MONTAgOVIAN DEFINITE
CLAUSE GRAMMAR
R.I. Bainbridge,
Dept. of Computer Science,
Teesside Polvtechnic,
Middlesbrough, Cleveland, England.
Abstract
This paper reports a completed stage of
ongoing research at the University oF
York. Landsbergen's advocacy of analyt-
ical
inverses
~or
compositional suntax
rules encourages the application of Defin-
ite Clause Grammar techniques to the cons-
t~uction
of
a parser returning Montague
analysis trees. A parser MDCg is pres-
ented which implements an augmented
Friedman - Warren algorithm permitting
post referencing, and interfaces with a
language oF intensional logic translator
LILT so as to display the dorivattonal
history of corresponding reduced IL Form-
ulae. Some familiarity with Montague's
PTG and the basic DCQ mechanism is
assumed.
Keqwords Compositional Semantics, Definite
Clause O~amma~, Friedman Warren Algorithm,
Intensional Logic, Mont•gue Orammmar, Nat-
ural
Language Processing, PROLOG.
I Introduct~op
Consideration is given by Landsbergen
(20] to the global algorithmic structure
of a top down parser demonstrably equi-
valent to a compositional grammar such as
that
of PTG [223. The method is
as
Follo~s:
1. Formulate the original grammar in
te~ms of indexed compositional-M rules
of
form:
If syntax trees ~Sk Sn) satisf~
condition C then combine
<Sk Sn~ into Sj
such that the compositional history may be
represented on a derivation tree (i.e. a
skeletal analgsis tree lacking node
labels).
~. Subject to specified restraints evolve
inverse analytical-M Rules of form:
If Sj conforms to condition C"
then analgse Sj into <Sk Sn~.
3. P~ove that the composltional and ana-
lytical M ~ules are equivalent.
4. Construct a two stage parser:
(i) Parse • sentence using a context
free grammar (CFg) thus deriving a
syntax tree.
(ii) Traverse the svntax tree in
postorder [19] under the guidance of
the analytical-M rules, constructing
the derivation tree which reflects
the reverse order application of the
inverse
rules.
An
abstract algorithm describing the
parser is given in the Form
of
procedural
pseudo code, however the problem of
establlshing that an implementation con-
Forms to the algorithm is deferred, a
problem perh•ps aggravated bv the absence
Of
•
Formal
notation for M rules which
might otherwise have suggested
appropriate
data structures.
The postorder traverse in (ii) of a
preorder ere•tiDe involves a duplication
which may be •voided by •dopting the
PROLOG DefiniteClause grammar (DCg)
formalism, (C28] of. [3],
£4],
C5], [21]),
which, as has been observed [32] virtually
forces the methodology of syntax directed
translation coupled with compositional
semantics. A DCG may be ingenuously
characterised as a CFQ having category
sumbols augmented by argument places, and
cont•ining supplementary goals not limited
in function to input consumption Logical
variables in argument places permit
synthesised and inherited attributes (18]
to be handled with equal Facilit U. The
clauses of • DC~ may be directlu executed
by a PROLOG interpreter, hence if combined
CFg+analytical-M rules are presented in
the form of
Definite
Clauses, the problem
of mapping algorithm to implementation
does not arise: the algorithm and program
are one and the s•me.
The pa~sers of both Landsbergen (20]
and Friedman ~ Warren [9] generate only
skeletal trees, other details being
recoverable from the le•ves and operation
indices: however the tedium of such
recover v may properl~ devolve on the comp-
uter, and For ped•gogical purposes at
25
least the production of Full analgsis
trees ~ould be advantageous. This pape~
outlines a DCO implementation of a
version
o~
the
compositional suntax o~ PTG ~hich
~etu~ns Full Montague analusis trees in
the Form of vine d;agrams modified at most
b~ additional .~eatu~e marking on vari-
ables.
Given an input sentence, MDCg
returns sets oF trees, optionally passing
members
to
a language
of
intensional logic
t~anslator (LILT) ~hich generates corres-
ponding IL Formulae. The tndete~minacg of
PRQLOg implies that a
DCO
written with
circumspection mau also be used in
reverse, but it remains to be investigated
~hether the model could be so modified as
to achieve the recent obJectives of
Friedman [8].
To
handle quantification
MDCO
emplous
a variation
oF
the
Friedman-Warren algorithm (FWA) [9].
The programs are implemented in
Universit~ oF Edinburgh DEC-IO PROLQG and
~un on the Universitu of York DEC-IO comp-
uter
~ Imolied Modifications to PT~
The
version
o~ PTO grammar implemented
;n MDCg has both significant and cosmetic
changes. As ~egards the First, Partee
observes ((24], C25]) that a version of
51~ which inse~ts labelled bracketing, and
a version oF $4 sensitive to such
bracketing and generalised to add subject
- agreement to the first verb in each
conjunct of a conjoined verb phrase, is
needed in o~dey.to distinguish (1) ~rom
(2).
