EFFICIENT INCREMENTAL PROCESSING WITH CATEGORIAL GRAMMAR 
Abstract 
Some problems are discussed that arise for incremental pro- 
cessing using certain flezible categorial grammars, which in- 
volve either undesirable parsing properties or failure to allow 
combinations useful to incrementality. We suggest a new cal- 
culus which, though 'designed' in relation to categorial inter- 
pretatious of some notions of dependency grammar, seems to 
provide a degree of flexibility that is highly appropriate for in- 
cremental interpretation. We demonstrate how this grammar 
may be used for efficient incremental parsing, by employing 
normalisation techniques. 
Introduction 
A range of categorial grammars (CGs) have been 
proposed which allow considerable flexibility in the 
assignment of syntactic structure, a characteristic 
which provides for categorial treatments of extrac- 
tion (Ades & Steedman, 1982) and non-constituent 
coordination (Steedman, 1985; Dowty, 1988), and 
that is claimed to allow for incremental processing 
of natural language (Steedman, 1989). It is this lat- 
ter possibility that is the focus of this paper. 
Such 'flexible' CGs (FCGs) typically allow that 
grammatical sentences may be given (amongst oth- 
ers) analyses which are either fully or primarily left- 
branching. These analyses have the property of des- 
ignating many of the initial substrings of sentences 
as interpretable constituents, providing for a style of 
processing in which the interpretation of a sentence 
is generated 'on-line' as the sentence is presented. 
It has been argued that incremental interpretation 
may provide for efficient language processing -- by 
both humans and machines -- in allowing early fil- 
tering of thematically or referentially implausible 
readings. The view that human sentence processing 
is 'incremental' is supported by both introspective 
and experimental evidence. 
In this paper, we discuss FCG approaches and 
some problems that arise for using them as a ba- 
sis for incremental processing. Then, we propose a 
grammar that avoids these problems, and demon- 
strate how it may be used for efficient incremental 
processing. 
Mark Hepple 
University of Cambridge Computer Laboratory, 
New Museums Site, Pembroke St, Cambridge, UK. 
e-mail : mrhQuk, a¢. cam. ¢i 
Flexible Categorial Grammars 
CGs consist of two components: (i) a categorial lex- 
icon, which assigns to each word at least one syn- 
tactic type (plus associated meaning), (ii) a calculus 
which determines the set of admitted type combina- 
tions and transitions. The set of types (T) is defined 
recursively in terms of a set of basic types (To) and 
a set of operators (\ and/, for standard bidirectional 
CG), as the smallest set such that (i) To C T, (ii) 
if x,y E T, then x\y, x/y E T. 1 Intuitively, lexi- 
cal types specify subcategorisation requirements of 
words, and requirements on constituent order. The 
most basic (non-flexible) CGs provide only rules of 
application for combining types, shown in (1). We 
adopt a scheme for specifying the semantics of com- 
bination rules where the rule name identifies a func- 
tion that applies to the meanings of the input types 
in their left-to-right order to give the meaning of 
the result expression. 
(1) f: X/Y + Y =~ X (where f= AaAb.(ab)) 
b: Y + X\Y =~ X (where b = AaAb.(ba)) 
The Lambek calculus 
We begin by briefly considering the (product-free) 
Lambek calculus (LC - Lambek, 1958). Various for- 
mulations of the LC are possible (although we shall 
not present one here due to space limitations). 2 
The LC is complete with respect to an intuitively 
sensible interpretation of the slash connectives whereby 
the type x/y (resp. x\y) may be assigned to any 
string z which when left-concatenated (resp. right- 
concatenated) with any string y of type y yields 
a string x.y (resp. y.x) of type x. The LC can 
be seen to provide the limit for what are possible 
1 We use a categorial notation in which x/y and x\y are 
both functions from y into x, and adopt a convention of 
left association, so that, e.g. ((s\np)/pp)/np may be writ- 
ten s\np/pp/np. 
2See Lambek (1958) and Moortgat (1989) for a sequent 
formulation of the LC. See Morrill, Leslie, Hepple & Barry 
(1990), and Barry, Hepple, Leslie & Morrill (1991) for a natu- 
ral deduction formulation. Zielonka (1981) provides a LC for- 
mulation in terms of (recursively defined) reduction schema. 
Various extensions of the LC are currently under investiga- 
tion, although we shall not have space to discuss them here. 
