| { |
| "paper_id": "W95-0204", |
| "header": { |
| "generated_with": "S2ORC 1.0.0", |
| "date_generated": "2023-01-19T04:36:58.484153Z" |
| }, |
| "title": "Locally Tree-shaped Sentence Autom ata and Resolution of Am biguity", |
| "authors": [ |
| { |
| "first": "Jussi", |
| "middle": [], |
| "last": "Piitulainen", |
| "suffix": "", |
| "affiliation": { |
| "laboratory": "", |
| "institution": "Research Unit for Multilingual Language Technology Department o f General Linguistics University o f Helsinki", |
| "location": {} |
| }, |
| "email": "jpiitula@ling.helsinki.fi" |
| } |
| ], |
| "year": "", |
| "venue": null, |
| "identifiers": {}, |
| "abstract": "T h e fr a m e w o r k o f fin ite s ta te in te rs e ctio n g r a m m a r s is d ir e c te d to w a rd s p r a c tic a l r e p re s e n ta tio n a n d p a rs in g o f r u n n in g te x t. A p r o b le m in pa rs in g h a s b e e n th a t in te r s e c tio n w ith su ccessiv e ru le a u t o m a t a ca n p r o d u c e p r o h ib it iv e ly la rg e in te r m e d ia te sen te n ce a u to m a t a. See (K o s k e n n ie m i,", |
| "pdf_parse": { |
| "paper_id": "W95-0204", |
| "_pdf_hash": "", |
| "abstract": [ |
| { |
| "text": "T h e fr a m e w o r k o f fin ite s ta te in te rs e ctio n g r a m m a r s is d ir e c te d to w a rd s p r a c tic a l r e p re s e n ta tio n a n d p a rs in g o f r u n n in g te x t. A p r o b le m in pa rs in g h a s b e e n th a t in te r s e c tio n w ith su ccessiv e ru le a u t o m a t a ca n p r o d u c e p r o h ib it iv e ly la rg e in te r m e d ia te sen te n ce a u to m a t a. See (K o s k e n n ie m i,", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Abstract", |
| "sec_num": null |
| } |
| ], |
| "body_text": [ |
| { |
| "text": "T a p a n a in e n a n d V o u tila in e n 1 9 9 2 ) a n d (K o s k e n n ie m i 19 90 ) a n d (V o u tila in e n a n d T a p a n a in e n 1 9 9 3 ).", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "", |
| "sec_num": null |
| }, |
| { |
| "text": "T h is p a p e r sk etch es a d a ta s tru ctu re a n d p a rs in g m e t T h e r e are fiv e a lte r n a tiv e w o rd b o u n d a r y la b e ls ; QQ b e g in s a n d e n d s a sen ten ce, e m b e d d e d fin ite cla u ses are e n clo s e d b e tw e e n \u00ae< a n d # > , a b o u n d a r y w h ere a fin ite cla u se e n d s a n d a n ew o n e s ta rts is la b e le d w ith \u00ae /, a n d \u00ae is u sed fo r oth e rs. ", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "", |
| "sec_num": null |
| }, |
| { |
| "text": "0-0- NON SG H^O --O -O v 8SUBJ INF VFIN 8MV PCP2^X )\\ \u00ae S C <passed>Dass 88 _ V t > -^O M V F", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "", |
| "sec_num": null |
| }, |
| { |
| "text": "In th e n ew rep re s e n ta tio n , se n ten ce a ", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Initialization and propagation", |
| "sec_num": "4.1" |
| }, |
| { |
| "text": "In th e n ew m e t h o d , th e se n ten ce a u t o m a t o n c a n n o t g ro w w h ile re a d in g s a re re ", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Expanding the sentence automaton locally", |
| "sec_num": "4.3" |
| }, |
| { |
| "text": "W h e n th e se n te n ce a u t o m a t o n is e x p a n d e d b e tw e e n S i a n d 5 i + 2i a ", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Expansion", |
| "sec_num": "6.3" |
| } |
| ], |
| "back_matter": [ |
| { |
| "text": "S p e c ia l th a n k s t o A n s s i Y li-J y r \u00e4 , K a ri G r a n \u00f6 , A t r o V o u tila in e n a n d K im m o K o sk e nn ie m i, a n d th a n k s t o T i m o L a h tin e n , T i m o J \u00e4 rv in e n , M ik k o L o u n e la an d th e tw o referees, fo r r e a d in g a n d c o m m e n ts . A t r o V o u tila in e n a n d P a si T a p a n a in e n . ", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "annex", |