(1) ~ohn t~ies to valk and talks.
(2) ~ohn tri#s to ~alk and talk.
Without
labelled bracketing, PTG has dtFF-
4~
if
if
if
if
if
if
if
if
4~
4~
if
4~
if
and then constrains the predicate to be a
conjunction of one or mo~e verb phrases
identifiable as commencing ~ith concordant
Finite Forms. Likewise the p~ocedure
~h|ch pa~ses infinitival complements in
accordance with $8 accepts a conjunction
of one or more verb phrases starting ~ith
infinitives. MDCG successFull~ generates
the trees illustrated in Fig i, thus
tacltlu assuming compositional counter-
parts adopting modifications such as
the b~acketin9 o~ Partee ((24], [~5])0
or the headverb Flagging
convention
of
Bennett [2]. Bennett's simplified sem-
antic tuping, ~hich results F~om t~eating
IV
and CN as primitive categories, is also
exploited in LILT as illustrated in the
appendix.
The MDCG post referencing Facilitq
requires the admission oF alternative
caoitaltse~ variables, and an amended #I0
~hich undertakes the replacement bQ term T
OF the earlier o#:
Ca) the First uncapitalised variable
with index n
or (b) the last occurring variable ~ith
index n.
Whethe~ capitaltsed : va~iables would prove
popular ~ith advocates OF the "well
Formedness constraint" [~7] is uncertain
Feature matching, ~hich is achieved b9
PROLOg's c~oss - 9oal variable instantiat-
ion conventions, plainlg affords a simple
mechanism, From the suntactic viewpoint,
Fo~ handling numbe~ concord and selection-
al restrictions on the basis o~ Feature
marked lexicel entries. Indeed since the
alternative operations licenced bU 52 a~e
also identified in the lexicon, MDCO has
the #acilitu without amendment to produce
analusis trees For plural sentences such
if4~4~4~4~4~4~4~4~41-~4~4~4~4i 4~4k41-g 4k4kakak4~4~4~ 4Hl~4~4kak4~4~.4~4~ak4~akak.aka~ ~.~~ ~.~.~ ~
(a)
#4:4 john tries to ~alk and talks
el: m john
#12:8 trg to ~alk and talk
#8:6 tr U to walk
#l:m
walk
#l:u talk
(b)
#4:4 JOhn tries to walk and talk
#1: ~ JOhn
#8:6 trg to malk and talk
el: ~ tr U
#12:8 ~lk and talk
#I:~ walk
el:= talk
if
if
if
if
tk
t
if
* fig 1.
icult~ identifying head verbs, but since a
DCg works top down it encounters no such
problems. The MDCG analogue 'of
identifies the
aS;
(3) The men have not eaten the Fishes.
$4 First
Features of the subject,
given a Further determiner clause in the
lexicon introducing a definite article
26
paired with an additional operation number
and marked with the features Cdof, pl].
The principle
of
composltlonalitq [I0]
demands that this syntactical facilitg
remain ofFiciallq untriggared pending the
int~oduction oF appropriate plural dater-
miner
interpretation clauses in
LILT~
however its introduction for experimental
purposes allows HOCO and LILT
to
p~ovido a
testbed for the investigation of senses
For
additional quantlflers.
The cosmetic variatian involves the
introduction of further feature marking on
variables, but since variables receive
semantic interpretation only in leaf
position where PTG and HI)CO are equi-
valent, the change has no semantic
significance. Variables as leaves are in
the range heO he~, but whereas PT@
introduces onl~ accusative marking as a
side effect of combination, MI~O adds
markings For gender (and If needed
number). Amendments to PT@ to reflect
these innovations would ba purely
decorative.
S2 would mark
its
outpu&
with
a number Featur( derived #row the
quantifier,
while .both £;4 and 85 would,
like 52, licorice alternative operations
such that f4.0 and
fS. 0
would be
restricted to cases where the input T wore
not a variable, and f4.1 F4.4, fS.l f~.4
would generate ha~ IV thauR IV, TV
him E TV them~ ~espoctivel V. Since the
translation rules ~T4 and TD refer to the
value of the Snout;term of a Function in
the F4,
F5
series these would be
unaffected. Rules in the range S3n,
Sl4n
516n
would apply on
condition
that
the
input sentence did not include a variable
with
index n having discordant features.
IF plural Forms became available, the
subJeCt agreement clause o~ 94 would need
generalisin9, and S13 would, Like Sll and
$12, gain access to FS, marking
its
output
with the number of its First argument in
case the operation were FS, or
with
[+plural] otherwise.