See Hepple (1990), Morrill (1990) and Moortgat (1990b). 
79 
type combinations -- the other calculi which we 
consider admit only a subset of the Lambek type 
combinations, s 
The flexibility of the LC is such that, for any com- 
bination xl,..,x, ==~ x0, a fully left-branching deriva- 
tion is always possible (i.e. combining xl and x2, 
then combining the result with x3, and so on). How- 
ever, the properties of the LC make it useless for 
practical incremental processing. Under the LC, 
there is always an infinite number of result types 
for any combination, and we can only in practice ad- 
dress the possibility of combining some types to give 
a known result type. Even if we were to allow only 
S as the overall result of a parse, this would not tell 
us the intermediate target types for binary combi- 
nations made in incrementally accepting a sentence, 
so that such an analysis cannot in practice be made. 
Comblnatory Categor|al GrRmmar 
Combinatory Categorial Grammars (CCGs - Steed- 
man, 1987; Szabolcsi, 1987) are formulated by adding 
a number of type combination and transition schemes 
to the basic rules of application. We can formulate a 
simple version of CCG with the rules of type raising 
and composition shown in (2). This CCG allows 
the combinations (3a,b), as shown by the proofs 
(4a,b). 
(2) T: x ::~ y/(y\x) (where T - AxAf.(fz)) 
B: x/y + y/z =:~ x/z 
(where B = 
(3) a. np:z, s\np/np:f =~ s/np:Ay.fyz 
b. vp/s:f, np:z =~ vp/(s\np):Ag.f(gz) 
(4) (a) np s\np/np (b) vp/s np T T 
s/(s\np) \]3 s/(s\nP)B 
s/np vp/(s\np) 
The derived rule (3a) allows a subject NP to com- 
bine with a transitive verb before the verb has com- 
bined with its object. In (3b), a sentence em- 
bedding verb is composed with a raised subject NP. 
Note that it is not clear for this latter case that the 
combination would usefully contribute to incremen- 
tal processing, i.e. in the resulting semantic expres- 
sion, the meanings of the types combined are not di- 
rectly related to each other, but rather a hypothet- 
ical function mediates between the two. Hence, any 
3In some frameworks, the use of non-Lambek-valid rules 
such as disharmonic composition (e.g. x/y + y\z ::~ x\z) 
has been suggested. We shall not consider such rules in this 
paper. 
requirements that the verb may have on the seman- 
tic properties of its argument (i.e. the clause) could 
not be exploited at this stage to rule out the re- 
sulting expression as semantically implausible. We 
define as contentful only those combinations which 
directly relate the meanings of the expressions com- 
bined, without depending on the mediation of hy- 
pothetical functions. 
Note that this calculus (like other versions of CCG) 
fails to admit some combinations, which are allowed 
by the LC, that are contentful in this sense -- for 
example, (5). Note that although the seman- 
tics for the result expression in (5) is complex, 
the meanings of the two types combined are still di- 
rectly related -- the lambda abstractions effectively 
just fulfil the role of swapping the argument order 
of the subordinate functor. 
(5) x/(y\z):f, y/w\z:g ~ x/w:Av.f(Aw.gwv) 
Other problems arise for using CCG as a basis 
for incremental processing. Firstly, the free use of 
type-raising rules presents problems, i.e. since the 
rule can always apply to its own output. In practice, 
however, CCG grammars typically use type specific 
raising rules (e.g. np =~ s/(s\np)), thereby avoiding 
this problem. Note that this restriction on type- 
raising also excludes various possibilities for flexible 
combination (e.g. so that not all combinations of the 
form y, x\y/z =~ x/z are allowed, as would be the 
case with unrestricted type-raising). 
Some problems for efficient processing of CCGs 
arise from what has been termed 'spurious ambigu- 
ity' or 'derivational equivalence', i.e. the existence 
of multiple distinct proofs which assign the same 
reading for some combination of types. For exam- 
ple, the proofs (6a,b) assign the same reading for 
the combination. Since search for proofs must be 
exhaustive to ensure that all distinct readings for a 
combination are found, effort will be wasted con- 
structing proofs which a....~ ~he same meaning, 
considerably reducing the elficiency of processing. 
