[PDF] - Proceedings of the 34th Annual Meeting of the ACL, Santa PDF Document

SLIDE 1 Proceedings

f

the 34th Annual Meeting

f

the ACL, Santa Cruz, June 1996, pp. 79-86. Ecien t Normal-F

rm

P arsing for Com binatory Categorial Gramm ar

Jason

Eisner Dept.

f

Computer and Information Science Univ ersit y

f

P ennsylv ania 200 S. 33rd St., Philadelphia, P A 19104-6389, USA jeisner@l inc .ci s.u pe nn. edu Abstract Under categorial gramma rs that ha v e p

w-

erful rules lik e comp

sition,

a simple n-w

rd

sen tence can ha v e exp

nen

tially man y parses. Generating all parses is inecien t and

bscures

whatev er true seman tic am biguities are in the input. This pap er addresses the problem for a fairly general form

f

Com binatory Categorial Grammar, b y means

f

an ecien t, correct, and easy to implem en t normal-form parsing tec h- nique. The parser is pro v ed to nd exactly

ne

parse in eac h seman tic equiv- alence class

f

allo w able parses; that is, spurious am biguit y (as carefully dened) is sho wn to b e b

th

safely and completely eliminated. 1 In tro duction Com binatory Categorial Grammar (Steedman, 1990), lik e

ther

\exible" categorial gramma rs, suers from spurious ambiguity (Witten burg, 1986). The non-standard constituen ts that are so crucial to CCG's analyses in (1), and in its accoun t

f

in to- national fo cus (Prev

st

& Steedman, 1994), remain a v ailable ev en in simpler sen tences. This renders (2) syn tactically am biguous. (1) a. Co

rdination:

[[John lik es ] S/NP , and [Mary pretends to lik e] S/NP ], the big galo

t

in the corner. b. Extraction: Ev eryb

dy

at this part y [whom [John lik es] S/NP ] is a big galo

t.

(2) a. John [lik es Mary] SnNP . b. [John lik es ] S/NP Mary . The practical problem

f

\extra" parses in (2) b e- comes exp

nen

tially w

rse

for longer strings, whic h can ha v e up to a Catalan n um b er

f

parses. An

This

material is based up

n

w

rk

supp

rted

under a National Science F

undation

Graduate F ello wship. I ha v e b een grateful for the advice

f

Ara vind Joshi, Nob

Komagata,

Seth Kulic k, Mic hael Niv, Mark Steedman, and three anon ymous review ers. exhaustiv e parser serv es up 252 CCG parses

f

(3), whic h m ust b e sifted through, at considerable cost, in

rder

to iden tify the t w

distinct

meanings for further pro cessing. 1 (3) the NP/N galo

t

N in (NnN)/NP the NP/N corner N that (NnN)/(S/NP) I S/(SnNP) said (SnNP)/S Mary S/(Sn NP) pretends (Sn NP)/(S inf nNP) to (S inf nNP)/(S stem nNP) lik e (S stem nNP)/NP This pap er presen ts a simple and exible CCG parsing tec hnique that prev en ts an y suc h explosion

f

redundan t CCG deriv ations. In particular, it is pro v ed in x4.2 that the metho d constructs exactly

ne

syn tactic structure p er seman tic reading|e.g., just t w

parses

for (3). All

ther

parses are sup- pressed b y simple normal-form constrain ts that are enforced throughout the parsing pro cess. This approac h w

rks

b ecause CCG's spurious am biguities arise (as is sho wn) in

nly

a small set

f

circum- stances. Although similar w

rk

has b een attempted in the past, with v arying degrees

f

success (Kart- tunen, 1986; Witten burg, 1986; P aresc hi & Steed- man, 1987; Bouma, 1989; Hepple & Morrill, 1989; K

nig,

1989; Vija y-Shank er & W eir, 1990; Hepple, 1990; Mo

rtgat,

1990; Hendriks, 1993; Niv, 1994), this app ears to b e the rst full normal-form result for a categorial formalism ha ving more than con text- free p

w

er. 2 Denitions and Related W

rk

CCG ma y b e regarded as a generalization

f

con text- free grammar (CF G)|one where a grammar has innitely man y non terminals and phrase-structure rules. In addition to the familia r atomic non ter- minal categories (t ypically S for sen tences, N for 1 Namely , Mary pretends to lik e the galo

t

in 168 parses and the corner in 84. One migh t try a statis- tical approac h to am biguit y resolution, discarding the lo w-probabilit y parses, but it is unclear ho w to mo del and train an y probabili tie s when no single parse can b e tak en as the standard

f

correctness.

