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HAL Id: inria-00070335https://hal.inria.fr/inria-00070335
Submitted on 19 May 2006
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Combining Data Structures with Nonstably InfiniteTheories using Many-Sorted Logic
Silvio Ranise, Christophe Ringeissen, Calogero Zarba
To cite this version:Silvio Ranise, Christophe Ringeissen, Calogero Zarba. Combining Data Structures with NonstablyInfinite Theories using Many-Sorted Logic. [Research Report] RR-5678, INRIA. 2005, pp.39. inria-00070335
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Thème SYM
INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE
Combining Data Structures with Nonstably InfiniteTheories using Many-Sorted Logic
Silvio Ranise — Christophe Ringeissen — Calogero Zarba
N° 5678
Septembre 2005
Unité de recherche INRIA LorraineLORIA, Technopôle de Nancy-Brabois, Campus scientifique,
615, rue du Jardin Botanique, BP 101, 54602 Villers-Lès-Nancy (France)Téléphone : +33 3 83 59 30 00 — Télécopie : +33 3 83 27 83 19
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K¥ cRmVσ = varsσ(Γ)
¤§pryσ ∈ ΣS ¤r|X*VqvcRm V =
⋃
σ∈ΣS Vσ¥\ekst*nc
ΓsvlNl|Xm-svl$³|Xo8qvc¤
m-_*cRyccK·Tsvl-mlV|Σ¢¹stkmcRy-~8ycRm|Xm-svpr
Bl|Xm-svl$§jkst8
Γ¥Ic © stqtq!w*lc Γ
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TΣ≈
¢¹stgmcRy-~8ycRm|Xm-svprA¥
¼*pryσ ∈ S
¤8cR*prmc © stm-_ ∼σm-_*ccnukw8st r|XqvcR*ncycRq®|Xm-svpr¬p cRy¸m-_*c
Σ¢¹mcRy-bdl¸p cRy
VpX
lpry-mσ¤8cK³**cnokj
s ∼σ ts sB = tB
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¯cK·Tm¤0qvcRmTσ
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Γ¥ 9ÁÃpry`|lpry-m
σ © c¸_|n cTσ = ∅
¤8³8·£|§ycnl-_£ |Xy-s®|Xo8qvcxσ
0
pX\lpry-mσ¥
cRmAocµm-_*c
Σ¢¹stgmcRy-~8ycRm|Xm-svpr²p cRy
V|X*m-_*c` r|Xy-s®|Xo8qvcnl
xσ0
npr*l-m-y-w*RmcnU|rl\Ãprqtqvp © lº¥¼styl-m¤ © c9qvcRm
Aσ =
Tσ/ ∼ ,s¦Tσ 6= ∅ ,
[xσ0 ]∼ ,
prm-_*cRy © svlc .|X*
xA = [x]∼ ,Ãpryc|r_F |Xy-s®|Xo8qvc
x .
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Σ¤³8· |X»cRqvcRbdcRgm
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§pry`§w8*Rm-svpr¾l-j0bVoprqvl f pX¡|Xy-stmÁj σ1 × · · · × σn → σK
fA([t1]∼, . . . , [tn]∼) =
[f(t1, . . . , tn)]∼ ,s¦f(t1, . . . , tn) ∈ Tσ ,
aσ0 ,
prm-_*cRy © svlc ,
§pry~8ycnTsv|Xmcl-j0bVoprqvl p pX!|Xy-stmËj σ1 × · · · × σnK
([t1]∼, . . . , [tn]∼) ∈ pA ⇐⇒(
tB1 , . . . , tBn
)
∈ pB .
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¢¹stgmcRy-~8ycRm|Xm-svpr¤\|X*¬m-_|XmAσ = [varsσ(Γ)]A
¤Ãpryc|r_¾lpry-m
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C Å X!C É Å ËÉ x = y
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C Å X!C É Å ÁÉ x = f(y1, . . . , yn)
È Å ÅfA É Æ [ É ! É 8C Æ
σ1 × · · · × σn → σ Ic9_| cK
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f(y1, . . . , yn) ∈ Tσ
= fA(
yA1 , . . . , yAn
)
.
C Å X!C É Å ÁÉ p(y1, . . . , yn)
ʬp(y1, . . . , yn)
È Å ÅpA= Å ÊAT Å
Æ É ! É 8 ® Æσ1 × · · · × σn
w*l-mpro*lcRy- c9m-_|Xm.K(
yB1 , . . . , yBn
)
∈ pB ⇐⇒ ([y1]∼, . . . , [yn]∼) ∈ pA
⇐⇒(
yA1 , . . . , yAn
)
∈ pA .