Hepple & Morrill (1989) suggest a solution to this 
problem that involves specifying a notion of nor- 
mal form (NF) for CCG proofs, and ensuring that 
the parser returns only NF proofs. 4 However, their 
method has a number of limitations. (i) They con- 
sidered a 'toy grammar' involving only the CCG 
rules stated above. For a grammar involving fur- 
ther combination rules, normalisation would need 
to be completely reworked, and it remains to be 
shown that this task can be successfully done. (ii) 
4Normalisation has also been suggested to deal with the 
problem of spurious ambiguity as it arises for the LC. See 
K6nig (1989), Hepple (1990) and Moortgat (1990). 
80 
The NF proofs of this system are right-branching 
-- again, it remains to be shown that a NF can be 
defined which favours left-branching (or even pri- 
marily left-branching) proofs. 
(6) (a) x/y y/z - (b) x/y y/z f B 
y x/z 
f f 
x x 
Meta-Categorial Grammar 
In Meta-Categorial Grammar (MCG - Morrill, 1988) 
combination rules are recursively defined from the 
application rules (f and b) using the metarnles (7) 
and (8). The metarules state that given a rule 
of the form shown to the left of ==~ with name ~, 
a further rule is allowed of the form shown to the 
right, with name given by applying tt or L to ¢ as 
indicated. For example, applying It to backward 
application gives the rule (9), which allows com- 
bination of subject and transitive verb, as T and 
B do for CCG. Note, however, that this calculus 
does not allow any 'non-contentful' combinations 
-- all rules are recursively defined on the applica- 
tion rules which require a proper functional relation 
between the types combined. However, this calcu- 
lus also fails to allow some contentful combinations, 
such as the case x/(y\z), y/w\z =:~ x/w mentioned 
above in (5). Like CCG, MCG suffers from spurious 
ambiguity, although this problem can be dealt with 
via normalisation (Morrill, 1988; Hepple & Morrill, 
1989). 
(7) ¢:x+y:~z =:~ R¢:x+y/w=C,z/w 
(where R = ~g,~a~b,~c.ga(bc)) 
(8) ¢:x+y=~z ==~ L¢:x\w+y:C,z\w 
(where L = ag a bae g(ac)b) 
(9) Rb: y + x\y/z =~ x/z 
The Dependency Calculus 
In this section, we will suggest a new calculus which, 
we will argue, is well suited to the task of incremen- 
tal processing. We begin, however, with some dis- 
cussion of the notions of head and dependent, and 
their relevance to CG. 
The dependency grammar (DG) tradition takes 
as fundamental the notions of head, dependent and 
the head-dependent relationship; where a head is, 
loosely, an element on which other elements depend. 
An analogy is often drawn between CG and DG 
based on equating categorial functors with heads, 
whereby a functor x/yl../yn (ignoring directional- 
ity, for the moment) is taken to correspond to a head 
requiring dependents Yl..Yn, although there are sev- 
eral obvious differences between the two approaches. 
Firstly, a categorial functor specifies an ordering 
over its 'dependents' (function-argument order, that 
is, rather than constituent order) where no such or- 
dering is identified by a DG head. Secondly, the 
arguments of a categorial functor are necessarily 
phrasal, whereas by the standard view in DG, the 
dependents of a head are taken to be words (which 
may themselves be heads of other head/dependent 
complexes). Thirdly, categorial functors may spec- 
ify arguments which have complex types, which, by 
the analogy, might be described as a head being able 
to make stipulations about the dependency require- 
ments of its dependent and also to 'absorb' those 
dependency requirements. 5 For example, a type 
x/(y\z) seeks an argument which is a "y needing a 
dependent z" under the head/functor analogy. On 
combining with such a type, the requirement "need 
a dependent z" is gone. Contrast this with the use 
of, say, composition (i.e. x/y, y/z =~ x/z), where a 
type x/y simply needs a dependent y, and where 
composition allows the functor to combine with its 
dependent y while the latter still requires a depen- 
dent z, and where that requirement is inherited onto 
the result of the combination and can be satisfied 
later on. 