3
Tree
Structures and P@T~ino
Procedures
Nodes on an analysis tree are repres-
ented internally by analogues of
the
"syn"
structures of McCord C213, having the
form:
node(N,F,L,D)
where:
N : A rule number in the Form #Sqn:Fun,
#Sun: (Fun, Inx), or #i:= such that Sun
and Fun ere Man•ague syntax rule and
structural operation numbers, Inx is
a variable subscript, and elm
indicates Iexical inse~ian.
F = A list of Features intrinsic to the
node.
L = A node label in list Form.
D = In the case o~ • non-terminal node a
binary
list
of daughters both of
which are nodes, otherwise a struc-
ture of form:
sense(Item, Category)
used by LILT in the generation of IL
Formulas.
Procedures which parse grammatical cat-
egortss normally have ten arguments the
nature oF which will where necessary be
explained in subsequent sections. The
general form
is
as #alloys:
categoru(N,F,E,L, Ia, Iz, FVB, SA, SRa, SRz)
where
N m A node structure as described.
F m The features of the category - in -
context which may exceed the node
Features. For example case is not an
intrinsic noun phrase leaf feature,
but it constrains adoption to specif-
ied configurations.
E
m
The environment (preorder
predecessors) of the category relat-
ive to which the parse is aborted if
N is non unique.
L m The transmission label.
Za, ZZ m
String
buffers before end after
parsing.
m
Free variables below list
m Substitutions above list.
SRa, SRz
=
Substttuens. required
lists
before and after pa~stng.
4 Imslem~n~tno FMA
in
PROLOQ
The FWA handles the introduction and
subsequent binding of indexed variables on
n-ary
substitutes
for skeletal analysis
trees by the manipulation of two lists,
FVB (free variables below) and SA (sub-
etltuttons above). In order to implement
the algorithm in a PROLOQ DCQ directed
towards the production of strictly
Manta•avian trees, each clause responsible
For creating a node requires both FVB and
SA argument places, the First to act as an
output and the second as in input
parameter, with the proviso that the top
level "sentence" call set both to the
empty list.
A clause charged with the construction
of a T (=NP) node, provided that it does
wore than read a surface pronoun, must be
given the ootion of returning • default
node, or alternatively of binding the noun
phrase discovered to the next available
variable, adding th~ binding to the FVB
set, and returning a variable node
instead. In HDC@ a binding takes the Form
not OF a <variable, noun-phrase) pair but
af a structure:
bind(Var, Inx,Node)
where:
Vat = The indexed variable.
Ins a The subscript.
Node m The complete structure
node(NoF, L,D) for a T or, in case the
binding is performed under the S3
27
analogue, for a CN. The feature
field includes both gender and number
although presentl~ available deter-
miners constrain number to be
singular.
Clauses responsible #o~ returning
sentence and verb phrase nodes must like-
,is• construct • default node, but must be
permitted
to
substit~t e
fo~
it •
node
having this default as younger daughter, a
T node from a binding extracted from the
:u~rent FV~ as elder daughter, and the
structural operation flagged with the
binding index.
In all cases the FVB ~etu~ned to the
head goal must represent the union of the
FVBs o? those sub-goals ~hich construct
daughters (p~eo~de~ successors), plus an U
additions ~esulting
from
a specific c•11
~o ootion, or less any extractions
accomplished b~ a specific call to sub-
stitute The FVB of
a given
node
m•U
nevertheless contain bindings •pparentlu
introduced b~ a preorde~ predecessor
0•cause the effect of substttu~
is
to
#dopt elder sisters. Accordingl~ the
published constraints [9] on
quantification
ove~
variables remaining
Free in preorder predecessors must be
preserved. P~ior to extr•ction MDCG
verifies that the V•r field o~ • binding
does not appear as a label dominated bu
the Node ~ield of an~ other b|nding
available in the current
FVB.
Vacuously quantified relative clauses
("not there" cases [16]) are, surpris-
ingly, tolerated bU the o~iginal FMA,
requirement that in the top level
"sentence" call FVB must be []. The
latter requirement constitutes a final
tilter
as suggested,
albeit with
reservation, by d•nssen ~16] as a means of
ensuring syntactic conformity to the
"variable principle".
When a parsing p~ocedu~e is called
other than at top level, the SA is
initiallsed at the union o~ the SA
of
the
head goal and the FVB of an~ goal
constructing an elde~ sister. A noun
phrase parsing clause which reads a
surface
p~onoun
may
~eference
any binding
in the SA such that, where Node =
node(NoF, L,D), the features in F conform
with the p~onoun in numbe~ and gender. A
variable node having the indexed variable
from the binding in its L Field is
returned, thus achieving an antecedent
~e~e~ence,
Neithe~ LIFO nor FIFO lists suffice to
generate all permitted quantifier scope
variations. I~ FVB and SA a~e formed by
simple concatenation then ~bstitute must
be capable of extracting members ~andomly
Alternatively substitute may safely select
the next available item p~ovided that ~he
lists are formed in such a ~a~ that all
permutations emerge in due course.
