CHART PARSING LAMBEK GRAMMARS: 
MODAL EXTENSIONS AND INCREMENTALITY 
Mark Hepple 
Cambridge University Computer Laboratory, Cambridge, UK. 
Abstract 
This paper I describes a method for chart parsing Lam- 
bek grammars. The method is of particular interest 
in two regards. Firstly, it allows efficient processing of 
grammars which use necessity operators, mr extension 
proposed for handling locality phenomena. Secondly, 
the method is easily adapted to allow incremental pro- 
ceasing of Lambek grammars, a possibility that has 
hitherto been unavailable. 
Introduction 
Categorial Grammars (CGs) consist of two compo- 
nents: (i) a lexicon, which assigns syntactic types 
(plus an associated meaning) to words, (ii) a calculus 
which determines the set of admitted type combina- 
tions. The set of types (T) is defined recursivety in 
terms of a set of basic types (To) and a set of op- 
erators ({\,/} for standard bidirectional CG), as the 
smallest set such that (i) To C T, (ii) /f x,y E T, 
then x\y, x/y E T. 2 Intuitively, lexical types specify 
subcategorisation requirements of words, and require- 
ments on constituent order. We here address a partic- 
ular flexible CG, the (product-free) Lambek calculus 
(L: Lambek, 1958). The rules below provide a nat- 
ural deduction formulation of L (Morrill et al. 1990; 
Barry et al. 1991), where dots above a type represent 
a proof of that type. Proofs proceed from a number 
of initial assumptions, consisting of individual types, 
some of which may be "discharged" as the proof is 
constructed. Eact~ type in a proof is associated with 
a lambda expression, corresponding to its meaning. 
The elimination rule/E states that proofs of A/B and 
B may be combined to construct a proof of A. The 
introduction rule /l indicates that we may discharge 
an assumption B within a proof of A to construct a 
proof of A/B (square brackets indicating the assump- 
tion's discharge). There is a side condition on the in- 
troduction rules, reflecting the ordering significance of 
the directional sloshes. For /1 (resp. \1), the assump- 
t I am grateful to Esther KSnig for discussion of the paper. 
The work was done under a grant to the Cambridge Univer- 
";ty Computer Laboratory, "Unification.based models of lexical 
•ccvu and incremental interpretation", SPG 893168. 
~ln this notation, x/y and x\y are both functions from y 
into x. A convention of left association is used, Bo that, e.g. 
((t~np)/pp)/np may be written s\np/pp/np. 
tion discharged must be the rightmost (resp. leftmost) 
undischarged assumption in the proof. Elimination 
and introduction inferences correspond semantically 
to steps of functional application and abstraction, re- 
spectively. 
Hypothesis rule: 
Elimination rules: 
Introduction rules: 
A:x 
i i i : 
A/B:/ B:x B:x A\B:/ -/E --.\E 
A:/x A:/x 
... \[B:x\] i \[B:x\]i . : . 
A:/ A:\] 
A/B:)tx.\]'/li A\B:),x.____.....~\ I i 
Each proof demonstrates the po6sibility of combin- 
ing the types of its undischarged assumptions, in their 
given order, to yield the type at the bottom of the 
proof. The following proof of "simple forward compo- 
sition" illustrates the approach. 
a/b:= b/c:~ \[c:zli -/E 
b:(~# /E 
a/c:)~z.(x (yz).)/li 
Following Prawitz (1965), a normal form (NF) 
for proofs can be defined using the following mean- 
ing preserving contraction rule and its mirror image 
dual with \ in place of/, which, under a version of 
the Curry-Howard correspondence between proofs and 
lambda terms, are analogous to tile/~-contraction rule 
((,~x.P)Q t> P\[Q/z\]) for lambda expressions. 
': "\[B\]~ ! 
...B A : t> : 
--/l i 
A/B B A --./E 
A 
Under this system, every L proof has an equivalent 
'fl-NF' proof. Such fl-NF proofs have a straight- 
forward structural characterisation, that their main 
branch (the unique path from the proof's end-type to 
an assumption, that includes no types forming the ar- 
gument for an elimination step) consists of a sequence 
of (> 0) eliminations followed by a sequence of (> 0) 
introductions. 