Barry & Pickering (B&P, 1990) explore the view 
of dependency that arises in CG when the functor- 
argument relationship is taken as analogous to the 
traditional head-dependent relationship. A problem 
arises in employing this analogy with FCGs, since 
FCGs permit certain type transformations that un- 
dermine the head-dependent relations that are im- 
plicit in lexical type assignments. An obvious exam- 
ple is the type-raising transformation x =~ y/(y\x), 
which directly reverses the direction of the head- 
dependent relationship between a functor and its 
argument. B&P identify a subset of LC combina- 
tions as dependency preserving (DP), i.e. those com- 
binations which preserve the head-dependent rela- 
tions implicit in the types combined, and call con- 
stituents which have DP analyses dependency con- 
stituents. B&P argue for the significance of this 
notion of constituency in relation to the treatment 
of coordination and the comparative difficulty ob- 
served for (human) processing of nested and non- 
5Clearly, a CG where argument types were required to be 
basic would be a closer analogue of DG in not allowing a 
'head' to make such stipulations about its dependents. Such 
a system could be enforced by adopting a more restricted 
definition of the set of types (T) as the smallest set such that 
(i) To C T, (ii) if x E T and y E To, then x\y, x/y E T (c.f. 
the definition given earlier). 
81 
nested constructionsfi B&P suggest a means for 
identifying the DP subset of LC transformations 
and combinations in terms of the lambda expres- 
sions that assign their semantics. Specifically, a 
combination is DP iff the lambda expression speci- 
fying its semantics does not involve abstraction over 
a variable that fulfils the role of functor within the 
expression (c.f. the semantics of type raising in (2))ff 
We will adopt a different approach to B&P for 
addressing dependency constituency, which involves 
specifying a calculus that allows all and only the DP 
combinations (as opposed to a criterion identifying 
a subset of LC combinations as DP). Consider again 
the combination x/(y\z), y/w\z =~ x/w, not admit- 
ted by either the CCG or MCG stated above. This 
combination would be admitted by the MCG (and 
also the CCG) if we added the following (Lambek- 
valid) associativity axioms, as illustrated in (11). 
(10) a: x\y/z=~x/z\y 
a: x/y\z=~x\z/y 
(where a = ~f~a\]b.fba) 
(II) x/(y\z) y/w\z 
~a 
y\,/w 
Rf x/w 
We take it as self-evident that the unary trans- 
formations specified by these two axioms are DP, 
since function-argument order is a notion extrane- 
ous to dependency; the functors x\y/z and x/z\y 
have the same dependency requirements, i.e. depen- 
dents y and z. s For the same reason, such reordering 
of arguments should also be possible for functions 
that occur as subtypes within larger types, as in 
(12a,b). The operation of the associativity rules 
can be 'generalised' in this fashion by including the 
unary metarules (13), 9 which recursively define 
eSee Baxry (forthcoming) for extensive discussion of de- 
pendency and CG, and Pickering (1991) for the relevance of 
dependency to human sentence processing. 
7B&P suggest a second criterion in terms of the form of 
proofs which, for the natural deduction formulation of the 
LC that B&P use, is equivalent to the criterion in terms 
of laznbda expressions (given that a variant of the Curry- 
Howard correspondence between implicational deductions 
and lambda expressions obtains). 
s Clearly, the reversal of two co-directional arguments (i.e. 
x/y/z =~ x/z/y) would also be DP for this reason, but is not 
LC-valld (since it would not preserve linear order require- 
ments). For a unidirectional CG system (i.e. a system with a 
single connective/, that did not specify linear order require- 
ments), free reversal of axguments would be appropriate. We 
suggest that a unidirectional variant of the calculus to be 
proposed might be the best system for pure reasoning about 
'categorial dependency', aside from linearity considerations. 
9These unary metarules have been used elsewhere as part 
of the LC formulation of Zielonka (1981). 
new unary rules from tile associat, ivit.) axioms. 
(12) a. a\b/c/d ~ a/ckb/d 
b. x/(a\b/c) ~ x/Ca/c\b) 
(13) a. ¢: x=~y ==~ V¢: x/z ::~y/z 
¢: x=~y ==~ V¢: x\z =~y\z 
(where V =  f a b.f(ab)) 
b. ¢:x=~y ==~ Z¢: z/y=~z/x 
¢: x==~y ~ Z¢: z\y=~ z\x 
(where Z = 
(14) x/(a\b/c):f~ x/(a/c\b):~v./O~a~b.vba) 
Clearly, the rules {V,Z,a} allow only DP unary 
transformations. However, we make the stronger 
claim that these rules specify the limit of DP unary 
transformations. The rules allow that the given 
functional structure of a type be 'shuffled' upto the 
limit of preserving linear order requirements. But 
the only alternative to such 'shuffling' would seem 
to be that some of the given type structure be re- 
moved or further type structure be added, which, by 
the assumption that functional structure expresses 
dependency relations, cannot be DP. 
