ilg, irlg. · ] a b o ok? c. * [of which month] i do es john strik e his donk ey [every da y (3) a....

LinearIndexedGrammars

Formaldescriptionsofnon�local

dependencies

ChristianWartena

linguatecentwicklung&servicesGmbH

Hebelstraÿe14

69115Heidelberg

Germany

[email protected]

ChristianWartena,LinearIndexedGrammars

1

Survey

1.TheBasicIdea

2.VariationsontheInheritanceMechanism

�Basicvariants:IG,LIG,DIG.

�Restrictingtheruleformat:ILG,IRLG.

�RestrictionsonLIGs.

3.VariationsontheStorageStructure

�Control:makingdependenciesexplicit

�Usingabstractstorages

�Non�localdependenciesinTAG

�Compositestorages


2

1

TheIdea

NP

NP

thekennel

R

PP

P in

NP

which

S/PP

NP

Fido

VP/PP

Vtends

VP/PP

to

VP/PP

Vsleep

PP/PP

t


3

CP

NPi

wem

C'[i]

Vk

stehst

IP[ki]

NP

du

VP[ki]

t i

V'[k]

PP

innichts

V[k]

t k

V nach


4

2

VariationsontheInheritanceMechanism

IndexedGrammar(Aho1968)

Anindexedgrammarisa�vetupleG=(N;�;I;R;Ain)

N

nonterminalsymbols

�

terminalsymbols

I

indexsymbols

R

productionrulesoftheform:A�!

�

Ain

thestartsymbol

whereA2N,�2(NI

�

[�)�and�2I

�.

�) G�,if�=�A�#0�

A�!

X1� 1X2� 2:::Xk� k2R

�=�X1� 1#1X2� 2#2:::Xk� k#k�

where

�#i=#0ifXi2N,

�� i#i=�ifXi2�.


5

Example1

S

!

Tf

T

!

Tg

T

!

ABA

Af!

a

Bf!

b

Cf!

c

Ag!

aA

Bg!

bBCC

Cg!

bC Af

Af

Agf

a

aa

a

aa

Aggf

Aggf A

gf

b

b

b

b

b

Cf

Cf

Cf

Cf

Bf

b

b

b

Cgf

Cgf

Bgf

b

Bggf

Tggf

TgfT

fS

L=fanbn

2anjn�1g


6

LinearIndexedGrammar(Gazdar1988)

Anlinearindexedgrammarisa�vetupleG=(N;�;I;R;Ain)

N

nonterminalsymbols

�

terminalsymbols

I

indexsymbols

R

productionrulesoftheform:

A�!

�1B�!�2

A�!

w

Ain

thestartsymbol

whereA;B2N,�1;�22(NI

�

[�)�,w2�

�

and�;�2I

�

andwhere!isaspecialsymbolnotinN[�[I

�) G�,ifeither(1)or(2).

(1)

�= A�#Æ

A�!

�1B�!�22R

�= �1B�#�2Æ

(2)

�= A�Æ

A�!

w2R

�= wÆ

where

A;B2N,�1;�2; ;Æ2(NI

�

[�)�,�;�;#2I

�,w2�

�


7

CopyingofStacks(I)

VP[f]

IP[f]V

P[gf] t 2

t 3

NP

NP3

NP2

IP[gf]

V3

V2

V2[f]

V1V

1[gf]

VP[gf]

VP[f]

VP[]

NP1

IP

V3

V2

V1

NP

IP

VP[]

VP[f]

IP[f]

VP[f]

VP[gf] t

t

NP

NP

NP2

IP[gf]V

P[gf]

VerbraisinginIG

VerbraisinginLIG


8

CopyingofStacks(II)

Fidosleepsin

Marymade

Thekennel

and

S/NP

S/NP

S/NP

S/NP

which

R

NP

NP

Coordinationofconstituentswithgaps


9

DistributedIndexedGrammar(Staudacher1994)

Andistributedindexedgrammarisa�vetupleG=(N;�;I;R;Ain

N

nonterminalsymbols

�

terminalsymbols

I

indexsymbols

R

productionrulesoftheform:A�!

