Lecture 7
.
......
Syntactic Formalisms for Parsing
Natural Languages
Aleš Horák, Miloš Jakubíček, Vojtěch Kovář
(based on slides by Juyeon Kang)
ia161@nlp.fi.muni.cz
Autumn 2013
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Lecture 7
Outline
Applicative system
Combinators
Combinators vs. λ-expressions
Application to natural language parsing
Combinators used in CCG
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Lecture 7
Applicative system
CL (Curry & Feys, 1958, 1972) as an applicative system
CL is an applicative system because the basic unique operation
in CL is the application of an operator to an operand
Operator(Operand)
Operator Operand
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Lecture 7
Combinators
CL defines general operators, called Combinators.
Each combinator composes between them the
elementary combinators and defines the
complexe combinators.
Certains combinators are considered as the basic combinators
to define the other combinators.
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Lecture 7
Elementary combinators
I =def λx.x (identificator)
K =def λx.λy.x (cancellator)
W =def λx.λy.xyy (duplicator)
C =def λx.λy.λz.xzy (permutator)
B =def λx.λy.λz.x(yz) (compositor)
S =def λx.λy.λz.xz(yz) (substitution)
Φ =def λx.λy.λz.λu.x(yu)(zu) (distribution)
Ψ =def λx.λy.λz.λu.x(yz)(yu) (distribution)
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Lecture 7
β-reductions
The combinators are associated with the β-reductions in a
canonical form:
β-reduction relation between X and Y
X ≥β Y
Y was obtained from X by a β-reduction
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Lecture 7
β-reductions
Ix ≥β x
Kxy ≥β x
Wxy ≥β xyy
Cxyz ≥β xzy
Bxyz ≥β x(yz)
Sxyz ≥β xz(yz)
Φxyzu ≥β x(yu)(zu)
Ψxyzu ≥β x(yz)(yu)
.
......
Each combinator is an operator which has a certain number of arguments (operands);
sequences of the arguments which follow the comnator are called “the scope of
combinator”.
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Lecture 7
β-reductions
Intuitive interpretations of the elementary combinators are
given by the associated β-reductions.
The combinator I expresses the identity.
The combinator K expresses the constant function.
The combinator W expresses the diagonalisation or the
duplication of an argument.
The combinator C expresses the conversion, that is, the
permutation of two arguments of an binary operator.
The combinator B expresses the functional composition of two
operators.
The combinator S expresses the functional composition and the
duplication of argument.
The combinator Φ expresses the composition in parallel of
operators acting on the common data.
The combinator Ψ expresses the composition by distribution.
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Lecture 7
Introduction and elimination rules of combinators
Introduction and elimination rules of combinators can be
presented in the style of Gentzen (natural deduction).
Elim. Rules Intro. Rules
If f
- - - [e-I] - - - [i-I]
f If
Kfx f
- - - - - [e-K] - - - - [i-K]
f Kfx
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Lecture 7
Introduction and elimination rules of combinators
Elim. Rules Intro. Rules
Cfx xf
- - - [e-C] - - - [i-C]
xf Cfx
Bfxy f(xy)
- - - - - [e-B] - - - - [i-B]
f(xy) Bfxy
Φfxyz f(xz)(yz)
- - - - - [e-Φ] - - - - [i-Φ]
f(xz)(yz) Φfxyz
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Lecture 7
Combinators vs. λ -expressions
The most important difference between the CL and λ-calculus
is the use of the bounded variables.
Every combinator is an λ -expression.
Bfg ≡ λx.f(gx)
Tx ≡ λf.fx
Sfg ≡ λx.fx(gx)
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Lecture 7
Application to natural language parsing
John is brilliant
The predicate is brilliant is an operator which operate on the
operand John to construct the final proposition.
The applicative representation associated to this analysis is the
following:
(is-brillant)John
We define the operator John* as being constructed from the
lexicon John by
[John* = C* John].
1 John* (is-brillant)
2 [John* = C* John]
3 C*John (is-brillant)
4 is-brillant (John)
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Lecture 7
Application to natural language parsing
John is brilliant in λ-term
Operator John* by λ-expression
[John* = λx.x (John’)]
1 John*(λx.is-brilliant’(x))
2 [John* = λx.x (John’)]
3 (λx.x(John’))(λx.is-brilliant’(x))
4 (λx.is-brilliant’(x))(John’)
5 is-brillinat’(John’)
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Lecture 7
Passivisation
Consider the following sentences
a. The man has been killed.
b. One has killed him.
→ Invariant of meaning
→ Relation between two sentences
:a. unary passive predicate (has-been-killed)
:b. active transitive predicate (have-killed)
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Lecture 7
Definition of the operator of passivisation ’PASS’
[PASS = B
∑
C =
∑
◦ C]
where B and C are the combinator of composition and of
conversion and where
∑
is the existential quantificator which,
by applying to a binary predicate, transforms it into the unary
predicate.
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Lecture 7
Definition of the operator of passivisation ’PASS’
.
...... [PASS = B
∑
C =
∑
◦ C]
1/ has-been-killed (the-man) hypothesis
2/ [has-been-killed=PASS(has killed)] passive lexical predicate
3/ PASS (has-killed)(the-man) repl.2.,1.
4/ [PASS = B
∑
C ] definition of ’PASS’
5/ B
∑
C (has-killed)(the-man) repl.4.,3.
6/
∑
(C(has-killed))(the-man) [e-B]
7/ (C(has-killed)) x (the-man) [e-
∑
]
8/ (has-killed)(the-main) x [e-C]
9/ [x in the agentive subject position = one] definition of ’one’
10/ (has-killed)(the-man)one repl.9.,8., normal form
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Lecture 7
Definition of the operator of passivisation ’PASS’
We establish the paraphrastic relation between the passive
sentence with expressed agent and its active counterpart:
The man has been killed by the enemy
↓
The enemy has killed the man
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Lecture 7
Definition of the operator of passivisation ’PASS’
.
