Multiple Adjunction in Featue-Based Tee-Adjoining Gamma Claie Gadent, Shashi Naayan To cite this vesion: Claie Gadent, Shashi Naayan. Multiple Adjunction in Featue-Based Tee-Adjoining Gamma. Computational Linguistics, Massachusetts Institute o Technology Pess (MIT Pess), 2015, 41 (1), pp.29. <http://www.mitpessjounals.og/toc/coli/41/1>. <10.1162/COLI_a_00217>. <hal- 01249975> HAL Id: hal-01249975 https://hal.inia./hal-01249975 Submitted on 4 Jan 2016 HAL is a multi-disciplinay open access achive o the deposit and dissemination o scientiic eseach documents, whethe they ae published o not. The documents may come om teaching and eseach institutions in Fance o aboad, o om public o pivate eseach centes. L achive ouvete pluidisciplinaie HAL, est destinée au dépôt et à la diusion de documents scientiiques de niveau echeche, publiés ou non, émanant des établissements d enseignement et de echeche ançais ou étanges, des laboatoies publics ou pivés.
Multiple Adjunction in Featue-based Tee-Adjoining Gamma Claie Gadent CNRS, LORIA UMR 7503 Vandoeuve-lès-Nancy F-54500, Fance claie.gadent@loia. Shashi Naayan Univesité de Loaine, LORIA UMR 7503 Vandoeuve-lès-Nancy F-54500, Fance shashi.naayan@loia. In pasing with Tee-Adjoining Gamma (TAG), independent deivations have been shown by Schabes and Shiebe (1994) to be essential o coectly suppoting syntactic analysis, semantic intepetation and statistical language modelling. Howeve, the pasing algoithm they popose is not diectly applicable to Featue-Based TAGs(FB-TAG). We povide a ecognition algoithm o FB-TAG which suppots both dependent and independent deivations. The esulting algoithm combines the beneits o independent deivations with those o Featue-Based gammas. In paticula, we show that it accounts o a ange o inteactions between dependent vs. independent deivation on the one hand, and syntactic constaints, linea odeing, and scopal vs. nonscopal semantic dependencies on the othe hand. 1. Intoduction A Tee Adjoining Gamma(TAG,(Joshi and Schabes 1996)) consists o a set o elementay tees and two combining opeations, substitution and adjunction. Consequently, a TAG deivation can be descibed by a tee (called a deivation tee) speciying which elementay TAG tees wee combined using which opeations to yield that deivation. In this tee, each vetex is labelled with a tee name and each edge with a desciption o the opeation (node addess and opeation type) used to combine the tees labelling its end vetices. As we shall see in Section 3.2, in TAG, each deivation tee speciies a unique pase tee also called deived tee. In pevious wok, it has been agued that TAG deivation tees povide a good appoximation o semantic dependencies between the wods o a sentence(koch 1989; Rambow, Vijay-Shanke, and Wei 1995; Candito and Kahane 1998; Kallmeye and Kuhlmann 2012). As shown by Schabes and Shiebe (1994) howeve, thee ae seveal possible ways o deining TAG deivation tees depending on how multiple adjunction o seveal auxiliay tee at the same tee node is handled. The standad notion o deivation poposed by Vijay-Shanke (1987) obids multiple adjunction thus enocing dependent deivations. In contast, the extended notion o deivation poposed by Schabes and Shiebe(1992, 1994) allows multiple adjunction at a single node theeby yielding socalled independent deivations i.e., deivations whee the elation between the adjoining tees is let unspeciied. The dieence between the two types o deivations is illustated in Figue 1. While in the standad (dependent) deivation, one adjective tee is adjoined to the othe adjective tee which itsel is adjoined to the noun tee o peppe, in the extended(independent) deivation, both adjective tees adjoin to the noun tee. 2005 Association o Computational Linguistics
Computational Linguistics Volume xx, Numbe xx N peppe Adj N oasted N Adj ed N α peppe β oasted β ed Standad(dependent) Deivation N α peppe 0 α peppe β ed 0 0 0 β oasted β ed β oasted Extended(independent) Deivation Figue 1 AnexampleTAG with the altenative TAG deivationsothe phaseoasted edpeppe. α peppe, β ed and β oasted aethe elementayteesopeppe (initial tee),ed(auxiliaytee)and oasted (auxiliay tee) espectively. Schabes and Shiebe (1994) ague that allowing both o dependent and independent deivations bette elects linguistic dependencies. Making use o the distinction intoduced in TAG between pedicative and modiie auxiliay tees (Schabes and Shiebe (1994), Section 3.1), they deine a pasing algoithm which assigns dependent deivations to pedicative auxiliay tees but independent deivations to multiple modiie auxiliay tees adjoining to the same node. In case both pedicative and modiie auxiliay tees adjoin to the same node, thei pasing algoithm ensues that pedicative tees appea above the modiie tees in the deived tee. This pasing algoithm is deined o eatueless vaiants o TAG. In contast, in implemented TAGs(e.g., XTAG(The XTAG Reseach Goup 2001), SemXTAG(Gadent 2008) o XXTAG 1 (Alahvedzhieva 2008)) eatue stuctues and eatue uniication ae cental. They ae used to minimize the size o the gamma; to model linguistic phenomena such as veb/subject ageement; and to encode a uniication-based syntax/semantics inteace(c., e.g.,(gadent and Kallmeye 2003)). In this pape, we extend Schabes and Shiebe s poposal to Featue-Based TAG (FB-TAG) and we show that the esulting pasing algoithm natually accounts o the inteplay o dependent vs. independent deivation stuctues with syntactic constaints, linea odeing, and scopal vs. nonscopal semantic dependencies. The pape is oganized as ollows. In Section 2, we ecap the motivations o independent deivations put owad by Schabes and Shiebe (1994) and we biely discuss the inteactions that may aise between dependent and independent deivations. Section 3 summaizes thei appoach. In Section 4, we pesent the intuitions and motivations undelying ou poposal and we highlight the dieences with Schabes and Shiebe s appoach. Section 5 pesents ou poposal. Section 6 concludes. 2. Why ae Independent Deivations Desiable? We stat by summaizing Schabes and Shiebe s motivations o independent deivations. We then discuss the inteactions between dependent and independent deivations. 1 XXTAGstandso XMG (Cabbéetal. 2013) basedxtag. 2