We propose the system {L,R,V,Z,a,f,b} as a cal- 
culus allowing all and only the DP combinations and 
transformations of types, with a 'division of labour' 
as follows: (i) the rules f and b, allowing the estab- 
lishment of direct head-dependent relations, (ii) the 
subsystem {V,Z,a}, allowing DP transformation of 
types upto the limit of preserving linear order, and 
(iii) the rules tt and L, which provide for the inher- 
itance of 'dependency requirements' onto the result 
of a combination. We call this calculus the depen- 
dency calculus (DC) (of which we identify two sub- 
systems: (i) the binary calculus B : {L,R,f,b}, (ii) 
the unary calculus U : {V,Z,a}). Note that B&P's 
criterion and the DC do not agree on what are DP 
combinations in all cases. For example, the seman- 
tics for the type transformation in (14) involves ab- 
straction over a variable that occurs as a functor. 
Hence this transformation is not DP under B&P's 
criterion, although it is admitted by the DC. We 
believe that the DC is correct in admitting this and 
the other additional combinations that it allows. 
There is clearly a close relation between DP type 
combination and the notion of contentful combi- 
nation discussed earlier. The 'dependency require- 
ments' stated by any lexical type will constitute the 
sum of the 'thematically contentful' relationships 
into which it may enter. In allowing all DP com- 
binations (subject to the limit of preserving linear 
order requirements), the DC ensures that lexieally 
82 
originating dependency structure is both preserved 
and also exploited in full. Consequently, the DC is 
well suited to incremental processing. Note, how- 
ever, that there is some extent of divergence be- 
tween the DC and the (admittedly vague) criterion 
of 'contentful' combination defined earlier. Con- 
sider the LC-valid combination in (15), which is 
not admitted by the DC. This combination would 
appear to be 'contentful' since no hypothetical se- 
mantic functor intervenes between land g (although 
g has undergone a change in its relationship to its 
own argument which depends on such a hypothet- 
ical functor). However, we do not expect that the 
exclusion of such combinations will substraet signif- 
icantly from genuinely useful incrementality in pars- 
ing actual grammars. 
(15) x/(y/z):/, x:l(X .g(Xh.hv)) 
Parsing and the Dependency Calculus 
Binary combinations allowed by the DC are all of 
the form (16) (where the vertical dots abbrevi- 
ate unary transformations, and ¢ is some binary 
rule). The obvious naive approach to finding possi- 
ble combinations of two types x and y under the DC 
involves searching through the possible unary trans- 
forms of x and y, then trying each possible pairing 
of them with the binary rules of B, and then deriv- 
ing the set of unary transforms for the result of any 
successful combination. 
At first sight, the efficiency of processing using 
this calculus seems to be in doubt. Firstly, the 
search space to be addressed in checking for possible 
combinations of two types is considerably greater 
than for CCG or MCG. Also, the DC will suffer spu- 
rious ambiguity in a fashion directly comparable to 
CCG and MCG (obviously, for the latter case, since 
the above MCG is a subsystem of the DC). For ex- 
ample, the combination x/y, y/z, z ::~ x has both 
left and right branching derivations. 
However, a further equivalence problem arises due 
to the interderivability of types under the unary 
subsystem U. For any unary transformation x :=~ y, 
the converse y :~ x is always possible, and the se- 
mantics of these transformations are always inverses. 
(This obviously holds for a, and can be shown to 
hold for more complex transformations by a simple 
induction.) Consequently, if parsing assigns distinct 
types x and y to some substring that are merely 
variants under the unary calculus, this will engen- 
der redundancy, since anything that can be proven 
with x can equivalently be proven with y. 
(16) x y 
X 0 
Z 
Normalisation and the Dependency Calculus 
These efficiency problems for parsing with the DC 
can be seen to result from equivalence amongst terms 
occurring at a number of levels within the system. 
Our solution to this problem involves specifying nor- 
mal forms (NFs) for terms -- to act as privileged 
members of their equivalence class -- at three differ- 
ent levels of the system: (i) types, (ii) binary com- 
binations, (iii) proofs. The resulting system allows 
for efficient categorial parsing which is incremental 
up to the limit allowed by the DC. 