| "sec_num": null |
| }, |
| { |
| "text": "Proceedings of NODALIDA 1995", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "58", |
| "sec_num": null |
| } |
| ], |
| "bib_entries": {}, |
| "ref_entries": { |
| "FIGREF0": { |
| "type_str": "figure", |
| "text": "h o d th a t m a y h e lp t o c o m p u t e th e in te r s e c tio n o f a se n te n ce a u t o m a t o n w ith all rule a u t o m a t a w h ile k e e p in g th e size o f in te r m e d ia te se n te n ce a u t o m a t a under c o n t r o l. 1 Representation of sentence readings S e n te n ce r e a d in g s are fin ite se q u en ce s o f w o rd fo r m s , m o r p h o s y n t a c t ic la b e ls an d b o u n d a r y la b e ls (V o u tila in e n 1994, V o u tila in e n a n d T a p a n a in e n 1 9 9 3 ). T h e s tru c tu re o f a se n te n ce re a d in g is su ch th a t each w o rd fo r m re a d in g co n ta in s th e w o rd fo r m , a b a se fo r m , a few m o r p h o lo g ic a l la b e ls a n d o n e o r tw o s y n ta c tic la b els , an d w o rd fo r m r e a d in g s are s e p a ra te d b y b o u n d a r y la b e ls . F or e x a m p le , th e co rr e ct r e a d in g o f th e se n te n ce ' A n d tim e b e g a n seriou sly t o p a s s .' in fig u re 1 id en tifies ' t i m e ' as a s u b je c t CSUBJ, a n d ' b e g a n ' as a m a in v e rb in a m a in cla u se 8NV MAIHC8 , a n d s o o n . O th e r r e a d in g s m ig h t m is id e n tify ' t i m e ' as a v e rb o r ' t o ' as a p r e p o s itio n , o r c o r r e c t ly id e n tify ' t i m e ' as a n o u n b u t m is id e n tify it s y n ta c tic a lly as an o b je c t o r an a p p o s it io n . A n in c o r r e c t c h o ic e fo r a w o r d b o u n d a r y la b e l w o u ld in d ic a t e th e p re se n ce o f m o r e th a n o n e fin ite v e rb fo r m .", |
| "uris": null, |
| "num": null |
| }, |
| "FIGREF1": { |
| "type_str": "figure", |
| "text": "fin ite set o f re a d in g s is re p res e n te d as an a c y c lic fin ite s ta te a u to m a t o n c a lle d a se n te n ce a u t o m a t o n . W o r d fo r m s in th e se n te n ce h a ve a ll le x ic a lly p o s s ib le m o r p h o s y n t a c t ic re a d in g s as a lte rn a tiv e s, a n d ea ch w o rd b o u n d a r y is m a d e fo u r o r s o m e tim e s five w ays a m b ig u o u s . T h e in itia l n u m b e r o f re a d in g s in a se n ten ce a u t o m a t o n is th e p r o d u c t o f th e le x ic a l a n d w o rd b o u n d a r y a m b ig u itie s in th e sen te n ce . ig u re 1 A sen ten ce re a d in g T h e g ra m m a r is rep resen ted as a n u m b e r o f fin ite s ta te a u t o m a t a c a lle d ru le a u to m a t a . E a ch ru le a u t o m a t o n is c o n s tr u c te d t o a c c e p t all r e a d in g s th a t are t o b e g ra m m a tic a l an d t o re je ct s o m e re a d in g s th a t are t o b e u n g r a m m a tic a l. In o th e r w o rd s , th e g ra m m a r is ta k en t o b e th e in te r s e c tio n o f in d iv id u a l rules. S en te n ce a u t o m a t a w ill b e d ra w n w ith th e th e s ta r t s ta te o n th e le ft, th e d ir e c tio n o f e d g es fr o m le ft to rig h t, a n d th e fin a l s ta te as a d o u b le circ le o n th e righ t.2 The problem T h e ta sk is to c o m p u te th e in te rse ctio n 5 n i i o n -\u2022 n i 2 \" _ i o f a se n te n ce a u t o m a t o n S' w ith all ru le a u t o m a t a iio i \u2022 \u2022 \u2022, f i n -i -T h e m a in p r a c tic a l p r o b le m is th a t s o m e tim e s in te rse ctio n w ith su ccessiv e ru le a u t o m