(()# &+$?7)+&)# *6' c2
N!7# *'!*!#!"# 7") 67+') #!(7+$!" I N!7# *!7O!"# +'!7O)' ,) '(+6+ K"6( )"
(6"P6"+ (RPathcλe )⋆({Inv(α)}) #7' E”1 >Q!$' (6 KL7') 12M@2
5-;<='$'; )@%-(&A)1'$<@ ! "#$"%$#!& (RPathcλe )⋆({Inv(α)}) '%( E”1 )#(
!*&(+ *%&,$ -./&#,$$/ #% "0#),&(+ 123 ,$ +'& )*'',4$+ .+ &(#,&+( $+ )(*&*"*$+ .5 $+"&,*!
6(4*(+'"+!& #7#!& 2450)(*"+''%' +! 8*,!' .5%!+ '+"*!.+9
FGH RPathoI
c(⊥,⊥)→ o(⊥,⊥) !"#!%
c(o(x, y), z)→ o(o(x, y), z) !"#)%
o(x, o(y, z))→ o(x, o(y, z)) !"#*%
D" +)( .EFGH RPathoλe )#+ %&,'$+ ,$=%)##!7# I
R :;<$+' </!/(/+'
O(o(x, y))→ o(x, y) !"&+%
O(⊥)→ ⊥ !"&,%
H(c(x, y)) → o(O(x), y) !"&$%
H(c(x, y)) → o(x,O(y)) !"&#%
R :;<$+' .+ ",("%$#&,*!
H(c(x, y)) → c(H(x), y) !"&&%
H(c(x, y)) → c(x,H(y)) !"&'%
!"! #$%&'() *+%,,)((-.-&-/0 1234 &)( -$5%4-%$/( 666
!"#$% &$ '()*+ ,-$
Patho(x) → x !"#$%
Patho(x) → Patho(H(x)) !"#!%
!"#$ %"&&'()"($ * +,-./- 0# 1".."& '( -'&+2,),(. %3,4#' 5./4#'..' c +,- o
'. &"(.'- ("'#0 +,- ("'#06 7, +-"+-/5.5 ,8,(. #(' 5./4#'..' o * 2, -,%/(' '$. %'22'
4#/ ("#$ /(.5-'$$'6
E2 = (RPatho)⋆(E”1) = (R
Pathoλe )⋆(Inv(E”1)).
9/($/: ("#$ +-5$'(."($ 2'$ $8$.;&'$ %/<0'$$"#$ +"#- %3'-%3'- #(' +-"+-/5.5 ,8,(.
#(' 5./4#'..' o * 2, -,%/(' =
E”2 = (E′
2 ∩ {o(t1, t2)|t1, t2 ∈ T}).
9/($/: (ROλe)
⋆({O(α)}) ,>'% O(o(x, y)) → o(x, y)"
?'..' +-"+-/5.5 '$. '@,%.'&'(. %"&&' '( AB#-' C6D $,#E 4#' 2' %3'&/( 0'$
5./4#'..'$ c '$. -'&+2,%5 +,- %'2#/ 0'$ 5./4#'..'$ o6
70(3&/%/( )8104-9)$/%38 6" .* /0#/1#0*+ (RPathoλe )⋆({Inv(α)}) %12 E”1 '02
*(+2$ (1+)# 3&4+0)##4 01 /50')+2$ 678 )# $%+ '(%%)9#$ &$ +20)+$2 #$ '2(+(/(#$ &:.#$/+)(*
;29(2$%/$*+ 0<0*+ 2400'2(/$%%1% $* '#1% &$ &$1- =)*1+$%>
!"!" #$%&'$%( )*+,&%( +% -.(%/$0 +% 1%2-3
F,($ 2'$ &"0;2'$ 0' -5$',#@ 0' G'.-/: /2 E,#. 1/'( &,H.-/$'- 2'$ -;B2'$ 0' %/-%#2,<
./"( +"#- ,22'- * 2I'(0-"/. 0' 2, +2,%' /(0/4#5' '. .'$.'- 2' &,-4#,B' $"#3,/.56
!"!"!6 :04-;,%/-2$ *+-$5%4-%$/( *)( <=-&2(21=)(
J'$. 0I/(>,-/,(%' +"#- >5-/A'- 4#I/2 8 , ,#%#(' %"(AB#-,./"( .'22' 4#'
0'#@ >"/$/($ +-'(('(. 2, &K&' E"#-%3'..'6 7' +-"12;&' 05$"-&,/$ 0'>/'(. 0'
2,(%'- #( $8$.;&' 0' -5<5%-/.#-' RConflitλe +"#- .'$.'- 2'$ 0'#@ >"/$/($6 L/
(RConflitλe )
⋆({Conflit(α)}) E"#-(/. #( -5$#2.,. ("( >/0': ,2"-$ /2 '@/$.' #(' .'22' %"(<
AB#-,./"(6
Conflit(system(x, y))→ Sync1N(system(x, y)) !"#&%
Conflit(system(x, y))→ SyncIIPP (system(x, y)) !"#'%
SyncIIPP (system(x, y))→ system(SyncIIPP (x), y) !"()%
SyncIIPP (system(x, y))→ system(x, SyncIIPP (y)) !"(*%
SyncIIPP (system(x, y)) → system(TestI(x), T estIPP (y)) !"(+%
Sync1N(system(x, y)) → system(TestIPP (x), T estI(y)) !"(,%
TestI(system(x, y)) → system(x, TestI(y)) !"(#%
TestIPP (system(x, y)) → system(TestIPP (x), y) !"((%
TestI(local(x, y)) → local(x, LocalTestI(y)) !"($%
TestIPP (local(x, y)) → local(x, LocalTestIPP (y)) !"(!%
LocalTestI(x)→ Testhasright(x) !"(&%
LocalTestIPP (x) → Testhasleft(x) !"('%
! "#$%&'() *+ ,-(&./$'&01 %$( 23456
!"# $!%$% &'()($(*% &+"' ,-'(.!' %( /! &0(1+%+&0! 2 3(!4 &'(% "4! *+"'/0!$$! 524%
1! )2(4 62"/0! 75'+($!8 9
Testhasleft(hasleft(x, y)) → hasleft(Test(x), y) !"#$%
Testhasleft(fork(x, y))→ fork(x, Testhasleft(y)) !"#&%
Testhasleft(hasright(x, y)) → hasright(x, Testhasleft(y)) !"#'%
Testhasleft(waitright(x, y))→ waitright(x, Testhasleft(y)) !"#(%
Testhasleft(waitleft(x, y)) → waitleft(x, Testhasleft(y)) !"#)%
Testhasleft(idle(x, y)) → idle(x, Testhasleft(y)) !"#*%
Testhasleft(fork(x, y))→ fork(x, Testhasleft(y)) !"##%
Testhasright(hasright(x, y)) → hasright(Test(x), y) !"#!%
Testhasright(fork(x, y)) → fork(x, Testhasright(y)) !"#+%
Testhasright(hasleft(x, y)) → hasleft(x, Testhasright(y)) !"#,%
Testhasright(waitright(x, y))→ waitright(x, Testhasright(y)) !"!$%
Testhasright(waitleft(x, y))→ waitleft(x, Testhasright(y)) !"!&%
Testhasright(idle(x, y)) → idle(x, Testhasright(y)) !"!'%
Testhasright(fork(x, y)) → fork(x, Testhasright(y)) !"!(%
Test(1) → 1 !"!)%
5-789'$'7 ):%-(&;)1'$8: < ! "#$%$&'!# !(#)* ("#$% +,-#'$%%- '" ./'0$#)* 123
!("& &(44*& .'0'5%*& ,* 6-)$7*) 8"* 0')4$ 2350/$%(&(0/*&3 $% !9: ' '"."!* .(!;
7<")'#$(! #*%%* 8"* ,*"= 6($&$!& 8"$ 0)*!!*!# %' 4>4* ?(")./*##* *! "!* ,$@'$!* ,*
4$!"#*&A
*+!+!+! ,-(&./$'&01 =)7 &1>$(&$1'7 =8 %(0'0/09) =) 5081=?50@&1 A8'):
:+"% (4$-'!%%+4% -621!)!4$ ; "4 $!%$ 5<(4,2'(24/! ϕ &+"' ,-'(.!' ="<(1 > 2 "4!
%!"1! '!%%+"'/! 524% 1! )+5?1!@ A(4%(B 4+"% !%%2>+4% "4 $!%$ 5! 4-62$(+4 5<(4,2'(24/!