MDCG
adopts the latter choice, employing a
p~edicate:
mix(LI,LI, L3)
~hich, given successive calls, simulates
the scattering of the members of L1 within
L2 in a ~andom pattern on the assumption
that L2 is al~ead~ ~andom.
* #14:10:2 the man such that he loves her finds mary *
* #I= mary *
* #4:4 the man such that he loves HER~ finds her2 *
* #2:1 the man such that he loves HER~ *
* #1:= the *
* #3:3:1 man such that he loves HER~ *
* ,,
* fig
2.
*
***********************************************************
although a pa~allel test for variable
eligibility is plainly needed. In MDCG
the eligibility p~oceduPe includes a
mechanism suitable for eliminating vacuous
applications of S3: the selected variable
may not be dominated by any node in
another FVB binding, but it mus t be
dominated by the embedded sentence node.
The elimination of "left ove~s", is.
indexed variables remaining f~ee on the
top node of an analysis tree, is achieved
partly by the constraints on substitution
which prevent appearances outside the
scape of quantification, and partly by the
5 Auamentino FW~ •
Since the gramma~ of PTQ does not
generate post ~efe~encing pronouns, FWA is
not designed to accommodate them. In MDCg
an augmented FWA is introduced to handle
post referencing via capitalised variables
which a~e ale•us realised as surface
p~onouns. For example in response to the
input:
(4) The man such that he
loves he~ finds Ma~y.
the output includes a t~ee commencing as
in fig 2.
28
The augment requires parsing procedures
to accept two additional list holding
argument places, SRa and SRz (Substituens
Required at start and at end). When a
surface pronoun is encountered, a check is
First made both in SA (For an antecedent
~e~e~ent) and
in
SRa (in case
a
previous
post reference has been made) Fo~ •
binding with matching number and gender.
IF none is Found then
a
dummu binding,
with onlu the F Field of the node struc-
ture set, is created. The union of this
item and SRa becomes SRz, ~hilst the dumm U
is added to FVB. The SRa of an elder
daughte~ is the SRa of its parent, the SRa
of a younger daughter is the SRz of its
elder sister, and the SRz of the younger
daughter becomes the SRz oF the parent.
It is no~ required that whenever a noun
phrase making clause exercises its ootion
to introduce' a variable, • check must
First be made of the SR list, and if
possible a suitable dummu binding ex-
tracted and completed with no addition to
the FVB list. The behav|our of PROLOG
ensures that completion effects all
existing
occurrences of the dumm U. A con-
sty•ant on
substitution must now p~ohibit
the extraction From the FVB of anu binding
appea~ing
in
the SRz
list
returned to the
heed goal. In this waq not onlu maq no
qounge~ sister dominate quantification
ove~ a variable remaining Free in the
~amilq of an elde~ ~ister (the original
constraint), but
the elder
siste~ must
extend the same courtesv to her sibling.
b The Mechanics of MOCQ
b. I Handl~na Left Recursion
Fig 3 illustrates the MIDCG equivalent
is essentia11u left rscursive, which pres-
ents problems For a top-down, left-right,
depth First DCQ technique. Standard
methods
(343
For
eliminating left
recurs/on From a CFQ would be inapprop-
riate as thou result in onlu weakl~ equi-
valent
grammars.
The
MDCg solution is to
emplov
a
well Fo~med
subst~ing table
(WFST), (vide [17], [31], (33], (35]) and
assume that the recurring item has al~eadg
been Found, adding to the table the ~esult
of subsequent parsing given that it is
unique relative to its
environment.
Since the WFST must record the ~elative
position of entries, gramm•~ rule notation
(GRN) which insulates the programme~ f~om
lexic•l decomposition must be p~osc~ibed:
accordinglu MDCQ is written in ~aw PROLOG,
pairs of variables in the ~ange Ia. Iz
~epresenting st~ing buffers before and
after parsing.
6. ~
Res.tor•tive
Editina
Reflection on the behaviou~ of the
clause in Fig 3 during the parsing of:
(6) Woman such that a man loves he~.
reveals that pTior to parsing the embedded
sentence, the kth variable (k=Inx) 'in the
~ange heO he~ is generated and its
binding to CN passed on in a new S~ list.