a t a p r o d u c e s p r o h ib it iv e ly b ig in te r m e d ia te sen te n ce a u to m a t a . A n y m e t h o d th a t c o m p u te s th e s a m e fin a l resu lt ca n b e u sed in ste a d o f th e s tra ig h tfo rw a r d in te r s e c tio n ; th is p a p e r p r o p o s e s o n e su ch m e t h o d . T h e n u m b e r o f rea d in g s rep resen te d b y th e se n te n ce a u t o m a t o n d ecrea ses m o n o t o n ic a lly d u rin g th e p a rs in g p ro ce s s. T h e n u m b e r o f s ta te s in th e se n te n ce a u to m a t o n ten d s t o increcise at first; th e in itia l a u t o m a t o n ca n b e v ery c o m p a c t p re cise ly b e ca u s e it co n ta in s all a lte rn a tiv e s , a n d a sm a lle r se t m a y b e re p re s e n te d b y a b ig g e r a u to m a t o n . T h e r e m o v a l o f th e d a sh ed e d g e s in fig u re 2 lea ves tw o m in im a l a u t o m a t a o f w h ich th e b ig g e r rep resen ts th e sm a lle r set o f re a d in g s. T h is s h o w s h o w r e m o v a l o f re a d in g s ca n req u ire a d d itio n o f s ta te s a n d edges. T h e tw o a u t o m a t a in fig u re 2 rep resen t a set o f fo u r re a d in g s. T h e u p p e r a u to m a t o n is m in im a l an d th e low er a u t o m a t o n is (a lm o s t ) m a x im a l. C o n s id e r n o w a ru le th a t reje cts th e se q u e n ce ( x , a , y , a , z ) a n d a c c e p ts th e o th e r th re e. N o e d g e can b e r e m o v e d fr o m th e m in im a l a u t o m a t o n s in ce ev e ry e d g e b e lo n g s t o o n e o f ( x , a , y , b , z ) an d { x , b , y , a , z ) a n d ( x , b , y , b , z ) . In th e m a x im a l a u t o m a t o n , th e d a sh ed ed g es an d th e s ta te b etw e e n th e m b e lo n g o n ly t o th e r e je c te d s e q u e n ce an d ca n b e re m o v e d . T h e fin al se n ten ce a u t o m a t o n 5 PI ii o 0 \u25a0 \u2022 \u2022 PI i i n _ i rep resen ts th e set o f c o rr e ct re a d in g s fo r th e sen ten ce . I f th e g r a m m a r is a c c u r a te , th is a u t o m a t o n is a g a in q u ite sm a ll b e ca u se it rep resen ts a v ery s m a ll set o f re a d in g s. ig u re 2 A m in im a l a u t o m a t o n a n d a m a x im a l a u to m a t o n 3 New representation of sentence automata T h is s e c tio n in tr o d u c e s a n ew r e p re s e n ta tio n fo r se n te n ce a u to m a t a . T h e n ew d a ta stru c tu re is d e s ig n e d t o fa c ilita t e s e p a r a tio n o f tw o o p e r a t io n s th a t in te rse ctio n w o u ld c o m b in e , n a m e ly r e m o v a l o f re a d in g s a n d e x p a n s io n o f th e sen te n ce a u to m a to n . T h is s e p a r a tio n s h o u ld h elp keep th e size o f in te r m e d ia te a u t o m a t a u n d er c o n t r o l. E d g e s a n d s ta te s ca n b e r e m o v e d i f th e y are n o t p a rts o f an y re a d in g th a t s h o u ld re m a in . In a tre e -s h a p e d a u t o m a t o n , a ll r e m o v a b le re a d in g s ca n b e re m o v e d b y r e m o v in g su ch ed g es a n d sta te s. S in ce it is s t r ic t ly im p r a c t ic a l t o m a k e th e w h o le se n te n ce a u t o m a t o n tr e e -s h a p e d , th e a u t o m a t o n is m a d e tr e e -s h a p e d o n ly lo ca lly . In itia lly , th e lo c a l trees sp a n r e a d in g s o f w o rd fo r m s , b u t th e y ca n b e e x p a n d e d to sp a n lo n g e r in terva ls. T h e r e m u st b e s ta te s w ith m o r e th a n o n e im m e d ia te p re d e ce s so r. T h e s e are im m e d ia t e su cce s so rs o f th e lea ves o f lo c a l trees. S in ce th e trees in itia lly sp a n w ord fo r m s , th e w o rd b o u n d a r ie s c o r r e s p o n d t o c o lle c tio n s o f th