The
standard
set
of
rules
defined
in
Combinatory
Categorial
Grammar
(
CCG
)
fails
to
provide
satisfactory
analyses
for
a
number
of
syntactic
structures
found
in
natural
languages
.
These
structures
can
be
analyzed
elegantly
by
augmenting
CCG
with
a
class
of
rules
based
on
the
combinator
D
(
Curry
and
Feys
,
1958
)
.
We
show
two
ways
to
derive
the
D
rules
:
one
based
on
unary
composition
and
the
other
based
on
a
logical
characterization
of
CCG
's
rule
base
(
Baldridge
,
2002
)
.
We
also
show
how
Eisner
's
(
1996
)
normal
form
constraints
follow
from
this
logic
,
ensuring
that
the
D
rules
do
not
lead
to
spurious
ambiguities
.
1
Introduction
Combinatory
Categorial
Grammar
(
CCG
,
Steedman
(
2000
)
)
is
a
compositional
,
semantically
transparent
formalism
that
is
both
linguistically
expressive
and
computationally
tractable
.
It
has
been
used
for
a
variety
of
tasks
,
such
as
wide-coverage
parsing
(
Hock-enmaier
and
Steedman
,
2002
;
Clark
and
Curran
,
2007
)
,
sentence
realization
(
White
,
2006
)
,
learning
semantic
parsers
(
Zettlemoyer
and
Collins
,
2007
)
,
dialog
systems
(
Kruijff
et
al.
,
2007
)
,
grammar
engineering
(
Beavers
,
2004
;
Baldridge
et
al.
,
2007
)
,
and
modeling
syntactic
priming
(
Reitter
et
al.
,
2006
)
.
A
distinctive
aspect
of
CCG
is
that
it
provides
a
very
flexible
notion
of
constituency
.
This
supports
elegant
analyses
of
several
phenomena
(
e.g.
,
coordination
,
long-distance
extraction
,
and
intonation
)
and
allows
incremental
parsing
with
the
competence
grammar
(
Steedman
,
2000
)
.
Here
,
we
argue
that
even
with
its
flexibility
,
CCG
as
standardly
defined
is
not
permissive
enough
for
certain
linguistic
constructions
and
greater
incrementality
.
Following
Wittenburg
(
1987
)
,
we
remedy
this
by
adding
a
set
of
rules
based
on
the
D
combinator
of
combinatory
logic
(
Curry
and
Feys
,
1958
)
.
We
show
that
CCG
augmented
with
this
rule
improves
CCG
's
empirical
coverage
by
allowing
better
analyses
of
modal
verbs
in
English
and
causatives
in
Spanish
,
and
certain
coordinate
constructions
.
The
D
rules
are
well-behaved
;
we
show
this
by
deriving
them
both
from
unary
composition
and
from
the
logic
defined
by
Baldridge
(
2002
)
.
Both
perspectives
on
D
ensure
that
the
new
rules
are
compatible
with
normal
form
constraints
(
Eisner
,
1996
)
for
controlling
spurious
ambiguity
.
The
logic
also
ensures
that
the
new
rules
are
subject
to
modalities
consistent
with
those
defined
by
Baldridge
and
Kruijff
(
2003
)
.
Furthermore
,
we
define
a
logic
that
produces
Eisner
's
constraints
as
grammar
internal
theorems
rather
than
parsing
stipulations
.
2
Combinatory
Categorial
Grammar
CCG
functors
are
functions
over
strings
of
symbols
,
so
different
linearized
versions
of
each
of
the
combinators
have
to
be
specified
(
ignoring
S
here
)
:
The
symbols
{
*
,
o
,
x
,
•
}
are
modalities
that
allow
subtypes
of
slashes
to
be
defined
;
this
in
turn
allows
the
slashes
on
categories
to
be
defined
in
a
way
that
allows
them
to
be
used
(
or
not
)
with
specific
subsets
of
the
above
rules
.
The
rules
of
this
multimodal
version
of
CCG
(
Baldridge
,
2002
;
Baldridge
and
Krui-jff
,
2003
)
are
derived
as
theorems
of
a
Categorial
Type
Logic
(
CTL
,
Moortgat
(
1997
)
)
.