SLIDE 2 nouns, NP for noun phrases, etc.), CCG allo ws innitely man y slashe d categories. If x and y are categories, then x=y (resp ectiv ely xn y ) is the category

f

an incomplete x that is missing a y at its righ t (resp ectiv ely left). Th us v erb phrases are an- alyzed as sub jectless sen tences Sn NP, while \John lik es" is an

b

jectless sen tence

r

S/NP. A complex category lik e ((Sn NP)n (SnNP))/N ma y b e written as SnNPn (Sn NP)/N, under a con v en tion that slashes are left-asso ciativ e. The results herein apply to the T A G-equiv alen t CCG formalization giv en in (Joshi et al., 1991). 2 In this v ariet y

f

CCG, ev ery (non-lexical) phrase- structure rule is an instance

f
ne
f

the follo wing binary-rule templates (where n

0):

(4) F

rw

ard generalized comp

sition

>Bn: x=y y j n z n

j

2 z 2 j 1 z 1 ! x j n z n

j

2 z 2 j 1 z 1 Bac kw ard generalized comp

sition

<Bn: y j n z n

j

2 z 2 j 1 z 1 xn y ! x j n z n

j

2 z 2 j 1 z 1 Instances with n = are called applic ation rules, and instances with n

1

are called c

mp
sition

rules. In a giv en rule, x; y ; z 1 : : : z n w

uld

b e instan tiated as categories lik e NP, S/NP,

r

Sn NPn(Sn NP)/N. Eac h

f

j 1 through j n w

uld

b e instan tiated as either /

r

n. A xed CCG gramma r need not include ev ery phrase-structure rule matc hing these templates. In- deed, (Joshi et al., 1991) place certain restrictions

n

the rule set

f

a CCG grammar, including a re- quiremen t that the rule degree n is b

unded
v

er the set. The results

f

the presen t pap er apply to suc h restricted gramma rs and also more generally , to an y CCG-st yle grammar with a de cidable rule set. Ev en as restricted b y (Joshi et al., 1991), CCGs ha v e the \mildly con text-sensitiv e" expressiv e p

w

er

f

T ree Adjoining Gramm ars (T A Gs). Most w

rk
n

spurious am biguit y has fo cused

n

categorial formalism s with substan tially less p

w

er. (Hepple, 1990) and (Hendriks, 1993), the most rigorous pieces

f

w

rk,

eac h establish a normal form for the syn- tactic calculus

f

(Lam b ek, 1958), whic h is w eakly con text-free. (K

nig,

1989; Mo

rtgat,

1990) ha v e also studied the Lam b ek calculus case. (Hepple & Morrill, 1989), who in tro duced the idea

f

normal- form parsing, consider

nly

a small CCG fragmen t that lac ks bac kw ard

r
rder-c

hanging com- p

sition;

(Niv, 1994) extends this result but do es not sho w completeness. (Witten burg, 1987) assumes a CCG fragmen t lac king

rder-c

hanging

r

higher-

rder

comp

sition;

furthermore, his revision

f

the com binators creates new, conjoinable constituen ts that con v en tional CCG rejects. (Bouma, 1989) pro- p

ses

to replace comp

sition

with a new com bina- tor, but the resulting pro duct-grammar sc heme as- 2 This formalization sw eeps an y t yp e-raising in to the lexicon, as has b een prop

sed
n

linguistic grounds (Do wt y , 1988; Steedman, 1991, and

thers).

It also treats conjunction lexically , b y giving \and" the generalized category xnx=x and barring it from comp

sition.

signs dieren t t yp es to \John lik es" and \Mary pretends to lik e," th us losing the abilit y to conjoin suc h constituen ts

r

sub categorize for them as a class. (P aresc hi & Steedman, 1987) do tac kle the CCG case, but (Hepple, 1987) sho ws their algorithm to b e incomplete. 3 Ov erview

f

the P arsing Strategy As is w ell kno wn, general CF G parsing metho ds can b e applied directly to CCG. An y sort

f

c hart parser

r

non-deterministic shift-reduce parser will do. Suc h a parser rep eatedly decides whether t w

adjacen

t constituen ts, suc h as S/NP and NP/N, should b e com bined in to a larger constituen t suc h as S/N. The role

f

the gramm ar is to state whic h com bi- nations are allo w ed. The k ey to eciency , w e will see, is for the parser to b e less p ermissiv e than the gramm ar|fo r it to sa y \no, redundan t" in some cases where the grammar sa ys \y es, grammati cal." (5) sho ws the constituen ts that un trammeled CCG will nd in the course

f

parsing \John lik es Mary ." The spurious am biguit y problem is not that the gramma r allo ws (5c), but that the grammar allo ws b

th

(5f) and (5g)|distinct parses

f

the same string, with the same meaning. (5) a. [John] S/(Sn NP) b. [lik es] (Sn NP)/NP c. [John lik es] S/NP d. [Mary] NP e. [lik es Mary] Sn NP f. [[John lik es] Mary] S

to

b e disal lowe d g. [John [lik es Mary]] S The prop

sal

is to construct all constituen ts sho wn in (5) except for (5f). If w e sligh tly constrain the use

f

the gramma r rules, the parser will still pro duce (5c) and (5d)|constituen ts that are indisp ensable in con texts lik e (1)|while refusing to c

mbine

those constituen ts in to (5f). The relev an t rule S/NP NP ! S will actually b e blo c k ed when it attempts to construct (5f). Although rule-blo c king ma y eliminate an analysis

f

the sen tence, as it do es here, a seman tically equiv alen t analysis suc h as (5g) will alw a ys b e deriv able along some

ther

route. In general,

ur

goal is to disco v er exactly

ne

analysis for eac h <substring, meaning> pair. By prac- ticing \birth con trol" for eac h b

ttom-up

generation

f

constituen ts in this w a y , w e a v

id

a p

pulation

explosion

f

parsing

ptions.