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〈a1, . . . , an〉¤ © _*cRyc n ≥ 0|X*
a1, . . . , an ⊆ A¥Ic98cR*prmc © stm-_ A∗ m-_*c¨lcRm)pX\qtsvl-ml)p cRy A ¥^`_*cm-_*cnpry-jpXqtsvl-mlTlist
_|rl9|Ulst|Xm-w8ycΣlist
nprkm|Xst8st8¾|lpry-melem
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'! ( !"!<'.6
car¤8pX¡|Xy-stmÁj
list→ elem
cdr¤8pX¡|Xy-stmÁj
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cons¤8pX¡|Xy-stmÁj
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¢¹stgmcRy-~8ycRm|Xm-svprFlY|Xm-svl$¢§j0st8m-_*c6§prqtqvp © st8npr*Tstm-svpr*l.K
Alist = (Aelem)∗
nilA = 〈〉
carA(〈e1, . . . , en〉) = e1¤0Ãpry)c|r_
n > 0|X*
e1, . . . , en ∈ Aelem
cdrA(〈e1, . . . , en〉) = 〈e2, . . . , en〉¤8§pry)c|r_
n > 0|X*
e1, . . . , en ∈ Aelem
consA(e, 〈e1, . . . , en〉) = 〈e, e1, . . . , en〉¤g§pry¶c|r_
n ≥ 0|X*
e, e1, . . . , en ∈ Aelem
¥^`_*c 3 46:5 : /3 svlm-_*c¡~|Xsty Tlist = 〈Σlist,A〉
¤ © _*cRycAsvlm-_*c¶Rq®|rllpX|Xqtqkl-m|X**|Xy
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!5$ $ |Belem| = κ
'>6($ ' ')40 κ > |Aelem| *
5:: cRm Vσ = varsσ(Γ)¤§pry
σ ∈ elem, list¥ Icnpr*l-m-y-w*Rm9|
Tlist
¢¹stgmcRy-~8ycRm|Xm-svprBp cRy
Velem ∪ Vlist
|rl`Ãprqtqvp © l¥¼s¦·F|lcRm A′ l-w*_Fm-_|Xm |Aelem ∪A′| = κ
¤|X*=qvcRm
Belem = Aelem ∪A′ ,
|X*
eB = eA ,§pry)c|r_
elem¢¹ r|Xy-s®|Xo8qvc
e ∈ Velem ,
xB = xA ,§pry)c|r_
list¢¹ r|Xy-s®|Xo8qvc
x ∈ Vlist .
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Tlist
¢¹stkmcRy-~8ycRm|Xm-svpr[l-w*_Fm-_|Xm|Belem| = κ
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BlY|Xm-svl$³cnl|XqtqqtstmcRy|Xqvlst
Γ¥
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Tlist
svl³*8stmcRqtj © stm-*cnll|Xo8qvc © stm-_Fycnl~3cnRm)mp elem ¤© c © stqtqw*lc¸m-_*c|XwT·0stqts®|Xy-jAw8*Rm-svpr*l compress|X*
δ¥
st cRx ∈ A∗ |X* X ⊆ A
¤m-_*c¨Aw8*Rm-svprcompress
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X¥¡¼*pry-bU|Xqtqtj K
compress(〈α1, . . . , αn〉, X) =
〈〉 ,s¦n = 0 ,
〈α1〉 compress(〈α2, . . . , αn〉, X) ,s¦n > 0
|X*α1 ∈ X ,
compress(〈α2, . . . , αn〉, X) ,prm-_*cRy © svlc¤
© _*cRyc svl`m-_*c¨npr*|XmcR|Xm-svpr[pr~cRy|Xmprypº cRy)qtsvl-ml¥ st cRx, y ∈ A∗ ¤Tm-_*c¸§w8*Rm-svpr δ mcnlml © _*cRm-_*cRy x = y
pryx 6= y
¥!¼*pry-bU|Xqtqtj K
δ([α1, . . . , αn], [β1, . . . , βm]) =
∅ ,s¦n = 0
|X*m = 0 ,
α1 ,s¦n > 0
|X*m = 0 ,
β1 ,s¦n = 0
|X*m > 0 ,
α1, β1 ,s¦n,m > 0
|X*α1 6= β1 ,
δ([α2, . . . , αn], [β2, . . . , βm]) ,prm-_*cRy © svlc¥
% É ¡É C É '>')) ) !!x, y
$5 '))C'# $ ) !
x 6= y ' ) * δ(x, y) 6= ∅
' '*!X x 6= y
' δ(x, y) ⊆ X
$5compress(x,X) 6= compress(y,X) *
'* ' # ' ( #+ + ' ' C NÅ É Î © stm-*cnll§w8*Rm-svpr witness list
§prym-_*c¶m-_*cnpry-jTlist
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Γoc6|Vnprrz±w8*Rm-svpr£pX |Xm
Σlist
¢¹qtstmcRy|Xqvl`l-w*_m-_|Xm
varselem(Γ) 6= ∅¥<Ic¶qvcRm
witness list(Γ)oc¡m-_*c¶ycnl-w8qtm\pX|X~8~8qtjkst8¸mp
Γm-_*cÃprqtqvp © st8
m-y|X*l-§pry-bU|Xm-svpr*l.K
cR~8q®|rncdc|r_ªqtstmcRy|XqµpXµm-_*cÃpry-b e ≈ car(x)st
Γ © stm-_m-_*cÃpry-bVw8q®| x 6≈ nil →x ≈ cons(e, y′)
¤ © _*cRyc y′ svl|Aycnl-_ list¢¹ |Xy-s®|Xo8qvc¥
cR~8q®|rncdc|r_qtstmcRy|XqpX`m-_*c§pry-b x ≈ cdr(y)st
Γ © stm-_ªm-_*c§pry-bw8q®| x 6≈ nil →y ≈ cons(e′, x)
¤ © _*cRyc e′ svl|Aycnl-_ elem¢¹ |Xy-s®|Xo8qvc¥
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'! ( !"!<'.6
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x,y
|X*w′′
x,y
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¤8|X*qvcRm
v = vars(∆) \ vars(Γ)¥¡^`_*cR
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Aelem = [vars elem(Γ)]A ∪A′ ,
# $5 |A′| ≤ 2k *
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1 ⇒ 2
K¥ cRmVσ = varsσ(Γ)
¤Ãpryσ ∈ elem, list
¥£ekst*ncΓsvlTlist
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Tlist
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¢¹stkmcRy-~8ycRm|Xm-svprnpr*l-m-y-w*Rmcn¨okj9qvcRm-m-st8
Aelem = V Belem ∪
α ∈ δ(
xB, yB)
|m-_*c9qtstmcRy|Xq
x 6≈ ysvlst
Γ
,
|X*
eA = eB ,Ãpry)c|r_
elem¢¹ r|Xy-s®|Xo8qvc
e ∈ Velem ,
xA = compress(
xB, Aelem
)
,Ãpry)c|r_
list¢¹ |Xy-s®|Xo8qvc
x ∈ Vlist .