ACq'ES Dl! COLING-92, NANTES, 23-28 ho~r 1992 1 3 4 Paoc. oF COLING-92, NANTES, AUG. 23-28, 1992 
The main approach for parsing L ha:~ been sequeu~ 
calculus theorem proving. 3 Used naiwdy, this ap- 
proach is inefficient due to 'spurious ambiguity', i.e. 
the existence of multiple equivalent proofs tor combi- 
nations. K6nig (1989) and llepple (1990u) develop a 
solution to this problem baaed on detining a NF for se- 
quent proofs. These NF systems as yet cover only the 
basic calculus, and do not extend to various additions 
proposed to overcome the basic system's shortcomings 
ms a grammatical framework. 
Some importance has been attached to the proper- 
ties of flexible CGs in respect of incremental process. 
ing. These grammars typically allow sentences to be 
given analyses which are either fully or prinmrily left> 
branching, in which many sentenceinitiat ~ubstrings 
are interpretable constituents, providing for process- 
ing in which the interl)retation of a sentence is gener- 
ated 'on-line' as the sentence ix presented, lncremen- 
tality is characteristic of hmnan sentence processing, 
and might also allow more efficient machine processing 
of language, by allowing early filtering of semantically 
implausible analyses. It ix notable, however, that us 
methods have yet been proposed tbr incremental pars- 
ing of Lambek grammars. In what tbllows, 1 d~seribe 
a chart method for L and then show how it may bc 
modified to allow both inclusion of an operator \[3, used 
for handling locality constraints, and also to allow in- 
cremental parsing of L. 
Chart Parsing Lambek Grammars 
Standard chart *nethods are inadequate lot L because 
proving that some combination of types is possible 
may involve 'hypothetical reasoning', i.e. using addi- 
tional assumptions over and above just the types tttat 
are being combined. For example, the above proof of 
a/b,b/c ::~ a/c requires an additional assumption c, 
subsequently discharged. Standard chart parsing in- 
volves ordering the edges tbr lexical categories along 
a single dimension, and then adding edges for con. 
stituents that span wider substretches of this dimen- 
sion as constituents are combined. Tim problem fi3r 
L is that there is no place in this set up for addi- 
tional hypothetical eleme*lts. Placing edges tot them 
anywhere on the single dimension of a normal chart 
would simply be incorrect. 
K6nig (1990, 1991), in the only previous chart 
method for L, handles this probhun by placing hyt,o 
thetical elements on separate, independently ordered 
'minicharts', which are created ('emitted') in rc~aponue 
to the presence of edges ttmt bear 'higher order' func 
tor types (i.e. seeking argunlents having fimctional 
types), which may require 'hylmtheticul reasoning' in 
the derivation of their argmnent. Minicharts may '~tt- 
taeh' themselves into other charts (including other 
minicharts) at points where combinations are po~i. 
ble, so that 'chains of attachment' may arise. Somc 
3 Spa¢:e limits preclude diac~msion of I~cei*t proof *~t work. 
fairly complicated I)ook-keeping is required to keep 
track of ~:'hat has combined with what as a basin for 
ensuring correct 'discharge' of hypothetical elements. 
This information ix encoded into edgc~4 by replacing 
the simple indices (or vertices) of standard charts with 
'complex in(liccs'. Unfortunately, the complexity of 
thin method precludes a proper exposition here. How- 
ever, somc (lifl~rcnc~ between K6nig's method and 
the method to be proposed will hc mentioned at the 
end of the next section. 
A New Chart Approach 
I next present a new chart parsing method for L. Its 
most striking diflbrence to tim standard approach is 
that there ix typically more than one ordering govern- 
ing the association of edges in a chart. These orderings 
intersect and overlap, making a chart a 'nmltidimen- 
sioual object'. A uecond difl~rence in that the basic 
unit we adopt ior ~q,ecifying the orderings of the chart 
is pr*ntilivc iuteroals, rather than point-like vertices, 
wi~ere the relative order of the primitive intervals that 
make up an ordering must be explicitly defined. The 
span of edg?'~ is specilicd cxtcnsionally tm tim concate- 
nated sum of some number of primitive intervals. The 
method i, pcfliaps most easily explained by example. 
