Combinatory Categorial Grammar
LCTG—Notes Part 7: Combinatory Categorial Grammar
• (1) S
→
NP VP
VP
→
TV NP
TV
→
{proved, finds, . . .}
• (2) proved := (S\NP)/NP
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2
Slash Modalities
The Application Rules
• Rules and function categories are “modalized,” as indicated by a subscript on
slashes.
• Function categories may be restricted as to the rules that allow them to
combine with other categories, via slashes typed with four basic modalities: ?,
×, , and ·. The ? modality is the most restricted and allows only the most
basic applicative rules; permits order-preserving associativity in derivations;
× allows limited permutation; and · is the most permissive, allowing all rules
to apply.
?
×
.
• (3) The functional application rules
a. X/?Y
b. Y
⇒ X
Y
X\ ?Y
⇒
(>)
(<)
X
• ? modality means “this rule applies to all categories, even ?”.
• A Derivation:
(4) a. Marcel
NP
completeness b. Marcel proved completeness
NP
V
NP
(S\NP)/NP
NP
>
VP
S\NP
proved
<
S
• By convention, the most permissive slash-types /· and \· are abbreviated as
categories like (2) as (S\NP)/NP, rather than (S\·NP)/·NP
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4
S
Semantics
(5) proved := (S\NP3s )/NP :
Notation: Left-Associativity Convention over Logical Forms
λx.λy.prove0 xy
(6) Functional application
a. X/?Y : f
Y :a
b. Y : a X\ ?Y : f
(7)
⇒
⇒
X : fa
(>)
X : fa
Marcel
(<)
proved
completeness
NP3sf : marcel0 (S\NP3s )/NP : λx.λy.prove0 xy NP : completeness0
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6
Coordination
Composition
(11) Forward composition (>B)
X/Y : f Y /Z : g ⇒B X/Z : λx. f (gx)
(9) The Conjunction Category
and := (X\ ?X)/?X
NP
• A nonordered form of the traditional VP is reflected at the level of
propositional logical form.
<
S : prove0 completeness0 marcel0
(10) Marcel conjectured
b. prove ’ completeness’ marcel ’
• Such logical forms therefore preserve traditional c-command.
>
S\NP3s : λy.prove0 completeness0 y
(8) a. (prove0 completeness0 )marcel0
and
proved
(S\NP)/NP (X\ ?X)/?X
completeness
(S\NP)/NP
((S\NP)/NP)\ ?((S\NP)/NP)
prove
completeness
VP/NP
: prove0
NP
: completeness0
>B
>
<
S
might
(S\NP)/NP
: λxλy.might0 (prove0 x)y
<
S\NP
and
NP
(S\NP)/NP (X\?X)/?X (S\NP)/VP
: and 0
: might0
: marcel0 : conjecture0
NP
>
(S\NP)/NP
(12) Marcel conjectured
>
((S\NP)/NP)\?((S\NP)/NP)
: λtvλxλy.and 0 (might0 (prove0 x)y)(tv xy)
(S\NP)/NP
<
: λxλy.and 0 (might0 (prove0 x)y)(conjecture0 xy)
>
S\NP
: λy.and 0 (might0 (prove0 completeness0 )y)(conjecture0 completeness0 y)
<
S : and 0 (might0 (prove0 completeness0 )marcel0 )(conjecture0 completeness0 marcel0 )
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8
• Type raising is restricted to primitive argument categories, NP, PP etc., and
over primitive functors like verbs, resembling the traditional notion of case.
We can do it lexically
Type-Raising
• (13) Forward type-raising (>T)
X : a ⇒ T/i (T\i X) : λf .fa
• The subscript i on the slashes means that they both have the same modality as
what ever function T\i X the raised category is applied to. T is a metavariable
over categories. If instantiated as S, it allows the following derivation:
(14)
Marcel
proved
NP
>T
and
I
(S\NP)/NP (X\ ?X)/?X
S/(S\NP)
S/NP
disproved completeness
(S\NP)/NP
NP
NP
>T
S/(S\NP)
>B
S/NP
(S/NP)\ ?(S/NP)
S/NP
>B
>
<
S
>
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Universal Grammar
• Only rules which obey the following principles are allowed:
(15) The Principle of Adjacency:
Combinatory rules may only apply to finitely many phonologically realized and string-adjacent entities.
(16) The Principle of Consistency:
All syntactic combinatory rules must be consistent with the directionality of the principal function.