¯prmcm-_|XmAsvl| © cRqtq¦¢Á8cK³**cn Tlist
¢¹stgmcRy-~8ycRm|Xm-svpr¤|X*m-_|XmAelem = V A
elem ∪A′ ¤Ãpry
|lcRmA′ l-w*_Fm-_|Xm |A′| ≤ 2k
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Γ¥
C Å X!C É Å ËÉ e1 ≈elem e2
Êe1 6≈elem e2
9ÁbbdcnTs®|Xmc¥
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C Å X!C É Å ËÉ x ≈list y
Ic¸_|n c
xA = compress(
xB, Aelem
)
= compress(
yB, Aelem
)
= yA .
C Å X!C É Å ËÉ x 6≈list y
µj²x!ypr~pl-stm-svpr5B © stm-_ X = Aelem
K¥
C Å X!C É Å ËÉ x ≈ nil
Ic9_|n c
xA = compress(
xB, Aelem
)
= compress(〈〉, Aelem)
= 〈〉 .
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Ic¸_|n c
xA = compress(
xB, Aelem
)
= compress(⟨
eB⟩
yB, Aelem
)
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eB⟩
compress(
yB, Aelem
) l-st*nceB ∈ Aelem
=⟨
eA⟩
yA
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vars elem(Γ) 6= ∅¥ Ic¨qvcRm
witness ′listoc¨m-_*cVycnl-w8qtm¸pX¶|X~8~8qtjkst8²mp
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Γ © stm-_ªm-_*c§pry-bw8q®| x 6≈ nil →y ≈ cons(e′, x)
¤ © _*cRyc e′ svl|Aycnl-_ elem¢¹ |Xy-s®|Xo8qvc¥
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§prycRqvcRbdcRkml¤|²lpry-m
index§pry¸st*Tsvncnl¤|X*¬|²lpry-m
array§pry9|Xy-y|njTl¤´~8qtw*l6m-_*cÃprqtqvp © st8£m © p²Aw8*Rm-svpr
ljkbo3prqvl.K
read¤8pX\lpry-m
array × index→ elem
write¤8pX\lpry-m
array × index× elem→ array¥
FÉ T É st cRa : I → E
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|X*e ∈ E
¤ © c8cK³**c ai7→e : I → E|rlÃprqtqvp © l.K
ai7→e(i) = e|X*
ai7→e(j) = a(j)¤8Ãpry
j 6= i¥
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¢¹stgmcRy-~8ycRm|Xm-svprlY|Xm-svl$Aj0st8m-_*c6Ãprqtqvp © st8npr*Tstm-svpr*l K
Aarray = (Aelem)Aindex
readA(a, i) = a(i)¤*§pry)c|r_
a ∈ Aarray
|X*i ∈ Aindex
writeA(a, i, e) = ai7→e¤0Ãpry)c|r_
a ∈ Aarray
¤i ∈ Aindex
¤*|X*e ∈ Aelem
¥^`_*c 3 46:5 : 8;5"58 svlm-_*cV~|Xsty Tarray = 〈Σarray,A〉
¤ © _*cRycAsvlm-_*cRq®|rllpX|Xqtq
lm|X**|Xyarray
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!J-/1 * 24$5 $5 > !!&Tarray 1 6 '
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!"$ $ |Belem| = |Aelem|
' |Bindex| = κ
>6 $ ')4
κ > |Aindex| *
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5:: cRm Vσ = varsσ(Γ)¤¶Ãpry
σ ∈ elem, index, array¥ IcFnpr*l-m-y-w*Rml|
Tarray
¢stkmcRy-~8ycRm|Xm-svpr
Bpº cRy
Velem ∪Vindex∪Varray
|rl¡Ãprqtqvp © l¥!¼s¦·|VlcRm A′ l-w*_Um-_|Xm |Aindex ∪A′| = κ
¤*|X*£qvcRm
Bindex = Aindex ∪A′ ,
Belem = Aelem ,
|X*
iB = iA ,Ãpry)c|r_
index¢¹ r|Xy-s®|Xo8qvc
i ∈ Vindex ,
eB = eA ,Ãpry)c|r_
elem¢¹ |Xy-s®|Xo8qvc
e ∈ Velem .