\John lik es Mary" has

nly
ne

reading seman tically , so just

ne
f

its analyses (5f){(5g) is disco v ered while parsing (6). Only that analysis, and not the

ther,

is allo w ed to con- tin ue

n

and b e built in to the nal parse

f

(6). (6) that galo

t

in the corner that thinks [John lik es Mary] S F

r

a c hart parser, where eac h c hart cell stores the analyses

f

some substring, this strategy sa ys that

SLIDE 3 all analyses in a cell are to b e seman tically distinct. (Karttunen, 1986) suggests enforcing that prop ert y directly|b y comparing eac h new analysis seman tically with existing analyses in the cell, and refusing to add it if redundan t|but (Hepple & Morrill, 1989)

bserv

e briey that this is inecien t for large c harts. 3 The follo wing sections sho w ho w to

btain

eectiv ely the same result without doing an y semantic in terpretation

r

comparison at all. 4 A Normal F

rm

for \Pure" CCG It is con v enien t to b egin with a sp ecial case. Sup- p

se

the CCG grammar includes not some but al l instances

f

the binary rule templates in (4). (As alw a ys, a separate lexicon sp ecies the p

ssible

categories

f

eac h w

rd.)

If w e group a sen tence's parses in to seman tic equiv alence classes, it alw a ys turns

ut

that exactly

ne

parse in eac h class satises the follo wing simple declarativ e constrain ts: (7) a. No constituen t pro duced b y >Bn, an y n

1,

ev er serv es as the primary (left) argumen t to >Bn , an y n

0.

b. No constituen t pro duced b y <Bn, an y n

1,

ev er serv es as the primary (righ t) argumen t to <Bn , an y n

0.

The notation here is from (4). More collo quially , (7) sa ys that the

utput
f

righ t w ard (left w ard) com- p

sition

ma y not comp

se
r

apply

v

er an ything to its righ t (left). A parse tree

r

subtree that satises (7) is said to b e in normal form (NF). As an example, consider the eect

f

these restrictions

n

the simple sen tence \John lik es Mary ." Ig- noring the tags {ot, {f c, and {bc for the mom en t, (8a) is a normal-form parse. Its comp etitor (8b) is not, nor is an y larger tree con taining (8b). But non- 3 Ho w inecien t? (i) has exp

nen

tially man y seman- tically distinct parses: n = 10 yields 82,756,612 parses in

20

10

=

48,620 equiv alence classes. Karttunen's metho d m ust therefore add 48,620 represen tativ e parses to the appropriate c hart cell, rst comparing eac h

ne

against all the previously added parses|of whic h there are 48,620/2

n

a v erage|to ensure it is not seman tically redundan t. (Additional comparisons are needed to reject parses

ther

than the luc ky 48,620.) Adding a parse can therefore tak e exp

nen

tial time. (i) n z }| { : : : S=S S=S S=S S n z }| { SnS SnS SnS : : : Structure sharing do es not app ear to help: parses that are group ed in a parse forest ha v e

nly

their syn tactic category in common, not their meaning. Karttunen's approac h m ust tease suc h parses apart and compare their v arious meanings individua ll y against eac h new candi- date. By con trast, the metho d prop

sed

b elo w is purely syn tactic|just lik e an y \ordinary" parser|so it nev er needs to unpac k a subforest, and can run in p

lynomial

time. standard constituen ts are allo w ed when necessary: (8c) is in normal form (cf. (1)). (8) a. S{ot S/(Sn NP){ot John SnNP{ot (Sn NP)/NP{ot lik es NP{ot Mary b. forwar d applic ation blo cke d by (7a) (e quivalentl y, not p ermitte d by (10a)) S/NP{f c S/(Sn NP){ot John (Sn NP)/NP{ot lik es NP{ot Mary c. NnN{ot (Nn N)/(S/NP){ot whom S/NP{f c S/(Sn NP){ot John (Sn NP)/NP{ot lik es It is not hard to see that (7a) eliminates all but righ t-branc hing parses

f

\forw ard c hains" lik e A/B B/C C

r

A/B/C C/D D/E/F/G G/H, and that (7b) eliminates all but left-branc hing parses

f

\bac kw ard c hains." (Th us ev ery functor will get its argumen ts, if p

ssible,

b efore it b ecomes an argumen t itself.) But it is hardly

b

vious that (7) eliminates al l

f

CCG's spurious am biguit y . One migh t w

rry

ab

ut

unexp ected in teractions in v

lving

crossing comp

sition

rules lik e A/B Bn C ! AnC . Signican tly , it turns

ut

that (7) really do es suce; the pro

f

is in x4.2. It is trivial to mo dify an y sort

f

CCG parser to nd

nly

the normal-form parses. No semantics is necessary; simply blo c k an y rule use that w

uld

violate (7). In general, detecting violations will not h urt p erformance b y more than a constan t factor. Indeed,

ne

migh t implemen t (7) b y mo di- fying CCG's phrase-structure grammar. Eac h

rdi-

nary CCG category is split in to three categories that b ear the resp ectiv e tags from (9). The 24 templates sc hematized in (10) replace the t w

templates
f

(4). An y CF G-st yle metho d can still parse the resulting spuriosit y-free grammar, with tagged parses as in (8). In particular, the p

lynomial-tim

e, p

lynomial
space

CCG c hart parser

f

(Vija y-Shank er & W eir, 1993) can b e trivially adapted to resp ect the constrain ts b y tagging c hart en tries.

