F rom Zen to Aum

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Computational Linguistics

F rom Zen to Aum

G ´erard Huet

Chalmers Univ ersit y , Ma y 21st, 2004

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The Zen to olkit - Generic tec hnology

Afewspecificapplicativetechniques:

•Localprocessingoffocuseddata

•Sharing

•Lexicaltrees

•Differentialwords

•Finitetransducersaslexiconmorphisms

•Searchbyresumptioncoroutines

•Multisetorderingconvergence

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Automata Mista - AuM

Werepresentfinite-stateautomatabyamixedstructure-adeterministicskeletondecoratedbynon-deterministictransitions.

Thefirstcomponentisaforestoflexicaltrees,usedascoveringtreesofthestatetransitionsgraph.Therestofthetransitionsisrepresentedasannotationsstatingthatonacertaininput(awordpossiblyempty,allowing-transitions),theautomatongoestoastatedesignatedbyavirtualaddress.Therearetwokindsofaddresses,localandglobal.Aglobaladdressisgivenbyaninteger(indexingintotheforestarray)andaword.Alocaladdresshasthesamestructure,butnowactsasadifferentialword.Itsfirstcomponentindexesintoanarrayrepresentingtheaccesspathinthecurrenttree(necessarybecauseofsharing).

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Differen tial w ords

typedelta=(int*word);

Adifferentialwordisanotationpermittingtoretrieveawordwfromanotherwordw 0sharingacommonprefix.Itdenotestheminimalpathconnectingthewordsinatree,asasequenceofupsanddowns:ifd=(n,u)wegoupntimesandthendownalongwordu.

Wecomputethedifferencebetweenwandw 0asadifferentialword diffww 0=(|w1|,w2)wherew=p.w1andw 0=p.w2,withmaximalcommonprefixp.

Theconverseofdiff:word->word->deltais

patch:delta->word->word:w 0mayberetrievedfromwand d=diffww 0asw 0=patchdw.

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The automaton structure

typeinput=word;

typedelta=(int*word)andaddress=[Globalofdelta|Localofdelta];

typeauto=[Stateof(bool*deter*choices)]anddeter=list(letter*auto)andchoices=list(input*address);

typeautomaton=(arrayauto*delta);

typebacktrack=(input*delta*choices)andresumption=listbacktrack;(*coroutineresumptions*)

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Completeness

Everynon-deterministicautomaton(possiblywithtrasitions)mayberepresentedasaflataum(withemptydeterministicstructure).

EverydeterministicautomatonmayberepresentedasanaumwhosechoiceannotationsState(b,[],[([],address)])donotgiverisetobacktrack.

Everyaumhasaminimalrepresentation,obtainedbymaximalsharing.N.B.Sharingthelocalvirtualadressesdoesnotnecessarilycorrespondbyequivalencebybisimulation.

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The transducer structure

typeinput=wordandoutput=word;

typedelta=(int*word)andaddress=[Globalofdelta|Localofdelta];

typetrans=[Stateof(bool*deter*choices)]anddeter=list(letter*trans)andchoices=list(input*output*address);

typetransducer=(arraytrans*delta);

typebacktrack=(input*output*delta*choices)andresumption=listbacktrack;(*coroutineresumptions*)

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foret

pile a1

an a1

an k 1k AuM

dag courant mot dag

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Memorisation of the curren t access

Theaccessstack[sn;sn−1;...s0]isnecessary,tointerpretlocalvirtualaddresses.Itmaybeconvenienttostoreaswellthecurrentaccess wordword=[an;...a1],stackedandunstackedalongthelocalaccesses.Wemaythusdistinguishtwooutputconstructors:

AbsoluteofwordetRelativeofdelta.Inthelastcase,outputiscomputedbypatchappliedtoword.

Applications:

•Inflectedformsdictionaryusedaslemmatizer(regularplural:(δ=(1,[ 0s 0]))

•Unglue(δ=(0,[]))

•Segment(δ=(0,u))

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Mo dular aums

Anaumisgivenbyapairin(arrayauto*delta).

Wemakethemmodularbymakingtheglobaladdressesrelocatable,andpossiblyinterpretingsuccessstatesbycontinuations.Continuationsareimplementedas-transitions,i.e.extrachoices,withemptyinput.

Nowitiseasytocompileregularexpressionsintoaums,asfollows:

•Thebasecaseisanyaum,itssizethesizeofitsarray

•ifA=(arrayA,deltaA)isofsizeaandB=(arrayB,deltaB)isofsizeb,A·BisobtainedbyrelocatingBbya,continuingAbya+deltaB,startingatdeltaA,ofsizea+b.

•ifA=(arrayA,deltaA)isofsizeaandB=(arrayB,deltaB)isofsizeb,A+BisobtainedbyrelocatingBbya,startingata+b+1,whereweput

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State(False,[],[([],deltaA);([],a+deltaB)]),ofsizea+b+1.

•ifA=(arrayA,deltaA)isofsizea,thenA∗isobtainedbycontinuingAbydeltaA,makingitsstartingnodeaccepting,ofsizea.

Thesetransformationsoughttobeeffectedbeforesharing.

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Conclusion

Automatamistaofferanelegantapplicativesolutiontomanyfinite-stateprocessingproblems,typicallythetreatmentoflexiconrepresentation,phonology,morphologyandsegmentationincomputationallinguistics.Thedeterministicspanningtreeoftheirstatespaceisthennaturallythedictionaryofinflectedformsofwords,whichisthusplacedatthecenterofthecomputertreatmentoflanguage.

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