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Underspecified Representations

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### UnderspecifiedRepresentations

Underspecified Representations

The Issue

- Every boxer loves a woman
- Ax(BOXER(X) => Ey(WOMAN(Y) & LOVE(X,Y))
- Ey(WOMAN(Y) & Ax( BOXER(X) =>LOVE(X,Y))

- Reading 1: every boxer has scope over or outscopes a woman
- Reading 2: a woman has scope over or outscopes every boxer
- Cause is semantic not syntactic

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4 Approaches

- Do nothing
- Montague’s original method
- Robin Cooper’s stores
- Keller Storage
- Hole semantics

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Do Nothing

- Is it really such a problem?
- Given
- Ax(BOXER(X) => Ey(WOMAN(Y) & LOVE(X,Y))
- Ey(WOMAN(Y) & Ax( BOXER(X) =>LOVE(X,Y))
Couldn’t we just choose the weaker reading and argue that because that is entailed by the stronger reading, it is the ‘real’ reading?

Then a method would be to always generate the weakest reading and construct the stronger reading via pragmatics

- Which is the weaker reading?

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The Problem Some of these are logically equivalent, namely {1,2}, {8,9}, {6,7}, {10,11}, {13,14,16,17} If we take these equivalences into account there are 11 distinct readings Moreover if we examine these readings closely we discover they are partitioned into two distinct groups

- Every owner of a hash bar gives every criminal a big kahuna burger
- There are 18 readings
- Ax((Ey(HBAR(y) & OF(x,y)) & OWNER(x)) => Az(CRIM(x) => Eu(BKB(u) & GIVE(x,z,u))))
- Ax(CRIM(x) => Ay((Ex (HBAR(z) & OF(y,z)) & OWNER(y)) => Eu(BKB(u) & GIVE(y,x,u))))
- [..]
- Ex(BKB(x) & Ay(CRIM(y) => Ex(HBAR(z) & Au((OF(u,z) & OWNER(U) => VIVE(u,y,x))))

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Groups of Readings

{8,9}

{13,14,15,16}

{4} {3}

{12} {15} {18}

{10.11} {6,7}

{1,2}

NB arrows represent logical implication

{5}

Underspecified Representations

Doing Nothing: The Problem

- In general there may not be a unique weakest reading
- Even when a weakest reading does exist, there is no guarantee that it will be generated by the methods discussed so far.
- Even in the simple case presented first, semantic construction generated by the parse tree yields the stronger reading

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Montague’s Approach

- Motivated in part by quantifier scope ambiguities Montague had introduced quantifier raising
- Instead of directly combining syntactic entities with the quantifying NP, we are permitted to introduce an “indexed pronoun” and combine the syntactic entity with it.
- Such indexed pronouns are placeholders for the quantifying NPs
- When this placeholder has moved high enough in the tree to give the scoping we want, we replace it by the quantifying NP of interest.

Underspecified Representations

Parse Tree with Logical Forms

Every boxer loves her-3 (S)

Ax(BOXER(x) => LOVE(x,z3)

loves her-3 (VP)

y.LOVE(y,z3)

Every boxer (NP)

u.Ax(BOXER(x) => [email protected])

loves (TV)

v.y.([email protected]x.LOVE(y,x))

her-3 NP

w.([email protected])

a woman

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Placeholder Pronouns

- Key point: this tree is totally normal
- Instead of combining loves with the quantifying term a woman we have combined it with the placeholder pronoun her-3.
- her-3 has a semantic representation which is familiar – just like a proper noun except that the name is an indexed variable instead of a constant
- [her-3] = w.([email protected])
- [vincent] = w.([email protected])

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Next Step

- Aim: a woman must outscope every boxer
- By using the placeholder pronoun, we have so far delayed introducing a woman into the tree.
- Now we introduce it using the following rule:
- Given a quantifying NP (a woman) and a sentence containing a placeholder pronoun (every boxer loves her-3), we can construct a new sentence by substituting the QNP for the placeholder.
- i.e. we can extend the previous tree as follows

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Extending the Tree

Every boxer loves a woman (S)

a woman (NP)

u.Ey(WOMAN(y)& [email protected])

Every boxer loves her-3 (S,3)

Ax(BOXER(x) => LOVE(x,z3)

previous

tree

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Getting the Semantics to Work (1)

u.Ey(WOMAN(y)& [email protected]) @ Ax(BOXER(x) => LOVE(x,z3))

Ey(WOMAN(y)& Ax(BOXER(x) => LOVE(x,z3))@y)

[stop]

- The problem is that if we apply a woman to every boxer loves her3 directly, no further reduction is possible.
- We need to perform lambda abstraction over every boxer loves her3, i.e. from
- Ax(BOXER(x) => LOVE(x,z3)) to
- z3.Ax(BOXER(x) => LOVE(x,z3)) to

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Getting the Semantics to Work (2)

u.Ey(WOMAN(y)& [email protected]) @ z3.Ax(BOXER(x) => LOVE(x,z3))

Ey(WOMAN(y)& z3.Ax(BOXER(x) => LOVE(x,z3))@y)

Ey(WOMAN(y)& Ax(BOXER(x) => LOVE(x,y)))

[stop - success]

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This is a solution, but ….

