over innite words
Marie-PierreBeal 1
andOlivierCarton 2
1
InstitutGaspardMonge,
UniversitedeMarne-la-Vallee
Marie-Pierre.Bealuniv-mlv.f r
http://www-igm.univ-mlv.fr/~be al/
2
InstitutGaspardMongeandCNRS
UniversitedeMarne-la-Vallee
Olivier.Cartonuniv-mlv.fr
http://www-igm.univ-mlv.fr/~ar ton/
Abstrat. We study the determinization of transduers over innite
words. Weonsider transduers with all their states nal. We give an
eetiveharaterizationofsequentialfuntionsoverinnitewords.We
alsodesribeanalgorithmtodeterminizetransduersoverinnitewords.
1 Introdution
Theaimofthispaperisthestudyofdeterminizationoftransduersoverinnite
words, that is of mahines realizing rational transdutions.A transduer is a
nite state automaton (or a nite state mahine) whose edges are labeled by
pairsofwordstakeninnitealphabets.Therstomponentofeahpairisalled
theinputlabel.Theseond onetheoutput label.Therationalrelationdened
by a transduer is the set of pairs of words whih are labels of an aepting
pathinthetransduer.Weassumethattherelationsdenedbyourtransduers
are funtions whih eah string of the domain to a string.This is adeidable
property[8℄.
The study of transduers has many appliations. Transduers are used to
modelodingshemes(ompressionshemes,onvolutionalodingshemes,od-
ing shemes for onstrained hannels, for instane). They are widely used in
omputer arithmeti [7℄ and in natural languageproessing [13℄. Transduers
are also used in programs analysis [6℄. The determinization of a transduer is
theonstrutionofanothertransduerwhihdenesthesamefuntion andhas
adeterministi (orrightresolving)input automaton.Suh transduersallowa
sequentialenodingandthus arealledsequentialtransduers.
Theharaterizationofsequentialfuntionsonnitewordswasobtainedby
Chorut[4,5℄.Hisproofontainsimpliitlyanalgorithmfordeterminizationof
atransduer. Thisalgorithm hasalsobeendesribed by Mohri[11℄and Rohe
and Shabes [13, p. 223{233℄. In this paper, we address the same problem for
statesarenalisthattheaseoftransduerswithnalstatesseemstobemuh
more omplex. Indeed, the determinization of automata over innite words is
already verydiÆult [14℄.In partiular, it is nottrue that any rational set of
innitewordsis reognized by adeterministiautomatonwithnal statesand
Buhiaeptaneondition.Otheraeptingonditions,astheMullerondition
forinstane,mustbeused.
Werstgiveaneetiveharaterizationofsequentialfuntionsoverinnite
words. This haraterization extends to innite words the twinning property
introdued by Chorut [4℄. We prove that a funtion is sequential if it is a
ontinuous mapwhose domain anbe reognizedby adeterministiBuhi au-
tomaton, and suh that the transduer obtained after removing some speial
stateshasthetwinningproperty.Theseonditionsanbesimpliedin thease
where thetransduer hasnoyling pathwith anemptyoutput label. Weuse
this haraterization to desribean algorithm hekingwhether afuntion re-
alized by atransduer issequential.This algorithm beomes polynomialwhen
thetransduerhasnoylingpathwithanemptyoutputlabel.Finally,wegive
analgorithmtodeterminizeatransduer.Thealgorithmismuhmoreomplex
thanin theaseofnitewords.
Thepaper is organizedasfollows. Setion 2 is devoted to basinotions of
transduers andrationalfuntions. Wegivein Set. 3aharaterization ofse-
quentialfuntionswhile thealgorithmfordeterminizationof transduersisde-
sribedin Set.4.
2 Transduers
In the sequel, A and B denote nite alphabets. The set of nite and right-
innitewordsoverA arerespetivelydenoted byA
and A
!
