Generalizing Dimensionality in Combinatory Categorial Grammar
Geert-Jan M. Kruijff
Computational Linguistics
Saarland University
Saarbr¨ucken, Germany
gj@coli.uni-sb.de
Jason Baldridge
ICCS, Division of Informatics
University of Edinburgh
Edinburgh, Scotland
jbaldrid@inf.ed.ac.uk
Abstract
We extend Combinatory Categorial Grammar
(CCG) with a generalized notion of multi-
dimensional sign, inspired by the types of rep-
resentations found in constraint-based frame-
works like HPSG or LFG. The generalized
sign allows multiple levels to share information,
but only in a resource-bounded way through
a very restricted indexation mechanism. This
improves representational perspicuity without
increasing parsing complexity, in contrast to
full-blown unification used in HPSG and LFG.
Well-formedness of a linguistic expressions re-
mains entirely determined by the CCG deriva-
tion. We show how the multidimensionality
and perspicuity of the generalized signs lead to
a simplification of previous CCG accounts of
how word order and prosody can realize infor-
mation structure.
1 Introduction
The information conveyed by linguistic utterances
is diverse, detailed, and complex. To properly ana-
lyze what is communicated by an utterance, this in-
formation must be encoded and interpreted at many
levels. The literature contains various proposals for
dealing with many of these levels in the description
of natural language grammar.
Since information flows between different levels
of analysis, it is common for linguistic formalisms
to bundle them together and provide some means
for communication between them. Categorial gram-
mars, for example, normally employ a Saussurian
sign that relates a surface string with its syntactic
category and the meaning it expresses. Syntactic
analysis is entirely driven by the categories, and
when information from other levels is used to affect
the derivational possibilities, it is typically loaded as
extra information on the categories.
Head-driven Phrase Structure Grammar (HPSG)
(Pollard and Sag, 1993) and Lexical Functional
Grammar (LFG) (Kaplan and Bresnan, 1982) also
use complex signs. However, these signs are mono-
lithic structures which permit information to be
freely shared across all dimensions: any given di-
mension can place restrictions on another. For ex-
ample, variables resolved during the construction of
the logical form can block a syntactic analysis. This
provides a clean, unified formal system for dealing
with the different levels, but it also can adversely af-
fect the complexity of parsing grammars written in
these frameworks (Maxwell and Kaplan, 1993).
We thus find two competing perspectives on com-
munication between levels in a sign. In this paper,
we propose a generalization of linguistic signs for
Combinatory Categorial Grammar (CCG) (Steed-
man, 2000b). This generalization enables different
levels of linguistic information to be represented but
limits their interaction in a resource-bounded man-
ner, following White (2004). This provides a clean
separation of the levels and allows them to be de-
signed and utilized in a more modular fashion. Most
importantly, it allows us to retain the parsing com-
plexity of CCG while gaining the representational
advantages of the HPSG and LFG paradigms.
To illustrate the approach, we use it to model var-
ious aspects of the realization of information struc-
ture, an inherent aspect of the (linguistic) meaning
of an utterance. Speakers use information struc-
ture to present some parts of that meaning as de-
pending on the preceding discourse context and oth-
ers as affecting the context by adding new content.
Languages may realize information structure us-
ing different, often interacting means, such as word
order, prosody, (marked) syntactic constructions,
or morphological marking (Vallduv´ı and Engdahl,
1996; Kruijff, 2002). The literature presents vari-
ous proposals for how information structure can be
captured in categorial grammar (Steedman, 2000a;
Hoffman, 1995; Kruijff, 2001). Here, we model the
essential aspects of these accounts in a more per-
spicuous manner by using our generalized signs.
The main outcomes of the proposal are three-
fold: (1) CCG gains a more flexible and general
kind of sign; (2) these signs contain multiple levels
that interact in a modular fashion and are built via
CCG derivations without increasing parsing com-
plexity; and (3) we use these signs to simplify pre-
vious CCG’s accounts of the effects of word order
and prosody on information structure.
2 Combinatory Categorial Grammar
In this section, we give an overview of syntactic
combination and semantic construction in CCG. We
use CCG’s multi-modal extension (Baldridge and
Kruijff, 2003), which enriches the inventory of slash
types. This formalization renders constraints on
rules unnecessary and supports a universal set of
rules for all grammars.