- Although this is a solution of a kind we had to modify the grammar in order to introduce, and then eliminate the placeholder pronoun.
- Bad use of syntax to control semantics
- Situation worsens (more rules required) to handle, e.g., interaction between negation and quantifier scope ambiguities.

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Cooper Storage

- Technique invented by Robin Cooper to handle quantifier scope ambiguities
- Key idea is to associate each node of the parse tree with a store containing
- core semantic representations
- quantifiers associated with lower nodes

- Scoped representations are generated after the sentence is parsed.
- The particular scoping generated depends on the order in which quantifiers are retrieved from the store

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The Store

- A store is an n-place sequence
- first item is always the core semantic representation i.e. a -expression F
- subsequent items are pairs (B,i) where B is the semantic representation of an NP (another -expression and i is an index which picks out a certain variable in F.
- <F,(B,j), ...,(B’,k)>

Underspecified Representations

Using Cooper Storage

- If <F,(B,j), ...,(B’,k)> is a semantic representation for an NP, then the store
<u.([email protected]), (F,i), (B,j), ...,(B’,k)>

where i is some unique index, is also a representation of that NP

- KEY POINT: The index i associated with F is identical with the subscript on the free variable in u.([email protected])
- When we encounter an NP, we are faced with a choice.

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Using Cooper Storage

- When we encounter a quantified NP, we can either pass on <F, ..other pairs..>
- or else we can pass on <u.([email protected]), (F,i), ..other pairs.. >
- In the second case the effect is to ‘freeze’ the quantifier F for later use.
- NB storage rule is not recursive. We just get the two choices.

Underspecified Representations

Parse Tree with Logical Forms

Every boxer loves a woman (S)

<LOVE(z6,z7),

(u.Ax(BOXER(x)=>[email protected]),6),

(u.Ey(WOMAN(y)& [email protected]),7)>

loves a woman (VP)

< u.LOVE(u,z7),

(u.Ey(WOMAN(Y)&[email protected]),7)>

Every boxer (NP)

< w.([email protected]),

(u.Ax(BOXER(x) => [email protected],6)>

loves (TV)

<z.u.([email protected]v.LOVE(u,v))>

a woman NP

<w.([email protected]),

(u.Ey(WOMAN(y)& [email protected]),7)>

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Remarks

- Note first of all that the two noun phrases are associated with 2-place stores
- Why is this?
- In the pre-storage era we had a woman:u.Ey(WOMAN(y) & [email protected]
- In the storage era this would be<u.Ey(WOMAN(y) & [email protected]>
- But now we have the choice of using
<w.([email protected]), (u.Ey(WOMAN(y) & [email protected],7)>

Underspecified Representations

Combining Stores

- If a functor node is associated with
<F,(B,j), ..., (B,k)>

- and an argument node is associated with
<G,(C,m), ..., (C,n)>

- The the store associated with the result of applying the first to the second is:
<[email protected], (B,j), ..., (B,k) ,(C,m), ..., (C,n)>

- It may be possible to do beta reduction on [email protected]

Underspecified Representations

Retrieval

- We now have an unscoped abstract representation
- We want to extract an ordinary scoped representation from it.
- That is the task of retrieval
- Retrieval removes one of the elements from the store and combines it with the core representation to form a new core representation.

Underspecified Representations

Cooper Retrieval Rule

- Let s1 and s2 be (possibly empty) sequences of binding operators.
- If the store
<F,s1,(B,i),s2>

is associated with an expression of category S, then the store

<[email protected]zi.F, s1,s2>

is also associated with this expression

Underspecified Representations

Embedded NPs

Every piercing that is done with a gun goes against the entire idea behind it

Mia knows every owner of a hash bar

Both of these are ambiguous

Both contain sub-NPs

Underspecified Representations

(u.Ay(OWNER(y) & OF(y,z1) => [email protected]), 2),

(w.Ex(HASHBAR(x) & [email protected]),1) >

- Now we have a choice as to which item in the store to use
- Suppose we choose to take the Universal quantifier first

Underspecified Representations

Taking the Universal first …

< KNOW(MIA,z2),

(u.Ay(OWNER(y) & OF(y,z1) => [email protected]), 2),

(w.Ex(HASHBAR(x) & [email protected]),1) >

<u.Ay(OWNER(y) & OF(y,z1) => [email protected])@

z2. KNOW(MIA,z2),

(w.Ex(HASHBAR(x) & [email protected]),1) >

Underspecified Representations

(u.Ay(OWNER(y) & OF(y,z1) => [email protected]), 2),

(w.Ex(HASHBAR(x) & [email protected]),1) >

<Ay(OWNER(y) & OF(y,z1) => KNOW(MIA,y),

(w.Ex(HASHBAR(x) & [email protected]),1) >

Underspecified Representations

….. It works

<Ay(OWNER(y) & OF(y,z1) => KNOW(MIA,y),

(w.Ex(HASHBAR(x) & [email protected]),1) >

<w.Ex(HASHBAR(x) & [email protected])