. Theemptyword
is denoted by ". The set A
!
is endowed with the usual topology indued by
thefollowingmetri:thedistane d(x;y) isequal2 n
where nistheminimum
minfijx
i 6=y
i
g.Inthispaper,afuntionfromA
!
toB
!
issaidtobeontinuous
iitisontinuouswithrespettothistopology.
Atransduer overAB is omposed of aniteset Qof states,a setE
QA
B
Qofedges andasetI Qofinitial states.Anedgee=(p;u;v;q)
fromptoqisdenoted byp ujv
!q.Thewordsuandv arealledtheinputlabel
and the output label. Thus, a transduer is the same objet asan automaton,
exept that thelabels ofthe edgesare pairsof wordsinstead ofletters. Inthe
literature, transduers also have a set of nal states. In this paper, we only
onsider transduers all of whih states are nal and with Buhi aeptane
ondition. Any innitepath whih starts at an initial state is then suessful.
Weomitthesetofnalstatesinthenotation.
Aninnitepath inthetransduerAisaninnitesequene
q u0jv0
!q u1jv1
!q u2jv2
!q
0 1 2
outputlabelisthewordy=v
0 v
1 v
2
:::.Thepathissaidtostart at q
0 .
Aninnitepathisthensuessful ifitstartsataninitialstate.Apair(x;y)
of innitewords is reognized by the transduer if it labels a suessful path.
A transduer denes then a relationR A
!
B
!
. Thetransduer omputes
a funtion if for any word x 2 A
!
, there exists at most one word y 2 B
!
suh that (x;y) 2R . We allit the funtion realized by the transduer. Thus
atransduer an beseenas amahine omputing nondeterministiallyoutput
wordsfrominputwords.Wedenotebydom(f)thedomainofthefuntionf.A
transduer that realizesafuntion anbetransformedin aneetiveway ina
transduerlabelledinAB
thatrealizesthesamefuntion.Thesetransduers
aresometimes alledrealtimetransduers.
LetT beatransduer.Theunderlyinginputautomaton ofT isobtainedby
omittingtheoutputlabelofeah edge.A transduerT issaidto be sequential
ifitislabeledinAB
andifthefollowingonditionsaresatised.
{ ithasauniqueinitialstate,
{ theunderlying inputautomatonisdeterministi.
These onditionsensurethat foreahwordx2A
!
, there isat mostone word
y2B
!
suhthat(x;y)isreognizedbyT.Thus,therelationomputedbyT is
apartialfuntionfromA
!
intoB
!
.Afuntionissequential ifitanberealized
byasequentialtransduer.
0 1 2
0j0
1j0
1j1
0j0
0j1
1j1
Fig.1.TransduerofExample1
Example 1. Let A = f0;1g be the binary alphabet. Consider the sequential
transduerT pituredinFig.1.Iftheinnitewordxisthebinaryexpansionof
arealnumber2[0;1),theoutputorrespondingtoxinT isthebinaryexpan-
sionof =3.Thetransduer T realizesthe divisionby 3onbinaryexpansions.
Thetransduerobtainedbyexhangingtheinputandoutputlabelsofeahedge
realizesof ourse themultipliation by 3.However,this new transdueris not
Inthissetion,weharaterizefuntionsrealizedbytransduerswithallstates
nal that anbe realizedby sequentialtransduers. Thisharaterizationuses
topologialpropertiesof thefuntion andsometwinning propertyofthetrans-
duer.ItextendstheresultofChorut[4,5℄to innitewords.
Theharaterizationofthesequentialityisessentiallybasedonthefollowing
notion introdued by Chorut [5, p. 133℄ (see also [3, p. 128℄). Two states q
andq 0
ofatransdueraresaidtobetwinned ifandonlyifforanypairofpaths
i uju
0
!q vjv
0
!q
i 0
uju 00
!q 0
vjv 00
!q 0
;
where i and i 0
are two initial states, the output labels satisfy the following
property. Either v 0
= v 00
= " or there exists anite wordw suh that either
u 00
= u 0
w and wv 0 0
= v 0
w, or u 0
= u 00
w and wv 0
= v 0 0
w. The latter ase is
equivalenttothefollowingtwoonditions:
(i) jv 0
j=jv 00
j,
(ii) u 0
v 0
!
