Bidirectional Ascendant Parsing for Natural Language Processing

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Proessing

Stefan ANDREI

Faulty of Computer Siene, \Al.I.Cuza" University

Str. Berthelot,nr. 16,6600, Iasi,Rom^ania.

E-mail: stefaninfoiasi.ro

Abstrat

The paper emphasizes some sublasses of ontext free grammars for whih there exists a parallel

approah useful for solving the membership problem. We ombine the lassial style of LR parsers

attahedtoagrammarGwitha\mirror"proessforG.Theinputwordwillbeanalysedfrombothsides

usingtwoproessors. Wepresent thegeneralbidiretionalparser(foranyontext freelanguage)using

anondeterministidevie. AgeneralMIMDmodelfordesribingthebidiretionalparsingispresented.

Ourgeneral bidiretional parser anbe also usedas a deterministi modelfor the knownLR (k) and

R L(k) parsers. Aordingly, the membership problem may be solved in linear time omplexity with

a parallel algorithm. Finally, we present anexample of a ontext free grammaruseful for desribing

syntatidependeniesfor Englishlanguage.

1 Introdution

Inreentyears,reognitionalgorithms,bothsequentialandparallel,forontextfreegrammars(with

appliationstoparsingprogramminglanguages)orfornonontextfreelanguages(withappliations

to omputational linguisti) havebeen the subjet of study by many omputer sientists ([7, 11℄).

In([11℄) wasdesribed aformal framework forbidiretional tabularparsingof general ontext free

languages,andsomeappliationstonaturallanguageproessingwerestudied. Moreover,analgorithm

forhead-drivenparsingandageneralalgorithmforisland-drivenparsingwerestudied.

Themainpurposeofthispaperistoprovideparallelparsingfor(sublassesof)ontextfree

gram-marsusingtwoproessors. Similarideasonerningparsingof theinputwordfrombothsideswere

presentedin[9,8℄. However,thementionpapersdonotonstruteetivelyanyparsersanddonot

mentionanyparallel modelofomputersforsuhomputation.

Inthis paper, wedene anew parser for the lass of ontext freelanguages. Theinput word is

analysedfrombothsides,buttheparsestrategyisinup-to-upsense(LRandR Lstylesareombined

forthisparallelstrategy). Someofthetheoretialresultsusedinthispaperwerereportedpreviously

in[3,4℄.

Theassoiatedderivationtreeorrespondingto theinputwordwisdownto uptraversedbyboth

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3

ST

1

ST

2

Figure1. Up-to-up bidiretionalparsing

The\derivation forests"ST

1

andST

2

will beparsedin parallel,and, nally, thesubtreeST

3 will

beparsedin asequentialway.

Theseond setion ontainsthe denition of thegeneral up-to-up bidiretionalparser. The

or-retnessof thisparser isproven asthe main resultofthis setion (Theorem2.1). The third setion

presentsaveryonvenientparallelapproahfordesribingthegeneralup-to-upbidiretionalparsing

strategy. ThehosenmodelisaMIMDomputer. TwoproessorsP1andP2analysetheinputword

frombothsides. Theorems3.1and3.2ensurethenitenessandorretnessofourparallelalgorithm.

Thenextsetionpointsoutthedenitionofsomenewsublassesofontextfreegrammars. Wealled

them LR R Lgrammars. These grammars anbe viewed assomeombinations betweenlassial

LRgrammars and themirror ones. The fth setion presents an exampleof asimple ontext free

grammarusefulfordesribingsyntatidependeniesforEnglishlanguage. Someopenproblemsare

emphasizedin Conlusions.

2 The General Bidiretional Parser

We suppose the reader familiar with the notions as ontext free grammars, words ([5, 10℄). Let

=

otherwise

and

otherwise.

Notation2.1 Let G =(V

: Thisfuntion an beeasily extendedtoan homomorphism h:(V [V 0

beingthe atenationoperation).