When the p~onoun is encountered, the
binding with index k m•U .be extracted, a
leaf node with he~ as label c~eated, and a
Fo~m marked For number, gende~ and case
returned as transmission label to the
immediatelq dominating node. The value o~
Lb (the embedded sentence label) ~ill in
due course be ~etu~ned as:
(b) a man loves her~.
Before this ma U be p~efixed w~th the
common noun plus "such that" to become the
4. .If
* common(Node, Ft, E,L,I•,Zz,FVB, SA,~Ra, SRz) "- *
* wFst(common(CN, Ft, E, La, Ia, Ib, FVB•, SA,
SRa,
SRb) ), *
sc•n([such, that2, Zb, Ic), *
* gensqm(he, He, lnx, SuFFix), *
* join( (bind(He, Inx, CN)IFVBa],SA, SAa), *
* join(E, CN, El), *
* sentence(S, (dell, El, Lb, Ic, Iz, FVBb, SA•, SRb, SRz), *
* eligible(bind(He,
Inx, CN),FVBb, [3, (3), *
* dominated (He. S). *
* makevars(Nom,_,Acc,_,SuFFJx, Subj,Obj,Ft), .
* editline(Nom, Ace, Sub j, Ob j, Lb, Lc), *
* join(L•, [such, thatlLc],Ld), *
* mix (FVB•, FVBb, FVBc ), *
* substitute(on, node(#3: (3: Inx),Ft, Ld, [CN, S]), *
* Node, Ld,L, [3, [],FVBc,FVB, [],SRz), •
* reco~dz (wFst (common(Node, *
* Ft, ~, L, Ia, I z, FVB, SA, SRa, SRz ) ) ). *
* Fig 3. .
oF Montague's ~ule $3. The inverse of $3 default label Ld it must be edited so as
29
to restore all variables with index k to
appropriate surface Forms. Samples OF
eligible variables
(i.e. k-variables of
appropriate numbep and gender) are created
by makeva~s, whet,after editli~q achieves
the
~equired
restoration.
b. 3 Node and Transmission Labels
The label o$ a leaf node is invariabl~
a
root #orm, but
a morphological variation
is very often required as transmission
label Non-leaf nodes may also be so
cha~acte~ised. When a vl~bph~ase is ex-
tracted F~om the WFST in fig 4, which ill-
~.4 Calls to "substitute" an~ "option"
Fig 4 includes a call to substitute
while a call to ootion occurs in Fig 5
which
illustrates
the
MDCg
clause
responsible #or parsing proper names. The
Form of a substit~tl call is as Follows:
substitute(T, Node, Nodel, T1,Tll,N1,
NL1, FVB, FVB1,Sk, SR)
~hore:
T
=
The t~pe of node involved (s=SEN,
vpmIV, cnmCN).
Node = The default node constructed.
Nodal - The replacement node (Nod.l-Node
if no substitution is made).
TI,TI1 = Default and replacement trans-
* verbphrase(nodo(NO, FO, LO, DO),VF, E,L, Ia, Iz, FVB, SA, SRa, SRz) *
* wfst(vel'bphrase(node(Nl, Flo LI° D1), VF, E, La, Ia, Ib° *
• FVBa0 SAo SRao SRb) )° *
* mix (FVBa, SA, SAa ), •
* join(E, node(N1, Fl° LI° D1 ), El),
* vpadvorb (VPADV, AV, El, Lb, Ib, I z, FVBb, SAa, SRa° SRz ), *
•
Join(L,, Lb, Lc),
•
• Join(Ll° Lb, LI), *
• mix (FVBa° FVBb, FVBc ), •
* substttute(vp, node(#10: 7, VF0 LI,
*
* [VPAI)V, node(N1, F1, L1,91) 3)~ *
• node(NO, FO, LO, DO), *
• Lc, Lo L~, LO, FVBc, FVB, SA, r], SRa), *
• ~ecord z (wFst (ve~bphrase (node (NO, FO, LO, DOt, VF, E, L, *
* ]a, Iz, FVB, SA, SRa, SRz ) ) ). *
* fig 4. *
ust~ates the MDCG equivalent of $10, the mission labels (TII=TI if no substit-
node label L1 must contain the bare ution made).
infinitive o~ the head verb while La Nl°N11 m Default and ~eplacement node
contains a finite Form. Having processed labels (NllmN1 if no substitution
the adverb, a Pa~T of new labels must made, and N1,NLI-[] iS T=s or T=cn
* nounphrase(Node, [g, (C, Num)],E,L, Ia, Iz,FVB, SA, SRa, SRz) "- *
* scan(Pn, Ia, Iz),
* propor(Pn, [O, (Num) ], *
* option(node(#1: "=', [O, (Num)], [Pn], [sense(Pn, [pn])]), *
* [g, (C, Num)], Node, [Phi, Lo [], FVB0 SRa, SRz ), *
* recordz(wFst(nounphrase(Node, [g, (C, Num)],E, *
•
L,
la,
Iz, FVB, SA, SRa, SRz) ) ).