Gadent, Naayan Multiple Adjunction in Featue-Based Tee Adjoining Gamma 2.1 Motivations o Independent Deivations Schabes and Shiebe (1994) give thee main motivations o independent deivations. The ist motivation concens the inteaction o vebs with multiple modiies. Conside sentences 2 in(1)and(2). (1) a. Richad Pake and Pi wandeed the Algae Island yesteday though the meekats. b. Richad Pake and Pi wandeed the Algae Island yesteday. c. Richad Pake and Pi wandeed the Algae Island though the meekats. (2) a. TheOangutanemindedPi o his motheyestedaythoughthemeekats. b. The Oangutan eminded Pi o his mothe yesteday. c. TheOangutanemindedPi o his mothethoughthemeekats. Movement vebs such as to wande allow o diectional modiies such as though the meekats wheeas vebs such as to emind do not. In TAG, such estictions can be modeled using selective adjoining constaints to speciy which modiie tee may o maynotbeadjoinedatapaticulanodeinagiventee.theeoeitispossibletolicense (1) and to ule out (2c). In (2a)howeve, unde the dependent notion o adjunction, the tee o the diectional advebial though the meekats will adjoin to the modiie tee o yesteday which itsel will adjoin to the tee selected by eminded. Thus constaints placed by the veb on its modiies must be passed though by modiie tees (hee the tee o yesteday) to also ule out sentences such as (2a). Popagating selective adjunction constaints in TAG would lead to a omalism o which deivation tees ae no longe context-ee(schabes and Shiebe 1994). The second motivation o independent adjunction stems om pobabilistic appoaches. Stochastic lexicalized TAG speciies the pobability o an adjunction o a given auxiliay tee at a given node in anothe elementay tee (Schabes 1992; Resnik 1992). Thus unde the standad notion o deivation, the oveall pobability o the sting oasted ed peppe would be emined by the pobability o ed adjoining to peppe and the pobability o oasted adjoining to ed. In contast, independent adjunction would esult in a deivation such that the oveall pobability o the sting oasted ed peppe would be emined by the pobability o both ed and oasted adjoining to peppe. Schabes and Shiebe (1994) ague that it is plausible that the most impotant elationships to chaacteize statistically ae those between modiie and modiied, athe than between two modiies. A thid motivation comes om semantics and moe paticulaly, om scope ambiguities involving modiies. Given a sentence such as (3) whee the elative scope o themodiiestwiceandintentionallyisambiguous 3,Shiebe(1994)showsthat,unhe extended deinition o adjunction, a synchonous TAG modelling the elation between syntactic tees and logical omulae can account o both eadings. (3) John blinked twice intentionally. The account cucially elies on multiple independent adjunction o the two modiie tees to the tee o blink: depending on which ode the auxiliay tees o twice and intentionally adjoins to blink, the logical omula built will be eithe intentionally(twice(blink)) o twice(intentionally(blink)) thus captuing the ambiguity. 2 Thechaactes in thesesentencesae boowed omyann Matel s book Lie o Pi. 3 The sentence can descibe eithe a single intentional act o blinking twice o two intentional acts each o single blinking (Shiebe 1994). 3
Computational Linguistics Volume xx, Numbe xx 2.2 Dependent, Independent and Mixed Deivations To captue the dieent types o semantic dependencies and mopho-syntactic constaints which may hold between multiple auxiliay tees adjoining to the same entity, both dependent and independent deivations ae needed. As agued above, because thee ae no constaints o semantic elation holding between each o them, multiple intesective modiies applying to the same entity (e.g., 4) ae best modeled using an independent deivation. (4) a. The tall black meekat slept. (Independent deivation) In contast, because they may involve stong scopal and mopho-syntactic constaints, stacked pedicative vebs (i.e., vebs taking a sentential complement, 5a) and non-intesective modiies (e.g., 5c) equie dependent deivations. Conside sentences (5a-b) o instance. I pedicative tees wee assigned an independent deivation, sentence (5a) would be judged ungammatical (because want equies an ininitival complement but would adjoin to the inite veb slept) and convesely, sentence (5b) would incoectly be judged gammatical (because both want and ty equie an ininitival complement). Similaly, in example (5c), the chuch is Syian Othodox, not Syian and Othodox. Assigning a dependent athe than an independent deivation to such cases staightowadly captue the distinction between intesective and non intesective modiies. (5) a. John wanted to assume that Pete slept. (Dependent deivation) b. John wantedpete tiestowalk. c. The meekat admied the Syian Othodox chuch. (Dependent deivation) Finally, some multiple adjunctions may involve both dependent and independent deivations, e.g., when multiple modiies and pedicative vebs adjoin to the same veb (e.g.,6a)ointhecaseoadeivation(e.g.,6b)involvingbothintesective(old)andnonintesective (i.e., Syian in Syian Othodox) modiies. (6) a. Yann said that John knows that Richad Pake and Pi wandeed the Algae Island yesteday though the meekats. (Mixed deivation) b. The meekat admied the old Syian Othodox chuch. (Mixed deivation) As we shall see in Section 5.3, the pasing algoithm we popose licenses dependent, independent and mixed deivations but is esticted to appopiately distinguish between vaious types o modiies. Moeove, the eatue inomation encoded in the gamma uthe esticts the deivation stuctues poduced theeby accounting o the inteactions between adjunction, linea odeing and mopho-syntactic constaints. 3. Multiple Adjunction in Tee Adjoining Gammas Vijay-Shanke and Wei(1991) intoduce a compilation o TAG to Linea Indexed Gammas (LIG, (Gazda 1988)) which makes the deivation pocess explicit. Schabes and Shiebe(1994) modiy this compilation to allow both o dependent and o independent deivations. The esulting LIG is uthe exploited to speciy a pasing algoithm which ecoves those deivations. In this section, we summaize Schabes and Shiebe s poposal. We stat(section 3.1) with an inomal desciption o thei appoach. In Section 3.2, we intoduce odeed deivation tees. Section 3.3 gives a bie intoduction to LIG. Section 3.4 summaizes the TAG-to-LIG compilation poposed by Vijay-Shanke and Wei(1991). Finally, Section 3.5 4