TheourreneofA

b;e

signiesthatAisthelabelofthederivationsubtreeofrootv intheforests

ST

1 orST

2

(Figure 1), andthefrontierhastheorrespondingrightmostderivation [r

b

In the following denition, we shall desribe our up-to-up bidiretional parser whih ontains 14

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Denition2.1 LetG=(V

ongurations,where#isaspeialharater(anewterminalsymbol). Thegeneralup-to-up

bidi-retional parser (denotedby G

u

gC isalledthesetof initialongurations. The rstomponent isthe state,the seond

(orderedfromright toleft)andthelastbutone(orderedfromlefttoright)omponentsofa

ongura-tionareforstoringthepartialsyntatianalysis. Thelastomponentisforstoringthenalsyntati

analysis. The third andthe fthomponents arethe work- staks. Thefourth omponent represents

the urrent ontent of the input word. The transition relation(`CC, sometimesdenoted by

GuBP(G)

) between ongurationsisgiven by:

1 0

Shift-Shift: (s

1

Redue-Shift: (s

1

Shift-Redue: (s

1

Redue-Redue: (s

1

Shift-Stay: (s

1

Redue-Stay: (s

1

Stay-Shift: (s

1

Stay-Redue: (s

1

Possible-aept: (s

1

Shift-Terminal: (s

2

Shift-Nonterminal: (s

2

Redue: (s

2

Rejet: (s

1

sitionsof type1 0

,2 0

,...,13 0

an beapplied.

Ourmodelisatwo-stakmahine([5℄). Thedierenesonsistintheexisteneoftwoheads(instead

ofonlyone)whihmayreadtheinputtape,andtwooutputtapeswhihanbeaessedonlyinwrite

style,andhasonlytwostates. Therefore,ourmodelansimulate aTuringmahine.

Theorem2.1 ([3 ℄) Let G be a ontext free grammar, S being its start symbol. Then

(s

3 Parallel Approah for General Bidiretional Parsing

In this setion, we present avery onvenient parallel approah for desribing the general up-to-up

bidiretionalparsingstrategy. OurmodelisaMIMD(multipleinstrutionstreamandmultiple data

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retionalparsingstrategy. Infat,ourparallel algorithm (denotedby(PAR-UUBP))learly follows

Denition2.1ofthegeneralup-to-upbidiretionalparser. Ouralgorithmusesthefollowingvariables:

w2V

T

theinputword,n=jwj;

i1,i2twoountersfortheurrentpositions inw;

aeptabooleanvariablewhihtakesthetruevalueiw2L(G);

Stak1,Stak2twoworkingstaksforP1andP2;

Output_tape1, Output_tape2 the output tapes of P1 and P2 for storing the partial syntati

analysis;

Output_tapetheoutputtapeforstoringtheglobalsyntatianalysis;

exitabooleanvariable whihistrueiP1orP2detetthenon-aeptaneofw.

Thevariables w, i1, i2, aept,Output_tape,exit arestored in the ommonmemory. We use

somepredenedproedures,suhas:

pop(Stak,)- thevalueof will bethestringof lengthjj startingfrom therst symbol of

Stak;afterthat,thestringisremovedfromStak;

push_first(Stak,A)-addtotheontentofStakthesymbolA;AwillbethenewtopofStak;

push_last(Stak,)-addtotheontentofStak,startingfromthelastsymbolofStak,the

string;Stakwillhavethesametop.

Now, the method of parallel algorithm (PAR-UUBP) an be pointed out (we suppose that the

ontextfreegrammarG=(V

N ;V

T

;S;P)isalreadyread):

begin

read(n); read(w); i1:=1; i2:=n; aept:=false; exit:=false;

repeat in parallel

ation1(P1); ation2(P2)

until (i1>=i2) or (exit=true);

if (exit = true) then aept := false

else repeat ation3(P1, P2) until (exit = true);

if (aept = true) then

write('w is aepted and has the right syntatial analysis',Output_tape);

else write('w is not aepted');

end.