=u 00
v 0 0
!
Atransduerhasthetwinningproperty ifanytwostatesaretwinned.
Beforestatingthemainresult,wedeneasubsetofstateswhihplayapar-
tiular role in the sequel. We say that a stateq of atransduer is onstant if
allinnitepathsstartingatthisstatehavethesameoutput label.Thisunique
outputisanultimatelyperiodiword.Itshouldbenotiedthatanystateaes-
sible from aonstantstateisalso onstant.Wenowstatethe haraterization
ofsequentialfuntions.
Proposition1. Letf be afuntion realizedby atransduer T. Let T 0
be the
transduerobtainedbyremovingfromT allstateswhih areonstant.Thenthe
funtion f issequential if andonlyif thefollowing threepropertieshold:
{ thedomain off anbe reognizedby adeterministiB uhi automaton,
{ thefuntion f isontinuous,
{ thetransduerT 0
hasthe twinning property.
Sinethefuntionf isrealizedbyatransduer,thedomainoff isrational.
However,itisnottruethatanyrationalsetofinnitewordsisreognizedbya
deterministiBuhiautomaton.Landweber'stheoremstatesthatasetofinnite
words is reognized by a deterministi Buhi automaton if and only if it is
rationalandG
Æ
[16℄.ReallthatasetissaidtobeG
Æ
isitisequaltoaountable
unionofopensetsfortheusualtopologyofA
!
.
Itisworthpointingoutthatthedomainofafuntionrealizedbyatransduer
maybeanyrationalsetalthoughitissupposedthatallstatesofthetransduer
arenal.ThenalstatesofaBuhiautomatonanbeenodedintheoutputsof
transition p a
!q ofAbeomesp ajv
!q in T where v is emptyifpis notnal
andisequaltoaxedletterbifpisnal.Itislearthattheoutputofapathis
inniteifandonlyifthepathgoesinnitelyoftenthroughanalstate.Thusthe
domainofthetransduerT isthesetreognizedbyA.Forinstane,thedomain
of a transduer may be not reognizable by a deterministi Buhi automaton
asin thefollowingexample.Itis howevertruethatthedomain is losedifthe
transduerhasnoylingpathwithanemptyoutput.
0 1
aj"
bj"
bjb
bjb
Fig.2.TransduerofExample2
Example 2. Thedomainofthefuntionf realizedbythetransduerofFig.2is
theset (a+b)
b
!
of wordshavinganite numberof a.Thefuntion f annot
berealizedbyasequentialtransduersineitsdomainisnotaG
Æ set.
Itmustbealsopointedoutthatafuntionrealizedbyatransduermaybe
notontinuousalthoughitissupposedthatallstatesofthetransduerarenal
asitisshowninthefollowingexample.
Example 3. The image of aninnite word x bythe funtion f realized bythe
transduerofFig.3isf(x)=a
!
ifxhasaninnitenumberofaandf(x)=a n
b
!
ifthenumberofainx isn.Thefuntion f isnotontinuous.Forinstane,the
sequene x
n
= b n
ab
!
onverges to b
!
while f(x
n ) = ab
!
does not onverge
tof(b
!
)=b
!
.
Beforedesribingthealgorithmfordeterminization, werststudyaparti-
ularase.Itturnsoutthat thersttwoonditionsofthepropositionaredueto
thefat that thetransduer T mayhaveyling pathswith anempty output.
If the transduer T has no yling path with an empty output, the previous
propositionanbestatedin thefollowingway.