This
treats
CCG
as
a
compilation
of
CTL
proofs
,
providing
a
principled
,
grammar-internal
basis
for
restrictions
on
the
CCG
rules
,
transferring
language-particular
restrictions
on
rule
application
to
the
lexicon
,
and
allowing
the
CCG
rules
to
be
viewed
as
grammatical
universals
(
Baldridge
and
Kruijff
,
2003
;
Steedman
and
Baldridge
,
To
Appear
)
.
These
rules
—
especially
the
B
rules
—
allow
derivations
to
be
partially
associative
:
given
appropriate
type
assignments
,
a
string
ABC
can
be
analyzed
as
either
A
(
BC
)
or
(
AB
)
C.
This
associativity
leads
to
elegant
analyses
of
phenomena
that
demand
more
effort
in
less
flexible
frameworks
.
One
of
the
best
known
is
"
odd
constituent
"
coordination
:
(
4
)
Bob
gave
Stan
a
beer
and
Max
a
coke
.
(
5
)
I
will
buy
and
you
will
eat
a
cheeseburger
.
The
coordinated
constituents
are
challenging
because
they
are
at
odds
with
standardly
assumed
phrase
structure
constituents
.
In
CCG
,
such
constituents
simply
follow
from
the
associativity
added
by
the
B
and
T
rules
.
For
example
,
given
the
category
assignments
in
(
6
)
and
the
abbreviations
in
(
7
)
,
(
4
)
is
analyzed
as
in
(
8
)
and
(
9
)
.
Each
conjunct
is
a
pair
of
type-raised
NPs
combined
by
means
of
the
&gt;
B
-
rule
,
deriving
two
composed
constituents
that
are
arguments
to
the
conjunction
:
1
Similarly
,
I
will
buy
is
derived
with
category
s
/
np
by
assuming
the
category
(
6i
)
for
I
and
composing
that
with
both
verbs
in
turn
.
CCG
's
approach
is
appealing
because
such
constituents
are
not
odd
at
all
:
they
simply
follow
from
the
fact
that
CCG
is
a
system
of
type-based
grammatical
inference
that
allows
left
associativity
.
3
Linguistic
Motivation
for
D
CCG
is
only
partially
associative
.
Here
,
we
discuss
several
situations
which
require
greater
associativity
and
thus
cannot
be
given
an
adequate
analysis
with
CCG
as
standardly
defined
.
These
structures
have
in
common
that
a
category
of
the
form
x
|
(
y
|
z
)
must
combine
with
one
of
the
form
y
|
w
—
exactly
the
configuration
handled
by
the
D
schemata
in
(
1
)
.
3.1
Cross-Conjunct
Extraction
In
the
first
situation
,
a
question
word
is
distributed
across
auxiliary
or
subordinating
verb
categories
:
(
10
)
.
.
.
what
you
can
and
what
you
must
not
base
your
verdict
on
.
We
call
this
cross-conjunct
extraction
.
It
was
noted
by
Pickering
and
Barry
(
1993
)
for
English
,
but
to
the
best
of
our
knowledge
it
has
not
been
treated
in
the
1We
follow
(
Steedman
,
2000
)
in
assuming
that
type-raising
applies
in
the
lexicon
,
and
therefore
that
nominals
such
as
Stan
have
type-raised
lexical
assignments
.
We
also
suppress
semantic
representations
in
the
derivations
for
the
sake
of
space
.
CCG
literature
,
nor
noted
in
other
languages
.
The
problem
it
presents
to
CCG
is
clear
in
(
11
)
,
which
shows
the
necessary
derivation
of
(
10
)
using
standard
multimodal
category
assignments
.
For
the
tokens
of
what
to
form
constituents
with
you
can
and
you
must
not
,
they
must
must
combine
directly
.
The
problem
is
that
these
constituents
(
in
bold
)
cannot
be
created
with
the
standard
CCG
combinators
in
(
3
)
.
base
your
verdict
on
what
you
can
The
category
for
and
is
marked
for
non-associativity
with
*
,
and
thus
combines
with
other
expressions
only
by
function
application
(
Baldridge
,
2002
)
.