'lb parse the coml)ina~ion x/y, y/z, z =~- x, we re- 
quire a three clement ordering orde~cing (a.b. e) (a b 
and c being primitive interwdu). Tim three types give 
three edg(.~, each having thrc~: fields: (i) the edge's 
spun (her,, a primitive mtervat), (ii) its type (iii) the 
typc~8 'meaning' (here ~ m,ique constant). 
\[~t, :~/y, £1\] 
\[b, y/z, ~s2Zl 
\[c, z, t3J 
Edg~ are combined uodvr the lollowing chart rules, 
corre~po,diag to our elimination ru\]c~: 
if It, :~/'/, A\] ema I:j, Y, B\] 
aud i~a ~itboid(i. j ) 
i~ \[i, Y, B\] and \[j, %\Y, A\] 
~a~d isa uubord (i. j ) 
~hon \[i.j, X, (AB)\] 
The ruh~4 a\[luw two edges with appropriate types 
to combine provided tilat the concatenation of their 
spans is n gubstrhlg of uome defined ordering (a test 
made by the predicate iua ~ubord), Given these 
rul~, our chart will expt~nd to include the follow- 
ing two edges. The l)reaeuce of atL edge with type x 
that ~l)anu the full width ol- the ningle defined ordering 
show~ that x/y, y/z, z -~ x can be derived. 
\[b.c, y, (%2 %3):I 
\[a.b.~, x, (tl (t.9. t3))\] 
ACfES DE COLING-92, NANTI!S, 23-28 AO{;f 1992 I 3 5 P~o(:. oF C()\[,ING-92, NAN I'ES, AUG. 23-28, 1992 
emit(£H,T,_\] ) : 
if T = I/(Y\BI/F1 .... \Bn/Fm), 
then (add_edges: \[il,Bl,vl\] ...... \[in,Bn,vn\], \[jm,Fm,~n\] ...... \[ji,Fl,wl\] 
add_condltion: 
if ordering(P.H.~.R) and non_empty(G) then ordering(il...in.~.jm..,jl) 
add_condition: if \[(il...in.K.jm...jl),Y,S\] and isa_subord(H.K) 
then \[K, (Y\B1/FI .... \Bn/Fm), (wa@vn@ .... .l@vl@S)\]) 
else 
if T = X\CY\BI/FI .... \Bn/Fla), 
then (add edgoJ: \[il,Bl,vl\], ..... \[in,Bn,vn\], \[Jm,Fn,tm\] ...... \[ji,Fl,ul\] 
add_condition: 
if ordering(P.~.H.R) and non_empty(~) then orderingCtl...in.~.jm...jl) 
add_condition: if \[(il...in.K.jm...jl),Y,S\] and isa_subord(H.K) 
then \[K, (Y\BI/FI .... \Bn/Fm), (~a@vn@ .... .i@vl@S)\]) 
Figure 1: The EMIT procedure 
Our next example x/(y\p/q), y/z\p, z/q =~ x re- 
quires 'hypothetical reasoning' in its derivation, which 
is made possible by the pr~ence of the higher-order 
functor x/(y\p/q). In the natural deduction ap- 
proach, deriving the functor's argument y\p/q might 
involve introduction inference steps which discharge 
additional assumptions p and q occurring peripher- 
ally within the relevant subproof. To chart parse the 
same example, we require firstly the following three 
edges and require a three element ordering: 
ordering ( a. b. c) 
\[a, x/(y\p/q), tl\] 
\[b, y/z\p, t2\] 
\[c, z/q, t3\] 
As in KSnig's approach, an 'emit' step is performed 
on the edge which bears a higher-order type, giving 
various additions to the chart needed to allow for hy- 
pothetical reasoning. Firstly, this gives two new edges, 
which are assigned new primitive intervals: 
\[d, p, ell 
\[e, q, v2\] 
Some new orderings must be defined to allow these 
edges to combine. Since the higher-order func- 
tor is forward directional, possible arguments for 
it must occupy non-empty intervals It such that 
isa_subord(a.H). Hypothetical reasoning with the 
two new edges is useful only in so far as it contributes 
to deriving edges that occupy these spans. Hence, 
the required new orderings are (d.H.e) such that 
isa_subord(a.H). Such new orderings are most con- 
veniently created by including the following condition 
on ordcrings. 
