computational linguisticsnlpcl.kaist.ac.kr/~cs579_2020/slides/579-fall-2020-12.pdf · 2020. 10....

Computational LinguisticsCS579: Fall Semester 2020

School of ComputingKorea Advanced Institute of Science and Technology

Jong C. Park

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First-Order Logic• First-Order Logic• Three Inference Tasks• A First-Order Model Checker• First-Order Logic and Natural Language

Lambda Calculus• Compositionality• Two Experiments• The Lambda Calculus• Implementing Lambda Calculus• Grammar Engineering

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Review of Previous Lectures

Lambda Calculus• We showed how to define a grammar that is

modular, extensible and reusable, in a framework of grammar engineering.

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Review of the Last Lecture

Scope Ambiguities Montague’s Approach Storage Methods Hole Semantics

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Underspecified Representations

Goals Today• We examine the nature of scope ambiguities.• We introduce Montague’s approach to scope

ambiguities.

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Underspecified Representations

Observation• Sentences with scope ambiguities are often

semantically ambiguous but fail to exhibit any syntactic ambiguity.

• They may have at least two non-equivalent first-order representations, but have only one syntactic analysis.

Problem• If there is no syntactic ambiguity, we will be able

to build only one of the possible representations.

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Underspecified Representations

We investigate four different approaches to scope ambiguities.• Montague’s original method• Two storage based methods• Cooper storage• Keller storage

• A modern underspecification based approach• Hole semantics

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Underspecified Representations

SCOPE AMBIGUITIES

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Scope ambiguity is a common semantic phenomenon and can arise from many sources.• quantifier, coordination, negation, adverbs, etc.

We will be concerned primarily with quantifier scope ambiguities. • They arise in sentences containing more than one

quantifying noun phrase.• Every boxer loves a woman.

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Scope Ambiguities

Deriving a representation for the sentence:

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Every boxer loves a woman (S)∀x(boxer(x)→ ∃y(woman(y)∧love(x,y)))

Every boxer (NP)𝜆u. ∀x(boxer(x)→u@x)

loves a woman (VP)𝜆z. ∃y(woman(y)∧love(z,y))

loves (TV)𝜆v. 𝜆z. v@𝜆x.love(z,x))

a woman (NP)𝜆w. ∃y(woman(y) ∧ w@y

The reading corresponding to the constructed first-order formula • x(boxer(x) y(woman(y) love(x,y)))• For each boxer there is a woman that he loves.• “every boxer” has scope over (or out-scopes) “a

woman”. • In other words, “every boxer” has wide scope

and “a woman” has narrow scope.

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There is another reading for the sentence.• y(woman(y) x(boxer(x) love(x,y)))• There is one woman who is loved by all boxers.• “a woman” has scope over (or out-scopes)

“every boxer”. • In other words, “a woman” has wide scope and

“every boxer” has narrow scope.

Is this a genuine reading, distinct from the previous reading?

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∀x(boxer(x)→ ∃y(woman(y)∧love(x,y)))

Question• Are scope ambiguities really such a problem?

For instance, we can argue • that the first reading is sufficient to cover both

readings, in that it is ‘weaker’, • that the weaker reading is the ‘real’

representation of the sentence, and • that the stronger reading is pragmatically

inferred from the weaker reading with the help of the contextual knowledge.

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1. ∀x(boxer(x)→ ∃y(woman(y)∧love(x,y)))2. ∃y(woman(y)∧ ∀x(boxer(x)→ love(x,y)))

Another sentence• Every owner of a hash bar gives every criminal

a big kahuna burger. • The sentence has 18 readings:

1. ∀x(∃y(hbar(y) ⋀ of(x,y) ⋀ owner(x)) → ∀z(crim(z) →∃u(bkb(u) ⋀ give(x,z,u))))

2. ∀x(crim(x) → ∀y((∃z(hbar(z) ⋀ of(y,z))⋀ owner(y)) → ∃u(bkb(u) ⋀ give(y,x,u))))

3. ∀x((∃y(hbar(y) ⋀ of(x,y) ⋀ owner(x)) →∃z(bkb(z) ⋀ ∀u(crim(u) → give(x,u,z))))

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4. ∀x(crim(x) → ∃y(bkb(y) ∧ ∀z(∃u(hbar(u) ⋀ of(z,u) ⋀owner(z)) → give(z,x,y))))

5. ∀x(crim(x) → ∃y(hbar(y) ∧ ∀z((of(z,y) ⋀ owner(z)) →∃u(bkb(u) ⋀ give(z,x,u)))))

6. ∀x(crim(x) → ∃y(hbar(y) ⋀ ∃z(bkb(z) ⋀ ∀u of u, y⋀ owner(u)) → give(u,x,z)))))

7. ∀x(crim(x) → ∃y(bkb(y) ⋀ ∃z(hbar(z) ⋀ ∀u of u, z⋀ owner(u)) → give(u,x,y)))))∃x bkb x ⋀ ∀y ∃z hbar z ⋀ of y,z ⋀ owner y → ∀u crim u → give y,u,x

9. ∃x bkb x ⋀ ∀y crim y → ∀z ∃u hbar u ⋀ of z,u ⋀ owner z → give z,y,x

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Every owner of a hash bar gives every criminal a big kahuna burger.