Gadent, Naayan Multiple Adjunction in Featue-Based Tee Adjoining Gamma descibes the modiications intoduced by Schabes and Shiebe (1994) to allow both o dependent and o independent deivations. 3.1 Schabes and Shiebe s Poposal: Motivations and Intuitions Tee Adjoining Gamma distinguishes between two types o auxiliay tees namely, modiie vs. pedicative auxiliay tees (Joshi and Vijay-Shanke 2001). While pedicative tees ae assigned to vebs taking a sentential agument, modiie tees ae assigned to all othe auxiliay tees, e.g., vebal auxiliaies, adjectives, advebs, pepositions and emines. Moe geneally, the dieence between a pedicative and a modiie tee is that in a pedicative tee, the oot node, like the substitution nodes, coesponds to an agument node selected by its lexical ancho (i.e., the wod that selects that tee) while in a modiie auxiliay tee, the oot node is an open slot coesponding to the phase being modiied. When associating semantic entities with tee nodes (as poposed, e.g., by Joshi and Vijay-Shanke (2001) and Gadent and Kallmeye (2003)), this dieence canbeseenbynoting theentities associatedwithoot andoot nodes:these aedistinct in a pedicative tee but identical in modiie tees. In thei appoach, Schabes and Shiebe speciy a TAG to LIG convesion which systematically associates dependent deivations with pedicative auxiliay tees and independent deivations with modiie auxiliay tees. In addition, they intoduce two mechanisms to ensue that each deivation tee unambiguously speciies a linguistically plausible deived tee. Fist, they enoce odeing constaints between modiie tees adjoining at the same node (which ae thus ambiguous with espect to the deived tee they descibe) by assuming that deivation tees ae odeed and that linea pecedence (LP) statements can be used to constain the ode o siblings in a deivation tee. Fo instance, given the independent deivation shown in Figue 1, an LP statement stating that β ed must occu beoe β oasted in the deivation tee will ensue that β oasted appeas above β ed in the deived tee and theeoe that the esulting deived tee yields the phase oasted ed peppe athe than ed oasted peppe. Second, when both pedicative and modiie tees adjoin at the same addess, pedicative tees always occu above all modiie tees in the deived tee ( outemost pedication ). This ensues o instance, that unde the eading whee yesteday ees to the aiving athe than the saying i.e., when both say and yesteday adjoin to aive, (7a) is deivedbutnot (7b). (7) a. Pete says that yesteday John aived late. b. YestedayPetesays that Johnaivedlate. 3.2 Odeed Deivation Tees In the standad vesion o TAG, each deivation tee descibes a unique deived tee. In the case o a dependent deivation, unicity ollows om the act that dependent deivations speciy the ode in which adjunction takes place (e.g.,β 2 adjoins to β 1 and the esult to α). As a esult, i β 2 adjoins to β 1, thee is only one possible deived tee namely,ateewheeβ 2 appeasaboveβ 1. When allowing o independent deivations howeve, seveal deived tees ae possible depending on the ode in which the auxiliay tees ae adjoined. To ensue a unique mapping om deivation to deived tee, Schabes and Shiebe(1994) theeoe intoduce the notion o odeed deivation tees. Odeed deivation tees die om 5
Computational Linguistics Volume xx, Numbe xx standad TAG deivation tees in that (i) they may contain sibling edges labelled with the same addessand(ii) they speciyatotal odeonsuchsiblings. τ g τ β n g g g β i β 1 β i β n β 1 Figue 2 Odeeddeivationteeand coespondingdeivedtee.τ,β 1,β i and β n aeelementaytees. β 1,β i and β n aeauxiliayteeswhich all adjoinatthe addessg inthe elementayteeτ. Figue 2 shows an example odeed deivation tee and associated deived tee. As indicated by the shaed g addess on thei paent edge, auxiliay tees β 1,..., β n adjoin to the same node namely the node with addess g in the elementay tee τ. Because the deivation tee is odeed, β 1 will appea below β 2 in the deived tee which in tun will be below β 3, and so on. In shot, given a set o auxiliay tees all adjoining to the same tee node, the deived tee poduced om an odeed deivation tee ollowing an independent deivation will be identical to the deived tee poduced with the coesponding dependent deivation i.e., the dependent deivation whee β 1,..., β n appeain inceasingindex odeomtop tobottom. 3.3 Linea Indexed Gamma Like Context-Fee Gammas(CFG), Linea Indexed Gammas(LIG,(Gazda 1988)) ae sting ewiting systems whee stings ae composed o teminals and nonteminals. In a LIG howeve, nonteminal symbols may be associated with a stack o symbols, called indices. A LIG ule can thus be epesented as ollows: τ N[..µ] N 1 [µ 1 ]...N i 1 [µ i 1 ]N i [..µ i ]N i+1 [µ i+1 ]...N n [µ n ] (1) N and N i ae nonteminals while µ and µ i ae stings o stack symbols. The symbol.. stands o the emainde o the stack symbols. Note that the emainde o the stack symbols associated with the LHS is associated with only one o the nonteminal (namely,n i )on therhs. Linea Indexed Gammas (LIG) have been used in the liteatue (Wei and Joshi 1988; Vijay-Shanke and Wei 1991) to povide a common amewok o the extensions o context-ee gammas. In paticula, Vijay-Shanke and Wei (1991, 1993) showed a weak equivalence between LIGs, TAGs and CCGs(Combinatoy Categoial Gammas, (Steedman 2000)) and poposed a LIG based polynomial-time CYK ecognition algoithm o TAGs and CCGs. In what ollows, we show how Schabes and Shiebe (1994) use a LIG vaiant o TAGs to license both dependent and independent deivations. 6
Gadent, Naayan Multiple Adjunction in Featue-Based Tee Adjoining Gamma 3.4 TAG to LIG Compilation The TAG to LIG compilation poposed by Vijay-Shanke and Wei (1991) poduces LIG ules which simulate a tavesal o the deived tee poduced by the oiginal TAG gamma.inthese LIGules, eachnodeη oatagelementayteeisviewedas having both a top t[..η] and a bottom b[..η] component to account o the possibility o an adjunction. Figue 3 illustates the tavesal o the TAG deived tees speciied by the LIG esulting omvijay-shanke and Wei (1991) TAG to LIG compilation. t[ηη ] t[η] b[η] Type 3 Type 4 b[ηη ] β Type 1/2 t[η 1 ] t[η i ] t[η n ] Type 5 t[ηη ] b[ηη ] Figue 3 LIGvaiant otag othe Standad deivation.eachothe teenodesinthe gammais assigneda uniqueaddess.foexample,heeη, η 1,η i and η n pointto the distinctnodesinthe letelementayteewheeasη and η pointto the ootand the ootnodesothe shown auxiliayteeβ inthe gamma.t[..η] and b[..η] aethe top and bottom components othe tee nodeη inthe gamma. Figue4liststheLIGulesesultingomtheTAGtoLIGcompilationpocess.Each nonteminal (t[..η] o b[..