HAL Id: hal-00350490
https://hal.archives-ouvertes.fr/hal-00350490
Submitted on 6 Jan 2009
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Expressiveness of Updatable Timed Automata
Patricia Bouyer, Catherine Dufourd, Emmanuel Fleury, Antoine Petit
To cite this version:
Patricia Bouyer, Catherine Dufourd, Emmanuel Fleury, Antoine Petit. Expressiveness of Updatable Timed Automata. 25th International Symposium of Mathematical Foundation of Computer Science (MFCS’2000), 2000, Bratislava, Slovakia. pp.232-242, �10.1007/3-540-44612-5_19�. �hal-00350490�
http://www.lsv.ens−cachan.fr/Publis/
In Proc. 25th Int. Symp. Math. Found. Comp. Sci. (MFCS’2000), Bratislava, Slovakia, Aug. 2000.
volume 1893 of Lecture Notes in Computer Science, pages 232−242. Springer, 2000.
Expressiveness of Updatable Timed Automata
P.Bouyer,C.Dufourd, E.Fleury, A.Petit
LSV,UMR8643,CNRS&ENSdeCahan,
61Av.duPrésidentWilson,94235Cahanedex,Frane
{bouyer, dufourd, fleury, petit}lsv.ens-ahan .fr
Abstrat. SinetheirintrodutionbyAlurandDill,timedautomata
havebeenoneofthemostwidelystudiedmodelsforreal-timesystems.
The syntatiextension of so-alled updatable timed automata allows
more powerful updatesof loks than the resetoperation proposed in
theoriginalmodel.
Weprove that any languageaepted by anupdatable timed automa-
ton (from lasses where emptiness is deidable) is also aepted by a
lassial timed automaton. We propose even more preiseresults on
bisimilaritybetweenupdatableandlassialtimedautomata.
1 Introdution
Sine their introdution by Alur and Dill [2,3℄, timed automata have been
oneofthemoststudiedmodelsforreal-timesystems(see [4,1,16,8,12,17,13℄).
Inpartiularnumerousworksproposedextensionsoftimedautomata[7,10,11℄.
This paper fouses on one of this extension, the so-alledupdatable timed
automata,introduedinorderto modeltheATMprotoolABR[9℄.Updatable
timedautomata areonstrutedwithupdatesofthefollowingforms:
x:jx:y+ wherex;yareloks;2Q
+
and 2f<;;=;6=;;>g
In [5℄, the (un)deidability of emptiness of updatable timed automata has
beenharaterizedina preiseway(seeSetion 2 fordetailedresults).Wead-
dress here the open question of the expressive power of updatable timed au-
tomata(from deidable lasses).We solveompletely this problem by proving
that anylanguage aeptedbyanupdatabletimedautomatonisalsoaepted
by a lassial timed automaton with "-transitions. In fat, we propose even
morepreiseresultsbyshowingthatanyupdatabletimedautomatonusingonly
deterministi updates is stronglybisimilar to a lassialtimedautomaton and
thatanyupdatabletimedautomatonusingarbitraryupdatesisweaklybisimilar
(butnotstronglybisimilar)toa lassialtimedautomaton.
Thepaperisorganized asfollows.InSetion 2,wepresentupdatable timed
automata, generalizing lassial denitions of Alur and Dill. Several natu-
ralequivalenesofupdatable timedautomata areintroduedinSetion3.The
bisimulationalgorithmsarepresentedinSetion4.
For lakof spae,thispaperontainsonlysomeskeths ofproofs.Theyare
TimedWordsand Cloks
If Z is any set, let Z
(respetively Z
!
) be the set of nite (resp. innite)
sequenesofelementsinZ andletZ 1
=Z
[Z
!
.Weonsiderastimedomain
T the set of non-negative rational Q
+
and as nite set of ations. A time
sequeneoverT isa niteorinnitenondereasingsequene =(t
i )
i1 2T
1
.
Atimedword !=(a
i
;t
i )
i1
isanelementof(T) 1
.
We onsider an at most ountable set X of variables, alled loks. A lok
valuationoverX isamappingv:X !T thatassignstoeahlokatimevalue.
Lett2T, thevaluation v+tisdenedby(v+t)(x)=v(x)+t,8x2X.
ClokConstraints
Givena subsetofloksX X,weintroduetwosetsoflokonstraintsover
X.Themostgeneralone,denotedbyC(X),isdenedbythefollowinggrammar:
'::=xjx yj'^'j:'jtrue;withx;y2X;2Q
+
;2f<;;=;6=;;>g
ThepropersubsetC
df
(X)ofdiagonal-freeonstraintsinwhihtheomparison
betweentwoloksisnotallowed,isdenedbythegrammar:
'::=xj'^'j:'jtrue; withx2X; 2Q
+
and 2f<;;=;6=;;>g
Wewritevj='whenthelokvaluationv satisesthelokonstraint'.