SLIDE 4 (9) {f c

utput
f

>Bn, some n

1

(a forw ard comp

sition

rule) {bc

utput
f

<Bn, some n

1

(a bac kw ard comp

sition

rule) {ot

utput
f

>B0

r

<B0 (an application rule),

r

lexical item (10) a. F

rw

ard application >B0:

x=y

{bc x=y {ot

(

y {f c y {bc y {ot ) ! x{ot b. Bac kw ard application <B0: ( y {f c y {bc y {ot )

xny

{f c xny {ot

!

x{ot c. Fwd. comp

sition

>Bn (n

1):
x=y

{bc x=y {ot

(

y j n z n

j

2 z 2 j 1 z 1 {f c y j n z n

j

2 z 2 j 1 z 1 {bc y j n z n

j

2 z 2 j 1 z 1 {ot ) ! x j n z n

j

2 z 2 j 1 z 1 {f c d. Bwd. comp

sition

<Bn (n

1):

( y j n z n

j

2 z 2 j 1 z 1 {f c y j n z n

j

2 z 2 j 1 z 1 {bc y j n z n

j

2 z 2 j 1 z 1 {ot )

xny

{f c xny {ot

!

x j n z n

j

2 z 2 j 1 z 1 {bc (11) a. Syn/sem for >Bn (n

0):

x=y m f y j n z n

j

2 z 2 j 1 z 1 m g ! x j n z n

j

2 z 2 j 1 z 1 m c 1 c 2 : : : c n :f (g (c 1 )(c 2 )

(c

n )) b. Syn/sem for <Bn (n

0):

y j n z n

j

2 z 2 j 1 z 1 m g xny m f ! x j n z n

j

2 z 2 j 1 z 1 m c 1 c 2 : : : c n :f (g (c 1 )(c 2 )

(c

n )) (12) a. A/C/F A/C/D A/B B/C/D D/F D/E E/F b. A/C/F A/C/E A/C/D A/B B/C/D D/E E/F c. xy :f (g (h(k (x)))(y )) A=C=F A=B B=C=D D=E E=F f g h k It is in teresting to note a rough resem blance b e- t w een the tagged v ersion

f

CCG in (10) and the tagged Lam b ek calculus L*, whic h (Hendriks, 1993) dev elop ed to eliminate spurious am biguit y from the Lam b ek calculus L. Although dierences b et w een CCG and L mean that the details are quite dieren t, eac h system w

rks

b y marking the

utput
f

certain rules, to prev en t suc h

utput

from serving as input to certain

ther

rules. 4.1 Seman tic equiv alence W e wish to establish that eac h seman tic equiv alence class con tains exactly

ne

NF parse. But what do es \seman tically equiv alen t" mean? Let us adopt a standard mo del-theoretic view. F

r

eac h leaf (i.e., lexeme)

f

a giv en syn tax tree, the lexicon sp ecies a lexic al interpr etation from the mo del. CCG then pro vides a derive d interpr etation in the mo del for the complete tree. The standard CCG theory builds the seman tics comp

sitionally

, guided b y the syn tax, according to (11). W e ma y therefore regard a syn tax tree as a static \recip e" for com bining w

rd

meanings in to a phrase meaning. One migh t c ho

se

to sa y that t w

parses

are seman tically equiv alen t i they deriv e the same phrase meaning. Ho w ev er, suc h a denition w

uld

mak e spurious am biguit y sensitiv e to the ne-grained seman tics

f

the lexicon. Are the t w

analyses
f

VP/VP VP VPnVP seman tically equiv alen t? If the lexemes in v

lv

ed are \softly kno c k t wice," then y es, as softly(t wice(kno c k)) and t wice(softly(kno c k)) ar- guably denote a common function in the seman tic mo del. Y et for \in ten tionally kno c k t wice" this is not the case: these adv erbs do not comm ute, and the seman tics are distinct. It w

uld

b e dicult to mak e suc h subtle distinc- tions rapidly . Let us instead use a narro w er, \in ten- sional" denition

f

spurious am biguit y . The trees in (12a{b) will b e considered equiv alen t b ecause they sp ecify the same \recip e," sho wn in (12c). No matter what lexical in terpretations f ; g ; h; k are fed in to the lea v es A/B, B/C/D, D/E, E/F, b

th

the trees end up with the same deriv ed in terpretation, namely a mo del elemen t that can b e determined from f ; g ; h; k b y calculating xy :f (g (h(k (x)))(y )). By con trast, the t w

readings
f

\softly kno c k

SLIDE 5 t wice" are considered to b e distinct, since the parses sp ecify dieren t recip es. That is, giv en a suitably free c hoice

f

meanings for the w

rds,

the t w

parses

can b e made to pic k

ut

t w

dieren

t VP-t yp e func- tions in the mo del. The parser is therefore conser- v ativ e and k eeps b

th

parses. 4 4.2 Normal-form parsing is safe & complete The motiv atio n for pro ducing

nly

NF parses (as dened b y (7)) lies in the follo wing existence and uniqueness theorems for CCG. Theorem 1 Assuming \pur e CCG," wher e al l p

s-

sible rules ar e in the gr ammar, any p arse tr e e

is

semantic al ly e quivalent to some NF p arse tr e e NF (). (This sa ys the NF parser is safe for pure CCG: w e will not lose an y readings b y generating just normal forms.) Theorem 2 Given distinct NF tr e es