η]) with the top o the stack symbol in a LIG ule coesponds to a unique node in some elementay tee o the gamma. The inne stack symbols ae used to keep tack o the nodes highe in the deived tee whee an auxiliay tee has been adjoined. Rules o Type1and 2captueimmediate dominance betweenthe bottom oanode η and the top o its immediate daughtes in two coniguations depending on whethe η dominates theoot node (Type1)o not(type2).rules otype3handlenodes which equie neithe substitution no adjunction. This ule handles cases whee no adjunction occus at a node by ewiting the top o this node to its bottom. Rules o Type 6 model substitution. Finally, ules o Type 4 and 5 handle adjunction. They speciy that, o any givennodeηandanyauxiliayteeβ whichmayadjointoη,thetopoη ewitestothe top o the oot node o β; and the bottom o the oot o β to the bottom o η. It ollows that theecanbeno multiple adjunction inthis LIGvesiono TAG. 3.5 Modiying the TAG to LIG Compilation to Allow o Multiple Adjunctions To associate pedicative tee adjunctions with dependent and multiple modiie adjunctions with independent deivations, Schabes and Shiebe(1994) modiy the compilation otagtoligpoposedbyvijay-shankeandwei(1991)assketchedinfigue5.type 4(a) ules apply to adjunctions involving pedicative tees. They ae identical to Type 4 ules in the Vijay-Shanke and Wei s appoach and theeoe enoce a standad(dependent) deivation o pedicative tees. In contast, Type 4(b) ules apply to adjunctions involving modiies and esult in an independent deivation. 7
Computational Linguistics Volume xx, Numbe xx Type 1: Immediate domination dominating oot. Fo each node η in the auxiliay tees which dominates the ootnodeandwithchildenη 1,...,η i,...,η n wheethechildη i alsodominatestheootnode,the ollowing LIG poduction ule is geneated. b[..η] t[η 1 ]...t[η i 1 ]t[..η i ]t[η i+1 ]...t[η n ] Type 2: Immediate domination not dominating oot. Fo each elementay tee node η which does not dominatetheoot nodeand withchilden η 1,...,η n,theollowing LIGpoduction ule is geneated. b[η] t[η 1 ]...t[η n ] Type 3: No adjunction. Fo each elementay tee node η that is not maked o substitution o obligatoy adjunction, the ollowing LIG poduction ule is geneated. t[..η] b[..η] Type 4: Stat oot o adjunction. Fo each elementay tee node η which allows the adjunction o the auxiliay teewith theoot nodeη,theollowing LIGpoduction ule is geneated. t[..η] t[..ηη ] Type 5: Stat oot o adjunction. Fo each elementay tee node η which allows the adjunction o the auxiliay teewith theoot nodeη,theollowing LIGpoduction ule is geneated. b[..ηη ] b[..η] Type 6: Stat substitution. Fo each elementay tee node η which allows the substitution o the initial tee with theoot nodeη,theollowing LIG poductionule is geneated(not shownin Figue 3). t[η] t[η ] Figue 4 LIG poduction ules o the Standad deivation. Type 4(a) Type 4(b) β ped β mod pedicative tee modiie tee Figue 5 LIG vaiant o TAG o Schabes and Shiebe s Extended deivation. The top and bottom components o the nodes ae pesented by. Type 4(a) tansitions suppot dependent, and Type 4(b) independent deivations. Type 4(a): Stat oot o adjunction o pedicative tees. Fo each elementay tee node η which allows the adjunction o the pedicative auxiliay tee with the oot node η,the ollowing LIGpoductionule is geneated. t[..η] t[..ηη ] (2) 8
Gadent, Naayan Multiple Adjunction in Featue-Based Tee Adjoining Gamma Type 4(b): Stat oot o adjunction o modiie tees. Fo each elementay tee node η which allows the adjunction o the modiie auxiliay tee with the oot node η, the ollowing LIG poduction ule is geneated. b[..η] t[..ηη ] (3) Note also that the outemost pedication constaint i.e., pedicative tees always occu above modiie tees adjoined at the same node, alluded to in Section 3.1 ollows om the inteactions between the Type 4(a) and Type 4(b) LIG ules. Schabes and Shiebe pove the weak-geneative equivalence o TAGs unde both standad and extended deivation using the LIG compilation. They also popose a ecognitionandapasingalgoithmwithcomplexityoo(n 6 )inthelengthothesting. 4. Multiple Adjunction in Featue-Based TAG In this section, we explain why a staightowad extension o Schabes and Shiebe s poposal to FB-TAG would not wok and we outline the intuitions and motivations undelying ou appoach. Section 5 will then intoduce the ails o ou poposal. 4.1 Featue-Based Tee Adjoining Gamma We stat by a bie desciption o FB-TAG and o the uniications peomed duing deivation. FB-TAG was intoduced by Vijay-Shanke (1987) and Vijay-Shanke and Joshi (1988, 1991) to suppot the use o eatue stuctues in TAG. Figue 6 shows a toy FB-TAG o illustation. NP [] [:all] Det NP* [] [] all NP [] [:the] Det NP* [:nil] [] the NP [] [] meekat β all β the α meekat Figue 6 A toy FB-TAG. Fo the sake o claity, eatue stuctues ae abbeviated. η η.t η.b α η o.t η o = η o η o.t η.t η.b η η.t η.b β η o η o.t η o.b = η o η o.t η.t η.b substitution η η.t η.b adjunction η o η.t η o.b η.b Figue 7 Featueuniications along substitutionand adjunction infb-tag.thenode η o insome elementayteeτ isthe opeationsiteoasubstitutionoan adjunction. Fothe sakeo claity, we onlyshow the opeationnodeη o. 9
Computational Linguistics Volume xx, Numbe xx NP [] [:all] NP [] [:all] β all NP* [] [] NP [] [:the] β the = NP* [:nil] [] NP [] [:all] β all NP [] [:the] β the NP [] [] α meekat = β all NP [] [:the] β the NP [:nil] [] NP* [:nil] [] α meekat Figue 8 Standad deivation o all the meekats. NP α meekat α meekat Det NP β the β all all Det NP the meekat Figue 9 The deived tee(let), the successul standad deivation tee(middle) and the ailed dependent deivation tee(ight) o all the meekats. β all β the An FB-TAG dies om a TAG in that tee nodes ae decoated with eatue stuctues. While nonteminal and oot nodes ae decoated with two eatue stuctues called top(t) and bottom(b), substitution nodes ae decoated with a single top eatue stuctue. Duing deivation, eatue stuctue uniication constains tee combination as illustated in Figue 7. Substitution uniies the top eatue stuctue o a substitution nodewiththetopeatuestuctueotheootnodeotheteebeingsubstitutedin.the adjunction o an auxiliay tee β to a tee node η o uniies the top and bottom eatue stuctues o η o with the top eatue stuctue o the oot node o β and the bottom eatue stuctue o its oot node espectively. Finally, at the end o the deivation, the top and bottomeatuestuctues oall nodes inthe deivedteeaeuniied. Figue 8 shows the standad deivation o the phase all the meekats using the gamma shown in Figue 6 while Figue 9 shows the coesponding deived and deivation tees. As can be seen, the eatue constaints encoded in the gamma coectly ensue that all the meekats can be deived (letmost deivation tee in Figue 9) but not the all meekats (ightmost in Figue 9). The incoect deivation is blocked by the eatue stuctue [ : nil] on the oot o the auxiliay tee β the which leads to a uniication ailue i β the is adjoined at the oot o β all with bottom eatue stuctue [ : the]. 4.2WhyaSimpleExtensionotheLIG FamewoktoFB-TAG willnotwok To motivate ou appoach, we stat by consideing a simple extension o Schabes and Shiebe s LIG amewok to FB-TAG whee each LIG ule enoces uniications mimicking those applied in FB-TAG. In paticula, let us assume that Type 3 ules 10