∃x hbar x ⋀ ∀y of y,x ⋀ owner y → ∀z crim z → ∃u bkb u → give y,z,u∃x hbar x ⋀ ∀y crim y → ∀z of z,x

⋀ owner z → ∃u bkb u ⋀ give z,y,u∃x hbar x ⋀ ∀y of y,x ⋀ owner y → ∃z bkb z ⋀ ∀u crim u → give y,u,z∃x hbar x ⋀ ∃y bkb y ⋀ ∀z of z,x ⋀ owner z →∀u crim u → give z,u,y∃x bkb x ⋀ ∃y hbar y ⋀ ∀z of z,y ⋀ owner z →∀u crim u → give z,u,x∃x hbar x ⋀ ∀y crim y → ∃z bkb z ⋀ ∀u of u,x ⋀ owner u → give u,y,z

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Every owner of a hash bar gives every criminal a big kahuna burger.

∃x hbar x ⋀ ∃y bkb y ⋀ ∀z crim z → ∀u of u,x ⋀ owner u → give u,z,y∃x bkb x ⋀ ∃y hbar y ⋀ ∀z crim z → ∀u of u,y ⋀ owner u → give u,z,x∃x bkb x ⋀ ∀y crim y → ∃z hbar z ⋀ ∀u of u,z ⋀ owner u → give u,y,x

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Every owner of a hash bar gives every criminal a big kahuna burger.

The logical equivalence of the readings leads to 11 distinct readings.• {1,2}, {8,9}, {6,7}, {10,11}, {13,14,16,17}• {3}, {4}, {5}, {12}, {15}, {18}

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The following diagram shows the logical relationships among the readings:

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13/14/16/17

12 15 18

10/11 6/7

5

8/9

4 3

1/2

Note that each group has a strongest reading ({13,14,16,17} and {8,9}) and a weakest reading ({5} and {1,2}), but there is no single weakest reading that covers all the possibilities.

In particular, it is hard to see how a pragmatic approach could account for this example.

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In addition, the idea of computing the weakest reading and relying on pragmatics for the rest faces difficulties even when the weakest reading exists.• Example• A boxer is loved by every woman.

• ∃y(boxer(y)∧ ∀x(woman(x)→ love(x,y)))• ∀x(woman(x)→ ∃y(boxer(y)∧love(x,y)))

• The stronger reading is what is generated by the direct approach to semantic construction.

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MONTAGUE’S APPROACH

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Scope ambiguities are a genuine problem. Montague introduced a rule of quantification

(called quantifier raising):• Instead of directly combining syntactic entities

with the quantifying noun phrase we are interested in, we are permitted to choose an ‘indexed pronoun’ and to combine the syntactic entity with the indexed pronoun instead.

• These indexed pronouns are ‘placeholders’ for the quantifying noun phrase.

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Montague’s Approach

• When this placeholder has moved high enough in the tree to give us the scoping we are interested in, we are permitted to replace it by the quantifying NP of interest.

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Example: the first stage• Every boxer loves a woman.

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Every boxer loves her-3 (S)∀x(boxer(x)→ love(x,𝑧 ))

Every boxer (NP)𝜆u. ∀x(boxer(x)→u@x)

loves her-3 (VP)𝜆y. love(y,𝑧 )

loves (TV)𝜆v. 𝜆y. v@𝜆x.love(y,x))

her-3 (NP)𝜆w. w@𝑧

Example: the second stage of substituting the quantifying NP

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Every boxer loves a woman (S)

a woman (NP)𝜆u. ∃y(woman(y) ∧ u@y

Every boxer loves her-3 (S,3)∀x(boxer(x)→ love(x,𝑧 ))

Lambda abstracting with respect to and -converting:

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Every boxer loves a woman (S)

a woman (NP)𝜆u. ∃y(woman(y) ∧ u@y

Every boxer loves her-3 (S,3)𝜆𝑧 .∀x(boxer(x)→ love(x,𝑧 )))

∃y(woman(y)∧ ∀x(boxer(x)→ love(x,y)))

Montague’s approach makes use of syntactic and semantic placeholders so that we can place quantifying NPs in parse trees at exactly the level required to obtain the desired scope relations.

A neat piece of ‘lambda programming’ ensures that the semantic information recorded by the placeholder is re-introduced into the semantic representation correctly.

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SUMMARY

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Scope ambiguities• A sentence with multiple quantifiers may have

genuine scope ambiguities. • We need a solution to generate multiple

readings.

Montague’s approach• Multiple readings can be derived by the use of

placeholders and ‘lambda programming’.

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Summary