6=
(on

the same se quenc e

f

le aves). Then

and
ar

e not semantic al ly e quivalent. (This sa ys that the NF parser is c

mplete:

generating

nly

normal forms eliminates al l spurious am biguit y .) Detailed pro

fs
f

these theorems are a v ailable

n

the cmp-lg arc hiv e, but can

nly

b e sk etc hed here. Theorem 1 is pro v ed b y a constructiv e induction

n

the

rder
f

, giv en b elo w and illustrated in (13):

F
r
a

leaf, put NF () = .

(<R;
;
>

denotes the parse tree formed b y com- bining subtrees

;
via

rule R.) If

=

<R;

;
>

, then tak e NF () = <R; NF ( ); NF ( )> , whic h exists b y inductiv e h yp

thesis,

unless this is not an NF tree. In the latter case, WLOG, R is a forw ard rule and NF ( ) = <Q;

1

;

2

> for some forw ard com- p

sition

rule Q. Pure CCG turns

ut

to pro- vide forw ard rules S and T suc h that

=

<S;

1

; NF (< T ;

2

;

>

)> is a constituen t and is seman tically equiv alen t to . Moreo v er, since

1

serv es as the primary subtree

f

the NF tree NF ( ),

1

cannot b e the

utput
f

forw ard com- p

sition,

and is NF b esides. Therefore

is

NF: tak e NF () =

.

(13) If NF ( ) not

utput
f

fwd. comp

sition,
=

R !

=

) R ! NF ( ) NF ( ) def = NF () else

=

R !

=

) R ! NF ( )

4

(Hepple & Morrill, 1989; Hepple, 1990; Hendriks, 1993) app ear to share this view

f

seman tic equiv alence. Unlik e (Karttunen, 1986), they try to eliminate

nly

parses whose denotations (or at least

terms)

are sys- tematically equiv alen t, not parses that happ en to ha v e the same denotation through an acciden t

f

the lexicon. = R ! Q !

1
2
=

) S !

1

NF T !

2
!

def = NF() This construction resem bles a w ell-kno wn normal- form reduction pro cedure that (Hepple & Morrill, 1989) prop

se

(without pro ving completeness) for a small fragmen t

f

CCG. The pro

f
f

theorem 2 (completeness) is longer and more subtle. First it sho ws, b y a simple induction, that since

and
disagree

they m ust disagree in at least

ne
f

these w a ys: (a) There are trees

;
and

rules R 6= R suc h that <R;

;
>

is a subtree

f
and

<R ;

;
>

is a subtree

f
.

(F

r

example, S/S SnS ma y form a constituen t b y either <B1x

r

>B1x.) (b) There is a tree

that

app ears as a subtree

f

b

th
and
,

but com bines to the left in

ne

case and to the righ t in the

ther.

Either condition, the pro

f

sho ws, leads to dieren t \imm ediate scop e" relations in the full trees

and
(in

the sense in whic h f tak es imm ediate scop e

v

er g in f (g (x)) but not in f (h(g (x)))

r

g (f (x))). Con- dition (a) is straigh tforw ard. Condition (b) splits in to a case where

serv

es as a secondary argumen t inside b

th
and
,

and a case where it is a primary argumen t in

r
.

The latter case requires consid- eration

f
's

ancestors; the NF prop erties crucially rule

ut

coun terexamples here. The notion

f

scop e is relev an t b ecause seman tic in terpretations for CCG constituen ts can b e written as restricted lam b da terms, in suc h a w a y that constituen ts ha ving distinct terms m ust ha v e dieren t in terpretations in the mo del (for suitable in terpretations

f

the w

rds,

as in x4.1). Theorem 2 is pro v ed b y sho wing that the terms for

and
dier

some- where, so corresp

nd

to dieren t seman tic recip es. Similar theorems for the Lam b ek calculus w ere previously sho wn b y (Hepple, 1990; Hendriks, 1993). The presen t pro

fs

for CCG establish a result that has long b een susp ected: the spurious am biguit y problem is not actually v ery widespread in CCG. Theorem 2 sa ys al l cases

f

spurious am biguit y can b e eliminated through the construction giv en in theorem 1. But that construction merely ensures a righ t-branc hing structure for \forw ard constituen t c hains" (suc h as A/B B/C C

r

A/B/C C/D D/E/F/G G/H), and a left-branc hing structure for bac kw ard constituen t c hains. So these familiar c hains are the

nly

source

f

spurious am biguit y in CCG. 5 Extending the Approac h to \Restricted" CCG The \pure" CCG

f

x 4 is a ction. Real CCG grammars can and do c ho

se

a subset

f

the p

ssible

rules.