Gadent, Naayan Multiple Adjunction in Featue-Based Tee Adjoining Gamma ( No adjunction ) uniy the top and the bottom eatue stuctues o nodes whee no adjunction occus while Type 4(b) ules( Stat oot o adjunction ) uniy the top eatue (η.t) o the node (η) being adjoined to with the top eatue stuctue (η.t) o the oot node (η ) o theauxiliay teebeing adjoined 4 : b[..η] t[..ηη ] η.t η.t (Type4b) (4) t[..η] b[..η] η.t η.b (Type3) (5) NP m NP m.t : [] NP m.b : [] α meekat (Type4(b)) (Type4(b)) NP NP the.t : [] (Type 3) the NP the.b : [ : the] β the NP NP all.t : [] (Type 3) all NP all.b : [ : all] β all NP NP all.t : [] all NP all.b : [] NP NP the.t : [ : nil] the NP the.b : [] Figue 10 Failed Deivation unde a Simple Extension o the Schabes and Shiebe s LIG amewok to FB-TAG:Type4(b) ulesuniynp m.t with both NP the.t and NP all.t while Type3ulesuniy NP the.t with NP the.b and NP all.t withnp all.b. Hence NP the.b and NP all.b shoulduniy. Howeve since thei values die, deivation ails.. As shown in Figue 10, this appoach can incoectly lead to deivation ailues in the case o an independent multiple adjunction. Intuitively, the eason o this is that, in the Schabes and Shiebe s appoach, multiple adjunction stats and ends om the bottom component o the node being adjoined to. This is ine when no eatues ae involved because the categoy o the node being adjoined to is always identical to the oot and oot node o the auxiliay tees being adjoined. When nodes cay eatue stuctues howeve, a uniication clash can occu which makes deivation ail. Thus in ou example, deivation incoectly ails because the bottom eatue stuctues o the oot node o the auxiliay tee o all and the bottom eatue stuctue o the oot node o the auxiliay tee o the should uniy but have conlicting value. As shown by the dependent deivation o all the meekats depicted in Figue 8, this is incoect. 4.3 Poposal: Intuition and Motivations As we just saw, in the case o multiple independent adjunctions, a staightowad extension o Schabes and Shiebe s LIG amewok to FB-TAG ails to coectly captue the uniication constaints encoded in the gamma. Moe geneally, when extending multiple independent adjunction to FB-TAG, it is cucial that the eatue constaints encoded by the linguist descibe the same set o deived tees no matte which deivation tee is poduced. We theeoe popose a pasing algoithm which, given seveal 4 Weassociate η.t η.t withthetype4(b)ulestomimictheadjunctioninfb-tagasshowninfigue7. 11
Computational Linguistics Volume xx, Numbe xx auxiliayteesβ 1,...,β n adjoining atthe samenode η o,peoms thesame uniications independently o whethe the deivation is dependent, independent o mixed dependent/independent. Figue11showstheuniicationsesultingomthemultipleadjunctionoβ 1 andβ 2 to a single node η o. While it depicts the uniications enoced by ou pasing algoithm o the deivation tee shown on the ight hand side namely o the independent adjunction o β 1 and β 2 to η o, these uniications ae in act exactly the same as those that would be enocedby adependentadjunction o β 2 intoβ 1 into η o. η 1 η 1.T η 1.B η o12 η o.t η 1.T η 2.T η 2.B β 1 η η 1.T 1 η 1.B η η 2.T 2 η 2.B β 2 η 2 η 2.T η 2.B η o η o.t η o.b = η 2 η 2.T η 2.B β 2 η 2 η 2.T η 2.B η o1 η o.t η 1.T η 1.B = η o1 η 1.T η o.b η 1.B η o12 η 2.T η 1.B η 2.B η o1 η 1.T η o.b η 1.B Deived Tee τ η 0 η 0 β 1 β 2 Odeed Deivation Tee Figue 11 Independent deivation and Featue Uniication. The uniications peomed by the independent adjunction o β 1 and β 2 to η o aethe sameas those that wouldbe peomedbyadependent adjunction o β 2 to β 1 and othe esultingdeivedteeto η o.cuciallyinthe independent deivation,although bothβ 1 and β 2 adjointo η o,the adjunction oβ 2 equiesaccess to the eatuestuctue othe ootoβ 1 (η 2.B η 1.B). One key point illustated by Figue 11 is that while multiple adjunction opeates on a single node (hee η o ), the uniication constaints o FB-TAG equie that the bottom eatue stuctue o the oot o an auxiliay tee which appeas highe in the deived tee (hee, β 2 ) uniies with the bottom eatue stuctue o the oot o the auxiliay tee appeaingimmediatelybelowitinthedeivedtee(heeβ 1 ) notwiththatotheooto thenoowhichitadjoins (heeη o ).Inothewods,whileamultipleadjunctiononη o opeatesonη o only,acoectimplementationofb-taguniicationconstaintsequies keeping tack o the eatue stuctues associated with the auxiliay tees successively adjoining toη o. In ou poposal, we captue this bookkeeping equiement by associating tee nodes not with eatue stuctues but with eeence vaiables pointing to eatue stuctues. The pasing algoithm is then speciied so as to suppot dependent, independent and mixed deivations while enocing the same uniications as would be peomed unde a dependent adjunction. 4.4 Compaison with Schabes and Shiebe s Appoach Beoe giving the technical ails o ou pasing algoithm(c. Section 5), we ist highlight some dieences between ou and Schabes and Shiebe s appoach. In paticula, we show (i) that wheeas Schabes and Shiebe esot to thee distinct mechanisms to account o wod ode constaints(namely, selective adjoining constaints, linea pecedence statements on deivation tees and a constaint on pasing), the FB-TAG appoach 12