e s ta te s th a t m a y h ave se v e ra l im m e d ia t e p re d e ce s so rs. T h e s e c o lle c tio n s w ill b e c a lle d slices. E a ch slice o f a sen te n ce a u t o m a t o n c o n ta in s e x a c tly o n e s ta te o f e v ery p a th b e tw e e n th e s ta r t s ta te a n d th e fin a l s ta te . T h e first s lice c o n ta in s th e s ta te th a t im m e d ia t e ly fo llo w s th e s ta r t s ta te , a n d th e la st s lice c o n ta in s th e fin a l s ta te . In i tia lly slice s c o n ta in o n ly o n e sta te . T h e tw o a u t o m a t a in fig u re 3 rep resen t a se t o f a few re a d in g s o f th e sen te n ce ' T im e p a s s e d ' . P u n c tu a t io n a n d s o m e la b e ls , n o ta b ly MAINC8, are o m it t e d t o save s p a c e . T h e u p p e r a u t o m a t o n is lo c a lly tr e e -s h a p e d ; th e low er a u t o m a t o n is an o r d in a r y m in im a l a u to m a t o n . T h e re cta n g le s in th e u p p e r a u t o m a t o n rep resen t th e th ree slices c o rr e s p o n d in g t o th e th ree w o rd b o u n d a r ie s in th e sen ten ce. E a ch s ta te in a s lice is a r o o t o f a lo c a l tree. T h e lea ves o f th e tree are im m e d ia te p re d e ce s so rs o f s o m e r o o t s in th e n e x t slice. N o te th a t a s ta te th a t is p r o p e rly b e tw e e n slice s h a s o n ly o n e im m e d ia te p re d e ce s so r, s o th a t th e trees d o n o t sh are s tru ctu re . T h e d a ta stru c tu re rep resen tin g a se n te n ce a u t o m a t o n s h o u ld g iv e ea sy a ccess to th e in fo r m a t io n n e e d e d in th e a lg o r ith m s t o b e d e s crib e d , in a d d itio n to th e u n d e r ly in g a u t o m a t o n s tru ctu re . F irst, th e c o n t a in in g slice, if any, s h o u ld b e a v a il a b le g iv e n a s ta te . S e c o n d , th e p r e v io u s s ta te o r s ta tes s h o u ld b e a v a ila b le g iv e n a 52 Proceedings of NODALIDA 1995", |
| "uris": null, |
| "num": null |
| }, |
| "FIGREF2": { |
| "type_str": "figure", |
| "text": "ig u re 3 L o c a lly tr e e -s h a p e d vs. m in im a l sta te . T h ir d , th e sta tes in slices a n d im m e d ia te p re d e ce s so rs o f s ta te s in slice s w ill b e a ssign ed sets o f ru le s ta tes th a t s h o u ld b e a v a ila b le g iv e n su ch a s ta te . It s h o u ld b e ea sy t o ite ra te ov e r all ed g es th a t lea ve a sta te .4 Operations on the new data structure T h e n ew d a ta s tru ctu re is d es ig n ed s o th a t r e m o v a l a n d a d d it io n o f s ta te s a n d e d g es are se p a ra te o p e r a t io n s . T h is is in c o n tra s t t o in te rs e ctio n th a t n o t o n ly re m o v e s re a d in g s b u t a lso e x p a n d s th e se n ten ce a u t o m a t o n in a w ay th a t is d ifficu lt to c o n tro l. In co n tra s t t o se n ten ce a u to m a t a , n e x t t o n o th in g is a s su m e d a b o u t th e re p re se n ta tio n o f ru le a u to m a t a . A ll th a t is re q u ire d is th a t it is p o s s ib le to c o m p u t e th e s ta te w h ere a g iv e n sta te ta k es a g iv e n s y m b o l, a n d t o d e c id e w h e th e r th ere is an y w ay t o rea ch a fin al s ta te fr o m a g iv e n sta te. A s ta te is c a lle d a sin k if th ere is n o w ay t o reach a fin a l s ta te fr o m it. A ru le ca n fa il b y g o in g in to a sink lo n g b e fo r e r e a ch in g th e e n d o f an in p u t s e q u en ce .", |
| "uris": null, |
| "num": null |
| }, |
| "FIGREF3": { |
| "type_str": "figure", |
