Compact Representation of Ambiguities. Generating Sentences by Grammar Specialization. Compact Representation of Ambiguities

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1 Compct Represettio of Ambiguities Geertig Seteces by Grmmr Speciliztio Jürge Wedekid Prsig Chrt: represets ltertive sytctic lyses pcked f-structures (cotexted costrit stisfctio, Mxwell & Kpl 91) Geertio Chrt-bsed pproches (Ky 96, Neum 98; bsed o Shieber 88) Chrcteriztio: 2 phses 1. phse: costructio of compct represettio of ltertive costituet structures/f-structures/strigs 2. phse: eumertio of prticulr lyses 1 2 Compct Represettio of Ambiguities Compct Represettio of Ambiguities Lg s forml chrcteriztio of the 2 phses (for cotext-free chrt prsig) 1. phse: grmmr speciliztio Give: CFG G & iput strig s Costruct: G s = chrt if s L(G) the L(G s ) = if s L(G) the L(G s ) = {s} d G s ssigs s the sme prse trees s G 2. phse: productio of prses by stdrd geertio lgorithm pplied to G s Our pproch to geertio (joit work with Ro Kpl) 1. phse: grmmr speciliztio Give: LFG G & cyclic f-structure f (iput) Costruct: G f (cotext-free) G f is speciliztio of the cotext-free bckboe of G G f derives ll strigs tht G reltes to f G f simultes ll derivtios of f i G 2. phse: productios of strigs by stdrd geertio lgorithm pplied to G f 3 4 LFG Represettios LFG Grmmr Setece, costituet structure d fuctiol structure Rules ( SUBJ)= S NP DET N studet c-structure VP V fell SUBJ studet NUM SG CASE NOM PAST fll<subj> f-structure NP DET N Cotext-free rules defie vlid c-structures ( )= studet Aottios re isttited t tree odes to give f(uctiol) descriptios = costrits o correspodig f-structure F-structure results from the miiml model of f-descriptio 6 1

2 NP S ( SUBJ)= ( )= studet NP S ( SUBJ)= ( )= studet DET N 6 studet 7 V DET N 6 studet 7 V 1. Well-formedess of costituet structure The whole structure is licesed by the rules 2. Isttitio of the metvribles 7 8 NP S ( SUBJ)= = ( )= studet NP S ( SUBJ)= = ( )= studet DET N 6 studet 7 V DET N 6 studet 7 V 2. Isttitio of metvribles by the licesed dughter 2. Isttitio of metvribles by the licesed dughter 9 10 NP S ( SUBJ)= = = = ( )= studet NP S ( SUBJ)= = = = = ( )= studet DET N 6 studet 7 V DET N 6 studet 7 V 2. Isttitio of metvribles by the licesed dughter 2. Isttitio of metvribles by the licesed dughter

3 NP S ( SUBJ)= = = = ( SPEC)=INDEF ( NUM)=SG = ( )= studet ( NUM)=SG ( SUBJ)= = = = ( SPEC)=INDEF ( NUM)=SG = ( )= studet ( NUM)=SG DET N 6 studet 7 V ( )= fll<subj> ( )=PAST V ( )= fll<subj> ( )=PAST 2. Isttitio of metvribles by the licesed dughter 3. Costructio of the f-descriptio = uio of the isttited descriptios of the licesig rules V {( SUBJ)=, = } { =, = } { = } ( SPEC)=INDEF, ( )= studet, { ( NUM)=SG { ( NUM)=SG ( )= fll<subj>, { ( } )=PAST } } S NP DET 6 N studet 7 V ( SUBJ)=, =, =, =, {( SUBJ)=, = } =, = ( 3 SPEC)=INDEF, ( NUM)=SG, ( )= studet, ( NUM)=SG, { =, = } { = } ( )= fll<subj>, ( )=PAST ( SPEC)=INDEF, ( )= studet, { ( NUM)=SG } { ( NUM)=SG } ( )= fll<subj>, { ( } )=PAST 3. Costructio of the f-descriptio = uio of the isttited descriptios of the licesig rules 3. Costructio of the f-descriptio = uio of the isttited descriptios of the licesig rules 1 16 ( SUBJ)=, =, =, =, ( SUBJ)=, = } =, S = ( DET N 3 SPEC)=INDEF, ( NUM)=SG, ( )= studet, ( NUM)=SG, { =, = } { = } ( )= fll<subj>, ( )=PAST NP T ( SPEC)=INDEF, ( )= studet, ( NUM)=SG } { ( NUM)=SG } DET N V ( )= fll<subj>, { ( } )=PAST 6 studet 7 ( SUBJ)=, =, =, =, {( SUBJ)=, = } =, = ( 3 SPEC)=INDEF, ( NUM)=SG, ( )= studet, ( NUM)=SG, { =, = } { = } ( )= fll<subj>, ( )=PAST ( SPEC)=INDEF, ( )= studet, { ( NUM)=SG } { ( NUM)=SG } ( )= fll<subj>, { ( } )=PAST SUBJ 3 4 b c d PAST fll<subj> SPEC NUM e f g studet INDEF SG miiml model M 4. Stisfibility test d costructio of the miiml model 4. Stisfibility test d costructio of the miiml model