Updates
Anupdateisafuntionwhihassignstoeahvaluationasetofvaluations.Here,
werestritourselvestoloalupdates whih aredened inthefollowingway. A
simple update overalokz isofoneofthetwofollowingforms:
up::=z:jz:y+d; where;d2Q
+
;y2X and 2f<;;=;6=;;>g
Whentheoperatoristheequality(=),theupdateissaidtobedeterministi,
non deterministi otherwise. Let v be a valuation and up be a simple update
over z. A valuation v 0
is in up(v) if v 0
(y) = v(y) for any lok y 6= z and if
v 0
(z) (v 0
(z)v(y)+dresp.)ifup=z:(up=z:y+dresp.)
Thesetlu(U)ofloalupdatesgeneratedbyasetofsimpleupdatesU isdened
asfollows.Aolletionup=(up
i )
1ik
isinlu(U)if,foreahi,up
i
isasimple
update of U over some lok x
i
2 X (notethat it ouldhappen that x
i
=x
j
for some i 6= j). Let v; v 0
2 T n
be two lok valuations. We have v 0
2 up(v)
if andonly if,for any i, thelok valuation v 00
dened byv 00
(x
i )=v
0
(x
i ) and
v 00
(y)=v(y)foranyy6=x
i
veriesv 00
2up
i (v).
Note that up(v) may be empty. For instane, theloal update (x:<1;x:>1)
leadstoanemptyset.Butifwetaketheloalupdate(x:>y;x:<7),thevalue
v 0
(x)hastosatisfy:v 0
(x)>v(y)^v 0
(x)<7.
ForanysubsetX ofX,U(X)istheset ofloalupdateswhihareolletionsof
simpleupdatesoverloksofX.Inthefollowing,U
0
(X)denotesthesetofreset
updates. Areset updateis anupdate upsuh that forevery lokvaluationv,
v 0
withv 0
2up(v)andanylokx2X, eitherv 0
(x)=v(x) orv 0
(x) =0.It is
AnupdatabletimedautomatonoverT isatupleA=(;Q;X;T;I;F;R ),where
is a nite alphabetof ations,Q a niteset of states,X X a nite set of
loks,T Q[C(X)[f"gU(X)℄Qanite setoftransitions,IQ
(F Q,RQresp.)thesubsetofinitial(nal,repeatedresp.)states.
Let C C(X) be a subset of lok onstraints and U U(X) be a subset of
updates,thelassAut
"
(C;U)isthesetofalltimedautomatawhosetransitions
only use lok onstraints of C and updates of U. The usual lass of timed
automata,denedin[2℄,isthefamilyAut
"
(C
df (X);U
0 (X)).
Apath inAisanite oraninnitesequeneofonseutivetransitions:
P =q
0
'1;a1;up1
!q
1
'2;a2;up2
!q
2
:::; where(q
i 1
;'
i
;a
i
;up
i
;q
i
)2T; 8i>0
Thepathissaidaepting ifq
0
2Iandeither itisniteanditendsinannal
state,or itisinniteandpassesinnitelyoftenthrougharepeatedstate.Arun
oftheautomatonthrough thepathP isa sequeneoftheform:
hq
0
;v
0 i
'
1
;a
1
;up
1
!
t
1 hq
1
;v
1 i
'
2
;a
2
;up
2
!
t
2 hq
2
;v
2 i:::
where =(t
i )
i1
isa timesequeneand (v
i )
i0
arelokvaluations suh that
8x2X;v
0
(x)=0and8i1;v
i 1 + (t
i t
i 1 )j='
i andv
i 2up
i (v
i 1 + (t
i t
i 1 ).
Remarkthat anysetup
i (v
i 1 +(t
i t
i 1
))ofarun isnonempty.
Thelabeloftherunis thesequene(a
1
;t
1 )(a
2
;t
2
)2(([f"g)T) 1
.The
timed word assoiated with this sequene is w = (a
i1
;t
i1 )(a
i2
;t
i2
)::: where
a
i1 a
i2
::: is thesequeneofationswhiharein (i.e.distintfrom ").Ifthe
pathP isaeptingthenthetimedwordwisaeptedbythetimedautomaton.
About Deidability of Updatable TimedAutomata
Forveriationpurposes,afundamentalquestionistoknowiftheemptinessof
(thelanguage aeptedby)anupdatable timedautomatonis deidableornot.
