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 Definite Clause 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) Definite clause 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 