Gadent, Naayan Multiple Adjunction in Featue-Based Tee Adjoining Gamma suppots a uniom teatment o wod ode and (ii) that ou appoach staightowadly accounts o mixed dependent/independent deivations which would equie some additional stipulation in Schabes and Shiebe s appoach. 4.4.1 Odeing constaints among modiie auxiliay tees. In TAG, emines and vebal auxiliaies ae modiie athe than pedicative auxiliay tees (c. Section 3.1). Because the Schabes and Shiebe s deinitions systematically associates modiies with independent deivations, all examples in(8a-d) undego an independent deivation and constaints must theeoe be povided to emine the ode o the sibling nodes in the esulting deivation tee. (8) a. The sonatas should have beenbeing playedby Saah. b. The sonatas haveshould beenbeing playedby Saah. c. All the meekats d. The allmeekats To speciy these constaints on simila cases(sot odeing constaints on adjectives and stict odeing constaints on tempoal and spatial advebial phases in Geman), Schabes and Shiebe (1994) suggest the use o linea pecedence constaints (LP) on deivation tee siblings. As illustated by the deivation o example (8c-d) in Figue 8 and 9, in the FB-TAG appoach, such additional constaints ae unnecessay: they simply all out o the eatue constaints encoded in the gamma. Note that even i emines and auxiliay vebs wee to be handled using dependent adjunction, the wod odeing constaints used by Schabes and Shiebe would ail to account o cases such as(9) whee auxiliay vebs ae inteleaved with advebs. (9) John has oten been selected o nomination. In this case, i the auxiliay vebs has and been wee teated as pedicative tees, Schabes and Shiebe s constaint that pedicative tees adjoin above modiie tees would peclude the deivation o (9) and incoectly pedict the deived sentence to be John has been oten selected o nomination. 4.4.2 Odeing constaints among pedicative tees. As discussed by Schabes and Shiebe (1994), auxiliay pedicative tees may impose dieent constaints on the type o sentential complement they accept. Thus example (10a) is coect but not example (10b) because want expects an ininitival complement (peviously shown in(5)). (10) a. Johnwanted toassume that Peteslept. b. John wantedpeteties towalk. While in the Schabes and Shiebe s appoach, selective adjoining constaints ae used to license (10a) and ule out (10b), in the FB-TAG appoach, this can be achieved using eatue constaints. 4.4.3 Odeing Constaints between Pedicative and Modiie Auxiliay Tees. In sentences such as(11a) whee both modiie and pedicative auxiliay tees adjoin to the same addess, the pedicative tees should geneally adjoin above any modiie tees so that the pedicative veb pecedes the modiie in the deived sting. (11) a. John pomised that Pete will leave tomoow. b. Tomoow Johnpomised thatpete will leave. 13
Computational Linguistics Volume xx, Numbe xx To ensue the appopiate lineaisation, the Schabes and Shiebe s appoach intoduces the outemost-pedication ule which stipulates that pedicative tees adjoin above modiie auxiliay tees. In contast, the FB-TAG appoach allows both odes and lets eatue constaints ule out ungammatical sentences such as (11b). This allows the appoach to diectly extend to a counte-example discussed by Schabes and Shiebe (1994)wheeamodiie (heeat whattime)must inactadjoinaboveapedicativetee. (12) At what time did Bockway say Haison aived? At what time did Bockway say Haison aived? aive say at Haison Did Bockway say at what time Haison aived? aive at say Haison Bockway what time what time Bockway Figue 12 Odeed deivation tees o the sentence (12). Dotted lines indicate substitutions and plain lines adjunctions. Schabes and Shiebe s Outemost Pedication pinciple ules out deivation tee on the let hand side. Figue 12 shows two possible deivation tees o the sentence (12) unde the intepetation whee it is the time o aiving (athe than the time o saying) which is questioned. These deivation tees show the two possible elative odeings o the (pedicative) auxiliay tee o say and the(modiie) auxiliay tee at what time. Because the outemost-pedication ule equies that pedicative tees adjoin above modiie tees (and thus occu outemost in the deivation tee), in Schabes and Shiebe s appoach, only the ight hand side deivation is possible thus ailing to deive sentence (12). In contast, since ou appoach does not explicitly constain the elative odeing o pedicative and modiie auxiliay tees adjoining to the same node, both deivations ae possible theeby licensing both (12) and the sentence Did Bockway say at what time Haison aived? 4.4.4 Mixed Dependent and Independent Multiple Adjunctions. In Schabes and Shiebe s appoach, all modiie auxiliay tees undego independent deivation. As shown in Section 2.2 howeve, non-intesective modiies aguably license a dependent deivation while some cases o multiple adjunction may involve both a dependent and an independent deivation. As we shall see in Section 5, ou FB-TAG appoach accounts o such cases by allowing both o independent and dependent deivations, by uling out dependent deivations o intesective modiies and by using eatue constaints to egulate the inteactions between multiply adjoining auxiliay tees. 5. Extending Schabes and Shiebe s LIG Famewok o FB-TAGs We now popose a compilation o FB-TAG to LIG which makes both dependent and independent deivations in FB-TAG explicit. We use this esulting LIG to speciy an Ealey algoithm o ecoveing multiple adjunctions in FB-TAG. This compilation dies in two main ways om that poposed by Schabes and Shiebe (1994). Fist, tee nodes ae associated with eeence vaiables pointing to eatue stuctues. Second, the LIG ules ae modiied and extended with uniication opeations. 14
Gadent, Naayan Multiple Adjunction in Featue-Based Tee Adjoining Gamma 5.1 Featue Stuctues and Reeence Vaiables To account o FB-TAG uniications while allowing o independent deivations, we eplace the eatue stuctues o FB-TAG with eeence vaiables pointing to those eatue stuctues. Each node in the elementay tees is decoated with two eeence vaiables:thetopeeencevaiablep T containstheeeencetothetopeatuestuctue T and the bottom eeence vaiable P B contains the eeence to the bottom eatue stuctue B. The top and the bottom eatue stuctues o a node η can be taced by val(η.p T ) and val(η.p B ) espectively whee P T and P B ae the top and the bottom eeence vaiables decoating the node η and the unction val(p) etuns the eatue stuctues eeed to by the eeence vaiable P. When speciying the pasing algoithm, we use eeence vaiables to ensue the appopiate uniications as ollows. In an independent deivation whee the node η o is adjoined to, ist by β 1 and second by β 2, the bottom eatue stuctue η o.b o η o (i) uniies with the bottom eatue stuctue η 1.B o the oot o β 1 and (ii) is eassigned (:=) to the bottom eeence vaiable η 1.P B o the oot o β 1. When β 2 is adjoined, its oot node will theeoecoectlybe uniied,not withthe bottom eatuestuctueo η o but withthat o η 1. 