A standard way of specifying NFs is based on 
the method of reduction, and involves defining a 
contraction relation (I>1) between terms, which is 
stated as a number of contraction rules of the form 
X !>1 Y (where X is termed a redez and Y its con- 
tractum). Each contraction rule allows that a term 
containing a redex may be transformed into a term 
where that occurrence is replaced by its contractum. 
A term is said to be in NF if and only if it contains 
no redexes. The contraction relation generates a re- 
duction relation (1>) such that X reduces to Y (X I> 
Y) iff Y is obtained from X by a finite series (pos- 
sibly zero) of contractions. A term Y is a NF of X 
iff Y is a NF and X 1> Y. The contraction relation 
also generates an equivalence relation which is such 
that X = Y iff Y can be obtained from X by a se- 
quence of zero or more steps, each of which is either 
a contraction or reverse contraction. 
Interderivability of types under U can be seen as 
giving a notion of equivalence for types. The con- 
traction rule (17) defines a NF for types. Since 
contraction rules apply to any redex subformula oc- 
curring within some overall term, this rule's do- 
main of application is as broad as that of the as- 
sociativity axioms in the unary calculus given the 
generalising effects of the unary metarules. Hence, 
the notion of equivalence generated by rule (16) is 
the same as that defined by interderivability un- 
der U. It is straightforward to show that the reduc- 
tion relation defined by (16) exhibits two impor- 
tant properties: (i) strong normalisation 1°, with the 
1°To prove strong normalisation it is sufficient to give a 
metric which assigns each term a finite non-negative integer 
score, and under which every contraction reduces the score 
for a term by a positive integer amount. The following metric 
suffices: (a) X ~ = 1 if X is atomic, (b) (X/Y) t = X ~ + Y~, 
(c) (X\Y)' = 2(X' + Y'). 
83 
consequence that every type has a NF, and (ii) the 
Church-Rosser property, from which it follows that 
NFs are unique. In (18), a constructive notion 
of NF is specified. It is easily shown that this con- 
structive definition identifies the same types to be 
NFs as the reduetive definition. 11 
(17) x/y\,. ~1 x\z/y 
(18) x\yl.-Yi/Yi+l..Yn 
where n _~ 0, x is a basic type and each yj 
(1 < j < n) is in turn of this general form. 
(19) ¢: x/ut,.u, + y =~ z ==~ 
L(n)¢: x\w/ul..U, + y =~ z\w 
(where L(n) ---- A#AaAbAc.#(Ava..vn.avl..vnc)b) 
We next consider normalisation for binary com- 
binations. For this purpose, we require a modified 
version of the binary calculus, called W, having the 
rules {L(n),R,f,b}), where L(n) is a 'generalised' 
variant of the metarule L, shown in (19) (where the 
notation X/Ul..Un is schematic for a function seek- 
ing n forward directional arguments, e.g. so that for 
n = 3 we have x/ux..un = X/Ul/U~/Us). Note that 
the case L(0) is equivalent to L. 
We will show that for every binary combination 
X + Y =~ Z under the DC, there is a correspond- 
ing combination X' + Y~ =* Z' under W, where X ~, 
Y' and Z' are the NFs of X, Y and Z. To demon- 
strate this, it is sufficient to show that for every 
combination under B, there is a corresponding W 
combination of the NFs of the types (i.e. since for 
binary combinations under the DC, of the form in 
(16), the types occurring at the top and bottom of 
any sequence of unary transformations will have the 
same NF). 
The following contraction rules define a NF for 
combinations under B ~ (which includes the combi- 
nations of B as a subset -- provided that each use 
of L is relabelled as L(0)): 
(20) IF w l>t w' THEN 
a. f: w/y + y :=~ w 1>1 f: w'/y + y =~ w' 
b. f: y/w + w ::~ y I>t f: y/w' + w' =~ y 
c. b: y+w\y=~w E>lb: y+w~\y=~w' 
d. b: w + y\w :=~ y !>1 b: w' + ykw' :=~ y 
e. L(i)¢: x\w/ul..Ui + y =~ z\w I>1 
L(i)¢: xkw'/ul..u/ + y =~ zkw t 
f. Re: x + y/w =~ z/w t>l 
Re: x + y/w' ::~ z/w' 
laThis NF is based on an arbitrary bias in the restruc- 
turing of types, i.e. ordering backward directional arguments 
after forward directional arguments. The opposite bias (i.e. 
forward arguments after backward arguments) could as well 
have been chosen. 