¬ϕ &+"' /0!'/0!' "4! /+4.6"'2$(+4 524% 12 ="!11! 9
C %+($ 2" )+(4% 5!"# &'+/!%%"% ="( "$(1(%!4$ /!$$! '!%%+"'/! !4 )D)! $!)&%
H(x)→ TestR(TestR(x)) !"!*%
TestR(systemGlobal(x, y)) → systemGlobal(x, TestR(y)) !"!#%
TestR(system(x, y)) → system(TestR(x), y) !"!!%
TestR(system(x, y)) → system(x, TestR(y)) !"!+%
TestR(local(x, y))→ local(x, TestR(y)) !"!,%
TestR(r(x, y)) → r(Test(x), y) !"+$%
TestR(psend(x, y))→ psend(x, TestR(y)) !"+&%
TestR(pload(x, y))→ pload(x, TestR(y)) !"+'%
TestR(pok(x, y)) → pok(x, TestR(y)) !"+(%
TestR(buffull(x, y))→ buffull(x, TestR(y)) !"+)%
TestR(bufidle(x, y)) → bufidle(x, TestR(y)) !"+*%
TestR(pask(x, y)) → pask(x, TestR(y)) !"+#%
TestR(pwait(x, y))→ pwait(x, TestR(y)) !"+!%
Test(1) → 0 !"++%
!"! #$%&'()* +),-$.)//)* 00"
!"#$ %&'&( )% $"&'*+,
H(x) → TestZ(x) !"#$%
TestZ(systemGlobal(x, y)) → systemGlobal(TestZ(x), T estZ(y)) !"$&%
TestZ(system(x, y)) → system(TestZ(x), T estZ(y)) !"$'%
TestZ(res(x, y))→ res(Test0(x), y) !"$(%
TestZ(local(x, y))→ local(x, TestZ(y)) !"$)%
TestZ(r(x, y)) → r(Test(x), y) !"$*%
TestZ(psend(x, y))→ psend(x, TestZ(y)) !"$+%
TestZ(pload(x, y))→ pload(x, TestZ(y)) !"$,%
TestZ(pok(x, y)) → pok(x, TestZ(y)) !"$!%
TestZ(buffull(x, y))→ buffull(x, TestZ(y)) !"$#%
TestZ(bufidle(x, y)) → bufidle(x, TestZ(y)) !"$$%
TestZ(pask(x, y)) → pask(x, TestZ(y)) !"'&&%
TestZ(pwait(x, y))→ pwait(x, TestZ(y)) !"'&'%
Test0(0) → 0 !"'&(%
1),2.'()* 03 !"# !$#%&'!(# )"% *%# #+#,-.%# */01%##"# #!(, "(% '%&#/!( 1%
Two2t %, No2t 1" 345 6!"& 7% 6&!,!*!7% 1% 8!"(1 8!$/( 9",%:; <(,"/,/'%.%(,=
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Two2t @No2tA;
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./(",%#;
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)#(1%#/+ 787 0"&/ 91'/#/+ 9+! 1.1(+3+($! !&#.#! &( !+&) '*+3#( 9+ '%)'&), -"&! %."(!
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%&$"3%$+ 9+ BC'*#, 7+ 0/"D)E3+ #'# +!$ 9+ '"(!$/&#/+ 9+! !$/&'$&/+! 9+ .1/#F'%$#"(
!G3D")#6&+ +( HI8JK L 0%/$#/ 9+! !01'#F'%$#"(! +( 787,
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7486&+&$6 9: 4",!.?,% 1% JK*E/
L!/, "(% ,?$7% 1% 'D&/,D 1% 7H?",!.?,% B ,&?(#/,/!(# A; 4/(#/= Rωλe %#, "( %(#%.$7%
1% &->7%# 1HCM38L *!..% #"/, N
TestChanget(ϕ(x, y))→ ϕ(TestChangesϕ−→s′
(x), y), t ∈ TA
!O TA %#, 7H%(#%.$7% 1% ,&?(#/,/!( 1% 7H?",!.?,% 1!((D;
! "#$%&'() *+ ,-(&./$'&01 %$( 23456
l t l
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B q ∨ (p ∧ next(pUq)) '. =;C22'".(.4#% Bq ∨ ¬now(pUq) *(%$ =( .(D=' *' )+64.+
2#//' $!4)(%. E
p q next(pUq) now(pUq) !!"#$%$&'(
!
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f = p U q ; 46'&(' -, $,&&, &6<5,= >,& 6"&!+6&, ,#& ',4'?#,%&? -6%# 56 @A"', B=C D6E=
,& ,#& (%&,'4'?&? 46' "% ,%#,+<5, -, '*A5,# Rωλe (%-(F"?# $,G-,##!"# H
TestChanget0 (pUq(x, y))→ pUq(Test1(x), y) !"#$%&
TestChanget1 (pUq(x, y)) → pUq(TestChange10(x), y) !"#$'&
TestChanget2 (pUq(x, y))→ pUq(Test0(x), y) !"#$(&
TestChanget3 (pUq(x, y)) → pUq(TestChange01(x), y) !"#$)&
TestChanget4 (pUq(x, y))→ pUq(Test1(x), y) !"#$!&
! "#$%&'(! )*+$#(,-+. !+/ )01/-&-+!# &! 2$),3! )! 2!/ ,1/$(,/! !4 56789. "1-+
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I% &,'+, -" 4'!-"(& #)%$9'!%(#? ,#& -, 56 :!'+,
ltl(tω, tM )
!J tω ,#& 5, $!-6A, -, 5.6"&!+6&,K tM ,#& 5, $!-6A, -" #)#&*+, ,& tω, tM ∈ T (Fbin)=
!"! #$%&'()* +),-$.)//)* 001
! "#$%&' $' "'( ')'*+,' %-'" ,.%/(#*%(' $' 01"23 $/ "4(5 &%/"2'6 '( ,' 787(9*'
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2*+, '. ,$)3$ &$ #4 &15.+,+*. ()1/1&$.,$6 78$.%$39#$ &$% )!"#$% &' ()*&'+, %-./0)*:
.+%1 RSyncλe $%, &15.+ (*') /04;'$ ()*()+1,1 ϕ /+:&$%%*'% <
Syncϕ(ltl(αω, βM ))→ ltl(TestChangesϕ−→s′
(αω), Sysϕ(βM ))
*=
> Rωλe $%, TestChange
sϕ−→s′
(αω)→ α′ω
> RMλe $%, '.$ /*3(*%+,+*. &$ ;'$#;'$% *(1)4,+*.% &$ /$ %-%,!3$ <
Sysϕ(βM )→ Succ(Invϕ(β′M )
4?$/ Succ #$ /4#/'# &$% 1,4,% %'+?4.,% &' %-%,!3$ Invϕ #$ 5#,)4"$ &$% 1,4,%
%4,+%@4+%4., ϕ $, αω, α′ω, βM , β′
M ∈ T (Fbin,X ) 6
B#/7 3,,/7(:#!7 ,' "#*+#:('*'!( $/ +:#$/3( 78!"2:#!375 +%: ,.')'*+,' 7/3-%!(=
78),-/) "6 $?$.*.% A #8$B$3(#$ ()1/1&$.,6 C()!% 4?*+) ?1)+51 &$% ()*()+1,1% $.
',+#+%4., InvtiD .*'% /*.%,)'+%*.% #8$.%$39#$ &$ )!"#$% RSyncλe (*') #$ ()*&'+, %-.:
/0)*.+%1 <
Sync0(ltl(x, y)) → ltl(TestChanget0 (x), Invt0 (y)) !"#$%&
Sync1(ltl(x, y)) → ltl(TestChanget1 (x), Invt1 (y)) !"#$'&
Sync2(ltl(x, y)) → ltl(TestChanget2 (x), Invt2 (y)) !"##$&
Sync3(ltl(x, y)) → ltl(TestChanget3 (x), Invt3 (y)) !"###&
Sync4(ltl(x, y)) → ltl(TestChanget4 (x), Invt4 (y)) !"##(&
E$%, &8+.?4)+4./$ RInvλe ⊆ RM
λe &4.% #4 5"')$ F6G H4I <
Invt0 (sys(x, y)) → Testp1(Testq0(sys(x, y)) !"##)&
Invt1sys(x, y)) → Testq1(sys(x, y) !"##*&
Invt2 (sys(x, y)) → Testq0(sys(x, y) !"##+&
Invt3 (sys(x, y)) → Testp0(Testq0(sys(x, y)) !"##,&
Invt4 (sys(x, y)) → Testq1(sys(x, y) !"##!&
9.$-$*&+&$5 : 7*"+;'$ 7E76 2*+, f '.$ @*)3'#$ 7E76 2*+, #84',*34,$ &$ JK/0+ &$
/$,,$ @*)3'#$ Rωλe 6 2*+, R
Mλe '. %-%,!3$6 C+.%+D %8+# $B+%,$ '. %-%,!3$ &$ )1:1/)+,')$
RSyncλe ,$# ;'$
(RSyncλe ∪ Rω
λe ∪ RMλe)
⋆({Sync(init)}) %= ∅
4#*)% M $%, ?1)+51 (4) #4 @*)3'#$ f 6
! "#$%&'() *+ ,-(&./$'&01 %$( 23456
7()89) : ;7()89) <8//&1/') =
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<+$'+3
(RSyncλe ∪ Rω
λe ∪ RMλe)
⋆({Sync(init)}) %= ∅
c.=.d. M !'( 41)+>1 5%) "% ?#)0*"! f ;
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f1 = F(hasLeft1 ∧ G(¬hasRight1))
f2 = (hasRight1 ∪ (Fork1 ∪ hasLeft2))
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M 934,.4 ('" *$3$" 3,,'""-54'" R⋆λe G (-",.$* (32" 43 "',$-!2 /)*,*('2$'8
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)*"'3.6 (' 7'$)- (!22*" N7!"$ ('A-'2$ 7)' '$ 7)' ('A-'2$ 7!"$O8 9' ,34,.4 '"$
(*:; (*H2- (32" 43 (*H2-$-!2 FP ; 43 "',$-!2 P8F8
0'" B!)&.4'" 910 $'44'" @.' EX '$ AX ".) .2 $')&' u (' Fbin "!2$ ,34,.4*'"
,-+('""!." G
M EXRλe (u) = R−1λe (u)
M AXRλe (u) = R⋆λe(u) \ (Rλe
−1(R⋆λe(u)\u))
7(0%0<&'&01 B H51)%(!*)' 5)+0+(+?' ,! "% "#I+J*! K87;
!"! #$%&'()* +),-$.)//)* 00
1/%$.&+2,) 3 !"#$" %& EURλe (u, v)
4.56$78&+&$7 9 u, v ∈ T (Fbin)' H ∈ FNT &( Rλe $) &)*&+,"& %& -./"&* %012345*
! a← ∅ 6
"! b← v 6
#! +:7+'() a %= b ;:&.)