Itremainstodesribetheproedures ation1(P1),ation2(P2)andation3(P1,P2).

proedure ation1(P1);

begin

ase

if (9r1 = A!h()2P; is inStak1starting from the top)then begin

/* redue ation */

let:=u

1 C

b;e :::u

m D

b 0

;e 0

0

,whereu

1 ;:::;u

m 2V

T ;

0

2V

V 0

;

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if ( does not ontainany symbol from V V) then b :=e

elseb 00

:=minfe 00

;bg;

pop(Stak1,);push_first(Stak1,A

b 00

;e 00

);push_first(Output_tape1,r1);

end;

if (i1 <= i2) then begin /* shift ation */

push_first(Stak1,w[i1℄);i1 := i1+1;

end

otherwise: begin /* baktrak is needed; */

if (all the baktrak stepsare over) and (still noredue or shift

ation ould be made)then exit := true;

end

end;

Thedesriptionofproedureation2(P2)is verysimilar toation1(P1).

proedure ation2(P2);

begin

ase

/* redue ation */

let :=u

1 C

b;e :::u

m D

b 0

;e 0

0

,whereu

1 ;:::;u

m 2V

T ;

0

2V

V 0

e 00

:=jOutput_tape2j+1;

if ( does not ontain anysymbol fromV 0

V)then b 00

:=e 00

elseb 00

:=minfe 00

;bg;

pop(Stak2,);push_first(Stak2,B

b 00

;e 00

);push_first(Output_tape2, r2);

end;

if (i1 < i2) then begin /* shift ation */

push_first(Stak2,w[i2℄);i2 := i2-1;

end

ation ould be made)then exit := true;

end

end;

Finally,wedesribetheproedureation3(P1,P2)inasequentialway. Thegoalistosimulatethe

transitions10 0

,11 0

,12 0

and13 0

forG

u

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resultsin Output_tape.

proedure ation3(P1,P2);

begin

ase

/* redue ation */

let =u

1 C

b;e :::

0

D

b 0

;e 0

00

;whereu2V

T ,

00

2V

, 0

2(V [V 0

)

;

let 0

1

from Output_tape1 suhthatj 0

1 j=e

0

b+1;

pop(Output_tape1, 0

1

);pop(Stak1,);push(Stak2,A);

push_first(Output_tape,r1);push_last(Output_tape, 0

1 );

end;

if (top of Stak2 is a terminal symbol)then begin

/* shift-terminal ation */

pop(Stak2,a),where a2V

T

;push_first(Stak1,a);

end;

if (top of Stak2 is from V')then begin/*shift-nonterminal ation*/

pop(Stak2,A

b;e

);push_first(Stak1,A);

let 0

2

from Output_tape2suh thatj 0

2

j=e b+1;

pop(Output_tape2, 0

2

);push_first(Output_tape, 0

2 );

end;

if (Output_tape1=;) and(Output_tape2=;) and(Stak1=S) and (Stak2=;)

then begin aept:=true; exit:=true end;

ation ould be made)then exit := true

end;

Theorem3.1 ([3 ℄)- nitenessof Algorithm(PAR-UUBP))

TheAlgorithm(PAR-UUBP)performsanite numberof stepsuntilterminatesitsexeution.

Theorem3.2 ([3 ℄)- orretness ofAlgorithm(PAR-UUBP))

For a given ontextfree grammar G=(V

N ;V

T

;S;P) and w2 V

T

asits input, Algorithm

(PAR-UUBP)givestheanswer"w is aeptedandhastherightsyntatial analysis"ifS

=)

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Deterministi(andlinear)parallelalgorithms(aspartiularasesofthegeneralup-to-upbidiretional

parsingalgorithm)for solvingthemembership problem, anbederivedfor some\ombinations" of

sublasses of ontext free languages. The deterministi up-to-upbidiretional parser has the same

devieasthegeneralmodel. Theonlydiereneistheuniquenessofhoosingtheprodutionrfrom

theset ofthegivenprodutionsoftheinputgrammar(i.e. nobaktrakstepisneeded). Weusethe

denitionsofLR (k)([6℄)grammars.

Denition4.1 LetGbeaontextfreegrammarandkbeanaturalnumber. WesaythatGisR L(k)

if e

Gisa LR (k)grammar. A languageL is R L(k) if thereexists aR L(k)grammarwhih generates

L.

In a similar way to Denition 4.1, we an dene the \mirror" of lassial sublasses of LR (k)

grammarsusefulin ompilertehniques. WeombinetheLR (k)and R L(k)stylesforobtainingthe

deterministiup-to-upbidiretionalparsingforontextfreelanguages.