if ordering(P.a.Q.R) and non_empty(Q) 
~hen ordering(d.Q, e) 
In general, such conditions may fire after tile emit step 
that creates the condition, when other new orderings 
are created that include the emitting edge's span. The 
above condition causes new orderings (d.b.e) and 
(d. b. c. e) to be defined, allowing combinations that 
yield the following edges: 
\[d.b, y/z, (t2 vl)\] 
It.e, z, (t3 v2)\] 
\['d.b.c.o, y, ((t2 vl)Ct3 v2))\] 
The final thing created by the emit process is the 
following condition on edges (where I@B represents 
tambda abstraction over I in B): 
if Cd.O.e, y, S\] and isa_subord(a.O) 
then \[0, y\p/q, v2@vl@S\] 
This condition has the effect that whenever an edge of 
a certain form is added to tile chart, another edge of 
a related form is also added. The condition completes 
the process of hypothetical reasoning, by syntactically 
and semantically abstracting over the hypothetical el- 
ements ('discharging' them) to derive the function re- 
quired as argument by the higher order functor. Note 
that, since the intervals d and a arc unique to the 
two edges created in the emit process, any edge span- 
ning an interval (d. 13. e) must have involved these two 
edges in its derivation. The condition 'fires' on the 
above edge spanning (d.b.c.e) to give the first of 
tile following edges, which by combination gives the 
second. This final edge demonstrates the derivability 
of original goal combination. 
\[b.c, y\p/q, v2@vl@((t2 vl)(t3 v2))\]. 
\[a.b.c, x, (tl v2@vl@((t2 v:t)(t3 v2)))\]. 
We have now seen all the basic ingredients re- 
quired for handling hypothetical reasoning in the 
new approach. Figure 1. shows a general (if still 
somewhat informal) statement of the emit procedure 
which is called on every edge added to the chart, 
but which only has an effect when an edge bears 
a higher-order type. The specific consequences de- 
pend on the functor's directionality. The notation 
(Y\B1/F1 .... \Bn/Fm) stands for a functional type re- 
quiring n backward directional arguments BI .... Bn 
AcrEs DE COLING-92, NAN'fE,% 23-28 AO{~ 1992 1 3 6 PROC. OF COLING-92, NANr~S, Auo. 23-28, 1992 
and m forward directional arguments F1 .... Fin in any 
order. 4 In each ease, the procedure simply adds an 
edge for each required hypothetical element, a condi- 
tion on orders (to create all required new orderings), 
and a condition on edges, which fires to produce an 
edge for the result of hypothetical reasoning, should 
it succeed. Note that edges produced by such condi~ 
tions are there only to be argument to some higher 
order functor, and allowing them combine with other 
edges as funclov wmdd be unnecessary work. I assume 
that such edges are marked, and that some mechanism 
operates to block such combinations, s 
A slightly modified emit procedure is required to al- 
low for deriving overall combinations that have a func- 
tional result type. I will not give a full statement of 
this procedure, but merely illustrate it. For example, 
in proving a combination F :¢, y\p/q, where an order- 
ing Q had been defined for the edges of the types F, 
emitting the result type y\p/q would give only a single 
new ordering (not a condition on orderings), a condi- 
tion on edges, and two new edges for the hypothetical 
elements as follows: 
ordering (a. Q. b) 
i~ \[a.Q.b, y, S\] 
then \[~, y\p/q, v2@(vl@S)\] 
Fa, p, eli 
I'b, q, v2\] 
That completes description of the new chart metimd 
for L. A few final comments. Although the method 
has been described for proving type combinations, it 
can also be used for parsing word strings, since lexi- 
cal ambiguity presents no problems. Note that defin- 
ing a new ordering may enable certain combinations 
of edges already present in the chart that were not 
previously allowed, tlowever, simply checking for all 
edge combinations that the new ordering allows will 
result in many previous combinations being redone, 
since new orderings always share some suborderings 
with previously defined orderings. One way to avoid 
this problem is to only check for corr~binations allowed 
by substrings of the new ordering that were not pre- 
viously suborderings. 
Concerning the soundness of this method, note that 
chart derivations can be easily translated into (cor- 
rect) natural deduction proofs, given a knowledge of 
4This notsttion il rather clumsy in that it anpears to mug- 
Kent the prettence of at lemU otie forward and one backward 
directional arguinent and also a relative ordering of these argu~ 
meats, wiles tteither of these impllcationa is intended. A similar 
point can be made about abstraction8 in the schematic semall- 
tics m~lvna .... wtlvllS, whose order and number will iu fact 
mitTor that of the corresponding syntactic arguments. A more 
satisfactory statement of the emit procedure could be made re- 
cursively, but tiffs would take up too nmch space. 