(17) The Principle of Inheritance:
If the category that results from the application of a combinatory rule
is a function category, then the slash defining directionality for a given
argument in that category will be the same as the one(s) defining directionality for the corresponding argument(s) in the input function(s).
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• The other two principles say that combinatory rules may not override, but
must rather “project,” the directionality specified in the lexicon:
• The Principle of Consistency excludes the following kind of rule:
(18) X\Y
Y
6⇒ X
• The Principle of Inheritance excludes rules like the following:
(19) X/Y
Y/Z
6⇒ X\Z
• These principles do allow rules such as the following:
(20) Forward crossing composition (>B× )
X/×Y : f Y \×Z : g ⇒B× X\×Z : λx. f (gx)
• The Principle of Adjacency is merely the definition of a combinator.
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12
Many Linguistic Predictions
• We correctly predict the fact that right-node raising is unbounded, as in a,
below, and also provide the basis for an analyis of the similarly unbounded
character of leftward extraction, as in b.
(21) a. [I conjectured]S/NP and [you think Marcel proved]S/NP completeness.
b. The result [which](N\N)/(S/NP) [you think Marcel proved]S/NP .
Linguistic Predictions (contd.)
• Since we have assumed Forward Type Raising and Composition, the
following construction is predicted on arguments of symmetry.
(23) give
Walt
the salt
<T
and
<T
Malcolm the talcum
<T
DTV TV\ DTV VP\TV (X\ ?X)/?X TV\ DTV
VP\ DTV
<B
VP\ DTV
(VP\ DTV)\ ?(VP\ DTV)
(22) a. a man who(m) [I think that]S/S [Keats likes]S/NP
b. *a man who(m) [I think that]S/S [likes Keats]S\NP
VP\ DTV
VP
• DTV = (VP/NP)/NP
<T
VP\TV
<B
>
<
<
TV = VP/NP
(24) *Three mathematicians [[in ten]PP [derive a lemma,]S\NP ] and [in a hundred prove completeness.]
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Gapping and the Order of Constituents
• The phenomenon of “argument cluster coordination” illustrated in (23) is an
example of a much broader cross-linguistic generalization due to Ross,
concerning a relation between basic word-order parameters such as verb
finality and constraints on deletion or gapping under coordination. While
Ross originally framed his generalization in terms of a then-orthodox
deep-structural word order, and on that basis it has been challenged, when
reformulated in terms of surface constituent order, it appears to hold.
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(traditionally SOV) which allow both leftward and rightward gapping also
have suspiciously mixed word-order, rendering the identification of “basic”
word-order moot.
• What Ross (1970) noticed was that direction of gapping (leftward or
rightward) depends on basic constituent order:
(25) a. SOV: *SOV and SO, SO and SOV
b. VSO: VSO and SO, *SO and VSO
c. SVO: SVO and SO, *SO and SVO
• Apparent exceptions such as Zapotec (traditionally VSO) and Dutch
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16
CCG is “Nearly Context Free”
• CCG categories can be viewed as their result category plus a stack-valued
feature identifying their arguments and the order of combination.
Crossed Dependencies in CCG
• The availability of crossed composition under the Principles of Consistency
and Inheritance allows crossed dependencies:
• Under the principles of inheritance and consistency, combinatory rules then
map one-to-one to LIG rules.
(26) zag:= ((S+SUB \NP)\NP)/×VP
zien, helpen:= (VP\NP)/×VP
voeren:= VP\NP
• Hence CCG is provably weakly equivalent to Tree-adjoining Grammar
(TAG), Head Grammar (HG), and Linear Indexed Grammar, a fact that gives
rise to a polynomial time worst-case complexity result (Vijay-Shanker and
Weir, 1994).
(27) Dutch Forward Crossing Composition I (> Bn× ):
X/×Y
(Y \×Z)$
⇒ Bn
X\×Z$
• This LIG equivalent class is not only mildly context sensitive. It is Nearly
Context Free.
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18
Hippo Sentences
Hippo Sentences (Contd.)
... omdat ik Cecilia de nijlpaarden
zag voeren.
... because I Cecilia the hippopotamuses saw feed
... omdat ik Cecilia Henk de nijlpaarden
zag helpen voeren.
... because I Cecilia Henk the hippopotamuses saw help feed
‘... because I saw Cecilia feed the hippopotamuses.’