9Ëpry8cRyVmp¾8cK³**cBpº cRyVm-_*c
array¢¹ |Xy-s®|Xo8qvcnl¤N³8·ª|XªcRqvcRbdcRgm
e0 ∈ Aelem
¥^`_*cR¤Ãpryc|r_
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a ∈ Varray
¤ © c9qvcRm
aB(i) =
aA(i) ,s¦i ∈ Aindex ,
e0 ,prm-_*cRy © svlc .
¶j npr*l-m-y-w*Rm-svpr¤Bsvl[|
Tarray
¢¹stgmcRy-~8ycRm|Xm-svpr l-w*_ m-_|Xm|Belem| = |Aelem|
|X*|Bindex| = κ
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Γ¥
C Å X!C É Å ÁÉ e1 ≈elem e2 e1 6≈elem e2 i ≈index j
Êi 6≈index j
9ÁbbdcnTs¦¢|Xmc¥
C Å X!C É Å ËÉ a ≈array b
cRmi ∈ Bindex
¥9¹i ∈ Aindex
m-_*cRaB(i) = aA(i) =
bA(i) = bB(i)¥/9Á
i /∈ Aindex
m-_*cRaB(i) = e0 = aB(i)
¥^`_gw*lº¤aB = bB
¥
C Å X!C É Å ÁÉ a 6≈array b
ekst*ncaA 6= bA
¤m-_*cRyccK·Tsvl-ml9|XÍst*8cK·i ∈ Aindexlw*_Fm-_|Xm
aA(i) 6= bA(i)¥ 9Ám`Ãprqtqvp © lm-_|Xm aB(i) 6= bB(i)
¤ © _8sv_Fstb~8qtsvcnl aB 6= bB¥
C Å X!C É Å ËÉ e ≈ read(a, i)
Ic¸_|n cK
eB = eA
= [read(a, i)]A
= aA(
iA)
= aB(
iB) lst*nc
iA ∈ Aindex .
éé äMNYïOYò
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C Å X!C É Å ËÉ a = write(b, i, e)
Ic²_|n caB(iB) = aA(iA) = eA = eB
¥¯6cK·Tm¤qvcRm
j ∈ Bindex
l-w*_¬m-_|Xmj 6= iB
¥>9¹j ∈ Aindex
m-_*cR © c_|n c aB(j) = aA(j) =bA(j) = bB(j)
¥ 9¹st*l-mc|rj /∈ Aindex
m-_*cRaB(j) = e0 = bB(j)
¥¶^`_gw*l¤aB = bB
iB 7→eB
¥
% É ¡É C É )( A
Tarray 01 7' !! *F '" 7'
Γ !
Σarray ) 16') ! !5$ $ Aelem
!-/01 * 24$ $5 '!!>Tarray 1 6 '
B! ! *' Γ
!5$ $ |Bindex| = |Aindex|
|Belem| = κ
>6 $κ > |Aelem| *
5:: ekstbstq®|Xympm-_*c9~8ypkpX!pX!x!ypr~3plstm-svpr0¥
% É ¡É C É ' É´É Å ' * * ! !'!S ⊆ elem, index
$$6'*
Tarray
! !.$ #$ 6! 6. S *
5:: µj=x!ypr~3plstm-svpr*lBB¤0¤*|X*¾T¥
'* ' # ' ( #+ + ' ' C NÅ É Î © stm-*cnll§w8*Rm-svpr witnessarray
Ãprym-_*c²m-_*cnpry-jTarray
|X»oc8cK³**cn|rl6Ãprqtqvp © lº¥I stm-_*prw8m9qvpll9pX¶cR*cRy|XqtstmËj¤qvcRm
Γoc|nprrz$w8*Rm-svprªpX |Xm
Σarray
¢qtstmcRy|Xqvl`l-w*_m-_|Xm
vars index(Γ) 6= ∅|X*
vars elem(Γ) 6= ∅¥ IcqvcRm
witnessarray(Γ)oc)m-_*c
ycnlw8qtm)pX!|X~8~8qtj0st8UmpΓm-_*c9§prqtqvp © st8dm-y|X*l$Ãpry-bU|Xm-svpr K
cR~8q®|rnc¬c|r_ qtstmcRy|Xq¨pXm-_*c§pry-ba 6≈array b
stΓ © stm-_ | qtstmcRy|Xq¨pXm-_*c[Ãpry-b
read(a, i′) 6≈ read(b, i′)¤ © _*cRyc i′ svl|§ycnl-_ index
¢¹ r|Xy-s®|Xo8qvc¥²Å ª8 & cRm
Γo3c|²nprrz±w8*Rm-svprpX |Xm
Σarray
¢¹qtstmcRy|Xqvl¤´qvcRm∆ = witnessarray(Γ)
¤|X*FqvcRm
v = vars(∆) \ vars(Γ)¥!^`_*cR
Γ|X*
(∃v)∆|Xyc
Tarray
¢Ácnukw8st r|XqvcRkm¥
% É ¡É C É'& )( Γ
E E'.' ! Σarray ) 16') !/!"$$
vars index(Γ) 6=∅
varselem(Γ) 6= ∅
' L '0 '* ) 16')/ $5 ' x 6≈array y *24$5
$ '))H# 3' @)C * Γ
!Tarray ! !- () *
* 24$5 '!! Tarray 01 7' A !! *' Γ
!"$$ Aindex = [vars index(Γ)]A'
Aelem = [vars elem(Γ)]A *
5:: 2⇒ 1K¥/9ËbbdcnTs®|Xmc¥
1⇒ 2
K¥ cRmVσ = varsσ(Γ)
¤0Ãpryσ ∈ elem, index, array
¥ekst*ncΓsvlTarray
¢Ál|Xm-svl-³|Xo8qvc¤m-_*cRyccK·0svl-ml¡|
Tarray
¢¹stgmcRy-~8ycRm|Xm-svprBlY|Xm-svl$Aj0st8
Γ¥JIc © stqtq*w*lc B stUpry8cRy!mp¨npr*l-m-y-w*Rm|XFpr~8~pry-m-w8*c
Tarray
¢¹stgmcRy-~8ycRm|Xm-svprA¥
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hpryc¨l~3cnRs¦³|Xqtqtj¤ © c9qvcRm A o3c¸m-_*cTarray
¢¹stkmcRy-~8ycRm|Xm-svpr[pº cRyVelem ∪ Vindex ∪ Varraynpr*lm-y-w*Rmcn¾|rl`Ãprqtqvp © l¥¡¼styl-m¤ © c9qvcRm
Aindex = V Bindex ,
Aelem = V Belem ,
|X*
iA = iB ,Ãpry)c|r_
index¢¹ r|Xy-s®|Xo8qvc
i ∈ Vindex ,
eA = eB ,Ãpry)c|r_
elem¢¹ |Xy-s®|Xo8qvc
e ∈ Velem .