This
ensures
that
each
conjunct
is
a
discrete
constituent
.
(
14
)
Gandeste-te
cui
ge
vrei
,
consider.imper.2s-refl.2s
who.dat
what
want.2s
§
i
cui
ge
pofi
,
sa
dai
.
"
Consider
to
whom
you
want
and
to
whom
you
are
able
to
give
what
.
"
It
is
thus
a
general
phenomenon
,
not
just
a
quirk
of
English
.
While
it
could
be
handled
with
extra
categories
,
such
as
(
s
/
(
vp
/
np
)
)
/
(
s
/
np
)
for
what
,
this
is
exactly
the
sort
of
strong-arm
tactic
that
inclusion
of
the
standard
B
,
T
,
and
S
rules
is
meant
to
avoid
.
3.2
English
Auxiliary
Verbs
However
,
this
type
is
empirically
underdetermined
,
given
a
widely-noted
set
of
generalizations
suggesting
that
auxiliaries
and
raising
verbs
take
no
subject
argument
at
all
(
Jacobson
,
1990
,
a.o.
)
.
Lack
of
semantic
restrictions
on
the
subject
;
iii
.
Inheritance
of
selectional
restrictions
from
the
subordinate
predicate
.
Two
arguments
are
made
for
(
16
)
.
First
,
it
is
necessary
so
that
type-raised
subjects
can
compose
with
the
auxiliary
in
extraction
contexts
,
as
in
(
18
)
:
Second
,
it
is
claimed
to
be
necessary
in
order
to
account
for
subject-verb
agreement
,
on
the
assumption
that
agreement
features
are
domain
restrictions
on
functors
of
type
s
\
np
(
Steedman
,
1992
,
1996
)
.
The
first
argument
is
the
topic
of
this
paper
,
and
,
as
we
show
below
,
is
refuted
by
the
use
of
the
D-combinator
.
The
second
argument
is
undermined
by
examples
like
(
19
)
:
the
average
age
of
retirement
]
]
.
In
(
19
)
,
appear
agrees
with
two
negative-polarity-sensitive
NPs
trapped
inside
a
neither-nor
coordinate
structure
in
which
they
are
licensed
.
Appear
therefore
does
not
combine
with
them
directly
,
showing
that
the
agreement
relation
need
not
be
mediated
by
direct
application
of
a
subject
argument
.
We
conclude
,
therefore
,
that
the
assignment
of
the
vp
/
vp
type
to
English
auxiliaries
and
modal
verbs
is
unsupported
on
both
formal
and
linguistic
grounds
.
Combining
(
20
)
with
a
type-raised
subject
presents
another
instance
of
the
structure
in
(
1
)
,
where
that
question
words
are
represented
as
variable-binding
operators
(
Groenendijk
and
Stokhof
,
1997
)
:
3.3
The
Spanish
Causative
Construction
The
aspect
of
the
construction
that
is
relevant
here
is
that
the
causative
verb
hacer
appears
to
take
an
object
argument
understood
as
the
subject
or
agent
of
the
subordinate
verb
(
the
causee
)
.
However
,
it
has
been
argued
that
Spanish
causative
verbs
do
not
in
fact
take
objects
(
Ackerman
and
Moore
,
1999
,
and
refs
therein
)
.
There
are
two
arguments
for
this
.
First
,
syntactic
alternations
that
apply
to
object-taking
verbs
,
such
as
passivization
and
periphrasis
with
subjunctive
complements
,
do
not
apply
to
hacer
(
Lujan
,
1980
)
.
Second
,
hacer
specifies
neither
the
case
form
of
the
causee
,
nor
any
semantic
entail-ments
with
respect
to
it
.
These
are
instead
determined
by
syntactic
,
semantic
,
and
pragmatic
factors
,
such
as
transitivity
,
word
order
,
animacy
,
gender
,
social
prestige
,
and
referential
specificity
(
Finnemann
,
1982
,
a.o
)
.