2.1 Categories and combination
Nearly all syntactic behavior in CCG is encoded in
categories. They may be atoms, like np, or func-
tions which specify the direction in which they seek
their arguments, like (s\np)/np. The latter is the
category for English transitive verbs; it first seeks
its object to its right and then its subject to its left.
Categories combine through a small set of univer-
sal combinatory rules. The simplest are application
rules which allow a function category to consume
its argument either on its right (>) or on its left (<):
(>) X/starY Y ⇒ X
(<) Y X\starY ⇒ X
Four further rules allow functions to compose
with other functions:
(>B) X/diamondmathY Y/diamondmathZ ⇒ X/diamondmathZ
(<B) Y\diamondmathZ X\diamondmathY ⇒ X\diamondmathZ
(>B×) X/×Y Y\×Z ⇒ X\×Z
(<B×) Y/×Z X\×Y ⇒ X/×Z
The modalities star, diamondmath and × on the slashes enforce
different kinds of combinatorial potential on cate-
gories. For a category to serve as input to a rule,
it must contain a slash which is compatible with
that specified by the rule. The modalities work
as follows. star is the most restricted modality, al-
lowing combination only by the application rules
(> and <). diamondmath allows combination with the appli-
cation rules and the order-preserving composition
rules (>B and <B). × allows limited permutation
via the crossed composition rules (>B× and <B×)
as well as the application rules. Additionally, a per-
missive modality · allows combination by all rules
in the system. However, we suppress the · modal-
ity on slashes to avoid clutter. An undecorated slash
may thus combine by all rules.
There are two further rules of type-raising that
turn an argument category into a function over func-
tions that seek that argument:
(>T) X ⇒ Y/i(Y\iX)
(<T) X ⇒ Y\i(Y/iX)
The variable modality i on the output categories
constrains both slashes to have the same modality.
These rules support the following incremental
derivation for Marcel proved completeness:
(1) Marcel proved completeness
np (s\np)/np np
>Ts/(s\np)
>Bs/np
>s
This derivation does not display the effect of us-
ing modalities in CCG; see Baldridge (2002) and
Baldridge and Kruijff (2003) for detailed linguistic
justification for this modalized formulation of CCG.
2.2 Hybrid Logic Dependency Semantics
Many different kinds of semantic representations
and ways of building them with CCG exist. We
use Hybrid Logic Dependency Semantics (HLDS)
(Kruijff, 2001), a framework that utilizes hybrid
logic (Blackburn, 2000) to realize a dependency-
based perspective on meaning.
Hybrid logic provides a language for represent-
ing relational structures that overcomes standard
modal logic’s inability to directly reference states
in a model. This is achieved via nominals, a kind of
basic formula which explicitly names states. Like
propositions, nominals are first-class citizens of the
object language, so formulas can be formed us-
ing propositions, nominals, standard boolean oper-
ators, and the satisfaction operator “@”. A formula
@i(p∧〈F〉(j ∧q)) indicates that the formulas p and
〈F〉(j ∧ q) hold at the state named by i and that the
state j is reachable via the modal relation F.
In HLDS, hybrid logic is used as a language
for describing semantic interpretations as follows.
Each semantic head is associated with a nominal
that identifies its discourse referent and heads are
connected to their dependents via dependency rela-
tions, which are modeled as modal relations. As an
example, the sentence Marcel proved completeness
receives the representation in (2).
(2) @e(prove ∧〈TENSE〉past
∧〈ACT〉(m∧Marcel)∧〈PAT〉(c∧comp.))
In this example, e is a nominal that labels the predi-
cations and relations for the head prove, and m and
c label those for Marcel and completeness, respec-
tively. The relations ACT and PAT represent the de-
pendency roles Actor and Patient, respectively.
By using the @ operator, hierarchical terms such
as (2) can be flattened to an equivalent conjunction
of fixed-size elementary predications (EPs):
(3) @eprove ∧ @e〈TENSE〉past ∧ @e〈ACT〉m
∧ @e〈PAT〉c∧ @mMarcel ∧ @ccomp.