| "text": "u t o m a t a c o n ta in tw o sp e cia l k in d s o f sta te s, n a m e ly th e r o o ts o f th e lo c a l trees a n d th e lea ves o f th e lo c a l trees. T h e r o o t s are in slices, a n d e a ch le a f is an im m e d ia te p re d e ce s so r o f a r o o t . ( T h e s ta r t s ta te is a lso th o u g h t o f as a le a f.) R e a d in g s are r e m o v e d in tw o s te p s c a lle d p r o p a g a t io n an d re m o v a l. T h e p r o p a g a t io n step assign s sets o f s ta te s o f a ru le a u t o m a t o n t o th e r o o t s an d leaves o f th e lo c a l trees. In itia lly , a set c o n t a in in g th e s ta r t s ta te o f th e ru le a u to m a t o n is a ssign ed to th e s ta r t s ta te o f th e se n te n ce a u t o m a t o n . T h e r e is o n ly o n e p a th fr o m th e sta rt sta te o f th e se n te n ce a u t o m a t o n t o th e first r o o t , th e o n e la b e le d 8 8 ; th e set c o n ta in in g th e s ta te t o w h ich th e s ta r t s ta te o f th e ru le a u t o m a t o n takes th e la b e l 88 is a ssign ed t o th e first r o o t. O n c e in itia liz e d , th e p r o p a g a t io n ste p co n tin u e s b y a s s ig n m e n t o f sets o f ru le sta tes t o th e r o o t s a n d lea ves o f ea ch lo c a l tre e in tu rn . F or e x a m p le , a ce rta in ru le 53 Proceedings of NODALIDA 1995 a u t o m a t o n g o e s t o a sink s ta te w h en it e n co u n te rs th e la b e l \u00aeMV a fter it has a lrea d y e n co u n te re d fiMV b u t n o t y e t e n co u n te re d a cla u se b o u n d a r y . L et s ta te 1 b e th e sta rt s ta te , s ta te 2 b e th e s ta te a fter th e first \u00aeMV a n d s ta te 5 b e th e sink sta te. In th e u p p e r a u t o m a t o n in fig u re 4 , th e r o o t h a s b e e n re a ch ed th r o u g h p a th s th a t con ta in o n e \u00aeMV a n d p a th s th a t d o n o t c o n ta in \u00aeMV. T h e re fo r e , th e r o o t is a ssign ed th e set o f tw o ru le sta tes {1,2}. W h e n th is set is p r o p a g a t e d fu rth e r, th e ru le a u to m a t o n sta y s in th ese tw o s ta tes u n til it e n co u n te rs th e \u00aeMV la b e l; th en th e s ta te set w o u ld b e {2 ,5 } b u t th e sin k s ta te 5 is s im p ly ig n o r e d . In th e lo w er a u t o m a t o n in fig u re 4 , th e r o o t h as b e e n re a ch ed o n ly th ro u g h p a th s th a t c o n ta in o n e \u00aeMV. T h e s ta te se t a fte r th e s e c o n d \u00aeMV co n ta in s o n ly th e sin k s ta te a n d b e c o m e s e ffe ctiv e ly em p ty . F ig u re 4 P r o p a g a t io n o f a ru le s ta te set T h e first r o o t has b een a ssign ed in th e in itia liz a tio n ste p . T h e r e is a u n iq u e p a th fr o m th e r o o t o f a lo c a l tree t o a le a f o f th a t tree. T h is p a th a n d th e rule s ta te s in th e r o o t d e te rm in e th e set o f rule s ta te s th a t are a ssign ed t o th e lea f. R u le s ta te s in th e le a f are th o se t o w h ich s o m e ru le s ta te in th e r o o t tak es th e p a th fr o m th e r o o t to th e lea f, e x c lu d in g sinks. W h e n all th e ru le s ta te s w o u ld b e sink s, th is se t is e m p ty . U n lik e lea ves, th e r o o t s in a slice ca n h a v e several p re d e ce s so rs. A r o o t is a ssig n e d th e set o f ru le sta tes th a t are re a c h a b le fr o m s o m e ru le s ta te in s o m e o f its p r e d e ce s s o rs b y s o m e p a th fr o m th e le a f t o th e r o o t.", |
| "uris": null, |
| "num": null |
| }, |
| "FIGREF4": { |
| "type_str": "figure", |