$! a← b 6
%! b+← (b ∪ (u ∩ EXRλe (b)))
&! <7 +:7'()
'! .)+$(.7). b
EXRλe (u) = R−1λe (u)
AXRλe (u) = R⋆λe(u)\(Rλe
−1(R⋆λe(u)\u))
4.)(=) 0> ?4.)(=) *(66&76+) @
M, s ² EXφ1 ⇐⇒ ∃π, s0 = s⇒M, s1 ² φ1! "#$%&$&'()(#$* #+%, +-$(#+#,
EXRλe (u) = R−1λe (u)
M, s ² AXφ1 ⇐⇒ ∀π, s0 = s⇒M, s1 ² φ1! .+%, /+%'+#, 0120%2(3 1%$3()(#$
(# %$&2&,1#$ ! 0+))( ,%&'1#$ 4
M, s ² AXφ1
⇐⇒ s %= s′ avec M, s ² EX¬φ1
⇐⇒ s %∈ S = {s′|∃π, s0 = s′ ⇒M, s1 ² ¬φ1}
56(0$&'()(#$* #+%, +-$(#+#,
AXRλe(u) = R⋆
λe(u)\(Rλe−1(R⋆
λe(u)\u))
7& 8"$*' )9$* $(:":*9)* "& #!"#$" %& 89:)( ;<& 89$- "&* =9-+$"&* 3> (&""&* ?$&
EU &( AU @
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"0!"/9-:(D+& E@
A AURλe (u, v) &*( "! 8&(:(& *9"$(:9) %& "0B?$!(:9) x = v∪ (u∩AXRλe (x))@ C9:-
"0!"/9-:(D+& FG@
A$7B)6+(.) > 7(, 128+3&$9)(, :* ;< ($ 2(, =+3)%2(, >?7 AU ($ EU!
EURλe (u, v) 0120%2@ A1#, 2B128+3&$9)( : (,$ 21 /($&$( ,+2%$&+# A( 2B@C%1$&+#
x = v ∪ (u ∩EXRλe (x))!
AURλe (u, v) 0120%2@ A1#, 2B128+3&$9)( ;< (,$ 21 /($&$( ,+2%$&+# A( 2B@C%1$&+#
x = v ∪ (u ∩AXRλe (x))!
! "#$%&'() *+ ,-(&./$'&01 %$( 23456
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<(-/01=&'&01 > u, v ∈ T (Fbin)' H ∈ FNT &( Rλe $) &)*&+,"& %& -./"&* %012345*
! a← ∅ 6
"! b← v 6
#! '$1'?@) a %= b A$&()
$! a← b 6
%! b+← (b ∪ (u ∩ AXRλe (b)))
&! .1 '$1?@)
'! ()'0@(1)( b
5):$(?@)B C ! "#$% &'%() *$( &'$+ &,#))-.'&+ /#+ 0 %)'$.() $& +1+%23( 4( )56
57)-%$)( "'&7%-'&&(! /'$) 3'45!-+() !(+ "')3$!(+ 89: AU (% EU;
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/)'/)-5%5 -&%5)(++#&%( (+% !,5%#% /)5754(&% 4( !,-&%()6=!'7#A( /)5+(&%5 /#) !# "')3$!(
89: EX(deadlock);
EX +$) 4(+ 5%#%+ #77(++-=!(+ (+% 4(+ 5%#%+ =!'*$#&%+ J
tsys1 = sys(t0local, t0local)
'F
t0local = local(idle(0,waitleft(1, hasleft(0, waitright(0,hasright(1, fork(0,⊥))))))
t1local = local(idle(0,waitleft(1, hasleft(0, waitright(0,hasright(1, fork(0,⊥))))))
K% !,5%#% +135%)-*$( J
t0local = local(idle(0, waitleft(0,hasleft(1,waitright(1, hasright(0, fork(0,⊥))))))
t1local = local(idle(0, waitleft(0,hasleft(1,waitright(1, hasright(0, fork(0,⊥))))))
EX2
+$) 4(+ 5%#%+ #77(++-=!(+ (+% $& (&+(3=!( 4( *$#%)( 5%#%+ !-+%5+ 7-64(++'$+ J
:( /)(3-() J
t0local = local(idle(0,waitleft(1, hasleft(0,waitright(1, hasright(0, fork(1,⊥))))))
t1local = local(idle(0,waitleft(1, hasleft(0, waitright(0,hasright(1, fork(0,⊥))))))
!"! #$%&'()* +),-$.)//)* 001
! "!#$%&'! (
t0local = local(idle(0,waitleft(1, hasleft(0, waitright(0,hasright(1, fork(0,⊥))))))
t1local = local(idle(0,waitleft(1, hasleft(0,waitright(1, hasright(0, fork(1,⊥))))))
)* +! *,-%.%&'! (
t0local = local(idle(0, waitleft(0,hasleft(1,waitright(1, hasright(0, fork(1,⊥))))))
t1local = local(idle(0,waitleft(1, hasleft(0,waitright(1, hasright(0, fork(0,⊥))))))
)/0/ +! 1#2*,%&'! (
t0local = local(idle(0,waitleft(1, hasleft(0,waitright(1, hasright(0, fork(0,⊥))))))
t1local = local(idle(0, waitleft(0,hasleft(1,waitright(1, hasright(0, fork(1,⊥))))))
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model(tprocess, tfifo)
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process 0
process
model
process process fifofifo
... ...
f i fo
...
process max
...
f i fo 0
...
f i fo max
...
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I )#,">.1-# (a == i)? N
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Test(a==i)?(b(x, y))→ b(Test(a==i)?(x), y)S
!" #$%&'()* +, -*). /0 1%02%2* &3/) 1% -4)'56%('30 &%) 789:;
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! "#$%&'( (a == b)? )
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DOM(b)
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Seta=i(b(x, y))→ b(x, Seta=i(y))/
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Seta=i(a(x, y))→ a(TestChangeji(x), y) *+,( %+,% j ∈ DOM(a)
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Seta++(b(x, y))→ b(x, Seta++(y))/
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! &'.,# a−− )
Seta−−(b(x, y))→ b(x, Seta−−(y))/
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! -#%&'( a = b )
Seta=b(x)→ Seta=i(Test(b==i)?(x)) *+,( %+,% i ∈ DOM(b)
! 0#(1,%# a < − > b )
Seta<−>b(x)→ Setb=i(Seta=b(Test(a==i)?(x))) *+,( %+,% i ∈ DOM(a)
! -#%&'( a = b op c +2 op ∈ {+,−, ∗, /} )
Seta=(b op c)(x) → Seta=(i op j)(Test(b==i)?(Test(c==j)?(x))) *+,( %+,% i ∈
DOM(b) #% j ∈ DOM(c)
<*. )=21*. >* ()%'(*?*0( >*. .()/6(/)*. >* (%'11* '050'* %#..#$ 3,# 4#$
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! recv ) D9*.';#1#8% 4#$ $+,$E%#(1#$ 4, ;<%9 4(+:% ', ;<%9 7',;>#
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GB+#( 2$*.+&( 6"+ %B!& &B/$/&'+ 3/# <+/"'!"3 /$+' '+((+ )+3)2#+&(/(*!& 0/*# &!&
3%"# #* !& /K!"(+ .+# '/)/'(,)+# 0/-*6"+#5
7/) +A+03%+J F (f(x, y)) → α si f = a +( F (f(x, y)) → β∀f ∈ Fbin, f %= a !L f
+#( %/ )+3)2#+&(/(*!& 3!") &B*03!)(+ 6"+% #40<!%+ <*&/*)+ ./&# Fbin M
Rλe =
F (f(x, y))→ α si f = a
F (f(x, y))→ β si f = b
F (f(x, y))→ β si f = c...