SLIDE 6 F

r

instance, to rule

ut

(14), the (crossing) bac k- w ard rule N/N Nn N ! N/N m ust b e

mitted

from English grammar. (14) [the NP/N [[big N/N [that lik es John] Nn N ] N/N galo

t

N ] N ] NP If some rules are remo v ed from a \pure" CCG grammar, some parses will b ecome una v ailable. Theorem 2 remains true ( 1 NF p er reading). Whether theorem 1 ( 1 NF p er reading) remains true dep ends

n

what set

f

rules is remo v ed. F

r

most linguistically reasonable c hoices, the pro

f
f

theorem 1 wil l go through, 5 so that the normal-form parser

f

x4 remains safe. But imagine remo ving

nly

the rule B/C C ! B: this lea v es the string A/B B/C C with a left-branc hing parse that has no (legal) NF equiv alen t. In the sort

f

restricted grammar where theorem 1 do es not

btain,

can w e still nd

ne

(p

ssibly

non- NF) parse p er equiv alence class? Y es: a dieren t kind

f

ecien t parser can b e built for this case. Since the new parser m ust b e able to generate a non-NF parse when no equiv alen t NF parse is a v ailable, its metho d

f

con trolling spurious am biguit y cannot b e to enforce the constrain ts (7). The

ld

parser refused to build non-NF constituen ts; the new parser will refuse to build constituen ts that are seman tically equiv alen t to already-built constituen ts. This idea

riginates

with (Karttunen, 1986). Ho w ev er, w e can tak e adv an tage

f

the core result

f

this pap er, theorems 1 and 2, to do Karttunen's redundancy c hec k in O (1) time|no w

rse

than the normal-form parser's c hec k for {f c and {bc tags. (Karttunen's v ersion tak es w

rst-case

exp

nen

tial time for eac h redundancy c hec k: see fo

tnote

x 3.) The insigh t is that theorems 1 and 2 establish a

ne-to-one

map b et w een seman tic equiv alence classes and normal forms

f

the pure (unrestricted) CCG: (15) Tw

parses

;

f

the pure CCG are seman tically equiv alen t i they ha v e the same normal form: NF () = NF ( ). The NF function is dened recursiv ely b y x 4.2's pro

f
f

theorem 1; seman tic equiv alence is also dened indep enden tly

f

the grammar. So (15) is meaningful and true ev en if ;

are

pro duced b y a restricted CCG. The tree NF () ma y not b e a legal p arse under the restricted gramma r. Ho w- ev er, it is still a p erfectly go

d

data structure that can b e main tained

utside

the parse c hart, to serv e 5 F

r

the pro

f

to w

rk,

the rules S and T m ust b e a v ailable in the restricted grammar, giv en that R and Q are. This is usually true: since (7) fa v

rs

standard constituen ts and prefers application to comp

sition,

most grammars will not blo c k the NF deriv ation while allo w- ing a non-NF

ne.

(On the

ther

hand, the NF parse

f

A/B B/C C/D/E uses >B2 t wice, while the non-NF parse gets b y with >B2 and >B1.) as a magnet for 's seman tic class. The pro

f
f

theorem 1 (see (13)) actually sho ws ho w to construct NF () in O (1) time from the v alues

f

NF

n

smaller constituen ts. Hence, an appropriate parser can compute and cac he the NF

f

eac h parse in O (1) time as it is added to the c hart. It can detect redundan t parses b y noting (via an O (1) arra y lo

kup)

that their NFs ha v e b een previously computed. Figure (1) giv es an ecien t CKY-st yle algorithm based

n

this insigh t. (P arsing strategies b esides CKY w

uld

also w

rk,

in particular (Vija y-Shank er & W eir, 1993).) The managemen t

f

cac hed NFs in steps 9, 12, and esp ecially 16 ensures that duplicate NFs nev er en ter the

ldNFs

arra y: th us an y alternativ e cop y

f

.nf has the same arra y co

rdinates

used for .nf itself, b ecause it w as built from iden tical subtrees. The function PreferableTo( ,

)

(step 15) pro- vides exibilit y ab

ut

which parse represen ts its class. PreferableTo ma y b e dened at whim to c ho

se

the parse disco v ered rst, the more left- branc hing parse,

r

the parse with few er non- standard constituen ts. Alternativ ely , PreferableTo ma y call an in tonation

r

discourse mo dule to pic k the parse that b etter reects the topic-fo cus divi- sion

f

the sen tence. (A v arian t algorithm ignores PreferableTo and constructs

ne

parse forest p er reading. Eac h forest can later b e unpac k ed in to in- dividual equiv alen t parse trees, if desired.) (Vija y-Shank er & W eir, 1990) also giv e a metho d for remo ving \one w ell-kno wn source"

f

spurious am biguit y from restricted CCGs; x 4.2 ab

v

e sho ws that this is in fact the

nly

source. Ho w ev er, their metho d relies

n

the grammati calit y

f

certain in ter- mediate forms, and so can fail if the CCG rules can b e arbitr arily restricted. In addition, their metho d is less ecien t than the presen t

ne:

it considers parses in pairs, not singly , and do es not remo v e an y parse un til the en tire parse forest has b een built. 6 Extensions to the CCG F

rmalism

In addition to the Bn (\generalized comp

sition")

rules giv en in x 2, whic h giv e CCG p

w

er equiv alen t to T A G, rules based

n

the S (\substitution") and T (\t yp e-raising") com binators can b e linguistically useful. S pro vides another rule template, used in the analysis