| "text": "T h e sets o f ru le s ta te s in th e lea ves h e lp to r e m o v e sta te s a n d ed g es. F irst, i f a le a fis a ssign e d th e e m p ty set o f ru le s ta te s, th ere is n o w ay fo r th e ru le t o reach th is lea f. T h e le a f ca n b e r e m o v e d , a n d ed g e s a n d sta te s p r e ce d in g it ca n b e r e m o v e d b a ck t o th e first s ta te w ith m o r e th a n o n e im m e d ia te su cce s sor. A ll b o u n d a r y e d g e s fo r w a rd ca n b e r e m o v e d , a n d i f th e fo llo w in g r o o t h a d n o o th e r im m e d ia te 54 Proceedings of NODALIDA 1995 p re d e ce s so r, r e m o v a l can co n tin u e fo r w a rd . T h is m e a n s th a t th e e d g e s la b e le d V VFIH (OMV QO in th e low er a u t o m a t o n in fig u re 4 ca n b e r e m o v e d . S e c o n d , if all th e rule s ta tes in s o m e le a f ta k e s o m e b o u n d a r y e d g e t o a sin k , th e e d g e ca n b e r e m o v e d . I f all th e ed g e s fr o m th e le a f t o a r o o t are r e m o v a b le th is w ay, re m o v a l ca n c o n tin u e b a ck w a r d a n d fo r w a rd . T h ir d , a n o n -fin a l s ta te in th e la st s lice ca n b e tr e a te d as i f it w ere a sin k . T h e re is n o w a y t o reach a n o th e r fin a l s ta te fr o m th e la st s ta te in th e se n te n ce a u to m a t o n . O n ly su ch re a d in g s are r e m o v e d as w o u ld b e r e m o v e d b y in te r s e c tio n . H ow e v er, n o t all su ch re a d in g s ca n b e r e m o v e d th is w ay. T o a llo w r e m o v a l o f fu rth e r re a d in g s, th e se n te n ce a u to m a t o n m a y n eed to b e e x p a n d e d .", |
| "uris": null, |
| "num": null |
| }, |
| "FIGREF5": { |
| "type_str": "figure", |
| "text": "m o v e d . A se p a ra te o p e r a t io n o f lo c a l e x p a n s io n is p r o v id e d t o o p e n u p r e m o v a l o p p o r tu n itie s . O b s e rv e th a t th e in itia l sen te n ce a u t o m a t o n is tre e -s h a p e d b e tw e e n an y p a ir o f su cce ssiv e slices. T h e e x p a n s io n m a k e s it tre e -s h a p e d (r a th e r fo r e s t-s h a p e d ) b etw een s o m e g iv e n p a ir o f slices. E a ch s ta te th a t is p r o p e r ly b etw e e n th e tw o slices w ill h a ve e x a c tly o n e im m e d ia te p red ece s sor. T h e e x a m p le se n ten ce ' T im e p a s s e d ' is so s h o r t th a t th ere is o n ly o n e w ay to ch o o s e a p a ir o f slices fo r e x p a n s io n . T h e in itia l a u t o m a t o n is s h o w n in fig u r e 3; th e e x p a n d e d a u t o m a t o n is sh o w n in fig u re 5. F or lo n g e r s e n te n ce s, su ch m a x im a l e x p a n s io n is n o t fea sib le. T h e first w o rd fo r m ' t i m e ' is th ree w a y s a m b ig u o u s a n d th e w o r d b o u n d a r y a fter it is fo u r w a y s a m b ig u o u s ; to g e th e r ' t i m e ' a n d th e b o u n d a r y are tw e lv e w ays a m b ig u o u s . E x p a n s io n m a k es tw elv e c o p ie s o f th e s ta te in th e s lice b e tw e e n th e re a d in g s o f ' t im e ' a n d ' p a s s e d ' , a n d th e tree o f ' p a s s e d ' is tu rn ed in to a fo r e s t o f tw elve id e n tica l trees. M u ch o f th e n ew s tru ctu re is easy t o r e m o v e . P a r ticu la rly in te re stin g are th e th ree trees o f ' p a s s e d ' th a t fo llo w th e b o u n d a r y s y m b o l f l / . T h is b o u n d a r y s y m b o l se p a ra te s fin ite clau ses: th ere s h o u ld b e th e la b e l VFIN b o t h b e fo r e a n d a fte r it. T h is co n s tra in t is n o t e n o u g h t o r e m o v e a n y th in g in th e a u t o m a t o n o f fig u re 3 sin ce VFIH o c cu rs in o n e o f th e rea d in g s o f ' t i m e ' , b u t in th e a u t o m a t o n o f fig u re 5 , tw o o f th e th ree trees ca n b e r e m o v e d . F or e x a m p le , a fte r QQ < t im e > t i m e N NOM SG OSUBJ 0 / th e ru le s h o u ld b e in a sink sta te . A n o th e r im p o s s ib ility is t o h a v e 0< a n d (0(D o r 0< a n d (0< as s u cce s siv e b o