.
3+"( N()+ 2')*( 3/) .+# *&$/)*/&(# '!00+ #"*$/&( +& 89:; M
Rparλ =
F (x)→ F1(InvaInvall,Invall
(x))
F1(x)→ α
F (x)→ F2(¬InvaInvall,Invall
(x))
F2(x)→ β
.
3+"( N()+ 2')*( 3/) .+# *&$/)*/&(# '!00+ #"*$/&( +& C89:; M
Rparλe =
F (f(x, y))→ F1(InvaInvall,Invall
(f(x, y)))
F1(f(x, y))→ α
F (f(x, y))→ F2(¬InvaInvall,Invall
(f(x, y)))
F2(f(x, y))→ β
.
D+ 3%"#J &!"# .21&*##!&# 2-/%+0+&( %/ #20/&(*6"+ .+# #40<!%+#555
!" #$%&'()* +, -*). /0 1%02%2* &3/) 1% -4)'56%('30 &%) 789:;
<450'('30 => !"# $% !&'&( ≫ )$& Fbin #(* +$(
fmax ≫ ...≫ fi ≫ fj ≫ ...≫ f1 ≫ f0
,
f+-f−. )!%# '/0%") 1!22( *( )324!*( )$"56%# -7&/1/'6%#. '6%) *8!&'&( 9
: )" f = fj , j < max 6*!&) f+ = fi: )" f = fi, j > 0 6*!&) f− = fj
!" #$#%&'# ( ...≫ 2≫ 1≫ 0 )* ...≫ c≫ b≫ a+++
, 0+ = 1- 1+ = 2- a+ = b- b+ = c+++
, 1− = 0- 2− = 1- b− = a- c− = b+++
.)*/ !0)1/ 23415 2#/ 3'3%#16/ 137#//!5"#/ &)*" 2#/ 89/6:%#/ 3'3%#16!5"#/
&!"!%36"3/+
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&)*" %!15&*'#" 2#/ 2)113#/ %5#*$ ?*@#1 ;;;+
<450'('30 =? !"# Fpar $% (%)(24*( '");!"%# '( Fbin (# '( FNT , <%( &=>*( '(
?@A 76&62/#&/( ()# '/0%"( 1!22( $%( B!%1#"!% F : T (Fbin) (→ 2Rpar 65(1
F ∈ Fpar (# Rpar = T (Fbin ∪ FNT ,X )× T (Fbin ∪ FNT ∪ Fpar,X ),
C!$) %!#!%) 1(##( &=>*( 76& FT (Fbin),
.)*/ "#%!"?*)1/ ?*# Rpar #/6 R 2# <=>8 /5 7#6 #1/#%A'# 2# ":F'#/ 1# 7)165#16
!*7*1 /9%A)'# 2# Fpar+
:@21*. A* B89:; &3/) A*. 3&4)%(*/). &)'C'('D. A* ED- F
, TESTa(⊥,⊥) G)* InvaInvall,InvallH #/6 *1 #1/#%A'# 2# ":F'#/ 7)%%# /*50!16/ (
Rparλe =
{
Test(a(x, y))→ a(x, y)}
.
, SETa(⊥,⊥) #/6 *1 #1/#%A'# 2# ":F'#/ 7)%%# /*56 (
Rparλe =
{
Set(f(x, y))→ a(x, y)}
.
, TESTV ALUEa(⊥,i(⊥,⊥)) #/6 *1 #1/#%A'# 2# ":F'#/ 7)%%# /*56 (
Rparλe =
TestV alue(f(x, y))→ A(TESTa(⊥,⊥)(f(x, y)))
A(f(x, y))→ a(TESTi(⊥,⊥)(x), y)
TestV alue(f(x, y))→ B(¬TESTa(⊥,⊥)(f(x, y)))
B(f(x, y))→ f(x, TestV alue(y))
B(f(x, y))→ f(TestV alue(x), y)
.
!"#" $%&'()*& +,+)*-'./0*& 1.0.)+'0+& 2345$& 67
! "#$% &! #''() *+&! #&%,( -.-%/0( *+12345 '#,#06%,6 TESTi(⊥,⊥)
7 TESTV ARa(⊥,b(⊥,⊥)) (-% &! (!-(08)( *( ,/9)(- :;00( -&$% <
Rparλe =
TestV ar(f(x, y))→ A(TESTa(⊥,⊥)(f(x, y)))
A(f(x, y))→ a(x,TESTVALUEb(⊥,Top(x))(y))
TestV ar(f(x, y))→ B(TESTb(⊥,⊥)(f(x, y)))
B(f(x, y))→ b(x,TESTVALUEa(⊥,Top(x))(y))
TestV ar(f(x, y))→ C(TEST¬(b∨a)(f(x, y)))
C(f(x, y))→ f(x, TestV ar(y)),
C(f(x, y))→ f(TestV ar(x), y)
.
! "#$% &! #''() *+&! #&%,( 12345 '#,#06%,6 TESTVALUEa(⊥,i(⊥,⊥))
#=(: )# ,/9)( Top(i(x, y)) → i(⊥,⊥) !( 9#,*( >&( )# )(%%,( )# ')&- ?#&%( *&
%(,0(@
7 SETV ALUEa(⊥,i(⊥,⊥)) (-% &! (!-(08)( *( ,/9)(- :;00( -&$% <
Rparλe =
SetV alue(f(x, y))→ A(TESTa(⊥,⊥)(f(x, y)))
A(a(x, y))→ a(SETi(⊥,⊥)(x), y)
SetV alue(f(x, y))→ B(¬TESTa(⊥,⊥)(f(x, y)))
B(f(x, y))→ f(x, SetV alue(y)),
B(f(x, y))→ f(SetV alue(x), y)
.
7 PLUSPLUSa(⊥,⊥) (-% &! (!-(08)( *( ,/9)(- :;00( -&$% <
Rparλe =
Plusplus(f(x, y))→ A(TESTa(⊥,⊥)(f(x, y)))
A(f(x, y))→ f(Plus(x), y)
Plus(f(x, y))→ f+(x, y)
Plusplus(f(x, y))→ B(¬TESTa(⊥,⊥)(f(x, y)))
B(f(x, y))→ f(x, P lusplus(y)),
B(f(x, y))→ f(Plusplus(x), y)
.
7 MINUSMINUSa(⊥,⊥) (-% &! (!-(08)( *( ,/9)(- :;00( -&$% <
Rparλe =
Minusminus(f(x, y))→ A(TESTa(⊥,⊥)(f(x, y)))
A(f(x, y))→ f(Minus(x), y)
Minus(f(x, y))→ f−(x, y)
Minusminus(f(x, y))→ B(¬TESTa(⊥,⊥)(f(x, y)))
B(f(x, y))→ f(x,Minusminus(y)),
B(f(x, y))→ f(Minusminus(x), y)
.
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!"!" #$%&'() *!"! +,-#.