Denition4.2 LetGbeaontextfreegrammarandk

1 ,k

2

2N:WesaythatGisaLR (k

1

) R L(k

2 )

grammar i G is both a LR (k

1

) andR L(k

2

) grammar. A language L is alled LR (k

1

) R L(k

2 ) if

thereexistsGaLR (k

1

) R L(k

2

)grammar for whih L=L(G):

Weanremark that G isaLR (k

1

) R L(k

2

) grammari Gis aR L(k

2

) LR (k

1

) grammar. Of

ourse,beausethelassofLR (k)languages,fork1;equalsto thelassofLR (1)languages([6℄),

thentheabovedenitionhaspratialinterestfork

1 ;k

2

2f0;1g.

Partiularly, thedeterministi up-to-upbidiretionalparsingis similar to the generalparallel

ap-proah, the only dierene being the lak of baktrak steps. The proedures ation1(P1) and

ation2(P2)arerelatedtolassialsequentialsyntatialanalysisalgorithmsforLR (1)(orthe

sub-lassesLR (0),SLR (1)andLALR (1))andR L(1). Wealsogetalinearrunningtimefortheproedures

ation1(P1)and ation2(P2) insteadof an exponentialsequentialrunning time (baktraksteps

arenomorerequired).

Obviously,the proedure ation3(P1,P2)hasno baktrakstepsfor thesublassesof grammars

giveninDenition4.2. Consequently,theproedureation3(P1,P2)isdeterministiandhasalinear

runningtime.

Theorretnessofthedeterministiparallelalgorithmsisensuredbytheorretnessofthegeneral

parallelalgorithmandtheorretnessofeah ofthesequentialsyntatianalysersforthementioned

sublassesofontextfreegrammars(suhasLR (1),R L(1)).

Theorem4.1 ([3 ℄)- the omplexityof the deterministiparallel algorithms)

Let us denote with T

1 (n); T

2

(n) and T

3

(n) the running time of the sequential proedures

ation1(P1),ation2(P2)andation3(P1,P2),wherenisthelengthof theinputword. Supposing

that the routing timeiszero, the parallel runningtime t(n)satises the relations:

minfT1(n);T2(n)g

2

+

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Wepresentamodiedontextfreegrammarsimilartotheonepresentedin[12℄fordesribingsyntati

dependeniesforEnglishlanguage.

The term 'grammatial ategory' overs the parts of speeh and types of phrases, suh asnoun

phraseandprepositionalphrase. Foronveniene,wewillabbreviatethem, sothat'NOUN'beomes

'N','NOUNPHRASE'beomes'NP','DETERMINER'beomes'D',et. Letusonsiderthefollowing

ontextfreegrammar:

G = (fS;NP;VP;D;A;N;PP;V;Pg; V

T

; S; Prod); where V

T

; being the English

alphabet andProdbeingthefollowingset ofprodutions:

1:S!NP VP 7:NP !N 13:D!the 21: N!hunter

2:NP !D AAN PP 8:VP!V NP PP 14:D!some 22: V !attak

3:NP !AAN PP 9:VP!V NP 15:A!big 23: V !ate

4:NP !D AN PP 10: VP !V PP 16:A!brown 24: V !wathed

5:NP !AN PP 11: VP !V 17:A!old 25: P!for

6:NP !D N 12: PP !P NP 18:N !birds 26: P!beside

19:N !fleas 27: P!with

20:N !dog

In[12℄, theprodutions1,...,12arealled rules,and13,...,27arealled lexion.

UsingaJAVAimplementationforourbidiretionalparsingalgorithm,wedeterminetheviableprex

automatonforgrammarGwhihhas88statesorrespondingtoLR (1)itemswithnoonits

(redue-redue,redue-shift),soGisaLR (1)grammar. Moreover,weonstruttheviableprexautomaton

forgrammar e

G whih has 70states orrespondingto R L(1) items withno onits (redue-redue,

redue-shift),so e

GisaR L(1)grammar.