(21) L(i)R¢: x\w/ul..ui + y/v =~ z/v\w t>l 
RL(i)¢: x\w/ul..ui + y/v ::~ zkw/v 
(22) L(o)f: x/w\v + w ~ x\v \[:>1 
f: x\v/w + w =~ x\v 
(23) L(i)f: xkw/ul..Ui + ui =*" x/ul..ui-t\w t>l 
f: x\w/ul..ul + ui ~ x\w/ul..u;_~ 
for i > O. 
(24) b: ~. + x/y\~, ~ x/y ~1 
Rb: z + x\z/y =~ x/y 
(25) L(i)¢: X/V\W/Ul..U i + y ~ Z\W E> 1 
L(i+I)¢: x\w/v/ul..ui + y ==~ z\w 
(26) IF ¢: x+y==~z 1>1 ¢': x'+y':=~z' 
THEN R¢:x+y/w:=~z/w I>l 
Re': x' + y'/w =~ z'/w 
(27) IF ¢: X/Ul..Ui + y :=~ z I>t 
¢~: x'/ul'..ul ~ + y' =~ z' 
THEN L(i)~b: x\w/ul..ui + y =~ z I>1 
L(i)¢': x'\w/ul'..ui' + y' ~ z' 
These rules also transform the types involved into 
their NFs. In the cases in (20), a contraction is 
made without affecting the identity of the particular 
rule used to combine the types. In (21-25), the 
transformations made on types requires that some 
change be made to the rule used to combine them. 
The rules (26) and (27) recursively define new 
contractions in terms of the basic ones. 
This reduction system can be shown to exhibit 
strong normalisation, and it is straightforward to ar- 
gue that each combination must have a unique NF. 
This definition of NF accords with the constructive 
definition (28). (Note that the notation R n rep- 
resents a sequence of n Rs, which are to be brack- 
eted right-associatively with the following rule, e.g. 
so that R~f = (R(Rf)), and that i takes the same 
value for each L(i) in the sequence L(i)"L) 
(28) ¢:x+y~z 
where x, y, z are NF types, and ¢ is (Rnf) 
or (RnL(i)mb), for n, m > 0. 
Each proof of some combination xl,..,xn =~ x0 
under the DC can be seen to consist of a number of 
binary 'subtrees', each of the form (16). If we sub- 
stitute each binary subtree with its NF combination 
in W, this gives a proof of Xlt,..,x~ ' =~ x0 t (where 
each xl ~ is the NF ofxi). Hence, for every DC proof, 
there is a corresponding proof of the combination of 
the NFs of the same types under B'. 
Even if we consider only proofs involving NF com- 
binations in W, we observe spurious ambiguity of 
the kind familiar from CCG and MCG. Again, we 
can deal with this problem by defining NFs for such 
84 
proofs. Since we are interested in incremental pro- 
cessing, our method for identifying NF proofs is 
based on favouring left-branching structures. 
Let us consider the patterns of functional depen- 
dency that are possible amongst sequences of three 
types. These are shown in (29). 12 Of these cases, 
some (i.e. (a) and (f)) can only be derived with 
a left-branching proof under B' (or the DC), and 
others (i.e. (b) and (e)) can only be derived with 
a right-branching proof. Combinations of the pat- 
terns (c),(d) and (g) commonly allow both right and 
left-branching derivations (though not in all cases). 
(29) (a) ~ (h) ( 
x y z x y z 
(c) (d) 
x y z x y z 
(e) , (f) • 
x y z x y z 
(g) 
x y z 
(30) (R"f): x/y + y/ul..un ~ x/ul..u. 
(31) (R"L(/)mb): 
x\wl..wm/ul..u, + y\(xlul..n,)lvl..v. 
=~ y\wl..wm/vl..v,~ 
NF binary combinations of the pattern in (28) take 
the two more specific forms in (30) and (31). 
Knowing this, we can easily sketch out the schematic 
form of the three element combinations correspond- 
ing to (29c,d,g) which have equivalent left and 
right branching proofs, as shown in Figure 1. 