*
Fig 5.
accordingly be constructed, one For the since the new~ node label is
default node and one for its transmission to be Tll).
label. Should a substitution then be FVB, FVB1 = The free variable below
made, twin labels For the introduced before and after an~ extraction.
higher node must likewise be maitained by Sk
the
substitut e
procedure.
SR
taken
lists
= Those bindings bipassed in ancestor
calls to substitute (At top level
S~m£]).
= The substituons requi~ed list
containing the constraints on sub-
stitution.
30
Similarly a call to 9otion appears in
the Form:
option(Node, FoNodel, T1, Tll, FVB, FVB1,
SR, SR1)
where:
Node, Nodel = The default and replacement
nodes.
F = The Features (gender and number) of
the node.
TI, TI1 = The default and transmission
labels.
FVB, FVB1 = The Free variables below lists
before and afte~ any addition.
SR, SRI = The substituens requi~ed lists
before and after any subtraction.
7 A Foretaste of LILT
Warren [32] suggests two possibilities
For encoding l•mbda te~ms in PROLOQ given
the desire to represent • full typed
lambda calculus0 the First portraying
lambda variables as PROLOO structures and
the second equating them with PROLOQ vari-
descriptive commentary similar to that
given bq Paste• [25] and Dowry [7]. This
is accomplished during a traverse in
"g•lile•n" posto~der of the analysis tree,
producing output o~ the Form illustrated
in
the appendix, From which
it will
be
apparent that, since PROLOg does not
recognise • lambd• expression Formed by
juxtaposition, the initial pairing of
operator and ope~•nd
is
achieved via a
convenience p~edicate "eval" and
subsquently evaluated.
Whereas d•nssen ([14], [15]) accomp-
lishes reduction
by
a process of
essentially localised tree transform-
• tions, the simplification algorithm of
LILT takes advantage o~ PROLOG's list
processing
capabilities to undertake
global list transformations whenever
necessary. MDCg - LILT exemplifies the
reorg•nised directed process approach
discussed by War~en and Friedman [33], ie.
LILT is (optionally) called after each
parse. The present objective of display-
• *
• sense(theo[d(sg)],l•mbd•(p:lambd•(q:exists(Y:all(X: *
• ('p(X)<=~equ•ls(X,Y))k('~(Y))))))) "- !.
*
e
• Fig
&.
*
************************************************************
* transl•te(node(N,F,L,
[sense(R,T)]),S) "- *
* !, sense(R,T,S),message(O, EL, S]). *
* translate(Tree, IL) :- *
* structure (Tree, node(N, F, L, _), Lsub, R sub ), *
* tranel•te(Rsub, Rnew), trans1•te(Lsub,Lnew), *
* construct(node(N, F, L, _), Lnewo Rnew, Tree1 ), *
* formulate(Tree1, ILl), *
*
message(N,
ILl), *
* simpliFq (ILl, IL). *
* Fig 7. *
****** ** *************************************************
ables. Since LILT is concerned only with
that subset of lamda calculus needed For
~epresenting Montague's language IL, a
simpler scheme becomes possible. In LILT
predicate variables are represented by
PROLOg atoms while PROLOG variables •re
used directly For individual variables
introduced by "sense*' clauses (other than
those anaphoric references •1ready con-
strained to be in the range xO x~).
The essence of this scheme may be ex-
tracted From Fig 6 which illustrates the
clause correlating singular definite art-
icle with its sense. The top level trans-
lation clauses are illustrated in Fig 7.
These constitute a recursive p~ocedure
which generates reduced IL formulae with
ing a conventional derivational history
makes the immediate return of logical
representations rather than syntactic sub
trees inappropriate. Were all parsing
p~ocedu~es to call a mute ,/e~sion of
translate locally0 it is predicted that a
semantic equivalence parse (up tit) would
result.
8 R~Fe?entes
[I] Ajdukiewicz K. (1935) Sy,tactic con-
nexion, in McCall S. (Ed.) Polish
Lpaic 1920-1939.
Clarendon,
Oxford,
1967.
[2] Bennett M. (197&)
A
variation and
extension of a Montague Fragment of
31
English. in ParSee (1976).
[3] Clocksin W.F. & Mellish C.S. (1981)
P~oaramminq ~n PROLOQ.
Springe~-Verlag, Berlin.
[4] Colme~auer A. (1975) MetamoPphosis
g~amma~s, in Bole L. (Ed.) Natural
Lanauaqe Communi~ation with ~o~p-
ute~_.___~s. Springe~-Ve~lag0 Berlin,
1978.