Proposition2. Let f be a funtion realized by a transduer T whih has no
0
0 1
bj"
bjb
Fig.3.TransduerofExample3
from T all stateswhih are onstant.Then the funtion f is sequential if and
only ifthe transduerT 0
has the twinningproperty.
Thepreviouspropositionanbediretlydeduedfrom Proposition1asfol-
lows.IfthetransduerT hasnoylingpathwithanemptyoutput,anyinnite
pathhasaninniteoutput.Thus,aninnitewordxbelongstothedomainoff
if and only if it is the input label of an innite path in T. The domain of f
is then alosed set.It isthen reognized by adeterministi Buhi automaton
whose allstatesare nal.This automatonanbeobtainedbytheusualsubset
onstrutionontheinputautomatonofT.Furthermore,ifthetransduerT has
noylingpathwithanemptyoutput,thefuntionf isneessarilyontinuous.
WenowstudythedeidabilityoftheonditionsofPropositions1and2.We
havethefollowingresults.
Proposition3. It is deidable if a funtion f given by a transduer with all
statesnalissequential.Furthermore,ifthetransduerhasnoylingpathwith
an emptyoutput,this anbedeided inpolynomialtime.
ABuhiautomatonreognizingthedomainofthefuntionanbeeasilydedued
from the transduer. It is then deidable if this set an be reognized by a
deterministiBuhi automaton[16,thm 5.3℄.However,thisdeisionproblemis
NP-omplete.
Itisdeidableinpolynomialtimewhetherafuntiongivenbyatransduer
with nal states is ontinuous [12℄. The twinning property of a transduer is
deidableinpolynomialtime[2℄.
4 Determinization of Transduers
Inthissetion,wedesribeanalgorithmtodeterminizeatransduerwhihsat-
isesthepropertiesofProposition1.Thisalgorithmprovesthat theonditions
ofthepropositionaresuÆient.Thealgorithmisexponentialinthenumberof
LetT =(Q;E;I)beatransduerlabelledinA B thatrealizesafuntionf.
Let T 0
be the transduer obtained by removing from T all states whih are
onstant. Weassume that T 0
has the twinning property. Wedenote byC the
setofstateswhihareonstant.ForastateqofC,wedenotebyy
q
,theunique
outputof qwhihisan ultimatelyperiodiword.Wesupposethat thedomain
of f isreognized bythedeterministiBuhi automatonA.This automatonis
used in the onstruted transduer to ensure that the output is innite only
whentheinputbelongsto thedomain ofthefuntion.
WedesribethedeterministitransduerDrealizingthefuntionf.Roughly
speaking,thistransdueristhesynhronizedprodutoftheautomatonAofthe
domain and ofan automatonobtained by avariant ofthe subsetonstrution
appliedonthetransduer.Intheusualsubsetontrution,astateofthedeter-
ministi automaton is asubsetof stateswhih memorizes allaessiblestates.
Inourvariantof thesubsetonstrution,astateis asubset ofpairsformed of
astateandawordwhihis eitherniteofinnite.