9Ë¿pry8cRy²mpª8cK³**cAp cRy²m-_*c
array¢¹ |Xy-s®|Xo8qvcnl¤`³8·¿|X
e0 ∈ Aelem
¥ ^`_*cR¤`Ãpryc|r_a ∈ Varray
|X*i ∈ Vindex
¤ © c9qvcRm
aA(
iA)
=
aB(iB) ,s¦aB(iB) ∈ Aelem ,
e0 ,prm-_*cRy © svlc .
¯prmcm-_|XmAsvl| © cRqtq¦¢Á8cK³**cn Tarray
¢¹stgmcRy-~8ycRm|Xm-svpr¤¡|X*Ím-_|XmAσ = [varsσ(Γ)]A
¤Ãpry
σ ∈ index, elem¥¯cK·Tm¤ © c¨l_*p © m-_|Xm
Al|Xm-svl$³cnl|XqtqNqtstmcRy|Xqvl)st
Γ¥
C Å X!C É Å ÁÉ e1 ≈elem e2 e1 6≈elem e2 i ≈index j
Êi 6≈index j
9ÁbbdcnTs¦¢|Xmc¥
C Å X!C É Å ÁÉ a ≈array b
cRmi ∈ Aindex
¥9ÁaB
(
iB)
∈ Aelem
m-_*cRaA
(
iA)
=aB
(
iB)
= bB(
iB)
= bA(
iA) ¥9¹\st*l-mc|r
aB(
iB)
/∈ Aelem
m-_*cRaA(iA) = e0 = bA(iA)
¥
C Å X!C É Å ËÉ e ≈ read(a, i)
Ic¸_|n cK
eA = eB
= [read(a, i)]B
= aB(
iB)
= aA(
iA) l-st*nc
aB(
iB)
∈ Aelem .
C Å X!C É Å ÁÉ a = write(b, i, e)
ekst*ncaB
(
iB)
= eB ∈ Aelem
¤ © cd_|n c © c_| c
aA(
iA)
= bB(
iB)
= eB = eA¥ ¯6cK·0m¤µqvcRm
j ∈ Aindex
lw*_m-_|Xmj 6= iA
¥ 9¹aB(j) ∈ Aelem
m-_*cRaA(j) = aB(j) = bB(j) = bA(j)
¥ 9Á¶st*l-mc|raB(j) /∈ Aelem
m-_*cRaA(j) = e0 = bA(j)
¥
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'! ( !"!<'.6
% É ¡É C É'& ® ÅÈ C NÅ 8 ! C Æ ' * *L!. !!S ⊆ elem, index
$$6'*
Tarray
! - 1&# "!E!() #$ 6"! 6 S *
5:: µj=x!ypr~3plstm-svpr*lBT¤*cRbU|Xy6D gT¤|X*£x¡ypr~3pl-stm-svpr B¥
*' # #+ + ÅkÉ Å & %6É ! C Å Å ' '* * ! !'!