4 The LFG Derivtio Reltio SUBJ 3 4 b c d PAST fll<subj> SPEC NUM e f g I. Abstrctio from the odes: M L AV studet INDEF SG II. Abstrctio from the uiverses: [M L AV ] Give rbitrry LFG grmmr G Δ G = { (s,f ) G ssigs to strig s the f-structure f } Prser: Pr G (s) = { f Δ G (s,f ) } 4. Stisfibility test d costructio of the miiml model Geertor: Ge G (f) = { s Δ G (s,f ) } Forml Properties Pr G (s) is fiite set, for every G d s. Proscriptio of obrchig domice chis i c- structure (off-lie prsbility) gurtees fiite umber of trees for ech strig, d ech tree supports fiite umber of f-structures Ge G (f) is ifiite, for some G d f. Loger d loger strigs with the sme f-structure: G: f: [X Y] S S A Ge G (f) = + S A A ( X)=Y S A S A S A Reductio of the f-descriptio Spce Give: -ry LFG rule r Ist(r,(t,t 1.. t )) Uio of the isttited ottios of r i.e. - ll occurreces of re substituted by t, - ll occurreces of i the ottio of the j-th dughter re substituted by t j r = ( SUBJ)= Ist(r,(, ( SUBJ) )) = {( SUBJ)=( SUBJ), =} Reductio of the f-descriptio Spce Reductio of the f-descriptio Spce Give: -ry LFG rule r j is m-defible i Ist(r,(t,.. )) iff Ist(r,(t,.. )) I _ j = (t σ) S ( SUBJ) NP 1 = ( SUBJ) = ( SUBJ)= ( SUBJ) = 2 joh V ADV ( SUBJ ( )= joh = 4 ( ( ADJ)=( ELE) = fell 6 quickly 7 ( ( )= fll<subj> ( ) = quickly ( ( )=PAST

5 Reductio of the f-descriptio Spce Reductio of the f-descriptio Spce Two descriptios D d D with miiml models M d M re equivlet iff M L AV M L AV Remiig odes: d ll dughters which re ot m- defible However: their deottio is biuique T j ( σ) ( j σ ) if j is ot m-defible the from the descriptio it cot follow: j = for Τ Suppose: j = the j = ( j σ ) d σ = 0 thus ( σ) = j 2 26 Reductio of the f-descriptio Spce Reductio of the f-descriptio Spce F c e g SUBJ ADJ b c d e fll<subj> PAST ELE C F Biuiquely reme ll dughters which re ot m-defible Reducig substitutio = compositio of the defiitios d the remig substitutio g f joh h quickly ( SUBJ) Itroduce coicl costts for ech elemet g tht is ot deoted by tomic feture vlue Nodes ot occurrig i the f-descriptio Reductio of the f-descriptio Spce Determitio of the f-descriptio Spce F c e g SUBJ ADJ b fll<subj> f c joh d PAST e g h ELE quickly C F ll terms which deote i the expsio (but do ot refer to the deottio of tomic feture vlue) C F d ( SUBJ) ( ADJ) ( g ELE) Appropritely isttited rules Give m-ry LFG rule r d terms t, t 1.. t m of T F (r, (t, t 1.. t m )) is ppropritely isttited iff (i) if t j = the j does ot occur i Ist(r,(t,.. m )) (ii) if j is m-defible i Ist(r,(t,.. m )) the Ist(r,(t,.. m )) I _ j = t j (iii) otherwise t j C F d t j = / t d t j = / t i for ll t i, i = / j plus ll terms tht re obtied from them by substitutig the ew costts by ( SUBJ) ( ADJ) ( ELE) plus T F 29 30