!"#$" % !"#"$%&%"' (
) α→{l→r} β ⇒ Fl→r(α)→⋆Rλ
β * +,-"' -. /01'%&%"'2 %- ,3%$&, 4', #"$%&%"' p /,
α ,& 4', $45$&%&4&%"' σ : X → T (Fbin) &,--,$ 64, α|p = lσ ,& β = α[rσ]p7 +"%&
α′ 4' &,!8, 9"'$&!4%& 9"88, -, $4%:.'& * α′ = α[Fl→r(α|p)]p7 ;-.%!,8,'&2
,' 4&%-%$.'& $,4-,8,'& -,$ !<=-,$ /, 9%!94-.&%"'2 c.>.d. -,$ !<=-,$ /, -. ?"!8,
Fl→r(a(x, y))→ a(Fl→r(x), y) ,& Fl→r(a(x, y))→ a(x, Fl→r(y))2 '"4$ .:"'$
Fl→r(α)→⋆Rλ
α′. !"#$
+,-"' -. @A#"&@<$,2 %- ,3%$&, 4', $45$&%&4&%"' σ : X → T (Fbin) &,--, 64, α|p =
lσ7 .! 9"'$064,'&2 '"4$ .:"'$ Fl→r(α|p) = Fl→r(lσ)7 B%'$%2 Fl→r(α|p) =
Fl→r(l)σ7 +,-"' -. 9"'$&!49&%"' /, α′2 %- ,3%$&, 4', !<=-, =0'0!0, /, Rλ2 c.>.d
Fl→r(l) → r2 4', $45$&%&4&%"' σ : X → T (Fbin) ,& 4', #"$%&%"' p &,--,$ 64,
α′|p = Fl→r(l)σ7 C' 9"'$064,'9,2 '"4$ #"4:"'$ 9"'$&!4%!, -, &,!8, α′[rσ] 64%
,$& 0=.- > β $,-"' -. 9"'$&!49&%"' /, α′7 D"4$ .:"'$ /"'9 *
α′ →Rλ β. !"%$
B%'$%2 ,' 4&%-%$.'& EB7(F ,& EB7GF2 '"4$ #"4:"'$ /0/4%!, 64, Fl→r(α)→⋆Rλ
β7
.! 9"'$064,'&2 α→{l→r} β ⇒ Fl→r(α)→⋆Rλ
β7
) α →{l→r} β ⇐ Fl→r(α) →⋆Rλ
β * +"%& R′λ -, HIJ+ &,- 64, R′λ = Rλ \
{Fl→r(l) → r |l → r ∈ R}7 CK,9&%:,8,'&2 '"4$ #"4:"'$ 8"'&!,! 64,
R′⋆λ ({Fl→r(α)}) = ∅7
C' 9"'$064,'9,2 9,-. %8#-%64, 64,2 #"4! "5&,'%! -, &,!8, β ∈ R⋆λ({Fl→r(α)})2
-. !<=-, Fl→r(l) → r /"%& L&!, .##-%640, .4 8"%'$ 4', ?"%$7 B9&4,--,8,'&2
'"4$ /,8.'/"'$ 64, 9,&&, !<=-, ,$& .##-%640, ,3.9&,8,'& 4', ?"%$7 CK,9&%:,M
8,'&2 #"4!$4%:"'$ ,' %'/49&%"' $4! -, 9@,8%' /, !0M09!%&4!, 64% '"4$ 8<',
> #.!&%! /, Fl→r(α) > β c.>.d %- ,3%$&, t0, . . . , tn ∈ T (Fbin ∪ FNT ) &,- 64,
t0 = Fl→r(α) →Rλ t1 →Rλ . . . tn−1 →Rλ tn = β7 D"4$ 8"'&!"'$ /N.5"!/ 64,
[email protected], ti . 4', #"$%&%"' p .4 #-4$ &,--, 64, ti(p) = Fl→r7 +"%& Pn -. #!"#"$%&%"'
$4%:.'&, * #"4! &"4& p, p′ ∈ Pos(tn)2 $% tn(p) = tn(p′) = Fl→r .-"!$ p = p′7
) P0 * 4%$64, t0 = Fl→r(α) ,& α ∈ T (Fbin)2 ε ,$& -N4'%64, #"$%&%"' /, t0 "O
Fl→r .##.!.P&7 P0 /"'9 ,$& :!.%7
!" #$$%&% #' ()%*+%,
Pn ⇒ Pn+1 ! "#$$%&%'& (#) Pn &%*+ ,-.*/ 0.- 1%'&2(#)'+3 tn &.+*&4.*+ 5
6. $-%$-*2+2 ! p, p′ ∈ Pos(tn)3 &* tn(p) = tn(p′) = Fl→r/ 7) $6#&3 $#*&(#)
tn →Rλ tn+13 *6 )8*&+) '21)&&.*-)9)'+ #') #'*(#) $%&*+*%' p :) tn +)66) (#)
tn(p) = Fl→r/ 0%#-&#*,%'& )' -2;21-*,.'+ 6<.'.6=&) :) 1.& !
Fl→r(a(x, y)) → a(Fl→r(x), y) )&+ .$$6*(#2) 5 6. $%&*+*%' p ! >6 :%'1 )8;
*&+) #') &#?&+*+#+*%' σ : X → T (Fbin) +)66) (#) tn|p = Fl→r(a(x, y))σ =
Fl→r(a(σ(x), σ(y)))/ 0.- 1%'&2(#)'+3 tn+1 = tn[a(Fl→r(x), y)σ] =
tn[a(Fl→r(σ(x)), σ(y))]/ 0#*&(#) σ(x), σ(y) ∈ T (Fbin) )+ ∀p′ ∈ Pos(tn)
+)66)& (#) p &= p′3 tn(p′) /∈ FNT 3 tn+1 &.+*&4.*+ [email protected])9)'+ 5 ∀p, p′ ∈
Pos(tn+1)3 &* tn+1(p) = tn+1(p′) = Fl→r/ 06#& $-21*&29)'+3 6<#'*(#) $%;
&*+*%' p′ :) tn+1 +)66) (#) tn+1(p) = Fl→r )&+ p′ = p.0/
Fl→r(a(x, y)) → a(x), Fl→r(y) )&+ .$$6*(#2) 5 6. $%&*+*%' p ! 6) 1.& :)
$-)#,) )&+ &*9*6.*-) .# $-212:)'+/
Fl→r(l)→ r )&+ .$$6*(#2) 5 6. $%&*+*%' p ! >6 :%'1 )8*&+) #') &#?&+*+#+*%'
σ : X → T (Fbin) +)66) (#) tn|p = Fl→r(l)σ = Fl→r(lσ)/ A' 1%'&2(#)'1)3
tn+1 = tn[rσ]/ 0#*&(#) r ∈ T (Fbin,X ) )+ σ : X → T (Fbin)3 rσ )&+
.*'&* #' +)-9) :) T (Fbin) )+ tn+1 ∈ T (Fbin)/ B*'&*3 tn+1 &.+*&4.*+ 5 6.
$-%$-*2+2 ! ∀p, p′ ∈ Pos(tn+1)3 &* tn+1(p) = tn+1(p′) = Fl→r/
C%#& ,)'%'& D#&+)9)'+ :) 9%'+-)- (#) 1E.(#) +)-9) :.'& 6) 1E)9*' :) -2;
21-*+#-) 1%'+*)'+ )8.1+)9)'+ #' &=9?%6) Fl→r )+ :F& (#) 6. -F@6) Fl→r(l)→ r
)&+ .$$6*(#2)3 #' +)-9) :) T (Fbin) )&+ %?+)'#/ G) HIJ" Rλ ') $)#+ :%'1 $6#&
K+-) .$$6*(#2)/ L)6. 9%'+-) 6. -),)':*1.+*%'/
B*'&*3 6) 1E)9*' :) -2;21-*+#-) '%#& 9)'.'+ 5 $.-+*- :) Fl→r(α) 5 β )&+
:) 6. 4%-9) &#*,.'+) ! t0 = Fl→r(α) →Rλ\{Fl→r(l)→r} t1 →Rλ\{Fl→r(l)→r}. . . tn−1 →{F (l)→r} tn = β/ B,)1 #') *':#1+*%' +-F& $-%1E) .# $-212:)'+3
'%#& $%#,%'& 9%'+-)- (#<*6 )8*&+) #') $%&*+*%' p :) α )+ #') &#?&+*+#+*%'
σ : X → T (Fbin) +)66)& (#) tn−1 = α[F (l)σ]p/ C%#& $%#,%'& [email protected])9)'+ 9%';
+-)- (#) tn−1|p.0 = α|p/ A' 1%'&2(#)'1)3
lσ = α|p. !"#$
7) $6#&3 tn = β = tn−1[rσ]p/ B*'&*3
β = α[rσ]p. !"%$
")6%' 6. 1%'&+-#1+*%' :) Rλ3 l → r ∈ R/ B*'&*3 :<.$-F& MB/NO )+ MB/PO3 *6
)8*&+) #') -F@6) l → r ∈ R3 #') $%&*+*%' p ∈ Pos(α) )+ #') &#?&+*+#+*%' σ :
X → T (Fbin) +)66)& (#) lσ = α|p )+ β = α[rσ]p/ 0.- 1%'&2(#)'+3 α→{l→r} β/
A' -2Λ '%#& .,%'& 9%'+-2 (#) α→{l→r} β ⇐ Fl→r(α)→⋆Rλ
β3 1%'16#.'+
6. $-)#,)/
!"! #$%&'()* +! ,-.(/0*. 1* 23435)'(6)* 7895('899*:. ";"
<)*6=* > !"#$%&' (
) R⋆(E) ⊆ R⋆λ(E
′) * +#,- α ∈ E '- β ∈ R⋆({α}). +'/#0 /1 2"30,-,#0 2'
R⋆4 ,/ '5,6-' l1 → r1, . . . ln → rn ∈ R '- t0, . . . , tn -'//'6 78' α →{l0→r0}
t0 →{l1→r1} . . . →{ln−1→rn−1} tn−1 →{ln→rn} tn = β. 9' : ;+ Rl0→r0 $'&<
=/,- /'6 >#02,-,#06 1--'028'6 6=">,3"'6 ? /1 =$#=#6,-,#0 (. @1$ >#06"78'0-4
6'/#0 /1 =$#=#6,-,#0 (4 t0 →l0→r0 t1 ,&=/,78' 78' Fl0→r0(t0) →⋆Rl0→r0
t1.