2.3 Semantic Construction
Baldridge and Kruijff (2002) show how HLDS
representations can be built via CCG derivations.
White (2004) improves HLDS construction by op-
erating on flattened representations such as (3) and
using a simple semantic index feature in the syntax.
We adopt this latter approach, described below.
EPs are paired with syntactic categories in the
lexicon as shown in (4)–(6) below. Each atomic cat-
egory has an index feature, shown as a subscript,
which makes a nominal available for capturing syn-
tactically induced dependencies.
(4) prove turnstileleft (se\npx)/npy :
@eprove ∧ @e〈TENSE〉past
∧ @e〈ACT〉x∧ @e〈PAT〉y
(5) Marcel turnstileleft npm : @mMarcel
(6) completeness turnstileleft npc : @ccompleteness
Applications of the combinatory rules co-index
the appropriate nominals via unification on the cat-
egories. EPs are then conjoined to form the result-
ing interpretation. For example, in derivation (1),
(5) type-raises and composes with (4) to yield (7).
The index x is syntactically unified with m, and this
resolution is reflected in the new conjoined logical
form. (7) can then apply to (6) to yield (8), which
has the same conjunction of predications as (3).
(7) Marcel proved turnstileleft se/npy :
@eprove ∧ @e〈TENSE〉past
∧ @e〈ACT〉m∧ @e〈PAT〉y ∧ @mMarcel
(8) Marcel proved completeness turnstileleft se :
@eprove ∧ @e〈TENSE〉past ∧ @e〈ACT〉m
∧@e〈PAT〉c∧@mMarcel ∧@ccompleteness
Since the EPs are always conjoined by the com-
binatory rules, semantic construction is guaranteed
to be monotonic. No semantic information can be
dropped during the course of a derivation. This pro-
vides a clean way of establishing semantic depen-
dencies as informed by the syntactic derivation. In
the next section, we extend this paradigm for use
with any number of representational levels.
3 Generalized dimensionality
To support a more modular and perspicuous encod-
ing of multiple levels of analysis, we generalize the
notion of sign commonly used in CCG. The ap-
proach is inspired on the one hand by earlier work
by Steedman (2000a) and Hoffman (1995), and on
the other by the signs found in constraint-based ap-
proaches to grammar. The principle idea is to ex-
tend White’s (2004) approach to semantic construc-
tion (see §2.3). There, categories and the mean-
ing they help express are connected through co-
indexation. Here, we allow for information in any
(finite) number of levels to be related in this way.
A sign is an n-tuple of terms that represent in-
formation at n distinct dimensions. Each dimension
represents a level of linguistic information such as
prosody, meaning, or syntactic category. As a repre-
sentation, we assume that we have for each dimen-
sion a language that defines well-formed representa-
tions, and a set of operations which can create new
representations from a set of given representations.1
For example, we have by definition a dimension
for syntactic categories. The language for this di-
mension is defined by the rules for category con-
struction: given a set of atomic categories A, C is a
category iff (i) C ∈ A or (ii) C is of the form A\mB
or A/mB with A,B categories and m ∈ {star,diamondmath×,·}.
The set of combinatory rules defines the possible
operations on categories.
This syntactic category dimension drives the
grammatical analysis, thus guiding the composition
of signs. When two categories are combined via
a rule, the appropriate indices are unified. It is
through this unification of indices that information
can be passed between signs. At a given dimen-
sion, the co-indexed information coming from the
two signs we combine must be unifiable.
With these signs, dimensions interact in a more
limited way than in HPSG or LFG. Constraints (re-
solved through unification) may only be applied
if they are invoked through co-indexation on cat-
egories. This provides a bound on the number of
indices and the number of unifications to be made.
As such, full recursion and complex unification as in
attribute-value matrices with re-entrancy is avoided.
The approach incorporates various ideas from
constraint-based approaches, but remains based on
a derivational perspective on grammatical analysis
and derivational control, unlike e.g Categorial Uni-
fication Grammar. Furthermore, the ability for di-
mensions to interact through shared indices brings
several advantages: (1) “parallel derivations” (Hoff-
man, 1995) are unnecessary; (2) non-isomorphic,
functional structures across different dimensions
can be employed; and (3) there is no longer a need
to load all the necessary information into syntactic
categories (as with Kruijff (2001)).