S ⊆ elem, indexL$5
$6'*Tarray
! ') 1#$ "! S *
5:: µj=x!ypr~3plstm-svpr*ld|X* gT¥
¸3
^`_*cVm-_*cnpry-jFpX¡lcRmlTset
_|rl|Ul-st|Xm-w8ycΣset
nprkm|Xst8st8=|Ulpry-melem
ÃprycRqvcRbdcRgml9|X*|lpry-m
setÃpry)lcRml)pX!cRqvcRbdcRgml¤8~8qtw*lm-_*c6§prqtqvp © st8l-j0bVoprqvl.K
m-_*c¨npr*l-m|Xkml-j0bVoprq ∅ ¤8pX\lpry-m set
m-_*c¸§w8*Rm-svpr¾l-j0bVoprqvl.K
·¤pX\lpry-m
elem→ set
∪¤∩¤*|X*
\¤8pX!lpry-m
set× set→ set
m-_*c9~8ycnTsv|Xmc¨l-j0bVoprq ∈ ¤8pX!lpry-m elem× set¥
-£Å C É'&& Î 3(8;1 8;5 set . /1 3 45"765"4,38 3"/:1 A svlV|Σset
¢¹stgmcRy-~8ycRm|Xm-svpr»l|Xm$¢svl-Aj0st8m-_*c6§prqtqvp © st8npr*Tstm-svpr*l.K
Aset = P(Aelem)
m-_*cVl-j0bVoprqvl ∅ ¤ · ¤ ∪ ¤ ∩ ¤ \ ¤|X* ∈ |Xyc¨stkmcRy-~8ycRmcn[|rnnpryTst8²mpm-_*cRsty6l-m|X**|XystgmcRy-~8ycRm|Xm-svprÍp cRylcRml¥^`_*c 3 46:5 : 4,3 svlm-_*cµ~|Xsty Tset = 〈Σset,A〉
¤ © _*cRycAsvl\m-_*c`Rq®|rll\pX|Xqtq8l-m|X**|Xy
set¢Ál-m-y-w*Rm-w8ycnl¥
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AE
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!
Σset )+1 ) ! L !"$ $ Aelem
! - 1 * 24$5 $56 '!! Tset 01 7' B!! *F Γ
!5$ $ |Belem| = κ
'>6($κ > |Aelem| *
5:: cRm Vσ = varsσ(Γ)¤Ãpry
σ ∈ elem, set¥ Icnpr*l-m-y-w*Rm¸|
Tset
¢¹stgmcRy-~8ycRm|Xm-svprBp cRy
Velem ∪ Vset
|rl`§prqtqvp © l¥¡¼s¦·F|lcRm A′ l-w*_Fm-_|Xm |Aelem ∪A′| = κ
¤*|X*£qvcRm
Belem = Aelem ∪A′ ,
|X*
eB = eA ,§pry)c|r_
elem¢¹ r|Xy-s®|Xo8qvc
e ∈ Velem ,
xB = xA ,§pry)c|r_
set¢¹ |Xy-s®|Xo8qvc
x ∈ Vset .
¶jFnpr*lm-y-w*Rm-svprBsvl|
Tset
¢¹stkmcRy-~8ycRm|Xm-svprl-w*_m-_|Xm|Belem| = κ
¥)h[prycnp cRy¤3stmsvlm-y-st 0s®|Xqmp_*cn D£m-_|Xm
BlY|Xm-svl$³cnl|XqtqqtstmcRy|Xqvlst
Γ¥
% É ¡É C É'& ' É´É Å 24$5 $56"*Tset
! !".$ #$ "! 6 $5 !'
elem *
5:: µj=x!ypr~3plstm-svpr*lBBV|X* gT¥
'* ' # ' ( #+ + ' ' C NÅ É Î © stm-*cnll§w8*Rm-svpr witness set
§pry´m-_*c!m-_*cnpry-jTset
|X¨o3c8cK³**cn|rlÃprqtqvp © l¥ I stm-_*prw8m`qvpll¶pXcR*cRy|XqtstmÁj¤TqvcRm
Γoc|nprrz$w8*Rm-svpr=pX |Xm
Σset
¢¹qtstmcRy|Xqvl`l-w*_m-_|Xm
varselem(Γ) 6= ∅¥<Ic¶qvcRm
witness set(Γ)o3c¶m-_*c¶ycnl-w8qtmpX|X~8~8qtjkst86mp
Γm-_*cÃprqtqvp © st8
m-y|X*l-§pry-bU|Xm-svpr K
cR~8q®|rncdc|r_ªqtstmcRy|XqµpXµm-_*cÃpry-b x 6≈set yst
Γ © stm-_ |£qtstmcRy|XqpXµm-_*cd§pry-b e′ ∈(x \ y) ∪ (y \ x)
¤ © _*cRyc e′ svl|§ycnl-_ elem¢¹ r|Xy-s®|Xo8qvc¥
²Å ª8 & cRmΓoc¸|nprrz$w8*Rm-svprFpX |Xm
Σset
¢¹qtstmcRy|Xqvlº¤TqvcRm∆ = witness set(Γ)
¤*|X*qvcRm
v = vars(∆) \ vars(Γ)¥¡^`_*cR
Γ|X*
(∃v)∆|Xyc
Tset
¢Ácnugw8st |XqvcRgm¥
% É ¡É C É'& )( Γ
' ! Σset ) 16') ! !"$ $
vars elem(Γ) 6=∅ ' ''0 '*A)+16')& $5 " x 6≈set y * 24$5 $ '))H'# 6
@') 0
éé äMNYïOYò
'! ( !"!<'.6
* Γ!Tset !!- )C *
* 24$5 '!! Tset 1 6 ' A ! ! *' Γ
!"$$ Aelem = [vars elem(Γ)]A *
5:: 2⇒ 1K¥/9ËbbdcnTs®|Xmc¥
1 ⇒ 2
K¥ cRmVσ = varsσ(Γ)
¤§pryσ ∈ elem, set
¥Fekst*ncΓsvlTset
¢ÁlY|Xm-svl$³|Xo8qvc¤m-_*cRyccK·Tsvl-ml|
Tset
¢¹stgmcRy-~8ycRm|Xm-svprBl|Xm-svl$§jkst8
Γ¥ Ic © stqtq¶w*lc B stªpry8cRyVmpnpr*l-m-y-w*Rm|X
pr~8~pry-m-w8*cTset
¢¹stkmcRy-~8ycRm|Xm-svprA¥
hprycl-~3cnRs¦³|Xqtqtj¤ © cqvcRm A ocm-_*cTset
¢¹stgmcRy-~8ycRm|Xm-svprFp cRyVelem∪Vset
npr*l-m-y-w*RmcnokjqvcRm-m-st8
Aelem = V Belem ,
|X*
eA = eB ,Ãpryc|r_
elem¢¹ |Xy-s®|Xo8qvc
e ∈ Velem ,
xA = xB ∩Aelem ,Ãpryc|r_
set¢¹ |Xy-s®|Xo8qvc
x ∈ Vset .