Aordingto Denition4.2,itfollowsthat grammarGisaLR (1) R L(1)grammar. Weonsider

nowtheinputword:

w=thebigbrown dogwithfleaswathedthebirdsbesidethehunter

Weshallsimulateapossibleexeutionofthebidiretionalparsingalgorithmforthisexample

(pro-essorsP1andP2worksynhronously). Wehaven=jwj andi1andi2twopointersbetween1and

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Ation: Initial Ation: Initial

Stak1=[℄ Stak2=[℄

i1=1 i2=12

Output_tape1=[℄ Output_tape2=[℄

Ation: Shift Ation: Shift

Stak1=[the℄ Stak2=[hunter℄

i1=2 i2=11

Output_tape1=[℄ Output_tape2=[℄

Ation: Redue Ation: Redue

Stak1=[D

1;1

℄ Stak2=[N

1;1 ℄

i1=2 i2=11

Output_tape1=[13℄ Output_tape2=[21℄

Stak1=[D

1;1

,big℄ Stak2=[the,N

1;1 ℄

i1=3 i2=10

Output_tape1=[13℄ Output_tape2=[21℄

Stak1=[D

1;1 ,A

2;2

℄ Stak2=[D

2;2 ,N

1;1 ℄

i1=3 i2=10

Output_tape1=[15, 13℄ Output_tape2=[21,13℄

Ation: Shift Ation: Redue

Stak1=[D

1;1 ,A

2;2

,brown℄ Stak2=[NP

1;3 ℄

i1=4 i2=10

Output_tape1=[15, 13℄ Output_tape2=[6,21,13℄

Ation: Redue Ation: Shift

Stak1=[D

1;1 ,A

2;2 ,A

3;3

℄ Stak2=[beside,NP

1;3 ℄

i1=4 i2=9

Output_tape1=[16, 15,13℄ Output_tape2=[6,21,13℄

Ation: Shift Ation: Redue

Stak1=[D

1;1 ,A

2;2 ,A

3;3

,dog℄ Stak2=[P

4;4 ,NP

1;3 ℄

i1=5 i2=9

Output_tape1=[16, 15,13℄ Output_tape2=[6,21,13, 26℄

Stak1=[D

1;1 ,A

2;2 ,A

3;3 ,N

4;4

℄ Stak2=[PP

1;5 ℄

i1=5 i2=9

Output_tape1=[20, 16,15,13℄ Output_tape2=[12,6,21, 13,26℄

Stak1=[D

1;1 ,A

2;2 ,A

3;3 ,N

4;4

,with℄ Stak2=[birds,PP

1;5 ℄

i1=6 i2=8

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Stak1=[D

1;1 ,A

2;2 ,A

3;3 ,N

4;4 , P

5;5

℄ Stak2=[N

6;6 ,PP

1;5 ℄

i1=6 i2=8

Output_tape1=[27, 20,16,15,13℄ Output_tape2=[12,6,21,13,26,18℄

Stak1=[D

1;1 ,A

2;2 ,A

3;3 ,N

4;4 , P

5;5

, fleas℄ Stak2=[the,N

6;6 ,PP

1;5 ℄

i1=7 i2=7

Output_tape1=[27, 20,16,15,13℄ Output_tape2=[12,6,21,13,26,18℄

Stak1=[D

1;1 ,A

2;2 ,A

3;3 ,N

4;4 , P

5;5 , N

6;6

℄ Stak2=[D

7;7 ,N

6;6 ,PP

1;5 ℄

i1=7 i2=7

Output_tape1=[19, 27,20,16,15,13℄ Output_tape2=[12,6,21,13,26,18,13℄

Stak1=[D

1;1 ,A

2;2 ,A

3;3 ,N

4;4 , P

5;5 , NP

6;7

℄ Stak2=[NP

6;8 ,PP

1;5 ℄

i1=7 Stak2=[NP

6;8 ,PP

1;5 ℄

Output_tape1=[7,19,27,20,16,15,13℄ Output_tape2=[12,6,21,13,26,6,18,

13℄

Ation: Redue Ation: Shift

Stak1=[D

1;1 ,A

2;2 ,A

3;3 ,N

4;4 , PP

5;8

℄ Stak2=[wathed,NP

6;8 ,PP

1;5 ℄

i1=7 i2=6

Output_tape1=[12, 7,19,27,20,16,15, Output_tape2=[12,6,21,13,26,6,18,

13℄ 13℄

Stak1=[NP

1;9

℄ Stak2=[V

9;9 ,NP

6;8 ,PP

1;5 ℄

i1=7 i2=6

Output_tape1=[2,12,7,19,27,20,16,15, Output_tape2=[12,6,21,13,26,6,18,

13℄ 13,24℄

Now,theproessorsP1and P2meetinthe\middle"oftheinputword,andonlyproessorP1will

be strongly ative. ProessorP2 only sends the data (i.e. Stak2,Output_tape2) to the internal

memoryofP1. Infat,weshallillustratetheexeutionstepsmadebytheproedureation(P1,P2).