AstateofDisapair(p;P)wherepisastateofAandP isasetontaining
twokindsofpairs.Therstkindarepairs(q;z)whereqbelongtoQnCandzis
anite wordoverB.Theseondkindarepairs(q;z)where qbelongstoC and
zisanultimatelyperiodiinnitewordoverB.Wenowdesribethetransitions
ofD.Let(p;P)beastateofDandletabealetter.LetRbeequalto theset
dened asfollows
R=f(q 0
;zw)jq 0
=
2C and9(q;z)2P; q2=C andq ajw
!q 0
2Eg
[f(q 0
;zwy
q 0
)jq 0
2C and9(q;z)2P; q2=Candq ajw
!q 0
2Eg
[f(q 0
;z)jq 0
2C and9(q;z)2P; q2C andq ajw
!q 0
2Eg
We now dene the transition from the state (p;P) input labeled by a. If R
is empty, there isno transition from (p;P)input labeled by a.Otherwise, the
output of this transition is the word v dened as follows. Let p a
! p 0
be the
transitioninAfromplabeledbya.Ifp 0
isnotanalstateofA,wedenev as
theemptyword.Ifp 0
isanalstate,wedenevastherstletterofthewords
z ifR only ontainspairs(q 0
;z)withq 0
2C and ifalltheinnitewordsz are
equal.Otherwise, wedene v as thelongestommon prex of allthe nite or
innitewordszfor(q 0
;z)2R .ThestateP 0
isthendened asfollows
P 0
=f(q 0
;z)j(q 0
;vz)2R g
There is then a transition (p;P) ajv
! (p 0
;P 0
) in D. The initial state of D is
the pair (i
A
;J)where i
A
is theinitial stateof A and where J = f(i;") j i 2
I andi2= Cg[f(i;y
i
)ji 2I andi2 Cg. Ifthe statep 0
is notnal in A, the
outputof thetransition from (p;P)to (p 0
;P 0
)isemptyandthewordsz ofthe
pairs(q;z)in P,mayhaveanonemptyommonprex.Weonlykeepin Dthe
aessiblepartfromtheinitialstate.ThetransduerDhasadeterministiinput
niteandthatitisequivalenttothetransduerT.Bothtransduersrealizethe
samefuntionoverinnitewords.
Proposition4. ThesequentialtransduerD hasa nitenumber ofstatesand
itrealizesthe samefuntion f asthe transduerT.
It isnotstraightforwardthat thetransduer Dhasatuallyanite numberof
states.Itmustbeprovedthatthenitewordswhihourasseondomponent
of the pairs in the states are bounded. It follows then that the innite words
ouring as seond omponent of the pairs are suÆxes of a nite number of
ultimatelyperiodiwords.Therefore,there arenitely manysuhwords.
ItmustalsobeprovedthatthetransduerDrealizesthesamefuntionasT.
This followsmainlyfrom thefollowinglemma whih statesthekeypropertyof
theedgesinD.
Lemma1. Let u be a nite word. Let (i
A
;J) ujv
! (p;P) be the unique path
in D with input label u from the initial state. Then, the state p is the unique
stateof Asuhthat i
A u
!pisapath inA andthe set P isequalto
P =f(q;z)j9i ujv
0
!q inT suhthat v 0
=vz if q2=C
v 0
y
q
=vz if q2Cg
Thisonstrutionisillustratedbythefollowingexample.
0 1
aja
bjb
aj"
aj"
jaa
Fig.4.TransduerofExample4
Example 4. Consider the transduer pitured in Fig. 4.A deterministiBuhi
automaton reognizing the domain is pitured in Fig. 5. If the algorithm for
determinization is applied to thistransduer, onegets thetransduer pitured
in Fig.6.
Thesedeterminizationsdonotpreservethedynamipropertiesofthetrans-
duersastheloalityofitsoutputautomaton.Reallthataniteautomatonis
loal ifanytwobiinnitepathswiththesamelabelareequal.Wementionthat
A B C D a
b
a
Fig.5.AdeterministiBuhiautomatonforthedomain
A
0;"
B
0;"
1;a
!
C
1;a
!
D
1;a
!
aja
aja
aj"
aj"
bjb
bjb
ja
ja ja
Fig.6.DeterminizationofthetransduerofFig.4
thathavearightlosinginput(orthataren-deterministiordeterministiwith
a nite delay in the input) and a loal output (see also [10, p. 143℄ and [1,
p.110{115℄).Thisalgorithmpreservestheloalityoftheoutput.Thesefeatures
areimportantforodingappliations.
Aknowledgments
The authorswould liketo thankJean Berstelfor veryhelpful suggestionsand
ChristianChorutandIsabelleFagnotfortheirrelevantommentsonaprelim-
inaryversionofthispaper.
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