6 Determitio of the f-descriptio Spce Comptible isttitios IR F = set of ll ppropritely isttited rules fiite Two ppropritely isttited rules (r, (t, t 1.. t m )), (r, (t, t 1.. t l )) re comptible iff ll dughters which re ot m-defible re pir-wise distict isttited IRD F = set of ll subsets IR of IR F such tht (i) the isttited descriptios of the rules i IR together describe f fiite i.e. they hve miiml model M with M L AV f (ii) ll rules i IR re pir-wise comptible Give: LFG G = (N,T,S,R) d iput f-structure [F] Costruct: CFG grmmr G F = (N F,T F,S F,R F ) S F = ew strt symbol (ot i N T) N F = {S F } {A:t:IR A N, t T F, IR is subset of set i IRD F } R F strt rules for every IR i IRD F, R F cotis rule S F S::IR T F = {:t: T, t T F } R F The other rules ( SUBJ) For every LFG rule r = A X 1.. X m R F cotis rules of the form D 1 D m S: S ( SUBJ)= NP Joh NP:( SUBJ) NP VP: 2 (, ) ( SUBJ)= = 2 (, ) (, ) SUBJ PAST joh fll<subj> A:t:IR X 1 :t 1 :IR 1... X m :t m :IR m such tht Joh: V: V 3 ( )= joh = (, ) (i) IR = {(r,(t,t 1.. t m ))} IR 1.. IR m (ii) ech isttited rule occurrig i {(r,(t,t 1.. t m ))} IR i or IR i IR j ist selfcomptible fell: ( )= fll<subj> ( )=PAST

7 ( SUBJ) ( SUBJ) (, ( ) SUBJ) ) ( SUBJ)= S: NP Joh (((, SUBJ) 3 ), ) NP:( SUBJ) VP: 2 (,, ) 4 ) (, ( SUBJ) ) ( SUBJ)= IR 1 IR 2 S::IR NP Joh (( SUBJ), ) IR 3 NP:( SUBJ):IR 1 VP::IR (, ) IR 4 ( ( 4,, ) ) Joh: V: 4 fell: 1. Replce isttitig odes by the vlues of the reducig substitutio Joh::IR 3 V::IR (, ) 4 fell::ir IR S:: (, ( SUBJ) ) ( SUBJ)= NP Joh (( SUBJ), ) (, ) (, ) S:: (, ( SUBJ) ) ( SUBJ)= NP Joh (( SUBJ), ) (, ) (, ) NP Joh (( SUBJ), ) NP:( SUBJ): (, ) 1 VP:: (, ) Joh:: (, ) term compoet simultes reducig substitutio V:: ( SUBJ) fell:: NP:( SUBJ): NP Joh (( SUBJ), ) Joh:: rule compoet cotis ll rules licesig the subderivtio (with the licesed odes replced by (, ) VP:: (, ) (, ) V:: fell:: (, ( SUBJ) ) ( SUBJ)= NP Joh (( SUBJ), ) (, ( SUBJ) ) ( SUBJ)= NP Joh (( SUBJ), ) S:: (, ) (, ) S:: (, ) (, ) NP:( SUBJ): NP Joh (( SUBJ), ) Joh:: (, ) VP:: (, ) NP:( SUBJ): NP Joh (( SUBJ), ) Joh:: (, ) VP:: (, ) rule compoet cotis ll rules licesig the subderivtio (with the licesed odes replced by (, ) V:: fell:: rule compoet cotis ll rules licesig the subderivtio (with the licesed odes replced by (, ) V:: fell::

8 S:: (, ( SUBJ) ) ( SUBJ)= NP Joh (( SUBJ), ) (, ) (, ) S:: (, ( SUBJ) ) ( SUBJ)= NP Joh (( SUBJ), ) (, ) (, ) NP:( SUBJ): NP Joh (( SUBJ), ) Joh:: (, ) VP:: (, ) NP:( SUBJ): NP Joh (( SUBJ), ) Joh:: (, ) VP:: (, ) rule compoet cotis ll rules licesig the subderivtio (with the licesed odes replced by (, ) V:: fell:: rule compoet cotis ll rules licesig the subderivtio (with the licesed odes replced by (, ) V:: fell:: Observtios d Cosequeces Observtios d Cosequeces Two phses of LFG geertio Costructio of the CFG grmmr G F for [F] Well-uderstood, fiite represettio of (ifiitely) my strigs Apply CF geertio lgorithm to produce strigs Use pumpig lemm to select short strigs Use PCFG techology to select most probble strig Select by other heuristics Decidble properties of Ge G (f) It is decidble whether Ge G (f) is empty, fiite, or ifiite. Decidbility of geertio does ot deped o proscribig obrchig domice chis (off-lie prsbility), ulike prsig Observtios d Cosequeces Observtios d Cosequeces Exteds to other forml devices: sets, fuctiol ucertity, fuctiol precedece, etc. (but ot to the restrictio opertor) Applies to other cotext-free bsed uifictio formlisms: PATR, ALE, perhps some versios of HPSG