Thus
,
there
is
neither
syntactic
nor
semantic
evidence
that
hacer
takes
an
object
argument
.
However
,
Spanish
has
examples
of
cross-conjunct
extraction
in
which
hacer
hosts
clitics
:
le
hicieron
barrer
la
verada
.
cl.dat.3ms
made.3p
sweep
the
sidewalk
"
They
not
only
ordered
him
to
,
but
also
made
him
sweep
the
sidewalk
.
"
hicieron
barrer
la
verada
The
preceding
data
motivates
adding
D
rules
(
we
return
to
the
distribution
of
the
modalities
below
)
:
To
illustrate
with
example
(
10
)
,
one
application
of
&gt;
D
allows
you
and
can
to
combine
when
the
auxiliary
is
given
the
principled
type
assignment
s
/
s
,
and
another
combines
what
with
the
result
.
The
derivation
then
proceeds
in
the
usual
way
.
Likewise
,
D
handles
the
Spanish
causative
constructions
(
29
)
straightforwardly
:
hice
dormir
The
D-rules
thus
provide
straightforward
analyses
of
such
constructions
by
delivering
flexible
constituency
while
maintaining
CCG
's
committment
to
low
categorial
ambiguity
and
semantic
transparency
.
4
Deriving
Eisner
Normal
Form
Adding
new
rules
can
have
implications
for
parsing
efficiency
.
In
this
section
,
we
show
that
the
D
rules
fit
naturally
within
standard
normal
form
constraints
for
CCG
parsing
(
Eisner
,
1996
)
,
by
providing
both
combinatory
and
logical
bases
for
D.
This
additionally
allows
Eisner
's
normal
form
constraints
to
be
derived
as
grammar
internal
theorems
.
4.1
The
Spurious
Ambiguity
Problem
CCG
's
flexibility
is
useful
for
linguistic
analyses
,
but
leads
to
spurious
ambiguity
(
Wittenburg
,
1987
)
due
to
the
associativity
introduced
by
the
B
and
T
rules
.
This
can
incur
a
high
computational
cost
which
parsers
must
deal
with
.
Several
techniques
have
been
proposed
for
the
problem
(
Wittenburg
,
1987
;
Karttunen
,
1989
;
Hepple
and
Morrill
,
1989
;
Eisner
,
1996
)
.
The
most
commonly
used
are
Karttunnen
's
chart
subsumption
check
(
White
and
Baldridge
,
2003
;
Hockenmaier
and
Steedman
,
2002
)
and
Eisner
's
normal-form
constraints
(
Bozsahin
,
1998
;
Clark
and
Curran
,
2007
)
.
Eisner
's
normal
form
,
referred
to
here
as
Eisner
NF
and
paraphrased
in
(
30
)
,
has
the
advantage
ofnot
requiring
comparisons
of
logical
forms
it
functions
purely
on
the
syntactic
types
being
combined
.
i.
C
is
not
the
argument
of
(
AB
)
resulting
from
application
of
&gt;
B1
+
.
A
is
not
the
argument
of
(
BC
)
resulting
from
application
of
&lt;
B1
+
.
The
implication
is
that
outputs
of
B1+
rules
are
inert
,
using
the
terminology
of
Baldridge
(
2002
)
.
Inert
slashes
are
Baldridge
's
(
2002
)
encoding
in
OpenCCG3
of
his
CTL
interpretation
of
Steedman
's
(
2000
)
antecedent-government
feature
.
Eisner
derives
(
30
)
from
two
theorems
about
the
set
of
semantically
equivalent
parses
that
a
CCG
parser
will
generate
for
a
given
string
(
see
(
Eisner
,
1996
)
for
proofs
and
discussion
of
the
theorems
)
(
31
)
Theorem
1
:
For
every
parse
tree
a
,
there
is
a
semantically
equivalent
parse-tree
NF
(
a
)
in
which
no
node
resulting
from
application
of
B
or
S
functions
as
the
primary
functor
in
a
rule
application
.
(
32
)
Theorem
2
:
If
NF
(
a
)
and
NF
(
a
'
)
are
distinct
parse
trees
,
then
their
model-theoretic
interpretations
are
distinct
.