5An alternative would be not entering 6uclt edgea at all, but 
instead have a condition on edges that creates tat edge for the 
restllt of combining the emitting higher-order functor with its 
implicitly derived argtulteltt, directly. 
which edges gave rise to which others, i.e. with bi- 
nary edge combinations corresponding to elimination 
inferences, and with the creation of an edge by a con- 
dition on edges corresponding to some sequence of in- 
troduction inferences. In fact, chart derivations all 
translate to fl-NF proofs, i.e. with introductions al- 
ways made after any eliminations on tile main branch 
of any subproof. This observation provides at least art 
informal indication of the completeness of the method, 
since ttle mechanisms described should allow for chart 
derivations corresponding to all possible/~-NF proofs 
of a given combination, which (as we noted earlier) 
are fully representative. 
Another issue is whether the method is minintal in 
tile sense of allowing only tt single chaxt derivation 
for each reading of a combination. This is not Bo, 
given that distinct but equivalent fl-NF proofs of a 
combination axe possible, due to a second source of 
equivalence for proofs analogous to y-equivalence of 
lambda expressions (i.e. that f "= Ax.f~:). For exani- 
pie, the combination a/(b/c), b/c ==~ a has two fl-NF 
proofs, one involving 'unnece~axy' hypothetical rea~ 
soning. However, having equivalent edges represented 
on the chart, and the undesirable consequences for 
subsequence derivations, can be avoided by a simple 
identity check on edge addition, provided that the 
meaning terms of edges produced by conditions on 
edges are subject to rt-normalisation. 
I will finish with some comparisons of the method 
to that of kSnig (1990, 1991). The importance of 
KSnig's method tm precursor for the new method can- 
not be overstated, tlowever, the new approach is, I be- 
lieve, conceptually much simpler than KSnig's. This 
is largely due to the use of 'conditions on edges' in the 
new approach to handle discharge of hypothetical ele- 
ments, which allows edges to be much simpler objects 
than in KSnig's approach, where edges instead have to 
encode the potentially complex information required 
to allow proper discharge in their 'complex indices'. 
The complex nature of KSnig's edges considerably ob- 
scures the nature of parsing as being simply reasoning 
about sequences of types, and also makes it difficult 
to see how the method might be adapted to allow for 
extensions of L involving additional operators, even 
ones that have straightforward sequent rules. 
A second difference of the new method is that order- 
ings that govern the association of edges are explicitly 
defined. There is a sense m which the multiple inter- 
secting orderings of the new approach can be seen to 
express the dimensions of the search space addressed 
in sequent calculus theorem proving, although collaps- 
ing together the parts of that search space that have 
common structure, ht K6nig's method, although the 
elements that belong together in a miniehart are rel- 
atively ordered, tt~e attachment of one minichaxt to 
another is allowed wherever relevant edges can com- 
bine (although subject to some constraints preventing 
infinite looping). This means that elements may be 
Ac~:s DE COLING-92, NANTES, 23-28 AO~' 1992 1 3 7 PROC. OV COLlNG-92, NAIVFES, AUG. 23-28, 1992 
combined that would not be in sequent calculus theo- 
rem proving or in the new chart method. The conse- 
quences of this difference for the relative complexity 
of the two chart methods is at present unknown. 
Parsing Modal Extensions 
Various extensions of the basic Lambek calculus have 
been proposed to overcome some of its limitations as n 
grammatical approach. Morrill (1989, 1990) suggests 
a unary operator o, for handling locality constraints 
on binding and reflexivisation. This has the following 
inference rules, which make it behave somewhat like 
necessity in the modal logic $4: 
OA A where every undischarged 
--OE ~OI assumption is a O-type A OA 
I will try to briefly suggest how o may help in han- 
dling locality constraints. Boundaries arise where lex- 
ieal functors seek a modal argument, i.e. are of the 
form x/\[:ly, the presence of the o making the argu- 
ment phrase potentially a bounded domain. In ad- 
dition, all lexical types are of the form Ox, i.e. have 
a single o as their outermost operator, which allows 
them to appear embedded within modal domains (c.f. 