......Relation between give-to and receive-from
z gives y to x
↕
x receives y from x
.
......
The lexical predicate “give-to” has a predicate converse associated to “receive-from”;
[receive-from z y x = give-to x y z]
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Lecture 7
Definition of the operator of passivisation ’PASS’
1/ (receive-from) z y x
2/ C((receive-from) z) x y
3/ BC(receive-from) z x y
4/ C(BC(receive-from)) z x y
5/ C(C(BC(receive-from)) x) y z
6/ BC(C(BC(receive-from))) x y z
7/ [give-to=BC(C(BC(receive-from)))]
8/ give-to x y z
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Lecture 7
Combinators used in CCG
Motivation of applying the combinators
to natural language parsing
Linguistic: complex phenomena of natural language applicable
to the various languages
Informatics: left to right parsing (LR)
ex: reduce the spurious-ambiguity
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Lecture 7
Parsing a sentence in CCG
Step 1: tokenization
Step 2: tagging the concatenated lexicon
Step 3: calculate on types attributed to the concatenated
lexicons by applying the adequate combinatorial rules
Step 4: eliminate the applied combinators (we will see how to do
on next week)
Step 5: finding the parsing results presented in the form of an
operator/operand structure (predicate -argument structure)
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Lecture 7
Parsing a sentence in CCG
Example: I requested and would prefer musicals
STEP 1 : tokenization/lemmatization → ex) POS Tagger,
tokenizer, lemmatizer
a. I-requested-and-would-prefer-musicals
b. I-request-ed-and-would-prefer-musical-s
STEP 2 : tagging the concatenated expressions → ex)
Supertagger, Inventory of typed words
I NP
Requested (S\NP)/NP
And CONJ
Would (S\NP)/VP
Prefer VP/NP
musicals NP
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Lecture 7
Parsing a sentence in CCG
STEP 3 : categorial calculus
c. apply the coordination rules Coordination: (< & >)
X conj X ⇒ X
b. apply the functional composition rules Forward Composition: (> B)
X/Y : f Y/Z : g ⇒ X/Z : Bfg
a. apply the type-raising rules Subject Type-raising (> T)
NP : a ⇒ T/(T\NP) : Ta
7/ S
6/ S/NP NP (>)
5/ S/(S\NP) (S\NP)/NP NP (>B)
4/ S/(S\NP) (S\NP)/NP NP (> Φ)
3/ S/(S\NP) (S\NP)/NP CONJ (S\NP)/NP NP (>B)
2/ S/(S\NP) (S\NP)/NP CONJ (S\NP)/VP VP/NP NP (>T)
1/ NP (S\NP)/NP CONJ (S\NP)/VP VP/NP NP
I- requested- and- would- prefer- musicals
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Lecture 7
Parsing a sentence in CCG
STEP 4 : semantic representation (predicate-argument
structure)
7/S: and’(will’(prefer’ musicals’) i’)(request’ musicals’ i’)
6/ :λy.and’(would’(prefer’ musicals’)y)(request’ musicals’ y)
5/ : λxλy.and’(will’(prefer’x)y)(request’xy)
4/ : λxλy.and’(will’(prefer’x)y)(request’xy)
3/ : λx.λy.will’(prefer’x)y
2/ :λf.f I’
1/ :i’ :request’ :and’ : will’ :prefer’ : musicals’
I requested and would prefer musicals
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Lecture 7
Semantic representation in term of the
combinators
I- requested and- would- prefer musicals
1/ NP (S\NP)/NP CONJ (S\NP)/VP VP/NP NP
2/ S/(S\NP) (S\NP)/NP CONJ (S\NP)/VP VP/NP NP (>T)
C*I requested and would prefer musicals
3/ S/(S\NP) (S\NP)/NP CONJ (S\NP)/NP NP (>B)
C*I requested and B would prefer musicals
4/ S/(S\NP) (S\NP)/NP NP (> Φ)
C*I Φ and requested (B would prefer) musicals
5/ S/NP NP (>B)
B((C*I)(Φ and requested (B would prefer))) musicals
6/ S (>)
B((C*I)(Φ and requested (B would prefer))) musicals
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Lecture 7
Semantic representation in term of the
combinators
.
...... I requested and would prefer musicals
S: B((C*I)(Φ and requested (B would prefer))) musicals
1/ B((C*I)(Φ and requested (B would prefer))) musicals
2/ (C*I)((Φ and requested (B would prefer))) musicals) [e-B]
3/ ((Φ and requested (B would prefer))) musicals) I [e-C*]
4/ (and (requested musicals) ((B would prefer) musicals)) I [e-Φ]
5/ ((and (requested musicals) (would (prefer musicals))) I ) [e-B]
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Lecture 7
Normal form
A normal form is a combinatory expression which is irreducible
in the sense that it contain any occurrence of a redex.
If a combinatory expression X reduce to a combinatory
expression N which is in normal form, so N is called the
normal form of X.
.
Example
..
......
Bxyz is reducible to x(yz).
x(yz) is a normal form of the combinatory expression Bxyz.
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Lecture 7
Normal form
.
Example
..
......
Prove xyz is the normal form of BBCxyz.
BBCxyz →β xyz
1/ BBCxyz
2/ C(Cx)yz [e-B]
3/ Cxzy [e-C]
4/ xyz [e-C]
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Lecture 7
Classwork
Give the semantic representation in term of combinators.
Please refer to the given paper on last lecture on CCG Parsing.
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