5.2 LIG Rules with Uniication Opeations Type 3(b) t[η] b[η] Type 4 t[ηη ] b[ηη ] Type 3(a) Type 1/2 t[η 1 ] t[η S ] t[η n ] Type 5 t[ηη ] b[ηη ] Figue 13 LIG vaiant o TAG o the extended deivation in FB-TAG. To suppot both dependent and independent deivations while enocing the coectuniications,wemodiythetagtoligcompilationinsuchawaythattheesulting LIG ules captue the tee tavesal depicted in Figue 13. Independent deivations ae accounted o by the act that adjunction stats and ends at the bottom component o the node being adjoined to (Type 4 and 5 ules). Ou LIG compilation automatically suppots dependent deivations by allowing sequential adjunctions at the oots o auxiliay tees. Type4: Stat oot o adjunction. Fo each elementay tee node η which allows the adjunction o the auxiliay tee with the oot node η, the ollowing LIG poduction ule is geneated. b[..η] t[..ηη ] val(η.p T ) val(η.p T ),η.p B := η.p B 15
Computational Linguistics Volume xx, Numbe xx Type5: Stat oot o adjunction. Fo each elementay tee node η which allows the adjunction o the auxiliay tee with the oot node η, the ollowing LIG poduction ule is geneated. b[..ηη ] b[..η] val(η.p B ) val(η.p B ) Type 6: Stat substitution. Fo each elementay tee node η which allows the substitution o the initial tee with the oot node η, the ollowing LIG poduction ule is geneated(not shown in Figue 13). t[η] t[η ] val(η.p T ) val(η.p T ) To peom multiple adjunction while enocing the appopiate eatue uniications (as depicted in Figue 11), we split Type 3 ules into two subtypes. Type 3(a) ules apply to the oot o auxiliay tees and peom no uniication. By no uniication, they ensue that eatue stuctues ae not blocked o the possibility o the adjunction o the ollowing auxiliay tee and allow o the coect uniications to be caied out o independent deivations. Type 3(b) ules unction as temination o multiple adjunction by uniying the top andbottom eatuestuctueso the node.it is applicabletoalltee nodes except oots o auxiliay tees. Type3(a): teminating adjunction at the oot o the auxiliay tee. Fo each oot node η o the auxiliay tees, the ollowing LIG poduction ule is geneated. t[..η] b[..η] Type3(b): teminating adjunction at any othe node. Fo each node η that is not a oot node o some auxiliay tee and is not maked o substitution, the ollowing LIG poduction ule is geneated. t[..η] b[..η] val(η.p T ) val(η.p B ) Given this set o ules, both dependent and independent deivations ae possible. Fo example, given two auxiliay tees β 1 and β 2 adjoining at the node η in an elementay tee τ, a dependent deivation will occu wheneve the Type 4 ule applies to pedict the adjunction o, e.g., β 2 at the oot o β 1. Convesely, i the Type 3(a) ule appliesattheootoβ 1,ecognitionwillmoveomthetopotheootoβ 1 toitsbottom allowing o Type 5 ule to complete the adjunction o β 1 at the node η and the Type 4 uleappliestopedicttheadjunctionoβ 2 atthenodeη oτ,egisteinganindependent deivation. 5.3 Pasing Algoithm In this section, we pesent ou pasing algoithm o FB-TAGs with dependent and independent deivations. We stat with an inomal desciption o how the algoithm handles the inteactions between uniication and independent deivations. We then go on to speciy the ineence ules making up the algoithm. We do this in two steps. Fist, we pesent a basic set o ules allowing o both dependent and independent deivations. Second, we show how to constain this algoithm to minimize spuious ambiguity. 16
Gadent, Naayan Multiple Adjunction in Featue-Based Tee Adjoining Gamma η o.p T η o.t η o η o.p B η o.b t b (9) (8) (4) (3) t b η η 1.P T η 1.T 1 η 1.P B η 1.B β 1 (7) t b η η 2.P T η 2.T 2 η 2.P B η 2.B β 2 (1) (2) t η η 1.P T η 1.T b 1 η 1.P B η 1.B (6) t η η 2.P T η 2.T b 2 η 2.P B η 2.B (5) (1). Type5,η o.p B F 1,η 1.P B F 1,F 1 = η o.b η 1.B (2). Type3(b),η 1.P T F 2,η 1.P B F 2,F 2 = η o.b η 1.T η 1.B (3). Type3(a),η 1.P T η 1.T,η 1.P B η 1.B (4). Type4,η o.p T F 3,η 1.P T F 3,F 3 = η o.t η 1.T,η o.p B η 1.B (5). Type5,η o.p B F 4,η 2.P B F 4,F 4 = η 1.B η 2.B (6). Type3(b),η 2.P T F 5,η 2.P B F 5,F 5 = η 1.B η 2.T η 2.B (7). Type3(a),η 2.P T η 2.T,η 2.P B η 2.B (8). Type4,η o.p T F 6,η 2.P T F 6,F 6 = η o.t η 1.T η 2.T,η o.p B η 2.B (9). Type3(b),η o.p T F 7,η o.p B F 7,F 7 = η o.t η 1.T η 2.T η 2.B Figue 14 Multipleindependentadjunction oβ 1 and β 2 to η o. Theuniications and eassignmentsae listed in the ode in which they ae peomed duing the ecognition pocess. 5.3.1 Independent Deivations and Featue-Stuctue Uniication. Beoe speciying the pasing algoithm, we illustate by means o an example the inteplay between multiple independent adjunction and eatue stuctue uniications. Figue 14 displays the eatue uniications and eassignment peomed duing the ecognition pocess o a multiple independent adjunction. The linea odeing o the equations elects the ode o the pasing completion opeations. Given the auxiliay tee β 1 and the adjunction site η o, the pictueshows that uniyingthebottomeatuestuctueotheootnodeoβ 1 withthebottomeatuestuctue o η o (Step 1: Type5,η o.b η 1.B) occus beoe the bottom eeence vaiable o η o is eassigned to the bottom eatue stuctue o the oot o β 1 (Step 4: Type4,η o.p B η 1.B). Also the eassignment ensues that the ollow up adjunction o β 2 at the node η o has access to the bottom eatue o the oot o the pevious auxiliay tee β 1 (Step 5: Type5,η 1.B η 2.B). At the end o the adjunction (Step 9), the Type3(b) ule ensues that the top and the bottom eatues o the oot o the last auxiliay tee (hee, β 2 ) adjoined aeuniied (η 2.T η 2.B). As we shall see below, this coect odeing between uniication and eassignment ollows om the poposed Ealey algoithm. Type 4 completo ules complete the pediction tiggeed at the oot o an auxiliay tee ( Stat oot o adjunction ) while Type 5 completo ules complete the pediction tiggeed at the oot node o an auxiliay 17