the D! rule's requirement of O-ed assumptions). For 
example, a lexica\] NP might be \[\]np, a transitive 
verb El(s\np/np), and a sentence-complement verb 
like believes type O(s\np/Os). In a standard flexi- 
ble CG treatment, extraction is handled by functional 
abstraction over the position of the missing (i.e. ex- 
tracted) element. The type of the abstracted element 
is determined by the type of the higher order lexical 
type that requires this abstraction, e.g. the relative 
pronoun type rel/(s/np) abstracts over np. Note that 
this relative pronoun type cannot extract out of an 
embedded modal domain, because it abstracts over a 
bare (i.e. non-modal) np, whose presence would block 
O1 rule's use in deriving the overall modal constituent. 
However, a relative pronoun rel/(s/\[\]np), which ab- 
stracts over a modal type Drip, can extract out of an 
embedded modal domain. 
Including this operator presents considerable prob- 
lems for efficient processing. Firstly, it excludes the 
use of the NF systems devised for the calculus (KSnig, 
1989; Hepple, 1990a). As noted above~ spurious ambi- 
guity makes ordinary (i.e. non-normal form) sequent 
theorem proving of L inefficient. This problem is 
greatly increased by inclusion of O, largely due to non- 
determinism for use of the OE rule. 6 
6 Conaldrr a sequent S = Ox ! ,l:\]x:h..., nXn ::~ x0, where the 
r~l~ted amquent S s ~ Xl,X2,... ,xn ~ x0 is a theorem. Non- 
determinism for use of \[\[\]L\] means that there are n{ different 
paths of inference from S to S t, so that there are at least n! 
proofs of S for each proof of S I. In fact, interaction of \[OLl with 
other inference ruler means that there is typically many more 
proofs than this. 
The new chart method is fairly easily adapted to 
allow for o, avoiding the non-determinism problem of 
the sequent system, so that parsing examples with \[\] 
is typically only slightly slower than parsing related 
examples without any Os. Firstly, it is crucial that we 
can always identify tbe parent edge(s) for some edge 
(i.e. the immediate edge(s) from which it is derived), 
and thereby an edge's more distant ancestors. I ig- 
nore here the precise details of how this is done. The 
following chart rule serves in place of the bE rule: 
if \[i, ~I. i\] then \[i, X, A\] 
For tile combination x/my, O(y/z), Oz =~ x, we would 
require the following ordering and first three edges. 
The next three edges then result from the operation 
of chart rules: 
ordering(a, b. c) 
\[a, x/~y, tl\] 
\[b, \[\](y/z), t2\] 
\[e. \[\]z, 13\] 
\[b, y/z, t2\] 
\[e, z, 13\] 
\[(b.e), y, (12 13)\] 
To allow completion, we must extend the emit proce- 
dure to also take action in cases where the type of an 
added edge seeks a modal argument. For the case at 
hand, the emit procedure would create a condition on 
edges as follows: 
if \[H, y, S\] and isa_aubord(a.H) 
and eheck_modal_hietory(\[H, y, S\]) 
then \[H, \[\]y, S\] 
The procedure check_modal_history used by this 
condition checks the edge's 'history' to see if it has ap- 
propriate ancestors to license the O-introduction step. 