(28) dat ik Cecilia de nijlpaarden
that I Cecilia
NP
NP
‘... because I saw Cecilia help Henk feed the hippopotamuses.’
zag
saw
the hippos
voeren
feed
(29)
dat ik Cecilia Henk de nijlpaarden
NP
NP
NP
NP
(S+SUB \NP)\NP
S+SUB \NP
<
S+SUB
“that I saw Cecilia feed the hippos”
<
<
(((S+SUB \NP)\NP)\NP)\×NP
(30)
dat ik Cecilia Henk de nijlpaarden
NP
NP
NP
NP
zag
helpen
>B2×
voeren
((S+SUB \NP)\NP)/
×VP (VP\NP)/
×VP VP\NP
>B2×
(((S+SUB \NP)\NP)\NP)/
×VP
(((S+SUB \NP)\NP)\NP)\×NP
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voeren
>B×
>B×
((S+SUB \NP)\NP)\×NP
helpen
(VP\NP)\×NP
((S+SUB \NP)\NP)/×VP VP\NP
NP
zag
((S+SUB \NP)\NP)/
×VP (VP\NP)/
×VP VP\NP
20
>B×
Hippo Sentences (Contd.)
(31)
dat
Jan
Cecilia
Henk
zag helpen voeren
T/(T\NP)
(((S+SUB \NP)\NP)\NP)\×NP
((S+SUB \NP)\NP)\NP
(S+SUB \NP)\NP
>
dat
Jan
Cecilia
Henk
T/(T\NP) T/(T\NP) T/(T\NP)
de nijlpaarden
zag helpen voeren
T/(T\NP)
(((S+SUB \NP)\NP)\NP)\×NP
>B
T/((T\NP)\NP)
T/(((T\NP)\NP)\NP)
>B
T/((((T\NP)\NP)\NP)\NP)
>
S+SUB
Jan
>
>
(S+SUB \NP)
dat
(32)
de nijlpaarden
T/(T\NP) T/(T\NP)) T/(T\NP)
(33)
Hippo Sentences (Contd.)
>B
S+SUB
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22
Hippo Sentences (Contd.)
Hippo Sentences (Contd.)
Cecilia
Henk
T/(T\NP) T/(T\NP) T/(T\NP)
de nijlpaarden
zag helpen voeren
T/(T\NP)
(((S+SUB \NP)\NP)\NP)\×NP
>B
T/((T\NP)\NP)
T/(((T\NP)\NP)\NP)
T/((((T\NP)\NP)\NP)\NP)
>B
• The Dutch “nonconstituent” coordinations in the last slide of the previous set
are therefore examples of constituent coordination in CCG, just like that in
example (23), and are further instances of Ross’s Generalization.
>B
S+SUB
23
>
>
24
CCG and LIG
CCG and LIG
• Joshi, Weir and Vijay-Shanker (1991) were the first to observe that there is a
close relation between linear indexed rules and the combinatory rules of CCG.
• Function categories like give and zag helpen voeren can be equated with
indexed categories bearing stack-valued features, as follows:
(34) give :=
(VP/NP)/NP
≡
VP[NP,NP]
zag helpen voeren := (((S\NP)\NP)\NP)\NP ≡ S[NP,NP,NP,NP]
• Note that the LIG categories no longer encode directionality—it is up to the
LIG rules to do that.
• Combinatory rules can be translated rather directly in terms of such categories
into sets of LIG productions of the form shown on the right of the
equivalences in (35) and (36).
• Since LIG categories do not capture directionality, the grammar for a
particular language will be made up of more specific instances of these
schemata involving just those categories that do in fact combine in the
specified order for that language.a
(35) X/Y Y
⇒ X
(36) X/Y Y/Z
0
≡ X[...]
⇒ X/Z
0
→ X[Y,...]
Y[ ]
0
≡ X[Z,...]
0
→ X[Y,...]
Y[Z]
a
Note also that the category X 0 in the LIG schemata will not in general be the same as the category X
in the corresponding instantiation of the categorial rule. If X is a function category, then X 0 is its result
and the stack . . . is its argument(s). In terms of the earlier $ convention, X is of the form X 0 $.
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CCG and LIG
• Rule (35) is forward application, realized as a binary LIG rule of the “push”
variety. Rule (36) is first-order forward composition, B, and involves both
pushing a Y and popping a Z.
• Crucially, the stack, represented as . . . , is passed to only one daughter.
Therefore this is a Linear Indexed Grammar (LIG).
• Provided higher order composition (Bn ) is kept to bounded n and type-raising
is restricted to lifting over functions of similarly bounded valency, the same
applies.
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