�

Ain

thestartsymbol

whereA2N,�2(NI

�

[�)�and�2I

�.

�) G�,if�=�A�#0�

A�!

X1� 1X2� 2:::Xk� k2R

�=�X1� 1#1X2� 2#2:::Xk� k#k�

where

�� i#i=�ifXi2�

�#1�#2�:::�#k=#0.


10

Example2

t

racontato

V

NP[j]

tPP[i]

V'[j]

VP[ji]

IP[ji]

NP

abiano

chestorie

CP[i]

midomando

acui

IP[i]

PP

CP

(1)a.Tuofratello,acuimidomandochestorieabbianoraccontato,

eramoltopreoccupato

b.Yourbrother,whomIwonderwhatstoriestheyhavetold,

wasveryworried


11

WeakGenerativeCapacity

context�sensitive

IG

DIGL

IGCFG

Figure1:Containmentrelationsofsomeclassesofindexed

grammars

LIGsareMildlyContextSensitive

1.LIGsonlygeneratelimitedcross-serialdependencies.

2.LIGshavetheconstantgrowthproperty.

3.LIGscanbeparsedinpolynomialtime.


12

CrossingDependencies

a

b

b

a

b

a

b

b

a

b

S[]

a

S[a]

b

S[ba]

b

S[bba]

a

S[abba]

b

T[babba]

b

T[abba]

a

T[bba] b

T[ba] b

T[a]

a

Figure2:CrossingdependenciesinastringandinaLIG�

derivationtree


13

VariationsontheRuleFormat

IndexedLinearGrammar

(DuskeandParchmann1984)

Anindexedlineargrammar

1

isanIG(DIG,LIG)

G=(N;�;I;R;Ain)whereeachproductionruleisoftheform:

A�!

�1B��2

whereA2N,B2N�,�1;�22�

�

and�;�2I

�.

Proposition1

fL(G)jGaCFGg�fL(G)jGaILGg�fL(G)jGaLIGg

Note.Foraproofnotethatfanbncndnjn2INgisanILLbut

fanxkykzkbncndnjn;k2INgisnot.

1IndexedLinearGrammars(indexedgrammarswithlinearrules)usuallyarecalledLinear

IndexedGrammars.HereIwillusethetermILGtodistinguishthemfromindexedgrammarswith

linearindexinheritance,thatarecalledLIGaswell.


14

IndexedRightLinearGrammar(Aho1968)

AnindexedrightlineargrammarisanILGG=(N;�;I;R;Ain)

whereeachproductionruleisoftheform:

A�!

�B�

whereA2N,B2N�,�2�

�

and�;�2I

�

Proposition2

fL(G)jGaIRLGg=fL(G)jGaCFGg

Note.

�ConstructaPDAwithNthestatesofthePDAandIthe

stackalphabetsuchthat:

Æ(A;a;f)=

f(B;g)jAf!

aBgg

�ForthereversedirectionconstructaCFGsuchthat

Af!

aBgif(B;g)2Æ(A;a;f)


15

RestrictionsonLIGs

(1)a.WhoididMarythink[thatJohnlovest i]?

b.WhoidoesMaryknow[afriendoft i]?

c.[Ofwhichmonth] idoesJohnhate[everydayt i]?

(2)a.*Whoidid[thatJohnlovest i]annoyMary?

b.*WhoididJohngive[afriendoft i]abook?

c.*[Ofwhichmonth] idoesJohnstrikehisdonkey[everyday

(3)a.WenidenktMaria,[daÿHansliebtt i]?

whothinksMaria

thatHansloves?

b.[VonPeter] ihatMarianoch[keinenFreundt i]kennengel e

ofPeter

hasMariayet

no�ACCfriend

knowlearn

(4)a.*WenibeunruhigtMaria,[daÿHansliebtt i]?

who�ACCannoysMariathatHansloves

b.*[VonPeter] ihatHansnoch[keinerFreundint i]eineRos

ofPeter

hasHansyet

no�DATfriend

arose


16

RightLinearIG(MichaelisandWartena1997)

ArightlinearindexedgrammarisanLIGG=(N;�;I;R;Ain)

whereeachproductionruleisoftheform:

A�!