Recall that the D! rule requires that the undischarged 
assumptions of the proof to which it applies are all O- 
types. The corresponding requirement for the chart 
system is that the edge must have ancestors with ~- 
types that together span the full width of the edge's 
span H (i.e. there must be a subset of the edge's an- 
cestor edges that have \[:l-types, and whose spans con- 
catenate to give H). The edge \[(b.c), y, (12 13)\] 
satisfies this requirement, and so the condition will 
fire, allowing the parse to proceed to successful com- 
pletion, as follows: 
\[(b.c), \[.\]y, (12 13)\] 
\[(a.b.c), x, (tl (t2 13))\] 
More complicated cases arise when an emitted func- 
tor seeks an argument type that is both functional and 
modal. As suggested above, a satisfactory statement 
of the emit process is best given recursively, but there 
is not sufficient space here. Hopefully, an example will 
adequately illustrate the method. Consider what is re- 
quired in emitting an edge \[a, w/(\[\] (x\y)/z), tl\], 
whose type seeks an argument (O(x\y)/z, i.e. a func- 
tion to a modal form of a further functional type. As 
before, emitting creates two new edges and a single 
condition on orderings: 
ACRES DE cOLING-92, NANTES, :23-28 AOI~T 1992 1 3 8 PROC. oF COLING-92, NANTES, Auo. 23-28. 1992 
\[i, y, vl\] 
\[j, z, vt\] 
if ordering(P.a.q.K) and non_empty(q) 
then ordering(i.Q, j) 
However, recursive decomposition of tile type 
((i3(x\y)/z) gives rise to three separate conditions Oil 
edges (which reflect the three aspects of the descrip- 
tion of this type as a 'function to a modal form of a 
further functional type'): 
if \[(i.H.j), x, S\] and isa_subord(a.H) 
then \[(H.j), x\y, vI@S\] 
if \[H.j, xky, S\] and isa_subord(a.B) 
and cheek_modal_history(\[H.j, x\y, S\]) 
then \[H.j, \[\](X\y), S\] 
if \[H.j, ~(x\y), S\] and isa_snbord(a.H) 
then \[H, \[\](x\y)/z, v2@S\] 
These three conditions 'chain' together to create edges 
with the type required by tbe emitted functor. Of 
course in practice, the.three conditions could be col- 
lapsed into a single condition, and such a move seems 
sensible from the viewlmint of efficiency. 
Incremental Parsing 
Despite considerable interest in the theoretical pos- 
sibility of incremental processing using the Lambek 
calculus, no incremental parsing methods have as 
yet been proposed. Indeed, most Lambek parsing 
work has been based around sequent theorem proving, 
which might be viewed as antithetical to incremental 
processing since it fundamentally involves reasoning 
about complete sequences of types. In fact it is fairly 
easy to modify the chart method to a|low some extent 
of incremental processing, i.e. so that scanning left-to- 
right through an input string (or type sequence), the 
chart will contain edges assigning types to substrings 
that would not otherwise receive types (luring parsing, 
including some for initial substrings of the input. 
The modification of the chart method involves al- 
lowing an additional extent of hypothetical reasoning 
over that so far allowed, so that edges for hypothet- 
ical types are added not only for higher-order func- 
tors, but also for first-order functors. This is allowe 
by a new procedure emit*, described below, emit* 
is called on every edge added to the chart, but only 
has an effect if the edge's type is functional, creating 
a new edge for a hypothetical type corresponding to 
the function's first argmnent, as well as a condition on 
orderings and one on edges. The condition on order- 
ings creates new orderings allowing the hypothetical 
edge to cmubine with its 'emitter', and tile result of 
that combination to he combined with filrther edges. 
(The requirement J \= i.K prevents the condition in- 
correctly reapplying to its own output.) Note that the 
new edge's interval is peripheral ill the new orderings 
that are defined since it is only in peripheral position 
that the new hypothesis cau be discharged (hence, we 
have (G. H. i) in the condition of tile first case rather 
than (0. H. i. J)). Such discharge is made by the new 
condition on edges. 
emit*( \[I1 ,T,_\] ) : 
i~ T = X/Y 
then add_edge: \[i, Y, v\] 
add_condit ion : 
if ordering(G.H,J) and J \= i.K 
then ordering(G.H, i) 
add eondit ion : 
if \[(q.i) ,Z,S\] and non_empty(q) 
then \[Q,Z/Y,v@S\] 
else 
if T = X\Y 
then add edge: \[i, Y, v\] 
add_ ¢ondit ion: 
if ordering(G.H.J) and G \= K.i 
then ordering(i.H. J) 
add_eondit ion: 
if \[(i.~) ,Z,S\] and non_empty(q) 
then \[q,zkY,v@S\] 
Let us look at some examples (where we limit our at- 
tention to just edges relevant to the discussion). Con- 
sider parsing the type sequence (x/y, y/z, z). Since 
the method should not depend on the parser know- 
ing the length of the input sequence in advauce, an 
ordering will be defined with each scanning step that 
covers just tile material so far scanned, and which ex- 
tends tile ordering of the previous scanning step by 
one. After scanning the first two types of the input, 
the chart will include at least tile following two edges 
and ordering: 
ordering (a.b) 
\[a, x/y, ill 
\[b, y/z, tt\] 
Applying emit* to the second edge (ignoring the first 
edge here) yields the following edge and conditions: 
\[i, z, v\] 
if ordering(G.b.J) and J \= i.K 
then orderiug(G .b. i) 
if \[(Q.i),T,S\] and non empty(q) 
then \[q,T/z, v@S\] 
The condition on orderings will fire ou the ordering 
(a.b) to produce a new ordering (a.b. i), which per- 
mits the first two of the following edges to be built, 
the third being generated from the second by the con- 
dition on edges. The type x/z this edge assigns to the 
initial substring (x/y, y/z) of the input (correspond- 
ing to the composition of the two functions) would not 
have been created during parsing with other Lambek 
parsing methods. 