f

parasitic gaps (Steedman, 1987; Sz- ab

lcsi,

1989): (16) a. >S: x=y j 1 z m f y j 1 z m g ! x j 1 z m z :f (z )(g (z )) b. <S: y j 1 z xn y j 1 z ! x j 1 z Although S in teracts with Bn to pro duce another source

f

spurious am biguit y , illustrated in (17), the additional am biguit y is not hard to remo v e. It can b e sho wn that when the restriction (18) is used to- gether with (7), the system again nds exactly

ne

SLIDE 7 1. for i := 1 to n 2. C [i

1;

i] := LexCats (word[i]) (* w

rd

i stretc hes from p

in

t i

1

to p

in

t i *) 3. for w idth := 2 to n 4. for star t := to n

w

idth 5. end := star t + w idth 6. for mid := star t + 1 to end

1

7. for eac h parse tree

=

< R;

;
>

that could b e formed b y com bining some

2

C [star t; mid] with some

2

C [mid; end] b y a rule R

f

the (restricted) grammar 8. :nf := NF () (* can b e computed in constan t time using the :nf elds

f
,
,

and

ther

constituen ts already in C . Subtrees are also NF trees. *) 9. existingNF :=

ldNFs

[:nf :rule ; :nf :leftchild :se qno ; :nf :rightchild :se qno ] 10. if undened(existi ngNF) (* the rst parse with this NF *) 11. .nf.se qno := (c

unter

:= c

unter

+ 1) (* n um b er the new NF & add it to

ldNFs

*) 12.

ldNFs

[:nf :r ul e; :nf :leftchild :se qno ; :nf :rightchild :se qno ] := .nf 13. add

to

C [star t; end] 14. .nf.currp arse :=

15.

elsif PreferableTo (; existingNF :cur r par se) (* replace reigning parse? *) 16. .nf := existingNF (* use cac hed cop y

f

NF, not new

ne

*) 17. remo v e .nf.currp arse from C [star t; end] 18. add

to

C [star t; end] 19. .nf.currp arse :=

20.

return(all parses from C [0; n] ha ving ro

t

category S) Figure 1: Canonicalizing CCG parser that handles arbitrary restrictions

n

the rule set. (In practice, a simpler normal-form parser will suce for most grammars.) parse from ev ery equiv alence class. (17) a. VP /NP (<Bx) VP 1 /NP (<Sx) VP 2 /NP led VP 1 n VP 2 /NP [without-reading] VP nVP 1 y esterda y b. VP /NP (<Sx) VP 2 /NP VP nVP 2 /NP (<B2) VP 1 nVP 2 /NP VP nVP 1 (18) a. No constituen t pro duced b y >Bn, an y n

2,

ev er serv es as the primary (left) argumen t to >S. b. No constituen t pro duced b y <Bn, an y n

2,

ev er serv es as the primary (righ t) argumen t to <S. T yp e-raising presen ts a greater problem. V ari-

us

new spurious am biguiti es arise if it is p ermit- ted freely in the grammar. In principle

ne

could pro ceed without grammatical t yp e-raising: (Do wt y , 1988; Steedman, 1991) ha v e argued

n

linguistic grounds that t yp e-raising should b e treated as a mere lexical redundancy prop ert y . That is, when- ev er the lexicon con tains an en try

f

a certain category X, with seman tics x, it also con tains

ne

with (sa y) category T/(Tn X) and in terpretation p:p(x). As

ne

migh t exp ect, this mo v e

nly

sw eeps the problem under the rug. If t yp e-raising is lexical, then the denitions

f

this pap er do not recognize (19) as a spurious am biguit y , b ecause the t w

parses

are no w, tec hnically sp eaking, analyses

f

dieren t sen tences. Nor do they recognize the redundancy in (20), b ecause|just as for the example \softly kno c k t wice" in x 4.1|it is con tingen t

n

a kind

f

lexical coincidence, namely that a t yp e-raised sub ject com- m utes with a (generically) t yp e-raised

b

ject. Suc h am biguiti es are left to future w

rk.

(19) [John NP left SnNP ] S vs. [John S/(Sn NP) left Sn NP ] S (20) [S/(SnNP S ) [SnNP S /NP O /NP I Tn(T/NP O )]] S/S I vs. [S/(SnNP S ) Sn NP S /NP O /NP I ] Tn(T/NP O )] S/S I 7 Conclusions The main con tribution

f

this w

rk

has b een formal: to establish a normal form for parses

f

\pure" Com- binatory Categorial Gramm ar. Giv en a sen tence, ev ery reading that is a v ailable to the grammar has exactly

ne

normal-form parse, no matter ho w man y parses it has in toto. A result w

rth

remem b ering is that, although T A G-equiv alen t CCG allo ws free in teraction among forw ard, bac kw ard, and crossed comp

sition

rules

f

an y degree, t w

simple

constrain ts serv e to eliminate all spurious am biguit y . It turns

ut

that all spuri-

us

am biguit y arises from asso ciativ e \c hains" suc h as A/B B/C C

r

A/B/C C/D D/En F/G G/H. (Wit-

SLIDE 8 ten burg, 1987; Hepple & Morrill, 1989) an ticipate this result, at least for some fragmen ts

f

CCG, but lea v e the pro

f

to future w

rk.