�B�!

A�!

w

whereA;B2N,�2(NI

�

[�)�;w2�

�

and�;�2I

�.

Proposition3

fL(G)jGaRLIGg=

fL(G)jGaCFGg

Note.

�EveryCFGinGreibachnormal-formisanRLIG.

�AnRLIG-derivationcanbesimulatedbyanPDA,sincethere

isno�wrapping�:

GeneralLIG-case

�= A�#Æ

A�!

�B��2R

�= �B�#�Æ

RLIG-case

�= A�#Æ

A�!

�B�2R

�= �B�#Æ


17

RelevanceofRLIGs

�Achievingamaximumofstronggenerativecapacitywitha

minimumofweakgenerativecapacity

�Howmuchcanwedowith(weakly)contextfreeformalisms?

�SimilaritywithTreeInsertionGrammars(SchabesandWa-

ters1995)

�LIGsarenotevenneededforcrossserialdependenciesin

Dutch:ERLIGs

ExtendedRightLIG(MichaelisandWartena1999)

AnextendedrightlinearindexedgrammarisanLIG

G=(N;�;I;R;Ain)whereeachproductionruleisoftheform:

A�!

�B�!w

A�!

w

whereA;B2N,�2(NI

�

[�)�;w2�

�

and�;�2I

�.


18

Proposition4

fL(G)jGaCFGg�fL(G)jGaERLIGg�fL(G)jGaLIGg

Note.ERLIGsarenotclosedundersubstitution,whileLIGs

are.

a a a

d d d x i

x i�

1 x 0

b b b

c c c

� �

Figure3:ERLIG�Structureoffa

nb

ncnwd

n

jn2IN;w2L

0g.


19

Example3

'...thatGijshearsherteachingthechildrentosingpsalms'

CP

C dat

IP

NP

Gijs

VP0

VP

0 0[f]

VP

0 0[gf]

IP[gf]

NP

haar

VP1[gf]

IP[f]

NP

dekinderen

VP2[f]

NP

psalmen

V2

tV1 tV

0

hoort

V1

leren

V2

zingen

Figure4:DutchVR-StructureasanERLIG�tree


20

3

VariationsontheStorageStructure

�LIGsusepushdownsofindices

�LIGshavederivationtreeswithcontextfreespines

�Dopushdownsconstituteanappropriatewaytostorein-

dices?

CP

NPi

wem

C'[i]

Vk

stehst

IP[ki]

NP

du

VP[ki]

t i

V'[k]

PP

innichts

V[k]

t k

V nach


21

LinearDistinguishedGrammar(Weir1992)

AnLDGisatupleG=(N;�;R;Ain)

N

nonterminalsymbols

�

terminalsymbols

Ain

2Nthestartsymbol

R

productionrulesoftheform:A!

�1X!�2

LinearControlledGrammar(Weir1992)

AnLCGisapairK=G=H,

G=(N;�;R;Ain)aLDG,

HalanguageoverR(�thecontrollanguage�)

��) G�if�= (A;!)Æ

r=A!

�1X!�22R

�= �

0 1(X;!r)�

0 2Æ

�L(G=H)=fa1a2:::anj(S;�))

�

G

(a1;!1)(a2;!2):::(an;

ai2�;!i2Hfor1�i�ng.


22

Storage

AStorageisatupleS=(C;c�;P;F;m)

C

Con�gurations

c �2CInitialoremptycon�guration

P

Predicatesymbols

F

Instructionsymbols

m

Meaningfunctionassociatingevery

p2Pwithamappingm(p):C!

ftrue;falseg

f2Fwithapartialfunctionm(f):C!