¶j¸npr*l-m-y-w*Rm-svpr¤AsvlN| © cRqtq¦¢Á8cK³**cn Tset
¢¹stgmcR~8ycRm|Xm-svprl-w*_Vm-_|XmAelem = [vars elem(Γ)]A
¥¯6cK·Tm¤ © c¨l-_*p © m-_|Xm
AlY|Xm-svl$³cnl|Xqtq´qtstmcRy|Xqvlst
Γ¥
C Å X!C É Å ËÉ e1 ≈elem e2
Êe1 6≈elem e2
9ÁbbdcnTs®|Xmc¥
C Å X!C É Å ËÉ x ≈set y
Ic¸_| cxA = xB ∩Aelem = yB ∩Aelem = yA
¥
C Å X!C É Å ËÉ x ≈ ∅
Ic9_| cxA = xB ∩Aelem = ∅ ∩Aelem = ∅
¥
C Å X!C É Å ÁÉ x ≈ e
Ic_| cxA = xB ∩Aelem =
eB
∩Aelem =
eB ¥
C Å X!C É Å ÁÉ x ≈ y ∪ z
Ic¸_|n cxA = xB ∩Aelem =
(
yB ∪ zB)
∩Aelem =(
yB ∩Aelem
)
∪(
zB ∩Aelem
)
= yA ∪ zA¥
C Å X!C É Å ÁÉ x ≈ y ∩ z
Ic¸_|n cxA = xB ∩Aelem =
(
yB ∩ zB)
∩Aelem =(
yB ∩Aelem
)
∩(
zB ∩Aelem
)
= yA ∩ zA¥
C Å X!C É Å ÁÉ x ≈ y \ z
Ic¨_|n cxA = xB ∩Aelem =
(
yB \ zB)
∩Aelem =(
yB ∩Aelem
)
\(
zB ∩Aelem
)
= yA \ zA¥
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C Å X!C É Å ËÉ e ∈ x
Êe 6∈ x
w*lmpro*lcRy- c9m-_|XmeA ∈ Aelem
¥
% É ¡É C É'& ' ® ÅÈ C NÅ 8 ! C Æ 24$5 $5 '"*Tset
!>-/1)+* # !E!)C#$ 6"! 6
elem *
5:: µj=x!ypr~3plstm-svpr BT¤*cRbU|Xy6D k0¤|X*Fx¡ypr~3pl-stm-svpr gT¥
*' # #+ + ÅkÉ Å %6É ! C Å Å 24$5$56'*
Tset
! ) 1#$ 6! 6. elem *
5:: µj=x!ypr~3plstm-svpr*l gd|X* gT¥
8 45+$
hw8qtm-svlcRml±i |Xqvlp Dk*p © |rl!o|Xl±i |XycnprqtqvcnRm-svpr*l¶m-_|Xm!bU|njnprgm|Xst²Tw8~8qtsv|Xmc)cRqvcRbdcRgml¥¼pry-bU|Xqtqtj¤|bVw8qtm-svlcRm
xsvl|§w8*Rm-svpr
x : A→ N¤0Ãpry)lprbdc¨lcRm
A¥
Icw*lc`m-_*c)l-j0bVoprq[[ ]]mp¨8cR*prmcm-_*c)cRb~8mËjbVw8qtm-svlcRm¥ I _*cR
n ≥ 0¤ © c © y-stmc [[e]](n)
mp8cR*prmcm-_*cbVw8qtm-svlcRm`nprgm|Xst8st8cK·*|rRm-qtjnpknRw8y-ycR*ncnlµpX
e|X*d*prm-_8st8cRqvlc¥JI¿_*cR
n < 0¤ © c9qvcRm [[e]]n = [[ ]]
¥ cRm
x, yo3c¸m © pdbw8qtm-svlcRml¥¡^`_*cR K
m-_*cRsty 7'x ∪ y
svlm-_*cbVw8qtm-svlcRmzl-w*_[m-_|Xm¤§pry6c|r_ÍcRqvcRbdcRgm
e¤m-_*ccnukw|XqtstmÁj
z(e) = max(x(e), y(e))_*prqv8l
m-_*cRsty !x ] y
svlVm-_*cbVw8qtm-svlcRmzl-w*_ m-_|Xm¤ÃpryVc|r_«cRqvcRbdcRkm
e¤m-_*cUcnukw|XqtstmÁj
z(e) = x(e) + y(e)_*prqv8l
m-_*cRsty 01"!67'x ∩ y
svlm-_*c=bw8qtm-svlcRmzl-w*_ m-_|Xm¤§pryc|r_¿cRqvcRbdcRgm
e¤¡m-_*c
cnugw|XqtstmËjz(e) = min(x(e), y(e))
_*prqv8lº¥^`_*c¡m-_*cnpry-jVpX8bVw8qtm-svlcRml
Tbag
_|rl|l-st|Xm-w8ycΣbag
nprgm|Xst8st8¨|lpry-mint
§prystkmcRcRyl¤|lpry-m
elemÃpry)cRqvcRbdcRgml¤3|X*¾|lpry-m
bagÃprybVw8qtm-svlcRml¤*~8qtw*lm-_*c9§prqtqvp © st8Ul-j0bVoprqvl.K
m-_*c¨npr*l-m|Xkml-j0bVoprqvl.K
0|X*
1¤*pX!lpry-m
int
[[ ]]¤pX\lpry-m
bag
m-_*c¸§w8*Rm-svpr¾l-j0bVoprqvl.K
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'! ( !"!<'.6