A1/!'8$'86'&'0-4 Fl0→r0(t0) &∈ E′. B'='0210-4 t0 = α '- α ∈ E. C,06,4
G(t0) ∈ E′. DE'>-,F'&'0-4 G(t0) →G(x)→G(Fl0→r0 (x)) G(Fl0→r0(α)). D0 >#0<
6"78'0>'4 G(t0) →⋆Rλ
G(t1) →G(x)→x t1 ,&=/,78' 78' t1 ∈ R⋆λ(E
′). D0
G"0"$1/,610- >' =$#>'66864 0#86 #H-'0#06 /' >!'&,0 2' $"<">$,-8$' >,<2'66#86 *
G(t0) →⋆Rλ
G(t1) . . . →⋆Rλ
G(tn) →G(x)→x tn = β. @1$ >#06"78'0-4 β ∈
R⋆λ(E
′).
C,06,4 0#86 =#8F#06 2"28,$' 78'
R⋆({α}) ⊆ R⋆λ({G(α)}). !"#$
) R⋆λ(E
′) ⊆ R⋆(E) * I-82,#06 R⋆λ({G(α)})
! "#$$%&%'& (#)#'* +,-.* /* .0 1%+2* G(x) → G(Fl→r(x)) *&3 0$4
$.5(#6* 75'&58 G(α) →G(x)→G(Fl→r(x)) G(Fl→r(α))) 9%3*: (#* G(x) →
G(Fl→r(x)) '* $*#3 $0& ;3+* 0$$.5(#6* /,& (#* .* &<2=%.* Fl→r 0$$0+0>3
"5 R⋆l→r(Fl→r(α)) = ∅ 0.%+& 0#?#' 3*+2* /* T (Fbin) $*#3 ;3+* 033*5'3 /*
Fl→r(α) 75'&58 $#5&(#* Rl→r $*+2*3 .0 /6@'535%' /* Rλ /0'& .* $+%$%&535%'
A8 '%#& $%#B%'& /6/#5+* (#* .0 +,-.* l → r ∈ R '* $*#3 $0& ;3+* 0$$.5(#6*
&#+ α
"5 R⋆l→r(Fl→r(α)) #= ∅ 0.%+& 5. *C5&3* #' 3*+2* /* t1 ∈ T (Fbin) (#5 $*#3
;3+* 033*5'3 /* Fl→r(α) D#5&(#* Fl→r(α)→⋆Rl→r
t18 '%#& 0B%'& G(α)→+Rλ
G(t1) E* $.#&8 /)0$+,& .0 $+%$%&535%' A8 α→{l→r} t1 9%3*: (#* t1 ∈ R⋆λ(E
′)
$#5&(#* .0 +,-.* G(x)→ x $*#3 ;3+* 0$$.5(#6* &#+ G(t1) 75'&58 ?* $+%?*&&#&
$*#3 ;3+* 536+6 *3 '%#& $%#B%'& ?%'&3+#5+* #'* &6(#*'?* 3*..* (#* G(α)→+Rλ
G(t1) →+Rλ
G(t2) . . . →+Rλ
G(tn) 0B*? ti ∈ R⋆λ(E
′) &5 .0 +,-.* G(x) → x *&3
0$$.5(#6* &#+ ?F0(#* 3*+2* G(ti) G?)*&3 $%&&5=.* $#5&(#* ti ∈ T (Fbin)H E*
$.#&8 &*.%' .0 $+%$%&535%' A8 '%#& 0B%'& ti →{l→r} ti+1 $%#+ i = 1, . . . , n−1
9%#& $%#B%'& /%'? /6/#5+* (#* tn ∈ R⋆({α}) I5'0.*2*'38 '%#& $%#B%'&
/6/#5+* (#*
R⋆({α}) ⊇ R⋆λ({G(α)}). !"#$
! "#$$%&%'& (#)0#?#'* +,-.* /* .0 1%+2* G(x) → G(Fl→r(x)) *&3 0$$.5(#6*
D0+ ?%'&6(#*'38 0#?#'* +,-.* /*
⋃
l→r∈RRl→r $*#3 ;3+* 0$$.5(#6*
J)#'5(#* 3*+2* 033*5'3 /* T (Fbin) *&3 /%'? %=3*'# *' 0$$.5(#0'3 .0 +,-.*
G(x)→ x 75'&58 α ∈ R⋆λ({G(α)}) E)0$+,& .0 /6@'535%'8 α ∈ R⋆({α})
D%#+ ?%'?.#&5%'8 $%#+ 3%#& α ∈ E8 /* G7 KH *3 G7 LH8 '%#& %=3*'%'& (#*
R⋆(E) = R⋆λ(E
′)
!" #$$%&% #' ()%*+%,
()%*+% - !""! #
$%&&'(')( %) *!+ ,-$ .')(*/%0* .'""! 12)( +! +!""! #3 $!+') +! *45'/6"! #7 )'%(
'8*!)')( R⋆λ(E
′) = R⋆(E)3
!"!# $%&'()* +!#! ,-./$0
()%*+% . /()%*+% ,*001$02% 3% 45 6)767,1217$ "8
! &'0)* .+5 !(* 1! .')(*/%0/! %) 9:,-$ Rλe ;%0 0"&+5"!)*! +! &/'.!((%( 1!
/5<5./0*%/!3
! 9:,-$ Rλe !(* .')(*/%0* .'""! +! (%0=2)* >
Rλe = Rl→rvisit ∪R
l→rTV ∪Rl→r
check ∪Rl→rσ ∪Rl→r
GS ∪Rl→rσ−apply
? α →{l→r} β ⇒ Fl→r(α) →∗Rλe
β > 9@!.*0=!"!)*7 (0 α →{l→r} β 2+'/( 0+
!A0(*! %)! &'(0*0') p 1! α !* %)! (%8(*0*%*0') σ : X → T (Fbin) *!++!( ;%!
α|p = lσ 2)1 β = α[rσ]p3 B &2/*0/ 1! +2 15C)0*0') 1! Rλe 7 )'%( &'%=')(
151%0/! ;%! Fl→r(α) →∗Rλe
α[Fl→r(α|p)]p3 B0)(07 !) 2&&+0;%2)* 1! +2 /6D+!
Fl→r(x)→ F l→rrewrite(F
l→rcheck(F
l→rcopy(F
εl→r(x)))) (%/ α[Fl→r(α|p)]p 7 )'%( '8*!)')(
α[F l→rrewrite(F
l→rcheck( F
l→rcopy(F
εl→r(α|p))))]p3 E%0(;%! σ !(* %)! (%8(*0*%*0') 1! X
*' T (Fbin)7 !* 1F2&/6( +F!)(!"8+! 1!( /6D+!( Rl→rTV ∪ R
l→rcheck ∪ R
l→rσ ∪ Rl→r
GS 7
α[F l→rrewrite(F
l→rcheck(F
l→rcopy(F
εl→r(α|p))))]p &!%* G*/! /5<5./0*( !) α[F
l→rrewrite(t)]p 'H
t !(* 15C)0 .'""! +! (%0=2)* >
? t(p) = σ(xi)7 t(p′) = ⊤X ∈ Fbin 2=!. p′ = (2.)i p = p′.1 !* xi !(* +2 ith
=2/028+! 1! l +% 1% .I*5 D2%.4! 1! l
? t(prmp) =⊥X∈ Fbin 2=!. prmp = (2.)m2 !* m !(* +! )'"8/! 1! =2/028+!(
2&&2/20((2)* 12)( l
B0)(07 +! *!/"! t /!&/5(!)*! .'""! %) *!/"! +2 (%8(*0*%*0') σ3 :0)2+!"!)*7
+! 9:,-$ Rl→rσ−apply )'%( &!/"!* 1! /!.')(*/%0/! r !* +2 =2+!%/ 1!( =2/0<
28+!( !(* .'&05! (J"8'+! &2/ (J"8'+!3 !( %)0;%!( *!/"!( 2..!((08+!( 1').
(')* rσ3 B0)(07 !) /5(%"57 F l→rrewrite(t) →
∗Rl→rσ−apply
rσ ∈ T (Fbin)3 9) .')<
(5;%!).!7 α[F l→rrewrite(t)]p →
∗Rl→rσ−apply
α[rσ]p ∈ T (Fbin) !* &%0(;%! α[rσ]p = β7
β ∈ R∗λe({Fl→r(α)}).