The paper[5℄ proposes a preise haraterization whih is summarized in the
piturebelow.Notethatdeidabilityandependonthesetoflokonstraints
that are used diagonal-free or not whih makes an important dierene
withlassial timedautomataforwhihitiswellknown thatthesetwokinds
of onstraints areequivalent.The tehniqueproposed in[5℄ showsthat allthe
deidabilityasesarePspae-omplete.
diagonal-freelokonstraints generallokonstraints
Deterministi updates
x:=;x:=y Deidable Deidable
x:=y+,2Q +
Deidable Undeidable
x:=y+,2Q Undeidable Undeidable
deterministi updates
x:<,2Q +
Deidable Deidable
x:>,2Q +
Deidable Undeidable
x:<y+,2Q +
Deidable Undeidable
x:>y+,2Q +
Deidable Undeidable
of the deidable lasses.To solvethis problem, we rst introduenatural and
lassialequivalenesbetweenupdatabletimedautomata.
3 Some Equivalenes of Updatable Timed Automata
Language Equivalene
Twoupdatabletimedautomataarelanguage-equivalent iftheyaeptthesame
timedlanguage.Byextension,two familiesAut
1
andAut
2
aresaidtobeequiv-
alent if any automaton of one of the families is equivalent to one automaton
of the other. Wewrite
`
inbothases.For instane, Aut
"
(C
df (X);U
0 (X))
`
Aut
"
(C(X);U
0
(X)), (seee.g.[7℄).
Bisimilarity
Bisimilarity[15,14℄ is stronger thanlanguage equivalene.It denes a stepby
steporrespondenebetweentwotransitionsystems.Twolabelledtransitionsys-
temsT =(S;S
0
;E;( e
!)
e2E
)and T 0
=(S 0
;S 0
0
;E;( e
!)
e2E
)are bisimilarwhen-
everthereexistsa relationRSS 0
whihmeetsthefollowingonditions:
initialization:
8s
0 2S
0 , 9s
0
0 2S
0
0
suhthat s
0 Rs
0
0
8s 0
0 2S
0
0 , 9s
0 2S
0
suhthat s
0 Rs
0
0
propagation: 8
>
>
>
<
>
>
>
: ifs
1 Rs
0
1 ands
1 e
!s
2
thenthereexists s 0
2 2S
0
suh thats 0
1 e
!s 0
2 ands
2 Rs
0
2
ifs
1 Rs
0
1 ands
0
1 e
!s 0
2
thenthereexists s
2 2S
suh thats
1 e
!s
2 ands
2 Rs
0
2
Strong and Weak Bisimilarity
Timed transition systems - Eah updatable timed automaton A =
(;Q;X;T;I;F;R )inAut
"
(C(X);U(X))denesatimedtransitionsystemT
A
=
(S;S
0
;E;( e
!)
e2E
)as follows:
S=QT X
,S
0
=fhq;vijq2I and8x2X;v(x)=0g,E=[f"g[Q
+
8a2[f"g,hq;vi a
!hq 0
;v 0
ii9(q;';a;up;q 0
)2T s.t.vj='andv 0
2up(v)
8d2Q
+ , hq;vi
d
!hq 0
;v 0
iiq=q 0
andv 0
=v+d
When"isonsideredasaninvisibleation,eahupdatabletimedautomatonA
inAut
"
(C(X);U(X))denesanothertransitionsystemT 0
A
=(S;S
0
;E 0
;( e
))
e2E )
asfollows:
S=QT X
,S
0
=fhq;vijq2I and8x2X;v(x)=0g,E 0
=[Q
+
8a2, hq;vi a
)hq 0
;v 0
iihq;vi
"
!
a
!
"
!
hq 0
;v 0
i
8d2Q
+ ,hq;vi
d
)hq 0
;v 0
iihq;vi
"
!
d
1
!
"
!
::: d
k
!
"
!
hq 0
;v 0
iandd= P
k
i=1 d
i
Two bisimilaritiesfortimedautomata- Twoupdatabletimedautomata Aand
B arestronglybisimilar,denoted A
s B,ifT
A andT
B
arebisimilar.Theyare
weaklybisimilar,denotedA
w B,ifT
0
and T 0
arebisimilar.
ilar. If the bisimulation R preserves the nal and repeated states, weakly or
stronglybisimilarupdatable timedautomatareognizethesamelanguage.
Let A a timedautomaton and be a onstant. We denote by A the timed
automatoninwhihalltheonstantswhihappeararemultipliedbytheonstant
. The proof of the following lemma is immediate and similar to the one of
Lemma4.1in[3℄.Thislemmaallowsustotreatonlyupdatabletimedautomata
where allonstants appearing in the lok onstraints and in the updates are
integer (andnotarbitraryrationals).
Lemma 1. Let A and B be two timed automata and 2 Q +
be a onstant.
ThenA
w
B () A
w
BandA
s
B () A
s B
4 Expressive Power of Deterministi Updates
Werstdealwithupdatabletimedautomatawhereonlydeterministiupdates
areused.Thefollowingtheoremisoftenonsideredas afolklore result.