[5] Dahl V. (1981) TPanslatlng spanish
into logic thPough logic. Ame~tcan
dou~nal
of
Computational Linguistics
Vol. 7 No. 3.
[b] Davis
S.
& Mithun M.
(Eds.) (1979)
Linauistics, Philosoohu, and Montao~e
gPammaP. Unive~sit@ oQ Texas,
Austin.
[7] Do~tq D.R., Wall R.E. & PetePs S.
(1981) Introduction
to Montaaue
Sem-
antics.
Reidel, DoPd~echt: Holland.
[8] F~iedman J. (1981) Expressing logic-
al
FoPmulas in natural language, in
gPoenendijk, danssen, & Sto~hoF
(1981).
~9] FPiedman d. & WaP~en D. 5. (1978) A
pa~sing method For Montague grammars.
Linguistics &
Philosoph~ 2.
[10] F~ege g. (1893)
On
sense and PeF-
e~ence,
in
geach P.
&
Black
M.
(Eds) Ph~losophica 1 Writ~nqs oF
~ottlob F~eg~. Dlackwell, OxFoPd,
19bb.
[II] g~oenendijk d.A.g., danssen T.M.V.; &
StokhoF M.B.d (Eds.) (1981) Formal
Methods
in
the Stud 4
Of
~qguaae I &
~
Mathematlsch CentPum, AmstePdam.
[12] Hintikka K.d.d., Mo~avcslk J.M.E. &
Suppes P. (Eds.) (1973) Ao~Poach~
t~ NatuPal Lanouaqff. Reade1,
Do~d~echt: Holland.
[13] Hobbs J.R. & Rosenschein
S.d.
(1978) Making computational sense of
Montague's lntenslonal logic.
A~i~-
icial Intelligence
9.
[14] danssen T.M.V. (1978) Simulation of
a Montague gPamma~. Annals of
Sqstems
ReseaPch 7.
[15] danssen
T.M.V. (1980)
Logical
investigations on PT@
a~Islr~g ~rom
p~ogramming requirements. Sqnthese
44
[16] danssen T.M.V. (1981) Compositional
semantics and Pelative clause Form-
ation in Montague g~ammaw, in g~oen-
endijk, danssen & StokhoF (1981).
[17] Kaplan R.M. (1973) A general s~ntac-
tic p~ocesso~, in Rustin (1973).
[18]
Knuth
D.E.
(1968)
Semantics
oF con-
text Free languages, Mathematical
S~stems Theor~ Vol. 2 No. 2.
[19] Knuth
D.E.
(1975) The AP~ oF ~9~P-
ute~
PPoqPammin@ Vol. I : Funda~e q-
tal Alao~ithm ~.
Addison
- Wesley,
Reading, Mass.
[~0] Landsbe~gen d. (1981) Adaptation of
Montague gPamma~
to the
~equi~ements
of paPsing, in gPoenendijk, danssen
& Stokho~ (1981).
[21] McCo~d M.
(1982)
Using
slots and
modifiers In logic g~ammaPs Fo~ nat-
uPal language. Artificial Intell-
igence 18.
[22] Montague R.M. (1972) The p~oper
tPeatment
of
quantification in ord-
inary English. in Hintikka et al
(1973) and Thomason (1974).
[23] PaPtee B.H. (1972) Comments on
Montague's papeP, in Hintikka et al
(1973).
[24] PaPtee B.H. (1973) Some transform-
ational extensions of Montague gram-
map. in ParSee (1976).
[25] ParSee B.H. (1975) Montague g~amma~
and t~ans~o~mational gPammar. Ling-
uistic Inquiry 6.
[26] Pa~tee B.H. (Ed.) (1976) Montaque
g~ammaP. Academic PPess, N.Y.
C27] ParSee
B.H.
(1977) ConstPaining
t~ansFoPmational Montague grammar: a
F~amewo~k and a Fragment. in Davis &
Mithun (1981).
[28] PePeira F.C.N. & Warren D.H.D.
(1980) Definiteclause grammars For
language analqsis. Artificial In-
telligence 13.
[29] Rustin R. (Ed.) (1973) Natural
Lanouaqe PPocess~q, Algorithmics
PPess,
N.Y.
[30] Thomason R.H. (1974) (Ed.) Formal
Philosoohu - Selected Papers of
Richard Montaque. Yale, New Ha~en
[31] Thompson
H.
(1981) Chart parsing and
Pule schemata in PSQ. Proceedings of
the 19th. annual meeting of the Ass-
ociation
Fo~
Computational Linguist-
ics 167-172.