? α →l→r β ⇐ Fl→r(α) →∗Rλe
β > 2 &/!%=! &'%/ .! .2( !(* &/'.4!
2% .2( (0"0+20/! */20*5 12)( +2 &/!%=! 1! +2 &/'&'(0*0') # 12)( +! (!)(
;%! K!( &2( 1! /5<5./0/! (')* '/1'))5( !* 15&!)12)*( 1% *!/"! 0)0*02+
c.L.d Fl→r(α)3 :0)2+!"!)*7 &%0(;%! +2 4J&'*46(! Fl→r(α) →∗Rλe
β7 1M
2%A 15C)0*0')( 1! Rl→rvisit !* Rl→r
TV 7 0+ !A0(*! %)! &'(0*0') p ∈ Pos(α)
*!++! ;%! Fl→r(α) →∗Rl→rvisit∪R
l→rTV
α[F l→rrewrite(F
l→rcheck(F
l→rcopy(F
εl→r(α|p))))]p !*
F l→rrewrite(F
l→rcheck(F
l→rcopy(F
εl→r(α|p)))) →
∗Rλe
β3 KM 2% &0+! 1!( (J"8'+!( 1! FNT
(%/ α|p7 )'%( &'%=')( "')*/!/ ;%! Fl→rrewrite(F
l→rcheck(F
l→rcopy(F
εl→r(α|p)))) →
∗Rλe
β ⇔ (Rl→rσ−apply)
∗((Rl→rGS )
∗((Rl→rσ )∗((Rl→r
check)∗(F l→r
rewrite(
F l→rcheck(F
l→rcopy(F
εl→r(α|p)))))))) = {β}. E2/ .')(5;%!)*7 (!+') 15C)0*0')(
1! Rl→rσ 7 Rl→r
GS !* Rl→rcheck7 &'%/ %) *!/"! t ∈ T (Fbin)7 )'%( &'%=')( "')*/!/
!"! #$%&'()* +! ,-.(/0*. 1* 23435)'(6)* 7895('899*:. ";<
!" #$ (Rl→rcheck)
∗({F l→rcheck(F
l→rcopy(F
εl→r(t)))}) #= ∅ %&'(# $& ")$#*" !+" #!,#*$*!*$'+
σ : X → T (Fbin) *"&&" !" lσ = t- .+ %//&$ !%+* 0'11" !+ (2#!&*%* 3" +'*("
2*!3" 3" 0%#4 $& ")$#*" σ : X → T (Fbin) *"& !" lσ = α|p- 5%# #"!&"1"+*
(Rl→rcheck)
∗({F l→rcheck(F
l→rcopy(F
εl→r(t)))}) #= ∅4 1%$# $& ")$#*" 26%&"1"+* !+ *"(1"
t′ 3" T (Fbin) *"& !" (Rl→rcheck)
∗({F l→rcheck(F
l→rcopy(F
εl→r(t)))}) = {t′}- 7" *"(1"
t′ "#* !+" ("/(2#"+*%*$'+ 3" σ4 "* &8")20!*$'+ 3" &8.9:;< Rl→rσ−apply #!( t′
+'!# /"(1"* 3" ("0'+#*(!$(" r "+ #!,#*$*!%+* 0=% !" >%($%,&" xi 3" r /%(
σ(xi)- 78!+$ !" *"(1" %00"##$,&" 3" t′ 3'+0 "#* rσ- ?$+#$4 "+ (2#!124 $&
")$#*" !+" /'#$*$'+ 3" α "* !+" #!,#*$*!*$'+ σ : X → T (Fbin) *"&&"# !"
α|p = lσ4 β = α[rσ]p "* Fl→r(α) →∗Rλe
β- @'!# /'!>'+# 323!$(" !"
α→l→r β ⇒ Fl→r(α)→∗Rλe
β !$ 0'+0&!* &% /("!>"-
=)*6>* ? :=2'(A1" B
C*%+* 3'++2 &8.9:;< Rλe =⋃
l→r∈R(Rl→rvisit ∪ R
l→rTV ∪ R
l→rcheck ∪ R
l→rσ ∪ Rl→r
GS ∪
Rl→rσ−apply) ∪ RFP %>"0 RFP = {G(a(x, y)) → G(Fl→r(a(x, y))) | l → r ∈ R ∧ a ∈
Fbin} ∪ {G(a(x, y)) → a(x, y) | a ∈ Fbin}4 &% /("!>" "#* *(A# /('0=" D 0"&&" 3!
*=2'(A1" E-
=)*6>* @ A=)*6>* .655'95(* 1* :% &)8&8.'('89 <B
C*%+* 3'++2 &8.9:;< Rλe 4 &% /("!>" "#* *(A# /('0=" D 0"&&" 3" &% /('/'#$*$'+ B-
7" /'$+* 0&2 "#* 3" 0'+#*(!$(" !+ .9:;< Rλe !$ $1/&21"+*" &" /('0"##!# 3"
(2F20($*!(" "+ 0'+#$32(%+* H 0'11" !+ #G1,'&" #/20$%& "+ Fbin "* "+ $6+'(%+* &"#
(A6&"# 3" 0$(0!&%*$'+-
7" .9:;< Rλe "#* 0'+#*(!$* 0'11" &" #!$>%+* H
Rλe = RH(t)→α
TV ∪RH(t)→α
check ∪RH(t)→ασ ∪R
H(t)→α
GS ∪RH(t)→α
σ−apply
!"!# $%&'()* +!#! ,-./,$0
=)*6>* ; A=)*6>* .655'95(* 1* :% &)8&8.'('89 +B
C*%+* 3'++2 &"# ;.9:;<# Rλe→I 4 Rλe→II 4 Rλe→III 4 Rλe→A "* Rλe→B4 &% /("!>"
"#* *(A# /('0=" D 0"&&" 3" &% /('/'#$*$'+ B-
7" /'$+* 0&2 "#* 3" 0'+#*(!$(" !+ ;.9:;< !$ $1/&21"+*" &" /('0"##!# 3" (2F
20($*!("-
7" ;.9:;< "#* 0'+#*(!$* 0'11" &" #!$>%+* H
Rλe→∪ = RH (a(x,y))→α
TV ∪RH (a(x,y))→α
check ∪ RH (a(x,y))→ασ ∪R
H (a(x,y))→α
σ−apply
'I
J RH (a(x,y))→α
TV = {FH (a(x,y))→α(x)→ FH (a(x,y))→αrewrite (
FH (a(x,y))→α
check (FH (a(x,y))→αcopy (F ε
H (a(x,y))→α(x))))} >% K*(" *(%3!$* "+
2C7D2,. 0'11" #!$* H
!" #$$%&% #' ()%*+%,
RH (a(x,y))→α
TV =
FH (a(x,y))→α(a(x, y))→ F 1TV (F
εH (a(x,y))→α
(a(x, y)))
F 1TV (a(x, y))→ F 2
TV (FH (a(x,y))→αcopy (a(x, y)))
F 2TV (a(x, y))→ F 3
TV (FH (a(x,y))→α
check (a(x, y)))
F 3TV (a(x, y))→ F 4
TV (FH (a(x,y))→αrewrite (a(x, y)))
F 4TV (a(x, y))→ a(x, y)
.
!"# ∀a ∈ Fbin$ R
H (a(x,y))→α
check %&' "( -./0-1 (!"& %#)%''*(' +% ,-#./%# &. "( '%#)% +%
T (Fbin) 0!(0!#+% *,%0 H (a(x, y))1
23* 4.0*'.!( +% 0%' -./0-1 %&' #%(,!5- *# 4* #-&%(0% +" &5)6!4%
(!(7'%#).(*4 F εH (a(x,y))→α
1 8'*(' +!((- "( '%#)% t ∈ T (Fbin) 0!(7
0!#+*(' *,%0 H (a(x, y))9 (RH (a(x,y))→α
check )∗({F εH (a(x,y))→α
(t)}) %&' {t′} !:
PosX (H(a(x, y))) = {11, 12} %' ⊕x,11,⊕y,12 ∈ Fbin ;
R=check
F εH (a(x,y))→α
(H(x))→ H(F 1H (a(x,y))→α
(x))
F 1H (a(x,y))→α
(a(x, y))→ a(F 11H (a(x,y))→α
(x), F 12H (a(x,y))→α
(y))
F 11H (a(x,y))→α
(x)→ ⊕x(x)
F 12H (a(x,y))→α
(y)→ ⊕y(y)
.