1In the context of this paper we assume operations are mul-
tiplicative. Also, note that dimensions may differ in what lan-
guages and operations they use.
4 Examples
In this section, we illustrate our approach on several
examples involving information structure. We use
signs that include the following dimensions.
Phonemic representation: word sequences, composi-
tion of sequences is through concatenation
Prosody: sequences of tunes from the inventory of
(Pierrehumbert and Hirschberg, 1990), composi-
tion through concatenation
Syntactic category: well-formed categories, combina-
tory rules (see §2)
Information structure: hybrid logic formulas of the
form @d[in]r, with r a discourse referent that has
informativity in (theme θ, or rheme ρ) relative to
the current point in the discourse d (Kruijff, 2003).
Predicate-argument structure: hybrid logic formulas
of the form as discussed in §2.3.
Example (9) illustrates a sign with these dimen-
sions. The word-form Marcel bears an H* accent,
and acts as a type-raised category that seeks a verb
missing its subject. The H* accent indicates that the
discourse referent m introduces new information at
the current point in the discourse d: i.e. the meaning
@mmarcel should end up as part of the rheme (ρ) of
the utterance, @d[ρ]m.
(9) MarcelH*
sh/(sh\npm)
@d[ρ]m
@mmarcel
If a sign does not specify any information at a
particular dimension, this is indicated by latticetop (or an
empty line if no confusion can arise).
4.1 Topicalization
We start with a simple example of topicalization in
English. In topicalized constructions, a thematic ob-
ject is fronted before the subject. Given the question
Did Marcel prove soundness and completeness?,
(10) is a possible response using topicalization:
(10) Completeness, Marcel proved, and sound-
ness, he conjectured.
We can capture the syntactic and information
structure effects of such sentences by assigning the
following kind of sign to (topicalized) noun phrases:
(11) completeness
latticetop
si/(si/npc)
@d[θ]c
@ccompleteness
This category enables the derivation in Figure 1.
The type-raised subject composes with the verb, and
the result is consumed by the topicalizing category.
The information structure specification stated in the
sign in (11) is passed through to the final sign.
The topicalization of the object in (10) only indi-
cates the informativity of the discourse referent re-
alized by the object. It does not yield any indica-
tions about the informativity of other constituents;
hence the informativity for the predicate and the Ac-
tor is left unspecified. In English, the informativity
of these discourse referents can be indicated directly
with the use of prosody, to which we now turn.
4.2 Prosody & information structure
Steedman (2000a) presents a detailed, CCG-based
account of how prosody is used in English as a
means to realize information structure. In the
model, pitch accents and boundary tones have an ef-
fect on both the syntactic category of the expression
they mark, and the meaning of that expression.
Steedman distinguishes pitch accents as markers
of either the theme (θ) or of the rheme (ρ): L+H*
and L*+H are θ-markers; H*, L*, H*+L and H+L*
are ρ-markers. Since pitch accents mark individual
words, not (necessarily) larger phrases, Steedman
uses the θ/ρ-marking to spread informativity over
the domain and the range of function categories.
Identical markings on different parts of a function
category not only act as features, but also as occur-
rences of a singular variable. The value of the mark-
ing on the domain can thus get passed down (“pro-
jected”) to markings on categories in the range.
Constituents bearing no tune have an η-marking,
which can be unified with either η, θ or ρ. Phrases
with such markings are “incomplete” until they
combine with a boundary tone. Boundary tones
have the effect of mapping phrasal tones into
intonational phrase boundaries. To make these
boundaries explicit and enforce such “complete”
prosodic phrases to only combine with other com-
plete prosodic phrases, Steedman introduces two
further types of marking – ι and φ – on categories.
The φ markings only unify with other φ or ι mark-
ings on categories, not with η, θ or ρ. These mark-
ings are only introduced to provide derivational con-
trol and are not reflected in the underlying meaning
(which only reflects η, θ or ρ).
Figure 2 recasts the above as an abstract speci-
fication of which different types of prosodic con-
stituents can, or cannot, be combined.2 Steedman’s
2There is one exception we should note: two intermediate
phrases can combine if a second one has a downstepped accent.