u n d aries. T h e fo r m e r c o n s tr a in t w o u ld a llow re m o v a l o f m u ch o f th e s tru c tu re in th e e x p a n d e d a u to m a t o n : th e la b e l OA w o u ld ta k e a ru le t o a sin k . P a rts o f th e low er fo u r trees o f ' p a s s e d ' are r e m o v a b le b y th e c o n s tr a in t th a t th e b o u n d a r y s y m b o l b etw e e n tw o fin ite v e rb fo r m s VFIH ca n n o t b e 8 , s o th a t th e s e c o n d VFIH w o u ld ta k e a ru le t o a sink. ig u re 5 A n e x p a n d e d se n ten ce a u to m a t o n 5 Com putation strategies T h e h igh level s tra te g y t o use is an o p e n q u e stio n . T h e o p e r a t io n s o f p r o p a g a t io n , re m o v a l a n d e x p a n s io n ca n b e used in d iffe ren t o rd e rs . In th is m e t h o d , rules n eed to b e u sed m o r e th a n o n c e , a n d th e o r d e r o f e x p a n s io n s a n d r e m o v a ls m a y a ffe ct e ffic ie n cy sig n ifica n tly . T h e s im p le m in d e d b u t p r o h ib it iv e ly e x p e n s iv e s tra te g y w o u ld b e t o first m a x im iz e th e se n te n ce a u t o m a t o n an d th en ju s t use ev e ry ru le in tu rn . A n o t h e r s tra te g y w o u ld b e t o e x e rcise all ru le a u t o m a t a o n a s h o r t in terva l in th e b e g in n in g o f th e sen tece a u t o m a t o n first, a n d th e n le n g th e n th e in te rva l. A th ir d s tra te g y w o u ld b e t o e x p a n d s h o r t in terva ls a n d th en p r o p a g a t e a n d rem ov e. M o r e in te llig e n t s tra te g ie s m ig h t ta k e in to a c c o u n t s p e cia l p r o p e rtie s o f each s e n te n ce a u t o m a t o n , o r s p e cia l p r o p e r t ie s o f s o m e rules. O r su ch in te rva ls m ig h t b e e x p a n d e d th a t c o n ta in re la tiv e ly few rea d in g s. K im m o K o s k e n n ie m i (p e r s o n a l c o m m u n ic a t io n ) su g g e s te d first p r o p a g a t in g a ru le in to th e se n te n ce a u t o m a t o n a n d th en s t u d y in g th e sets o f rule s ta te s to see w h e th e r th is ru le can a llo w re m o v a ls a fter e x p a n s io n . In th e p o s itiv e ca se, e x p a n d a n d th en p r o p a g a t e a g a in . 56 Proceedings of NODALIDA 1995 Some details T h is s e c tio n g iv e s a m o r e d e ta ile d d e s c r ip tio n o f th e o p e r a t io n s o n th e lo c a lly tre e s h a p e d a u t o m a t a th a n w as g iv e n b e fo r e . A ll a u t o m a t a h ere are d e t e r m in is tic fin ite s ta te a u to m a ta . A n a u to m a t o n is a fin ite set o f u n iq u e ly la b e le d s ta te s, p a r t it io n e d in to fin a l a n d n o n -fin a l s ta te s, w ith o n e s ta te d e s ig n a te d as th e s ta r t sta te . A ll s ta te s w ill b e seen as to t a l fu n c tio n s ta k in g se q u e n ces t o s ta te s , as fo llo w s . W h e n s a n d t are sta te s a n d a p a th fr o m s t o t is la b e le d b y a s e q u e n ce o f a lp h a b e t s y m b o ls w , th e n o t a t io n s w d e s ig n a te s th e s ta te t . T h e se t o f a ll p re fix e s o f su ch seq u e n ce s w th a t s tak es t o a fin al s ta te is c a lle d th e d o m a in o f s . A s ta te w ith e m p ty d o m a in is c a lle d a sin k ; i f w is n o t in th e d o m a in o f s , th en s w is a sink. E a ch s ta te s sa tisfies th e e q u a tio n s s Q -s a n d s (a , 6,...) = ( s ( a ) ) ( 6,...)for all a, 6, . . . in th e a lp h a b e t. In a m in im a l a u t o m a t o n , a sin k s is a n o n -fin a l s ta te th a t satisfies s w = s fo r all s e q u en ces w .", |
| "uris": null, |
| "num": null |
| }, |
| "FIGREF6": { |
| "type_str": "figure", |