2Two
parse
trees
are
semantically
equivalent
if
:
(
i
)
their
leaf
nodes
have
equivalent
interpretations
,
and
(
ii
)
equivalent
scope
relations
hold
between
their
respective
leaf-node
meanings
.
3http
:
/
/
openccg.sourceforge.net
Eisner
uses
a
generalized
form
Bn
(
n
&gt;
0
)
of
composition
that
subsumes
function
application
:
4
i.
If
a
is
a
lexical
item
,
then
a
is
in
Eisner-NF
.
ii
.
If
a
is
a
parse
tree
(
R
,
/
/
,
7
}
and
NF
(
/
/
)
,
NF
(
7
)
,
then
NF
(
a
)
.
NF
(
a
)
=
(
S
,
/
31
,
NF
(
&lt;
T
,
/
?
2,7
}
)
}
.
As
a
parsing
constraint
,
(
30
)
is
a
filter
on
the
set
of
parses
produced
for
a
given
string
.
It
preserves
all
the
unique
semantic
forms
generated
for
the
string
while
eliminating
all
spurious
ambiguities
:
it
is
both
safe
and
complete
.
Given
the
utility
of
Eisner
NF
for
practical
CCG
parsing
,
the
D
rules
we
propose
should
be
compatible
with
(
30
)
.
This
requires
that
the
generalizations
underlying
(
30
)
apply
to
D
as
well
.
In
the
remainder
of
this
section
,
we
show
this
in
two
ways
.
The
first
is
to
derive
the
binary
B
rules
from
a
unary
rule
based
on
the
unary
combinator
B
:
5
We
then
derive
D
from
B
and
show
that
clause
(
iii
)
of
(
35
)
holds
of
Q
schematized
over
both
B
and
D.
Applying
D
to
an
argument
sequence
is
equivalent
to
compound
application
of
binary
B
:
Syntactically
,
binary
B
is
equivalent
to
application
of
unary
B
to
the
primary
functor
A
,
followed
by
applying
the
secondary
functor
r
to
the
output
of
B
by
means
of
function
application
(
Jacobson
,
1999
)
:
4We
use
Steedman
's
(
Steedman
,
1996
)
"
$
"
-
convention
for
representing
argument
stacks
of
length
n
,
for
n
&gt;
0
.
5This
is
Lambek
's
(
1958
)
Division
rule
,
also
known
as
the
"
Geach
rule
"
(
Jacobson
,
1999
)
.
The
rules
for
D
correspond
to
application
of
B
to
both
the
primary
and
secondary
functors
,
followed
by
function
application
:
As
with
Bn
,
Dn-1
can
be
derived
by
iterative
application
of
B
to
both
primary
and
secondary
functors
.
Interpreted
in
terms
of
B
,
both
B
and
D
involve
application
of
B
to
the
primary
functor
.
It
follows
that
Theorem
I
applies
directly
to
D
simply
by
virtue
of
the
equivalence
between
binary
B
and
unary-B+FA
.
Eisner
's
NF
constraints
can
then
be
reinterpreted
as
a
constraint
on
B
requiring
its
output
to
be
an
inert
result
category
.
We
represent
this
in
terms
of
the
B-rules
introducing
an
inert
slash
,
indicated
with
"
!
"
(
adopting
the
convention
from
OpenCCG
)
:
The
binary
substitution
(
S
)
combinator
can
be
similarly
incorporated
into
the
system
.
Unary
substitution
S
is
like
B
except
that
it
introduces
a
slash
on
only
the
argument-side
of
the
input
functor
.
We
stipulate
that
S
returns
a
category
with
inert
slashes
:
T
is
by
definition
unary
.
It
follows
that
all
the
binary
rules
in
CCG
(
including
the
D-rules
)
can
be
reduced
to
(
iterated
)
instantiations
of
the
unary
combinators
B
,
S
,
or
T
plus
function
application
.
This
provides
a
basis
for
CCG
in
which
all
com-binatory
rules
are
derived
from
unary
BS
SS
,
and
T.