We deal with this exception at the end of the section.
completeness Marcel proved
si/(si/npc) sj/(sj\npm) (sp\npx)/npy
@d[θ]c
@ccompleteness @mMarcel @pprove ∧ @p〈ACT〉x∧ @p〈PAT〉y
>Bs
p/npy
@pprove ∧ @p〈ACT〉m∧ @p〈PAT〉y ∧ @mMarcel
>sp
@d[θ]c
@pprove ∧ @p〈ACT〉m∧ @p〈PAT〉c∧ @mMarcel ∧ @ccompleteness
Figure 1: Derivation for topicalization.
system can be implemented using just one feature
pros which takes the values ip for intermediate
phrases, cp for complete phrases, and up for un-
marked phrases. We write spros=ip, or simply sip if
no confusion can arise.
Figure 2: Abstract specification of derivational con-
trol in prosody
First consider the top half of Figure 2. If a con-
stituent is marked with either a θ- or ρ-tune, the
atomic result category of the (possibly complex)
category is marked with ip. Prosodically unmarked
constituents are marked as up. The lexical entries
in (12) illustrates this idea.3
(12) MARCEL proved COMPLETENESS
H* L+H*
sip/(sup\np) (sup\np)/np sip$\(sup$/np)
This can proceed in two ways. Either the marked
MARCEL and the unmarked proved combine to pro-
duce an intermediate phrase (13), or proved and the
marked COMPLETENESS combine (14).
(13) MARCEL proved COMPLETENESS
H* L+H*
sip/(sup\np) (sup\np)/np sip$\(sup$/np)
>s
ip/np
3The $’s in the category for COMPLETENESS are standard
CCG schematizations: s$ indicates all functions into s, such as
s\np and (s\np)/np. See Steedman (2000b) for details.
(14) MARCEL proved COMPLETENESS
H* L+H*
sip/(sup\np) (sup\np)/np sip$\(sup$/np)
<s
ip\np
For the remainder of this paper, we will suppress up
marking and write sup simply as s.
Examples (13) and (14) show that prosodically
marked and unmarked phrases can combine. How-
ever, both of these partial derivations produce cate-
gories that cannot be combined further. For exam-
ple, in (14), sip/(s\np) cannot combine with sip\np
to yield a larger intermediate phrase. This properly
captures the top half of Figure 2.
To obtain a complete analysis for (12), bound-
ary tones are needed to complete the intermediate
phrases tones. For example, consider (15) (based
on example (70) in Steedman (2000a)):
(15) MARCEL proved COMPLETENESS
H* L L+H* LH%
To capture the bottom-half of Figure 2, the bound-
ary tones L and LH% need categories which cre-
ate complete phrases out of those for MARCEL and
proved COMPLETENESS, and thereafter allow them
to combine. Figure 3 shows the appropriate cate-
gories and complete analysis.
We noted earlier that downstepped phrasal tunes
form an exception to the rule that intermediate
phrases cannot combine. To enable this, we not
only should mark the result category with ip (tune),
but also any leftward argument(s) should have ip
(downstep). Thus, the effect of (lexically) combin-
ing a downstep tune with an unmarked category is
specified by the following template: add marking
xip$\yip to an unmarked category of the form x$\y.
The derivation in Figure 5 illustrates this idea on ex-
ample (64) from (Steedman, 2000a).