Cassoziert.

Example4

Spd=(C;f�g;P;F;m)with

C��

�

forsome�nitealphabet�

P=ftop( )j 2�g

F=fpush( )j 2�g[fpopg[fidg

m(top( ))(a�)=(a= )

m(id)(a�)=a�

m(pop)(a�)=�

m(push( ))(a�)= a�


23

ContextFreeLinearSGrammar(Weir1994)

ACFL�S�Gisde�nedasatupleG=(N;�;S;R;Ain).

N

non�terminalsymbols

�

terminalsymbols

S

=(C;c0;P;F;m)astorage

Ain

2Nstartsymbol

R

productionrulesofthefollowingforms:

A!

if�then� 1Bf� 2

A!

if�thenw

where

A;B2N,�2BE(P),� 1;�22(N[�)�,f2F,w2�

�.

��) G�,ifeither(1)or(2).

(1)�=�(A;c)�

A!

if�then� 1Bf� 22

�=��

0 1(B;m(f)(c))�

0 2�

m(�)(c)=true

(2)�=�(A;c)�

A!

if�thenw2R

�=�w�

m(�)(c)=true

�L(G)=fwj(Ain;c�)) Gwg


24

S�Automaton

AnS�automatonisde�nedasatupleM

=(Q;�;S;Æ;q0;QF)

Q

states

�

inputalphabet

S

=(C;c�;P;F;m)astorage

q 0

2Qinitialstate

QF

�Q�nalstates

Æ

�Q��BE(P)�Q�F,transitionrelation

�(q1;xw;c1)`M

(q2;w;c2)if(q1;x;�;q2;f)2Æ

m(�)(c 1)=true

m(f)(c 1)=c 2

�LQ(M)=fwj(q0;w;c�)`

� M

(q;�;cf);q2Qfg

�LC(M)=fwj(q0;w;c�)`

� M

(q;�;c�)g

�LX(S)=fLjL=LX(M);M

isaS�automatong


25

Proposition5 fL

(G=H)jLaLDG;H2L(S)g

=

fL(G)jGaCFL�S�Gg


26

Ar 1

B r 2

Dr 3

d

Er 4

e

(d;r3)(e;r 1r 2r 4)

(q0;r1r 2r 4;c0)`

(q1;r2r 4;c1)`

(q2;r4;c2)`

(q4;�;c4)

(q0;r3;c0)`

(q3;�;c3)

(hA;q0i;c 0)

(hB;q1i;c 1)

(hD;q0i;c 0)

(hd;q3i;c 3)

d

(hE;q2i;c 2)

(he;q 4i;c 4)

e


27

(A;c0)

(B;c1)

(D;c0)

(d;c3)

d

(E;c2)

(e;c4)

e

Ar 1

B

r 2

Dr 3

d

Er 4

e

(d;r3)(e;r 1r 2r 4)

(qA;r1r 2r 4;c0)`

(qB;r2r 4;c1)`

(qE;r4;c2)`

(qf;�;c4)

(qD;r3;c0)`

(qf;�;c3)


28

TAG:AnotherwayofgeneratingCF�Spines

Proposition6

fL(G=H)jLaLDG;HaCFLg=

fSurf(G)jGaTAGg

d c

ba

�SNA

SSNA

DerivedTree

d c

ba

r 3r 2SNA

SSNA

AuxiliaryTree

�r 1S

InitialTree

LDG

r 1=

S

!

�

r 2=

S

!

aSd

r 3=

S

!

bSc

CFControlGrammar

Ain

!

Sr 1

S

!

�

S

!

r 2Sr 3


29

�Assumew.l.o.g.thattheunderlyingLDGhasonlyonenon�

terminalsymbol!

LDG

r 1=

S

!

aS

r 2=

S

!

bS

r 3=

S

!

Sa

r 4=

S

!

Sb

r 5=

S

!

�

CFControlGrammar

i:

S

!

