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We give an eetiveharaterizationofsequentialfuntionsoverinnitewords.We alsodesribeanalgorithmtodeterminizetransduersoverinnitewords

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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

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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

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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

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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

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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

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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

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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

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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

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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.

Referenes

1. Marie-PierreBeal. CodageSymbolique. Masson,1993.

2. Marie-PierreBeal,OlivierCarton, ChrhistophePrieur,andJaquesSakarovith.

Squaringtransduers. InLATIN'2000.

3. JeanBerstel. TransdutionsandContext-Free Languages. B.G.Teubner,1979.

4. ChristianChorut.Unearaterisationdesfontionssequentiellesetdesfontions

sous-sequentiellesentantquerelationsrationnelles.Theoret.Comput.Si.,5:325{

338, 1977.

5. Christian Chorut. Contribution a l'etude de quelques familles remarquables de

fontionsrationnelles. Thesed'

Etat,UniversiteParis VII,1978.

6. A.CohenandJ.-F.Collard.Instane-wisereahingdenitionanalysisforreursive

programsusingontext-freetransdutions. InPACT'98,1998.

7. ChristianeFrougny. Numerationsystems. InM.Lothaire,editor,AlgebraiCom-

(10)

22:135{140, 1986.

9. Brue Kithens. Continuity properties of fator maps in ergodi theory. Ph.D.

thesis,UniversityofNorthCarolina,ChapelHill,1981.

10. DougLindandBrianMarus.AnIntrodutiontoSymboliDynamisandCoding.

CambridgeUniversityPress,1995.

11. Mehryar Mohri. Onsomeappliationsofnite-stateautomatatheorytonatural

languagesproessing. JournalofNaturalLanguage Engineering,2:1{20,1996.

12. ChristophePrieur.Howtodeideontinuityofrationalfuntionsoninnitewords.

Theoret.Comput. Si.,1999.

13. EmmanuelRoheandYvesShabes. Finite-StateLanguageProessing,hapter7.

MITPress,Cambridge,1997.

14. ShmuelSafra. Ontheomplexityof!-automata. In29thAnnual Symposiumon

Foundations ofComputerSienes,pages24{29,1988.

15. Ludwig Staiger. Sequential mappingsof !-languages. RAIRO-Infor. Theor. et

Appl.,21(2):147{173,1987.

16. Wolfgang Thomas. Automata on innite objets. In J. van Leeuwen, editor,

Handbook of Theoretial Computer Siene,volumeB,hapter4,pages 133{191.

Elsevier,1990.

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