These normal-form results for pure CCG lead directly to useful parsers for real, restricted CCG grammars. Tw

parsing

algorithms ha v e b een presen ted for practical use. One algorithm nds

nly

normal forms; this simply and safely eliminates spurious am biguit y under most real CCG gramma rs. The

ther,

more complex algorithm solv es the spurious am biguit y problem for any CCG gramma r, b y using normal forms as an ecien t to

l

for grouping seman tically equiv alen t parses. Both algorithms are safe, complete, and ecien t. In closing, it should b e rep eated that the results pro vided are for the T A G-equiv alen t Bn (generalized comp

sition)

formalism

f

(Joshi et al., 1991),

ptionally

extended with the S (substitution) rules

f

(Szab

lcsi,

1989). The tec hnique eliminates all spurious am biguities resulting from the in teraction

f

these rules. F uture w

rk

should con tin ue b y eliminating the spurious am biguities that arise from grammati cal

r

lexical t yp e-raising. References Gosse Bouma. 1989. Ecien t pro cessing

f

exible categorial grammar. In Pr

c

e e dings

f

the F

urth

Confer enc e

f

the Eur

p

e an Chapter

f

the Asso ciation for Computational Linguistics, 19{26, Uni- v ersit y

f

Manc hester, April. Da vid Do wt y . 1988. T yp e raising, functional com- p

sition,

and non-constituen t conjunction. In R. Oehrle, E. Bac h and D. Wheeler, editors, Cate gorial Gr ammars and Natur al L anguage Structur es. Reidel. Mark Hepple. 1987. Metho ds for parsing com binatory categorial gramma r and the spurious am biguit y problem. Unpublished M.Sc. thesis, Cen tre for Cognitiv e Science, Univ ersit y

f

Edin burgh. Mark Hepple. 1990. The Gr ammar and Pr

c

essing

f

Or der and Dep endency: A Cate gorial Ap- pr

ach.

Ph.D. thesis, Univ ersit y

f

Edin burgh. Mark Hepple and Glyn Morrill. 1989. P arsing and deriv ational equiv alence. In Pr

c

e e dings

f

the F

urth

Confer enc e

f

the Eur

p

e an Chapter

f

the Asso ciation for Computational Linguistics, 10{18, Univ ersit y

f

Manc hester, April. Herman Hendriks. 1993. Studie d Flexibility: Cate- gories and T yp es in Syntax and Semantics. Ph.D. thesis, Institute for Logic, Language, and Compu- tation, Univ ersit y

f

Amsterdam. Ara vind Joshi, K. Vija y-Shank er, and Da vid W eir. 1991. The con v ergence

f

mildly con text-sensitiv e gramm ar formalism s. In F

undational

Issues in Natur al L anguage Pr

c

essing, MIT Press. Lauri Karttunen. 1986. Radical lexicalism. Rep

rt

No. CSLI-86-68, CSLI, Stanford Univ ersit y . E. K

nig.

1989. P arsing as natural deduction. In Pr

c

e e dings

f

the 27th A nnual Me eting

f

the As- so ciation for Computational Linguistics, V ancou- v er. J. Lam b ek. 1958. The mathematics

f

sen- tence structure. A meric an Mathematic al Monthly 65:154{169. Mic hael Mo

rtgat.

1990. Unam biguous pro

f

represen tations for the Lam b ek Calculus. In Pr

c

e e d- ings

f

the Seventh A mster dam Col lo quium. Mic hael Niv. 1994. A psyc holinguistically motiv ated parser for CCG. In Pr

c

e e dings

f

the 32nd A n- nual Me eting

f

the A CL, Las Cruces, NM, June. cmp-lg/9406031 . Remo P aresc hi and Mark Steedman. 1987. A lazy w a y to c hart parse with com binatory grammars. In Pr

c

e e dings

f

the 25th A nnual Me eting

f

the Asso ciation for Computational Linguistics, Stan- ford Univ ersit y , July . Scott Prev

st

and Mark Steedman. 1994. Sp ec- ifying in tonation from con text for sp eec h syn- thesis. Sp e e ch Communic ation, 15:139-153. cmp- lg/9407015. Mark Steedman. 1990. Gapping as constituen t co

r-

dination. Linguistics and Philosophy, 13:207{264. Mark Steedman. 1991. Structure and in tonation. L anguage, 67:260{296. Mark Steedman. 1987. Com binatory gramma rs and parasitic gaps. Natur al L anguage and Linguistic The

ry,

5:403{439. Anna Szab

lcsi.

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ris,

Dordrec h t. K. Vija y-Shank er and Da vid W eir. 1990. P

lyno-

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the Asso ciation for Computational Linguistics. K. Vija y-Shank er and Da vid W eir. 1993. P arsing some constrained gramma r formalism s. Compu- tational Linguistics, 19(4):591{636. K. Vija y-Shank er and Da vid W eir. 1994. The equiv- alence

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27:511{546. Ken t Witten burg. 1986. Natur al L anguage Pars- ing with Combinatory Cate gorial Gr ammar in a Gr aph-Unic ation-Base d F

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