Theorem 1. LetC C(X) be aset of lokonstraints andlet U lu(fx:=
djx 2 X andd 2 Q +
g[fx := yjx;y 2 Xg). Let A be in Aut
"
(C;U). Then
thereexistsB inAut
"
(C(X);U
0
(X))suhthatA
s B.
Thenexttheoremislosetothepreviousone.Noteneverthelessthatthistheo-
rembeomesfalseifweonsiderarbitrarylokonstraints,sineaswerealled
insetion2,theorrespondinglassisundeidable.
Theorem 2. Let C C
df
(X) be a set of diagonal-free lok onstraints. Let
U lu(fx := djx 2 X andd 2 Q +
g[fx:= y+djx;y 2X andd 2Q +
g).
Let A bein Aut
"
(C;U). Then there exists B in Aut
"
(C
df (X);U
0
(X)) suh that
A
s B.
5 Expressive Power of Non Deterministi Updates
In the ase of non deterministi updates, we rst show that it is hopeless to
obtainstrong bisimulationwith lassialtimedautomata.To this purpose, let
us onsider theautomaton C of Figure 1.It hasbeenproved in [7℄that there
is nolassialtimedautomatonwithout" transitionsthat reognizethesame
languagethanC.
Now, itisnotdiultto provethat theautomatonC reognizesthesame lan-
guagethantheautomatonBandthatBreognizesitselfthesamelanguagethan
A.IfAwasstronglybisimilartosomeautomatonDofAut
"
(C(X);U
0
(X)),this
automatonDwouldnotontainany" transition(sineAdoesnotontainsuh
transition). HeneL(D) would beequalto L(A)=L(C), inontradition with
theresultof[7℄realledabove.SineAbelongstothelassAut
"
(C(X);U
1 (X))
(whereU
1
(X)denotesthesetofupdatesorrespondingtotheellslabelledde-
idable inthe generallok onstraints olumn intabular of Setion 2), we
A
x=1^x=y 1;
a;x:=0
y=1^y=x 1;
a;y:=0
1<y<0;b;x:<0 0<x<1^x=y 1;b;y:<0
0<y<1^y=x 1;b;x:<0 x=1;
a;
x:=0
y=1;
a;
y:=0
B C
x=1;
a;
x:=0
0<x<1;
b;
x:=x 1
x=1;
a;
x:=0
0<x<1;b
x=1;";x:=0
Fig.1.TimedautomataA,B andC
Proposition 1. Aut
"
(C(X);U
1 (X))6
s Aut
"
(C(X);U
0 (X))
We now fous on weak bisimilarity. As it will appear, the onstrution of
an automaton of Aut(C(X);U
0
(X)) weakly bisimilar to a given automaton of
Aut(C(X);U
1
(X))is rathertehnial.AswerealledinSetion2,thedeidable
lassesofupdatabletimedautomatadependonthesetoflokonstraintsthat
areused.Weonsiderrsttheaseofdiagonal-freelokonstraints.
We rst propose a normal form for diagonal-free updatable automata. Let
(
x )
x2X
beafamilyofonstantsofN.Inwhatfollowswewillrestritourselves
tothelokonstraintsxwhere
x
.Wedene:I
x
=f℄d;d+1[j0d<
x
g[f[d℄j0d
x g[f℄
x
;1[g
A lokonstraint'issaidto betotalif'isa onjuntion V
x2X I
x
wherefor
eahlokx,I
x
isanelementofI
x
.Anydiagonalfreelokonstraintbounded
bytheonstants(
x )
x2X
isequivalenttoadisjuntionoftotallokonstraints.
WedeneI 0
x
=f℄d;d+1[j0d<
x g[f℄
x
;1[g.Anupdateup
x
iselementary
ifitisofoneofthetwofollowingforms:
- x:= orx2I 0
x withI
0
x 2I
0
x ,
- V
y2H
x:y+^x2I 0
x
with2f=;<;>g,I 0
x 2I
0
x
and8y2H,
x
y +.
Anelementaryupdate ((
V
y2H
x:y+)^x2I 0
x
)is ompatiblewith a total
guard V
x2X I
x
if, for any y 2 H, I
y
+ I 0
x
. By applying lassial rules
of propositionalalulus and splitting the transitions, weobtain the following
normalformfordiagonal-freeupdatable timedautomata.
Proposition 2. Any diagonal-free updatable timed automaton from
Aut
"
(C
df
(X);U(X)) is strongly bisimilar to a diagonal-free updatable timed
automaton from Aut
"
(C
df
(X);U(X))in whih forany transition (p;';a;up;q)
itholds:
'isatotalguard
up = V
x2X up
x
with for any x, up
x
is an elementary update ompatible
with'