Ar 5

ii:A

!

r 1Ar 3

iii:A

!

r 2Ar 4

iv:A

!

�

b

b

SNA

SSA(ii;iii)

SNA

iii:

SSA(ii;iii)

SNA

a

a

SNA

ii:

�SSA(ii;iii)

i:


30

storage

controllanguage

Gazdar'81

pushdown

CFL

Rambow'94multiset

??

Weir'88

pushdownofpushdowns

LIL

Wartena'99

concatenationofpushdownsERLIL

�Multiset

�Thewaylinguiststhinkaboutmovement

�Nocross�serialdependencies

�Verypowerfullwithoutfurtherrestrictions(e.g.bounded

size)

�PushdownofPushdowns

�Naturalhierarchy

�Nolinguisticinterpretation(??)

�ConcatenationofPushdowns

�Naturalhierarchy

�Linguisticmotivation


31

EmbeddingPushdowns

:::

abb

baaba

bab

aa

?r

6 w

6 w6 w6 w

6 w

6 w

Figure5:Con�gurationofathirdorderembeddedPDA

LinearPushdownofS

(Weir'94)

IfS=(C;c0;P;F;m)thenPlin(S)=(C

0;c

0 0;P

0;F

0;m

0)

C0

=

(��C)+forsome�nitealphabet�

c0 0

=

(�;c0)

P0

=

ftop( )j 2�g[ftest(p)jp2Pg

F0

=

fpush(�;f;i)j�2�+;f2F;1�i�j�jg[fpopg[

m0(push( 1::: i::: n;f;i))((�;c)�)=

( 1;c0):::( i�1;c0)( i;m(f)(c))( i+1;c0):::( n;c0)�

m0(pop)((�;c)�)=�


32

Proposition7

L(Plin(SPD))=fL(G)jGanLIGg

Note.InaderivationofanLIGeachgrammarsymbolisfol-

lowedbyastackofindices.


33

Concatenation

...RelativizedMinimalitymakestheblockinge�ectof

aninterveninggovernorrelativetothenatureofthe

governmentrelationinvolved.

Rizzi1990,p.2

Theapplicationof�ShortestMove�needstoberela-

tivizedtothetypeofconstituentsmovingandtothe

relevantlandingsite.

Marantz1995,p.355

Boundedconnectivityhypothesis:Thereisanaturalty-

pologyoflinguisticrelationssuchthatthepsychological

complexityofastructureincreasesquicklywhenmore

thanonerelationofanygiventypeconnectsa(partial)

constituent�(oranyelementof�)toanyconstituent

externalto�.

Stabler1994,p355


34

RestrictingTuplesofStacks(I)

�

b

a

a

b

a

b

a

b

a

a

a

?r

6 w

6 w

6 w

a.Concatenationw.r.t.reading(Cf.Breveglierietal.1996)

�

b

a

a

b

a

b

a

b

a

a

a

6 w

6 w

?r

?r

?r

b.Concatenationw.r.t.writing

Figure6:TuplesofStacks


35

RestrictingTuplesofStacks(II)

�Concatenationw.r.t.writingismoreplausiblesinceitsup-

portsclusteringof�landingsites�(asinmultiplewh-movement

orhead-movement)

�ProofsareeasierforConcatenationw.r.t.reading

�Concatenationw.r.t.readingcorrespondstoExtendedLeft

LinearIndexed(orDistinguished)Grammars

ProductofStorages(Wartena2000)

IfS1andS2arestoragesthentheirproductisde�nedas:

S1ÆS2=(C1�C2;c1 ��c2 �;P;F;m)where

P

=

f�pjp2P1g[f�pjp2P2g

F

=

f�fjf2F1g[f�fjf2F2g

m(�p)((c1;c2))=

m1(p)(c1)

m(�f)((c

1;c

2))=

(m1(f)(c1);c2)

m(�p)((c1;c2))=

m2(p)(c2)

m(�f)((c

1;c

2))=

(c1;m

2(f)(c2))