$ RH (a(x,y))→ασ %&' "( -./0-1 (!"& %#)%''*(' +% 0!(&'#".#% "(% 4.&'%
!#+!((-% +% '%#)%& .(+%<-& *# +%& ,*#.*64%& +% H(a(x, y))1 =%''% 4.&'%
%&' #% #-&%('-% *# "( '%#)% %' &-)*('.>"%)%(' *" 0!"#*('9 (!"& !",!(&
0!(&.+-#%# 0%''% 4.&'% 0!))% 4* &"6&'.'"'.!( #-&"4'*(' +" *& 0!(0!#7
+*(0%1 ?!.' t′ 4% '%#)% #-&"4'*(' +% (RH (a(x,y))→α
check )∗({F εH (a(x,y))→α
(t)})1
(RH (a(x,y))→ασ )∗(F
H (a(x,y))→αcopy (t′)) )@(% A "( "(.>"% '%#)% t′′ #% #-&%('*('
4* &"6&'.'"'.!( #-&"4'*(' +% 4* 0!(0!#+*(0% %('#% t %' H(a(x, y))1
$ RH (a(x,y))→α
σ−apply %&' "( -./0-1 & -0./*(' 43* 4.0*'.!( +% 4* &"6&'.'"7
'.!( 6.%(7B!#)-% #-&"4'*(' +% t %' H(a(x, y)) &"# 4% '%#)% α1 C.(&.9
(RH (a(x,y))→α
σ−apply )∗(FH (a(x,y))→αrewrite (t′′)) = {ασ}1
()%*+% 2 DE-!#@)% F
?%4!( 4%& #! !&.'.!(& G %' 4% 'E-!#@)% H1
!" #$%&'()* +! ,-%./%('01 2*3 343(56*3 7.76*1(%')*3
()%*+% 3 4()%*+% 5% 67 8)989,:;:9$ <=
(Rλe ∪ (R′λe→I ∪R
′λe→II))
⋆(E′) = R⋆λe(E
′).
*,%0 E ⊆ T (Fbin) %' E′ = {H(t)|t ∈ E}1
!"! #$%&'()* +! ,-./0'1%('-2 &%) 34567 89:
R⋆λe(E
′) ⊆ (Rλe ∪ (R′λe→I
∪R′λe→II
))⋆(E′)! "#$%& '()*$+& %$,-+ Rλe ⊆ (Rλe ∪
(R′λe→I
∪R′λe→II
))
(Rλe ∪ (R′λe→I
∪R′λe→II
))⋆(E′) ⊆ R⋆λe(E
′). .-/0%/)(-+% $+ &01)&1+& ,#1+1,2%$
*$ "1% 3
(Rλe ∪R′λe→I
)⋆(E′) ⊆ R⋆λe(E
′) 3
R′λe⋆H⋆ (a(x,y))→a(x,y)(E
′) ⊆ Rλe⋆H (a(x,y))→a(x,y)(E
′)
R′λe⋆H⋆ (a(x,y))→a(G⋆ (x),y)(E
′) ⊆ Rλe⋆H (a(x,y))→a(G (x),y)(E
′) "10
Hn (a(x, y)) →⋆H (a(x,y))→a(G (x),y) a(Gn (x), y) 1($" n %/4%155$+&
601+*7
8)5),1)0$5$+& 9-/0 ,$ 0$%&$7
(Rλe ∪R′λe→II
)⋆(E′) ⊆ R⋆λe(E
′) 3
8)5),1)0$5$+& 9-/0 ,$% 1/&0$% "1%7
;)*<=* 8> ?;)*<=* 1<@@'2@(* .* 0% &)-&-1'('-2 +A
(Rλe ∪ (R′λe→I ∪R
′λe→II) \ (Rλe→I ∪Rλe→II))
⋆(E′) = R⋆λe(E
′).
1($" E ⊆ T (Fbin) $& E′ = {H(t)|t ∈ E}7
:#190;% ,1 90-9-%)&)-+ < +-/% -=&$+-+%
(Rλe ∪ (R′λe→I ∪R
′λe→II))
⋆(E′) = R⋆λe(E
′).
1($" E ⊆ T (Fbin) $& E′ = {H(t)|t ∈ E}7
:#1/&0$ 910&! 1/ 5-5$+& *-++'! +-/% 9-/(-+% 90-/($0 >/#/+$ 0;6,$ $+ R′λe→I
$%& 199,)"1=,$ %$/,$5$+& %) ,1 0;6,$ "-00$%9-+*1+&$ $+ Rλe→I $%& 199,)"1=,$7 :$ 9,/%
,$ 0'%/,&1& *#/+ 91% *$ 0'?'"0)0$ *$ ,1 0;6,$ $+ R′λe→I
$%& )*$+&)>/$ @ "$,/) *$ ,1 0;6,$
"-00$%9-+*1+&$ $+ Rλe→I 7 8)5),1)0$ 9-/0 R′λe→II
1($" Rλe→II 7 A$,1 +-/% 9$05$&
*#-=&$+)0 ,$ 0'%/,&1& *#/+ 91% *$ 0'?'"0)0$ %1+% /&),)%$0 ,1 0;6,$ $+ (Rλe→I ∪Rλe→II)7
A$,1 5-+&0$ ,1 0$($+*)"1&)-+7
!" #$%&'()* +! ,-./0'1%('-2 &%) 34567
;)*<=* 88 ?;)*<=* 1<@@'2@(* .* 0% &)-&-1'('-2 9A
B/ 5-5$+& *-++'! +-/% 9-/(-+% 90-/($0 >/#/+$ 0;6,$ $+ Rλe $%& 199,)"1=,$ %/0 tm
%$/,$5$+& %) ,1 &01+%)&)-+ "-00$%9-+*1+&$ $+ T $%& C01+"D)%%1=,$ %/0 m7 EF$"&)($?
5$+&! +-/% 9-/(-+% 90-/($0 >/#), $G)%&$ /+$ C-+"&)-+ f *$ ,1 ",H&/0$ &01+%)&)($ R⋆λe
%/0 tm0 @ ,#$+%$5=,$ *$% 510>/16$% 1""$%%)=,$% *$ "$ 0'%$1/ .$&0)7
!8 #$%&'()* 9! :/)';<%('-2 &%) 34567
;)*<=* 8B ?;)*<=* 1<@@'2@(* .* 0% &)-&-1'('-2 C A
deadlock = R⋆λe(init)\R
−1λe (R
⋆λe(init))
deadlock ⊆ R⋆λe(init)\R
−1λe (R
⋆λe(init)) 3 I+&/)&)($5$+& %$,-+ deadlock ⊆
R⋆λe(init) $& deadlock ∩R
−1λe (R
⋆λe(init)) = ∅
!" #$$%&% #' ()%*+%,
R⋆λe(init)\R
−1λe (R
⋆λe(init)) ⊆ deadlock ! "#$%&'($)$*& +,-%$ ./$ 01'2 $3'0&$ t
,%%$00'42$ $& '2 *1$0& +,0 5,*0 R−1λe (R
⋆λe(init))6 ,27-0 '2 57'& 8&-$ /* 9&,& 427:
./,*&;
()%*+% - .()%*+% ,*//0$/1% 2% 34 5)656,0106$ 7 8
<$27* 2, %7*0&-/%&'7* 51/* &$2 0=0&>)$ 5$ -9:9%-'&/-$ RSyncλe 6 '2 (, 0/'(-$ 2$ %7)+7-&$:
)$*& 5/ 0=0&>)$ $* 0=*%?-7*'0,*& ,($% 2$ %?$)'* @A@ 5$ 21,/&7),&$ 5$ BC%?';
D'*0'6
(RSyncλe ∪ Rω
λe ∪ RMλe)
⋆({Sync(init)}) #= ∅
c.E.d. M $0& (9-'F9 +,- 2, G7-)/2$ f ;
()%*+% 9 .()%*+% ,*//0$/1% 2% 34 5)656,0106$ : 8
M, s ² EXφ1 ⇐⇒ ∃π, s0 = s⇒M, s1 ² φ1; "#$%&'($)$*&6 *7/0 74&$*7*0
EXRλe (u) = R−1λe (u)
M, s ² AXφ1 ⇐⇒ ∀π, s0 = s⇒M, s1 ² φ1; H7/0 +7/(7*0 %,2%/2$- ,/&-$)$*&
$* /&'2'0,*& ! %7))$ 0/'(,*& !
M, s ² AXφ1
⇐⇒ s #= s′ avec M, s ² EX¬φ1
⇐⇒ s #∈ S = {s′|∃π, s0 = s′ ⇒M, s1 ² ¬φ1}
"#$%&'($)$*&6 *7/0 74&$*7*0
AXRλe(u) = R⋆
λe(u)\(Rλe−1(R⋆
λe(u)\u))
!!"#" $
!"#$ #$ %&' #" ()*
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@H <7
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@H =7
;;;
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(1025# K@ (1023$!-$!24 #25# +7
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@H L7
L ?@ #: .A )$B 89($)1* (102#5 K@ (1024 #289($)12#5:6
(1025# K@ (1024 #289($)125#: +7
@H I7
;;;
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