| "text": "I f s { a ) = t fo r s o m e s y m b o l a , th en s is s a id t o b e an im m e d ia t e p re d e ce s so r o f t , a n d t is sa id t o b e an im m e d ia te su cce s so r o f s. In th e p r o ce s s t o b e s p e cifie d , a se n te n ce a u t o m a t o n S is rep re s e n te d as a se q u e n ce o f slices (5 o ,... , < S \"_ i). It m u s t b e re p la c e d b y s u cce s siv e s e n te n ce a u t o m a t a s o th a t th e fin al resu lt rep resen ts th e in te r s e c tio n o f th e o r ig in a l s e n te n ce a u t o m a t o n w ith a n u m b e r o f ru le a u t o m a t a R q , . . . , R t^i .", |
| "uris": null, |
| "num": null |
| }, |
| "FIGREF7": { |
| "type_str": "figure", |
| "text": "u le sta te s ca n b e p r o p a g a t e d fr o m S i t o . A n in te r m e d ia te s te p is t o p r o p a g a t e th e m to th e lea ves o f ea ch lo c a l tree w ith r o o t in S i . W h e n a p a th fr o m a r o o t t o a le a f is la b e le d w a n d th e se t o f ru le s ta te s a s sig n e d t o th e r o o t is R , th e set o f ru le sta te s in th e le a f w ill b e { r t u | r G i i } e x c lu d in g sinks. A ft e r p r o p a g a t io n t o th e lea ves, an im m e d ia te p re d e ce s so r p o f a s t a t e t in c o n ta in s th e c o rr e ct set R ' o f ru le sta tes. I f D is th e d o m a in o f p , th e se t o f rule sta te s in t w ill b e a su p e rse t o f { r (d ) | r G R ' a n d {d ) E D } e x c lu d in g sink s. O th e r im m e d ia te p red ece ssors o f t p r o d u c e o th e r sta tes. R u le s ta te s ca n b e p r o p a g a t e d fr o m th e sta rt s ta te t o e a ch s lice in tu rn . I f th e sta rt s ta te o f a ru le is r o , th e s ta te in th e first s lice w ill c o n t a in ju s t th e s ta te To {\u00ae #).", |
| "uris": null, |
| "num": null |
| }, |
| "FIGREF8": { |
| "type_str": "figure", |
| "text": "h e n rule sta tes h a v e b een p r o p a g a t e d fr o m S o t o S i , it m a y b e p o s s ib le t o r e m o v e ed g es a n d sta te s th a t p re ce d e s ta te s in <S,-. I f p is an im m e d ia te p red eces sor o f s o m e s ta te s in S i a n d th e set o f ru le sta tes in p is e m p ty , th en th e ru le takes t o a sin k all se q u e n ce s th a t la b e l p a th s f r o m th e sta rt s ta te to p. T h e s ta te p ca n b e re m o v e d to g e th e r w ith p r e ce d in g e d g e s an d sta te s b a ck t o th e first s ta te th a t has m o r e th a n o n e e d g e in its d o m a in . 57 Proceedings of NODALIDA 1995 I f th e set o f ru le s ta te s in p is n o t e m p ty b u t th ere are sev era l ed g es fr o m p t o s , a ll th e ru le s ta te s ca n ta k e s o m e o f th e e d g es t o a sink. S u ch ed g es ca n b e r e m o v e d .", |
| "uris": null, |
| "num": null |
| }, |
| "FIGREF9": { |
| "type_str": "figure", |
| "text": "n d s o m e s ta te s in <Sj+i h a s k im m e d ia te p re d e ce s so rs, th e s ta te s a n d its lo c a l tree are re p la ce d w ith k c o p ie s , e a ch w ith its o w n im m e d ia te p re d e ce s so r o u t o f th e im m e d ia te p red ecessors o f th e o r ig in a l s. 7 Summary In te r s e c tio n o f fin ite s ta te a u t o m a t a red u ces rea d in g s b u t m a y a d d s tru ctu re in the a u t o m a t a . S e p a r a tio n o f th e tw o e ffe cts m a y h elp to keep th e size o f a sen te n ce a u t o m a t o n u n d er c o n t r o l. K e e p in g th e se n ten ce a u t o m a t o n lo c a lly tre e -sh a p e d m a k e s th is s e p a r a tio n p o s sib le . F u tu re w ork w ill in v o lv e im p le m e n ta tio n o f th e pa rser a n d e m p ir ic a l s tu d y o f v a r io u s p r a s in g s tra teg ies.", |
| "uris": null, |
| "num": null |
| } |
| } |
| } |
| } |