+¤−¤max
¤*|X*min
¤8pX!lpry-mint× int→ int
[[·]](·)¤8pX\lpry-m
elem× int→ bag
∪¤]¤*|X*
∩¤TpX!lpry-m
bag× bag→ bag
count¤8pX\lpry-m
elem× bag→ int
m-_*c9~8ycnTsv|Xmc¨l-j0bVoprq < ¤8pX!lpry-m int× int¥
-£Å C É Î 38;1 8;5 bag . /1 3 45"765"4,38 3"/:1 A svl)|Σbag
¢¹stgmcRy-~8ycRm|Xm-svprÍl|Xm$¢svl-Aj0st8m-_*c6§prqtqvp © st8npr*Tstm-svpr*l.K
Aint = Z
Abag = NAelem
m-_*c=l-jkbo3prqvl 0¤1¤+¤−¤max
¤min
¤¡|X*<|XycstkmcRy-~8ycRmcn |rnnpryTst8¬mp¾m-_*cRsty
l-m|X**|Xy£stkmcRy-~8ycRm|Xm-svpr[pº cRym-_*c¸stgmcRcRyl
m-_*c¨l-j0bVoprq [[ ]] ¤ ∪ ¤ ∩ ¤ \ ¤ [[·]](·) |Xyc¸stkmcRy-~8ycRmcn[|rnnpryTst8dmpdm-_*cRsty)l-m|X**|Xy£stgmcRy$¢~8ycRm|Xm-svprpº cRy)bVw8qtm-svlcRml
countA(e, x) = x(e)¤T§pry)c|r_
e ∈ Aelem
|X*x ∈ Abag
¥^`_*c 3 4:;5&: !$ 3"/ 4,3 svl`m-_*c¸~|Xsty Tbag = 〈Σbag,A〉
¤ © _*cRycAsvl`m-_*c9Rq®|rllpX\|Xqtq
lm|X**|Xybag
¢Ál-m-y-w*Rm-w8ycnl¥
$% " # ++% É ¡É C É )(
AE
Tbag 01 6 !! *' ' 7'Γ
!
Σbag )+16') ! ' !5$ $ Aelem
! -/1 * 24$5 $6 !!&Tbag 01 6 B!! *F Γ
!5$ $ |Belem| = κ
'>6($κ > |Aelem| *
5:: cRm Vσ = varsσ(Γ)¤ºÃpry
σ ∈ elem, int, bag¥<Ic!npr*l-m-y-w*Rm|
Tbag
¢¹stkmcRy-~8ycRm|Xm-svprBp cRy
Velem ∪ Vint ∪ Vbag
|rl`Ãprqtqvp © lº¥¡¼s¦·£|lcRm A′ l-w*_¾m-_|Xm |Aelem ∪A′| = κ
¤|X*=qvcRm
Belem = Aelem ∪A′ ,
|X*
eB = eA ,§pry)c|r_
elem¢¹ r|Xy-s®|Xo8qvc
e ∈ Velem ,
uB = uA ,§pry)c|r_
int¢¹ |Xy-s®|Xo8qvc
u ∈ Vint .
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¼*w8y-m-_*cRy-bdpryc¤8ÃprycR cRy-jbag
¢¹ |Xy-s®|Xo8qvcx ∈ Vbag
|X*e ∈ Belem
¤ © c9qvcRm
xB(e) =
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Unité de recherche INRIA LorraineLORIA, Technopôle de Nancy-Brabois - Campus scientifique
615, rue du Jardin Botanique - BP 101 - 54602 Villers-lès-Nancy Cedex (France)
Unité de recherche INRIA Futurs : Parc Club Orsay Université - ZAC des Vignes4, rue Jacques Monod - 91893 ORSAY Cedex (France)
Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France)Unité de recherche INRIA Rhône-Alpes : 655, avenue de l’Europe - 38334 Montbonnot Saint-Ismier (France)
Unité de recherche INRIA Rocquencourt : Domaine de Voluceau - Rocquencourt - BP 105 - 78153 Le Chesnay Cedex (France)Unité de recherche INRIA Sophia Antipolis : 2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex (France)
ÉditeurINRIA - Domaine de Voluceau - Rocquencourt, BP 105 - 78153 Le Chesnay Cedex (France)
ISSN 0249-6399
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