\[(b.i), y, (t2 v)\] 
\[(a.b.i), x, (tl (t2 v))\] 
\[(a.b), x/z, ve(tl (t2 v))\] 
Acll~s OE COL1NG-92, NAN-I£S. 23-28 Aot~'r 1992 1 3 9 Paoc. ov COL1NG-92, NANTES, AU(i. 23-28, 1992 
As a second example, consider the stage of hav- 
ing scanned the first two types of the input sequence 
(y, x\y/z, z). Scanning yields the following ordering 
and the first two edges. Applying emit* to the second 
edge yields the third edge, and two conditions: 
ordering (a. b) 
Ca, x/y, ill 
\[b, y/z, t2\] 
ri, z, v2 
if ordering(G.b.J) and J \= i,K 
then ordering(G.b, i) 
if \[(G.i),T,SJ and non_empty(Q) 
then \[Q,T/z,v@SJ 
The ordering condition gives the following new order- 
ing, allowing creation of the subsequent new edges. As 
before, the last edge assigns a type to the combination 
of the first two input types which would not otherwise 
be expected during parsing. 
ordering(a.b, i) 
\[(b.i), x\y, (t2 v)\] 
\[(a.b.i), x, ((t2 v) tl)\] 
\[(a.b), x/z, v@((t2 v) tl)\] 
Although the method allows for a considerable de- 
gree of inerementality, some conceivable incremental 
constituents will not be created that would be in pars- 
ing with alternative categorial frameworks. For exam- 
ple, rules of type raising and composition in Combina- 
tory Catcgorial Grammar (Steedman, 1987; Szabolcsi, 
1987) would allow incremental combination of types 
vp/s, np ::~ vp/(s\np), not allowed by the present ap- 
proach. The modified chart method instead allows 
for the construction of incremental constituents in a 
manner that most closely relates to tim notion of de- 
pendency constituency argued for by Barry & Pick- 
ering (1990) (see also Hepple, 1991), although since 
the modified parser is still a complete parser for L 
it cannot be viewed as implementing a notion of de- 
pendency constituency. 7 Finally, it should be noted 
that the additional hypothetical reasoning allowed by 
emit* and combinations involving additional 'incre- 
mental constituents' result in many 'spurious' analy- 
ses, so that the incremental chart method is in general 
slower than the non-incremental chart method. 
Conclusion 
I have presented a chart parsing method for the Lam- 
bek calculus, which I would argue has several advan- 
tages over that of KSnig (1990, 1991). Firstly, I be- 
lieve that it is considerably conceptually clearer than 
KSnig's method, and more straightforwardly reflects 
intuitions about the nature of hypothetical reasoning 
:'However, some version of a chart parser that used only the 
kind of hypothetical reasoning allowed by the emit* procedure, 
and not that of the emit procedure, might well implement a 
notion of dependency constituency. 
in proving L combinations. Secondly, the relatively 
straightforward nature of the system with respect to 
reasoning about sequences of types should, I believe, 
make it easier to adapt the method to allow for addi- 
tional type-forming operators over those already pro- 
vided in the (product-free) Lambek calculus, partic- 
ularly where operators have fairly straightforward se- 
quent rules. We have seen how the method can be 
extended to allow for Morrill's Eloperator. We have 
also seen how the method may be modified to allow 
incremental parsing of Lambek grammars. 
Acres DE COLING-92, NANTES, 23-28 aotrr 1992 1 4 O PROC. OF COLING-92, NANTES, AUG. 23-28, 1992 

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