We can define a NF for proofs under B I (that use 
only NF combinations) by stating three contraction 
rules, one for each of the three cases in Figure 1, 
where each rule rewrites the right branching three- 
leaf subproof as the equivalent left branching sub- 
proof. This will identify the optimally left branch- 
ing member of each equivalence class of proofs as its 
NF exemplar. Again, it is easily shown that reduc- 
tion under these rules exhibits strong normalisation 
and the Church-Rosser property, so that every proof 
must have a unique normal form. However, it is not 
so easy to prove the stronger claim that there is only 
a single NF proof that assigns each distinct read- 
ing for any combination. 13 We shall not attempt 
12Note that various other conceivable patterns of depen- 
dency do not need to be considered here since they do not 
correspond to any Lambek-valid combination. 
~3 Thls holds if the contraction relation generates an equiv- 
to demonstrate this property, although we believe 
that it holds. We can identify the redexes of these 
three contraction rules purely in terms of the rules 
used to combine types, i.e. without needing to ex- 
amine the schematic form of the types, since the 
rules themselves identify the relevant structure of 
the types. In fact, the right-branching subproofs for 
cases (29c,g) collapse to the single schematic redex 
(32), and that for (29d) simplifies to the schematic 
redex (33). (Note that the notation ¢~ is used to 
represent any (NF) rule which is recursively defined 
on a second rule ~r, e.g. so that ~rb is any NF rule 
defined on b.) 
(32) x y zltm f 
w where n ~_ m 
v 
(33) x y z 
'~b(L(i}b) w where n ~ 1 
Ir b 
V 
Let us consider the use of this system for pars- 
ing. In seeking combinations of some sequence of 
types, we first begin by transforming the types into 
their NFs. 14 Then, we can search for proofs using 
only the NF binary combinations. Any proof that 
is found to contain a proof redexes is discontinued, 
so that only NF proofs are returned, avoiding the 
problems of spurious ambiguity. Any result types 
assigned by such proofs stand as NF exemplars for 
the set of non-NF types that could be derived from 
the original input types under the DC. We may want 
to know if some input types can combine to give a 
specific result type x. This will be the case if the 
parser returns the NF of x. 
Regarding incremental processing, we have seen 
that the DC is well-suited to this task in terms of al- 
lowing combinations that may usefully contribute to 
a knowledge of the semantic relations amongst the 
phrases combined, and that the NF proofs we have 
defined (and which the parser will construct) are 
optimally left-branching to the limit set by the cal- 
culus. Hence, in left-to-right analysis of sentences, 
the parser will be able to combine the presented 
material to the maximal extent that doing so use- 
fully contributes to incremental interpretation and 
the filtering of semantically implausible analyses. 
alence relation that equates any two proofs iff these assign 
extenslonally equivalent readings. 
14The complexity of this transformation is constant in the 
complexity of the type. 
85 
C~. (2s~): 
(a) x/y y/wa..w. W,/Vl..Vm gnf 
x/wa ..w, .R'nf 
x/wa ..wn-I/vl..vm 
C~ (2Sd): 
(~) w,\q~..qk/u,..us 
(b) x/y y/wl ..wn Wn/Vl..vm 
.I%mf 
y/wl ..Wn--1/Vl .-vmRm+n_l f 
x/wl..w,-a/va..v,, 
(b) w,\~..qk/ua..uj 
y\wl..Wn--l \(wn/ul..Uj)/vl..vi x\(y/vl..Vi)/tl..tm RmL(1)nb 
y\wl ..wn-a\q,..qk/v, ..vl 
x\wa ..wn-i \ql..~ltl ..tin 
Case (28g): 
(a) y\wl ..wj/ul ..ui x\(y/ul ..ui)/Vl ..Vm vm/ql--qn R'nL(i)~b 
X\Wl..Wj/Va..Vm Rnf 
x\wl..w~//vl..Vm-i/ql..qn 
(b) y\wl ..wj/ul ..ui x\(y/ul..ui)/vl ..vm vm/ql ..qn\]Ln f 
x\(ylul..Ui)/vz..vm-l/ql..qnam+n_ 1 L(i)Jb 
X\Wl..Wn-l\(wn/ul..uj)/tl..tm Rmg(j)kb 
x\wl ..w,-a \qu ..qk/ta..t,, 
y\wl ..w,-I \(wn/ul ..uj)/vl ..ViRiL.j.kb_() x\(y/vl ..vi)/tl ..tin 
x\wa..w~l,,l..v,,,-, lo~..qn 
RmL(1) k4n-I b 
Figure 1: Equivalent left and right-branching three-leaf subproofs 

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