4.3
A
Logical
Basis
for
Eisner
Normal
Form
The
previous
section
shows
that
deriving
CCG
rules
from
unary
combinators
allows
us
to
derive
the
D-rules
while
preserving
Eisner
NF
.
In
this
section
,
we
present
an
alternate
formulation
of
Eisner
NF
with
Baldridge
's
(
2002
)
CTL
basis
for
CCG
.
This
formulation
allows
us
to
derive
the
D-rules
as
before
,
and
does
so
in
a
way
that
seamlessly
integrates
with
Baldridge
's
system
of
modalized
functors
.
In
CTL
,
Bo
and
B
x
are
proofs
derived
via
structural
rules
that
allow
associativity
and
permutation
of
symbols
within
a
sequent
,
in
combination
with
the
slash
introduction
and
elimination
rules
of
the
base
logic
.
To
control
application
of
these
rules
,
Baldridge
keys
them
to
binary
modal
operators
o
(
for
associativity
)
and
x
(
for
permutation
)
.
Given
these
,
&gt;
B
is
proven
in
(
47
)
:
In
a
CCG
ruleset
compiled
from
such
logics
,
a
category
must
have
an
appropriately
decorated
slash
in
order
to
be
the
input
to
a
rule
.
This
means
that
rules
apply
universally
,
without
language-specific
restrictions
.
Instead
,
restrictions
can
only
be
declared
via
modalities
marked
on
lexical
categories
.
The
D
rules
are
also
theorems
of
this
system
.
For
example
,
the
proof
for
&gt;
D
applies
(
48
)
as
a
lemma
to
each
of
the
primary
and
secondary
functors
:
&gt;
D
ox
involves
an
associative
version
of
S
applied
to
the
primary
functor
(
50
)
,
and
a
permutative
version
to
the
secondary
functor
(
51
)
.
Rules
for
D
with
appropriate
modalities
can
therefore
be
incorporated
seamlessly
into
CCG
.
In
the
preceding
subsection
,
we
encoded
Eisner
NF
with
inert
slashes
.
In
Baldridge
's
CTL
basis
for
CCG
,
inert
slashes
are
represented
as
functors
seeking
non-lexical
arguments
,
represented
as
categories
marked
with
an
antecedent-governed
feature
,
reflecting
the
intuition
that
non-lexical
arguments
have
to
be
"
bound
"
by
a
superordinate
functor
.
This
is
based
on
an
interpretation
of
antecedent-government
as
a
unary
modality
ant
that
allows
structures
marked
by
it
to
permute
to
the
left
or
right
periphery
of
a
structure
:
6
Unlike
permutation
rules
without
ant
,
these
permutation
rules
can
only
be
used
in
a
proof
when
preceeded
by
a
hypothetical
category
marked
with
the
^ant
modality
.
The
elimination
rule
for
□
-
modalities
introduces
a
corresponding
-
marked
object
in
the
resulting
structure
,
feeding
the
rule
:
ant
nant
zJ
Re-introduction
of
the
[
a
h
ant
□
antz
]
k
hypothesis
results
in
a
functor
the
argument
of
which
is
marked
with
ant
□
ant
.
Because
lexical
categories
are
not
marked
as
such
,
the
functor
cannot
take
a
lexical
argument
,
and
so
is
effectively
an
inert
functor
.
In
Baldridge
's
(
2002
)
system
,
only
proofs
involving
the
ARP
and
ALP
rules
produce
inert
categories
.
In
Eisner
NF
,
all
instances
of
-
rules
result
in
inert
categories
.
This
can
be
reproduced
in
Baldridge
's
system
simply
by
keying
all
structural
rules
to
the
ant-modality
,
the
result
being
that
all
proofs
involving
structural
rules
result
in
inert
functors
.
As
desired
,
the
D-rules
result
in
inert
categories
as
well
.
For
example
,
&gt;
D
is
derived
as
follows
(
□
6Note
that
the
diamond
operator
used
here
is
a
syntactic
operator
,
rather
than
a
semantic
operator
as
used
in
(
16
)
above
.