Computational Linguistics Volume xx, Numbe xx tee ( Stat oot o adjunction ). Since completion opeates bottom-up, it ollows that Type 5 Rules apply beoe Type 4 ules. Thus, when adjoining an auxiliay tee β 1 to a node η o, the Type 5 completo ules, uniying the bottom eatue stuctue o the oot node o β 1 with the bottom eatue stuctue o the node η o, occus beoe the Type 4 completouleswhicheassignthebottomeeencevaiableoη o tothebottomeatue stuctueo the oot o β 1. 5.3.2 Ineence Rules. The pasing algoithm o FB-TAG is a modiication o the algoithm pesented by Schabes and Shiebe (1994). It is a chat-based pasing method based on the Ealey type deduction system. Each item in the chat is o the omat N[..η] Γ,i,j,k,l whee N is some LIG nonteminal (i.e., t o b) and, Γ and pesent the sequences o LIG nonteminals associated with stacks o node indices. The indicesi,j,k,andlaemakesintheinputstingshowingtheecognizedpotion 5 :the ecognized item stats in position i, ends in position l and i η dominates a oot node, the tee dominated by the oot node stats in j and ends in k. I the oot node is not dominated by the ecognized nonteminal sequence Γ, the values o j and k ae taken to be the dummy value. As in Ealey algoithms, the sepaates the nonteminal sequence Γ whichwas pasedomthe nonteminal sequence yet tobepased. The ist thee types o ules (Scanne, Pedicto and Type 1/2 Completo) ae identical to those intoduced by Schabes and Shiebe (1994) and do not involve any uniication opeations. The Type 3(b) completo ule enoces top and bottom uniication on all nodes which ae not the oot o an auxiliay tee while the Type 3(a) completo ule pevents top and bottom uniication at the oot o auxiliay tees. The Type 4 completo ule uniies the top eatue stuctue o the oot o the auxiliay tee with the top eatue stuctue o the adjunction site. In addition, it ensues that oncompletionoanadjunctionatnodeη,thebottomeatuestuctueoη iseassigned to the bottom eatue stuctue labelling the oot o the auxiliay tee. In this way, the uniications occuing in an independent deivation will mio those occuing in a dependent one in that any ollowing adjunction will induce uniications as i it wee happening at the ootnode η o the pecedingauxiliaytee(not atη). On completion o a oot node pediction (the tee dominated by the oot o the auxiliay tee has been ecognized), the Type 5 completo ule uniies the bottom eatue stuctue o the oot o the auxiliay tee with the bottom eatue stuctue o the adjunction site. Finally, the Type 6 completo uniies the top eatue stuctue o a substitution node with the top eatuestuctueo theoot o the teebeing substituted in. Scanne: b[..η] Γ w,i,j,k,l b[..η] Γw,i,j,k,l+1, w = w l+1, Iw (ateminal symbol) occus atposition l+1,the scanne ule ceatesa new itemwhose spanextendstol+1. 5 The indices i, j, k, l have been used in pevious pasing algoithms o tee-adjoining gammas (Vijay-Shanka and Joshi 1985a; Schabes and Joshi 1988; Schabes 1991). They delive the same unctionality hee. 18
Gadent, Naayan Multiple Adjunction in Featue-Based Tee Adjoining Gamma Pedicto: N[..η] Γ N [µ],i,j,k,l N, [µ] Θ,l,,,l Pedictoules aepoducedo all typeso poductionules. N and N aeligvaiablestaking the valueto b. Γ, andθaethe sequences o LIG nonteminals associated with stacks o node indices. µ is a sequence o node indices. Type1and2Completo: b[..η] Γ t[η 1 ],m,j,k,i t[η 1 ] Θ,i,j,k,l b[..η] Γt[η 1 ],m,j j,k k,,l, η 1 not a ootnode Type1and2Completo ules pemit completing Rules 1and2wheneve the top oachild node is ully ecognized.hee,t[η 1 ] has beenully ecognizedas the substing betweeniand l (i.e.,w i+1...w l ).Theeoe, t[η 1 ] canbecompleted inb[..η].i one o t[η 1 ] o b[..η] dominates the oot node othe tee,the inal b[..η]will haveindices associatedwith the substing ecognized by the oot subtee. The opeation is deined as ollows: x, iy =. y, ix =. x y = x, ix = y. undeined, othewise. Type3(a)Completo: t[..η] b[..η],i,,,i b[..η] Θ,i,j,k,l, t[..η] b[..η],i,j,k,l, η anauxiliayteeoot node This ule is used tocomplete the pedictiono anauxiliayteeootedinη. Once the auxiliay tee dominated by b[..η] has been ecognized, the auxiliay tee itsel is completely ecognized. As explained above, thee is inthis case noeatueuniication betweenthe top andthe bottom o the oot o the auxiliay tee. Type3(b)Completo: t[..η] b[..η],i,,,i b[..η] Θ,i,j,k,l, t[..η] b[..η],i,j,k,l val(η.p T ) val(η.p B ), η not anauxiliayteeootnode This completion ule ensues the uniication o the top and bottom eatue stuctueso allnodes that aenot the oot node oanauxiliaytee. 19
Computational Linguistics Volume xx, Numbe xx Type4Completo: b[..η] t[..ηη ],i,,,i t[..ηη ] Θ,i,j,k,l b[..η],j,p,q,k, b[..η] t[..ηη ],i,p,q,l val(η.p T ) val(η.p T ) η.p B := η.p B The auxiliayteeassociatedwiththe pedictedadjunction (t[..ηη ])atthe node η andthe subteedominated bythe node η (below b[..η]) ae completed, hence, b[..η] can be completely ecognized with this adjunction. The associated eatue uniication uniies the content o the top eeence vaiableothe adjoining node site η withthe content othe top eeence vaiableothe oot node η othe adjoined auxiliaytee.atethe successul adjunction o this adjoining tee, the bottom eeence vaiable o the adjoining nodesite η is eassignedto thecontent o the bottom eeencevaiableo theoot node η o theadjoined auxiliaytee. Type5Completo: b[..ηη ] b[..η],i,,,i b[..η] Θ,i,j,k,l, val(η.p B ) val(η.p B ) b[..ηη ] b[..η],i,i,l,l The oot node pediction can be completed when the adjunction has been peomedandthe bottom pato theadjoining node site η has been ecognized. The associated eatue uniication uniies the content o the bottom eeence vaiable o the adjoining node site η with the content o the bottom eeencevaiableo the oot node η o the auxiliayteebeing adjoined. Type6Completo: t[η] t[η ],i,,,i t[η ] Θ,i,,,l, val(η.p T ) val(η.p T ) t[η] t[η ],i,,,l This ule completes the substitution at the node η. The associated eatue uniication uniies the content o the top eeence vaiable o the node η with the content o the top eeencevaiableo the oot node η o the initial tee. Given these ineence ules, the ecognition pocess is initialized using axioms o the om t[η s ] Γ,0,,,0 oeachulet[η s ] Γwheeη s istheootnodeoaninitial tee labeled with the stat symbol. Given an input sting w 1...w n to be ecognized, the goal items inthe chataeothe om S t[η s ],0,,,n.Once atleast one goal item is ound in the chat, the ecognition pocess succeeds and the sting is successully accepted by the gamma, othewise it is ejected. We ee the eade to Appendix (c. Figue 2) o a ailed example o the ecognition o the sentence all the meekats using the poposed ineence system. Note also that while the ecognition algoithm we descibed uses uneduced ules i.e., geneated gamma ules maintaining the ull inomation o nonteminals and the 20