[3~] Wa~en D.S. (1983) Using lambda
calculus to Pep~esent meanings in
logic
gPammaps. P~oceedings of the
21st. Annual Meeting of the Assoc-
iation #o~ Computational Linguistics
[33] WaP~en D.S. & F~iedman d. (1982)
Using semantics in non context F~ee
paPsing oF Montague grammar
AmePican ~ou~nal of Computational
Linguistics 8.
[34] WinogPad T. (1983) Lanquaqe as a
Coanitive P~ocess. Addison-Wesle V,
Reading, Mass.
[35] Woods W.A. (1970) An expePimental
paPsing s~stem
~0~
tPansition network
g~ammaPs, in Rustin (1973).
32
Appendix : Sample Output
l:
mary believes chaC John
is
a man.
Parse No. 1
*************
#4:4 mary believes
that
john is a man
#1: = mary
#7:6 believe that John is a man
#1: - believe
#4:4 John is a man
#I:
" John
#5:5
be a man
#l: =
be
#2:2 a man
#1:
-
a
#l: - man
1? yes,
Composit£on
& Simplification
****************************
[0] ~rom Lexicon: Basic expression [man] ->
wan
It]
from Lexicon: Basic expression [a] =>
lambda(p:lambda(q:exists( 3423:('p(_3423)& "q(_3423)))))
[2] from [0,1]: Construction by T2 ->
eval(lambda(p:lambda(q:exiscs( 3423:(~p( 3423)&
'q(_3423))))),'man)
[3] from [2]: Instantiate variable
eval(" "man, 34231
[4] from [3]: Relational no~acion
• "man( 34231
[5] from [4]: Down-up ~onverslon
man(3423)
[6} from [2]: Lambd~converslon
lambda(q:exists( 3423:(man( 3423)& 'q(_3423))))
[7] from Lexicon: Basic expression [be] =~
lambda(sub:lambda( 4607:'sub('lambda(4608:
equals (. 4607 ,_.4608T) ) ) )
[8] from [6,7]: Construction by T5 ->
eval(lambda(sub:lambda(_.4607: "sub('lambda( 4608:
equals(4607, 4608))))),'lambda(q:exisCs( 3~23:
man(_3423)& "~(_3423)))))
[9]
from [8]:
Instantlate variable
eval(" "lambda(q:exists( 3423:(man( 3423)~ "q( 34231)11,
"lambda(4608:equals( 46~7,._4608)))
[I0] from [9]: Down-up conversion
eval(lambda(q:exiscs( 3423:(man( 3423)& "q(_3423)))),
"lambda (_4608 : equals (_ 4607 ,._4608~) )
33
[II] from [I0]: InscanClace variable
eval(" "lambda(4608:equals(_4607,_.4608)), 3423)
[12] from [ii]:
Down-up conversion
eval(lambda(.4608:equals(_4607,4608)),3423)
[13] from [12]: Laabda conversion
equals(4607,_3423)
[14] from [I0]: SubsclCuCe IdenCicals
man(_4607)
[15] from [I01: Lambda conversion
man(4607)
[16] from [8]:
Lamb~a conversion
lambda( 4607:man(4607))
[17] from Lexicon: BaSic expression [John] =>
lambda(p: "p(John))
[18]
from
[16,17]: Construction by T4 ->
eval(lambda(p: "p(John)),'lambda(_.4607:man(_4607)))
[19] from [18]: InscanCiace variable
eval(" "lambda(4607:man(_4607)),John)
[20]
from [19]: Down-up conversion
eval(lambda(_.4607:man(_4607)),John)
[21] from [20]: Lambda conversion
man(John)
[22] from [18]: Lambda conversion
man(John)
[23l from Lexicon: Basic expression [believe] ->
believe
[24] from [22,23]: Conscrucclon
by
T7 =>
eval(believe,'man(John))
[25] from [24]: RelaClonal noCaClon
belleve('man(John))
[26] from Lexicon: Basic expression [mary] ->
lambda(p: "p(mary))
[27]
from
[25,26]: Construction by
T4 =>
eval(lambda(p: "p(mary)),'belleve('man(John)))
[28] from [27]: InsCanclaCe variable
eval("
"believe(*man(John)),mary)
[29] from [28]: Relacional nocaclon
• "belleve(mary,'man(John))
[30] from [29]: Down-up conversion
belleve(mary,'man(John))
[31]
from
[27]: Lambda conversion
believe(mary,'man(John))
Logical
Form
believe(mary,'man(John))
34
. MONTAgOVIAN DEFINITE
CLAUSE GRAMMAR
R.I. Bainbridge,
Dept. of Computer Science,
Teesside. Fig 6 which illustrates the
clause correlating singular definite art-
icle with its sense. The top level trans-
lation clauses are illustrated in Fig