Ation: Shift-Nonterminal Ation: Shift-Nonterminal

Stak1=[NP

1;9

,V℄ Stak1=[NP

1;9

,V,NP,PP℄

Stak2=[NP

6;8 ,PP

1;5

℄ Stak2=[℄

Output_tape1=[2,12,7,19,27,20,16,15, Output_tape1=[2,12,7,19,27,20,16, 15,

13℄ 13℄

Output_tape2=[12, 6,21,13,26,6,18,13℄ Output_tape2=[℄

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Stak1=[NP

1;9

,V,NP℄ Stak1=[NP

1;9 ,VP℄

Stak2=[PP

1;5

℄ Stak2=[℄

Output_tape1=[2,12,7,19,27,20,16,15, Output_tape1=[2,12, 7,19,27,20,16,15,

13℄ 13℄

Output_tape2=[12, 6,21,13,26℄ Output_tape2=[℄

Output_tape=[6,18,13,24℄ Output_tape=[8,12,6,21,13,26, 6,18,

13,24℄

Thenaltransition is:

Ation: Redue

Stak1=[S℄

Stak2=[℄

Output_tape1=[℄

Output_tape2=[℄

Output_tape=[1,8,12,6,21, 13,26,6,18,13,24, 2,12,7,19,27,20,16,15,13℄

In onlusion, the word w is aepted by the bidiretional parser and it has the right syntati

analysis

rm

=[1;8;12;6;21;13;26;6;18;13;24;2;12;7;19;27;20;16;15;13℄:

Weanextend ourexampleaordingto agreementrestritionsbetween words andphrases. For

example, the NP \a dogs" is not orret English beause the artile \a" indiates a single objet

whilethenoun\dogs"indiates apluralobjet. Therearemanyotherformsofagreement(subjet

-verb,gender,et). Tousesuhagreementrestritions,theontextfreegrammarisextendedtoallow

onstituentsto havefeatures ([2℄). Forexample, we mightdene afeature NUMBER that may

take avalueof either \s" (forsingular)or\p" (for plural), andwethen mightwrite an augmented

ontextfreegrammarrulesuhas:

NP !DN onlywhenNUMBER (D)=NUMBER (N).

Thereisaninterestingissueofwhetheranaugmentedontextfreegrammarandesribelanguages

thatannot bedesribedbyasimpleontextfreegrammar. Theanswerdepends ontheonstraints

onwhatanbeafeaturevalue. Ifthesetoffeaturevaluesisnite,thenitwouldalwaysbepossibleto

reatenewonstituentategoriesforeveryombinationoffeatures. Thusitisexpressivelyequivalent

toaontextfreegrammar.Ifthesetoffeaturevaluesisunonstrained,however,thensuhgrammars

havearbitraryomputationalpower. Inpratie,evenwhenthesetofvaluesisnotexpliitlyrestrited,

thispowerisnotused, andthestandardparsingalgorithmsanbeusedongrammars thatinludes

features([2℄).

6 Conlusions

Wehaveseen that ourbidiretionalparser model isin fat atwo-stakmahine ([5℄) and therefore

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deterministiparallelalgorithm;tondfurtherlosureandinlusionpropertiesofLR R Lgrammars

andlanguages;toextendourbidiretionalapproahtononontextfreelanguages-withappliations

toomputationallinguistiandnaturallanguageproessing.

Wewanttothanktotheunknownrefereesfortheirusefulremarkswhihimprovethepaper.

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[2℄ Allen,P.NaturalLanguageUnderstanding.SeondEdition,TheBenjamin/CummingsPublishing

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