The
unary
modalities
used
in
CTL
describe
accessibility
relationships
between
subtypes
and
supertypes
of
particular
categories
:
in
effect
,
they
define
feature
hierarchies
.
See
Moortgat
(
1997
)
and
Oehrle
(
To
Appear
)
for
further
explanation
.
This
means
that
all
CCG
rules
compiled
from
the
logic
—
which
requires
ant
to
licence
the
structural
rules
necessary
to
prove
the
rules
—
return
inert
functors
.
Eisner
NF
thus
falls
out
of
the
logic
because
all
instances
of
B
,
D
,
and
S
produce
inert
categories
.
This
in
turns
allows
us
to
view
Eisner
NF
as
part
of
a
theory
of
grammatical
competence
,
in
addition
to
being
a
useful
technique
for
constraining
parsing
.
5
Conclusion
Including
the
D
-
combinator
rules
in
the
CCG
rule
set
lets
us
capture
several
linguistic
generalizations
that
lack
satisfactory
analyses
in
standard
CCG
.
Furthermore
,
CCG
augmented
with
D
is
compatible
with
Eisner
NF
(
Eisner
,
1996
)
,
a
standard
technique
for
controlling
derivational
ambiguity
in
CCG-parsers
,
and
also
with
the
modalized
version
of
CCG
(
Baldridge
and
Kruijff
,
2003
)
.
A
consequence
is
that
both
the
D
rules
and
the
NF
constraints
can
be
derived
from
a
grammar-internal
perspective
.
This
extends
CCG
's
linguistic
applicability
without
sacrificing
efficiency
.
Wittenburg
(
1987
)
originally
proposed
using
rules
based
on
D
as
a
way
to
reduce
spurious
ambiguity
,
which
he
achieved
by
eliminating
B
rules
entirely
and
replacing
them
with
variations
on
D.
Wittenburg
notes
that
doing
so
produces
as
many
instances
of
D
as
there
are
rules
in
the
standard
rule
set
.
Our
proposal
retains
B
and
S
,
but
,
thanks
to
Eisner
NF
,
eliminates
spurious
ambiguity
,
a
result
that
Wittenburg
was
not
able
to
realize
at
the
time
.
Our
approach
can
be
incorporated
into
Eisner
NF
straightforwardly
However
,
Eisner
NF
disprefers
incremental
analyses
by
forcing
right-corner
analyses
of
long-distance
dependencies
,
such
as
in
(
58
)
:
For
applications
that
call
for
increased
incremental-ity
(
e.g.
,
aligning
visual
and
spoken
input
incrementally
(
Kruijff
et
al.
,
2007
)
)
,
CCG
rules
that
do
not
produce
inert
categories
can
be
derived
a
CTL
basis
that
does
not
require
ant
for
associativity
and
permutation
.
The
D
-
rules
derived
from
this
kind
of
CTL
specification
would
allow
for
left-corner
analyses
of
such
dependencies
with
the
competence
grammar
.
An
extracted
element
can
"
wrap
around
"
the
words
intervening
between
it
and
its
extraction
site
.
For
example
,
D
would
allow
the
following
bracketing
for
the
same
example
(
while
producing
the
same
logical
form
)
:
Finally
,
the
unary
combinator
basis
for
CCG
provides
an
interesting
additional
specification
for
generating
CCG
rules
.
Like
the
CTL
basis
,
the
unary
combinator
basis
can
produce
a
much
wider
range
of
possible
rules
,
such
as
D
rules
,
that
may
be
relevant
for
linguistic
applications
.
Whichever
basis
is
used
,
inclusion
of
the
D-rules
increases
empirical
coverage
,
while
at
the
same
time
preserving
CCG
's
computational
attractiveness
.
Acknowledgments
Thanks
Mark
Steedman
for
extensive
comments
and
suggestions
,
and
particularly
for
noting
the
relationship
between
the
D-rules
and
unary
BS
.
Thanks
also
to
Emmon
Bach
,
Cem
Bozsahin
,
Jason
Eisner
,
Geert-Jan
Kruijff
and
the
ACL
reviewers
.