To relate prosody to information structure, we ex-
tend the strategy used for constructing logical forms
described in §2.3, in which a simple index feature
MARCEL proved COMPLETENESS
H* L L+H* LH%
sip/(s\np) (scp/scp$)\star(sip/s$) (s\np)/np sip$\(s$/np) scp$\starsip$
< <s
cp/(scp\np) sip\np
<s
cp\np
>scp
Figure 3: Derivation including tunes and boundary tones; (70) from (Steedman, 2000a)
Marcel PROVED COMPLETENESSL+H* LH% H* LL%
np (sip:p\npx)/npy scp$\starsip$ sip\(s/npc) (scp\scp$)\star(sip\s$)
@d[θ]p @d[ρ]c
@mMarcel @pprove ∧ @p〈ACT〉x∧ @p〈PAT〉y @ccompleteness
>T <s
ip/(sip\np) scp\(scp/npc)@
d[ρ]c@
mMarcel @ccompleteness>B
sip/np
@d[θ]p
@pprove ∧ @p〈ACT〉m∧ @p〈PAT〉y ∧ @mMarcel
<s
cp/npy@
d[θ]p@
pprove ∧ @p〈ACT〉m∧ @p〈PAT〉y ∧ @mMarcel
<s
cp@
d[θ]p ∧ @d[ρ]c@
pprove ∧ @p〈ACT〉m∧ @p〈PAT〉c∧ @mMarcel ∧ @ccompleteness
Figure 4: Information structure for derivation for (67)-(68) from (Steedman, 2000a)
on atomic categories makes a nominal (discourse
referent) available. We represent information struc-
ture as a formula @d[i]r at a dimension separate
from the syntactic category. The nominal r stands
for the discourse referent, which has informativity
i with respect to the current point in the discourse
d (Kruijff, 2003). Following Steedman, we distin-
guish two levels of informativity, namely θ (theme)
and ρ (rheme).
We start with a minimal assignment of informa-
tivity: a theme-tune on a constituent sets the infor-
mativity of the discourse referent r realized by the
constituent to θ and a rheme-tune sets it to ρ. This
is a minimal assignment in the sense that we do not
project informativity; instead, we only set informa-
tivity for those discourse referents whose realization
shows explicit clues as to their information status.
The derivation in Figure 4 illustrates this idea and
shows the construction of both logical form and in-
formation structure.
Indices can also impose constraints on the infor-
mativity of arguments. For example, in the down-
step example (Figure 5), the discourse referents cor-
responding to ANNA and SAYS are both part of the
theme. We specify this with the constituent that has
received the downstepped tune. The referent of the
subject of SAYS (indexed x) must be in the theme
along with the referent s for SAYS. This is satisfied
in the derivation: a unifies with x, and we can unify
the statements about a’s informativity coming from
ANNA (@d[θ]a) and SAYS (@d[θ]x with x replaced
by a in the >B step).
5 Conclusions
In this paper, we generalize the traditional Saus-
surian sign in CCG with an n-dimensional linguis-
tic sign. The dimensions in the generalized linguis-
tic sign can be related through indexation. Index-
ation places constraints on signs by requiring that
co-indexed material is unifiable, on a per-dimension
basis. Consequently, we do not need to overload the
syntactic category with information from different
dimensions.
The resulting sign structure resembles the signs
found in constraint-based grammar formalisms.
There is, however, an important difference. Infor-
mation at various dimensions can be related through
co-indexation, but dimensions cannot be directly
ANNA SAYS he proved COMPLETENESS
L+H* !L+H* LH%
npip:a (sip:s\npip:x)/sy s/(s\np) (sp\np)/np
@d[θ]a @d[θ]s ∧ @d[θ]x @d[θ](pron) @d[i]p
>Ts
ip/(sip\npip)@
d[θ]a
>Bs
ip/s@
d[θ]s ∧ @d[θ]a
>Bs
ip/(s\np)@
d[θ]s ∧ @d[θ]a ∧ @d[θ](pron)
>Bs
ip/np@
d[θ]s ∧ @d[θ]a ∧ @d[θ](pron) ∧ @d[i]p
Figure 5: Information structure for derivation for (64) from (Steedman, 2000a)
referenced. As analysis remains driven only by in-
ference over categories, only those constraints trig-
gered by indexation on the categories are imposed.
We do not allow for re-entrancy.
It is possible to conceive of a scenario in which
the various levels can contribute toward determin-
ing the well-formedness of an expression. For ex-
ample, we may wish to evaluate the current informa-
tion structure against a discourse model, and reject
the analysis if we find it is unsatisfiable. If such a
move is made, then the complexity will be bounded
by the complexity of the dimension for which it is
most difficult to determine satisfiability.
Acknowledgments
Thanks to Ralph Debusmann, Alexander Koller,
Mark Steedman, and Mike White for discussion.
Geert-Jan Kruijff’s work is supported by the DFG
SFB 378 Resource-Sensitive Cognitive Processes,
Project NEGRA EM 6.

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