36

ProductofStorages(Wartena2000)

Æ

r

asÆbutm(�pop)((c

1;c

2))onlyde�nedifc1=c �

Æ

w

asÆbutm(�push)((c1;c2))onlyde�nedifc1=c �

Proposition8

L(SPDÆ rSPD)=fL(G)jGanELLIGg

InversionofStorages

IfS=(C;c�;P;F;m)thenSinv=(C;c�;P;F;m

0)

withm

0(f)=m(f)�1

�L(Sinv)=L(S)R

�SPDÆ wSPD=(SPDÆ rSPD)inv

�fL(G)jGanERLIGg=f(L(G))R

jGanELLIGg

�L(SPDÆ wSPD)=fL(G)jGanERLIGg


37

Ar 1

a

Br 2

b

D1

d

r 3E

r 5e

D2

dr 4

Fr 6

f

a

(a;�)(b;�)(d;�)(e;r 1r 2r 3r 5)(d;�)(f;r4r 6)(a;�)

(q0;r1r 2r 3r 5;c0)`

(q1;r2r 3r 5;c1)`

(q2;r3r 5;c2)`

(q3;r5;c3)`

(q5;�;c�)

(q0;r4r 6;c0)`

(q4;r6;c4)`

(q6;�;c�)

a)DerivationinanELLDGcontrolledbyanS�automaton.

(hA;q0i;abdedfa;(c 0;�))`

(hB;q1i;bdedfa;(c 1;a))`

(hD1;q2i;dedfa;(c 2;D2a))`

(hE;q3i;edfa;(c 3;D2a))`

(h�;q 5i;dfa;(c �;D2a))`

(hD2;q0i;dfa;(c �;a))`

(hF;q4i;fa;(c 4;a))`

(h�;q 6i;a;(c �;a))`

(h�;q 0i;�;(c�;�))

b)SimulationbyanSÆ rSpdautomaton.


38

(q0;abcdefg;(c0;�))`

(q1;bcdefg;(c1;�))`

(q2;cdefg;(c1;A))`

(q3;defg;(c1;BA))`

(q4;efg;(c�;BA))`

(q5;fg;(c�;A))`

(q6;g;(c �;A))`

(q7;�;(c �;�))

a)ComputationofanSÆ rSpd�automaton

(q0;�;q7)

a

r 1

(q1;�;q7)

b

r 2

(q2;A;q7)

c

r 3

(q3;B;q5)

d

r 4

(q4;B;q5)

r 5e

(q5;A;q7)

f

r 6 (q6;A;q7)

r 7g

(q0;r1r 2r 3r 4r 5;c0)`

(q1;r2r 3r 4r 5;c1)`

(q2;r3r 4r 5;c1)`

(q3;r4r 5;c1)`

(q4;r5;c�)`

(q5;�;c�)

(q5;r6r 7;c0)`

(q6;r7;c0)`

(q7;�;c�)

b)SimulationbyanELLDGandancontrollingS�automaton


39

CP

NP0

wie

C'[][np]

C+Vaux

heeft

IP[vaux][np]

NP

Gijs

I'[vaux][np]

VP[][np]

VP[v2][np]

VP[v1,v2][np]

IP[v1,v2][np]

NP

debuurman

VP[v1,v2][np]

IP[v2][np]

NP0

t

VP[v2]

NP

Engels

V2

tV1 t

V0

horenV

1latenV

2

praten

I+Vaux

t

�WenhatGijsdenNachbarnEnglischsprechenlassengehört?�


40

FinalRemarks

�NobigsystemsorprojectsforLIG

�Usefullforthethestudyof�movement�,non�localdepen-

denciesandtheirformalimplications.

�Helpfullfortheunderstandingofothermildlycontextsen-

sitiveformalisms(varyingfromTAGtoSlotGrammar)


41

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ilg, irlg. · ] a b o ok? c. * [of which month] i do es john strik e his donk ey [every da y (3) a....

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