@z1.Ay(OWNER(y) & OF(y,z1) => KNOW(MIA,y)

Ex(HASHBAR(x) & z1…..OF(y,z1) … @ x

Ex(HASHBAR(x) &

Ay(OWNER(y) &

OF(y,x) => KNOW(MIA,y)

Underspecified Representations

Taking the Existential first …

< KNOW(MIA,z2),

(u.Ay(OWNER(y) & OF(y,z1) => [email protected]), 2),

(w.Ex(HASHBAR(x) & [email protected]),1) >

< w.Ex(HASHBAR(x) & [email protected])@

z1. KNOW(MIA,z2),

(u.Ay(OWNER(y) & OF(y,z1) => [email protected]), 2),>

Underspecified Representations

Taking the Existential first …

< w.Ex(HASHBAR(x) & KNOW(MIA,z2)),

(u.Ay(OWNER(y) & OF(y,z1) => [email protected]), 2),>

[…]

Ay(OWNER(y) & OF(Y,z1) =>

Ex(HASHBAR(X) & KNOW(MIA,y)))

- This is not what we wanted
- The result is a formula with a free variable

Underspecified Representations

What went wrong

- The Cooper storage mechanism ignores the hierarchical structure of the NP
- a hash bar contributes the free varable z1, but z1 has been moved out of the core representation and is put in the store.
- Hence lambda abstracting the core representation wrt z1 is not guaranteed to take into account z1’s contribution – which is made indirecty through the stored universal quantifier every owner.
- Everything is ok if we restore UQ first since that restores z1 to the core representation.

Underspecified Representations

What went wrong

- However, if we choose to retrieve the existential quantifier first then then we get a problem.
- Cooper storage does not impose enough discipline on storage and retrieval
- Keller (1988) suggests a solution: allow nested stores
- As before, nested stores are associated with a storage rule and a retrieval rule.

Underspecified Representations

Keller Storage Rule

- If the nested store
<F,s>

- s an interpretation for an NP, then the nested store
<u.([email protected]),(<F,s>,i)>

for some unique index i, is also an interpretation of that NP

Underspecified Representations

Parse Tree with Logical Forms

Every owner of a hash bar (NP)

(<u.Ay(OWNER(y)&OF(y,z1) => [email protected]),

(<w.Ex(HASHBAR(x) & [email protected])>,1)>,2)>

Owner of a hash bar (VP)

<u.OWNER(u)&OF(u,z1)),

(<w.Ex(HASHBAR(x)&[email protected])>,1)>

Every (DET)

<w.u.Ay([email protected] => [email protected])>

owner (N)

<x.OWNER(x)>

of a hash bar (PP)

<v .u.([email protected]&OF(u,z1)),

(<w.Ex(HASHBAR(x)&[email protected])>,1)>

Underspecified Representations

Keller Retrieval Rule

- Let s, s1 and s2 be (possibly empty) sequences of binding operators
- If the nested store
- <F,s1,(<G,s>,i),s2>
- is an interpretation for an expression of category S, then so is
- <[email protected]zi.F,s1,s,s2>

Underspecified Representations

Keller Retrieval

<F,s1,(<G,s>,i),s2>

<[email protected]zi.F,s1,s,s2>

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Keller Retrieval

- Any operators stored whilst processing G become accessible only after G has been retrieved, i.e.
- Nesting overcomes the problem of generating readings with free variables.

Underspecified Representations

Example of a Nested Store

Mia knows every owner of a hash bar

<KNOW(MIA,z2),

(<u.Ay(OWNER(y)&OF(y,z1)=>[email protected]),

(<w.Ex(HASHBAR(x) & [email protected])>,1)>,2)>

There is only one reading

Underspecified Representations

Keller Retrieval

<F,(<G,s>,2)> => <[email protected]z2.F,s>

<KNOW(MIA,z2),

(<u.Ay(OWNER(y)&OF(y,z1)=>[email protected]),

(<w.Ex(HASHBAR(x) & [email protected])

>,1)

>,2)>

=>

Underspecified Representations

Keller Retrieval

<u.Ay(OWNER(y)&OF(y,z1)=>[email protected])@

z2.KNOW(MIA,z2),

(<w.Ex(HASHBAR(x) & [email protected])>,1)>

<Ay(OWNER(y)&OF(y,z1)=>KNOW(MIA,y),

(<w.Ex(HASHBAR(x) & [email protected])>,1)>

(<w.Ex(HASHBAR(x) & [email protected])@

z1.Ay(OWNER(y)&OF(y,z1)=>KNOW(MIA,y)>,

Underspecified Representations

(<w.Ex(HASHBAR(x) & [email protected])@

z1.Ay(OWNER(y)&OF(y,z1)=>KNOW(MIA,y)>,

<Ex(HASHBAR(x) & Ay(OWNER(y)&OF(y,x)=>KNOW(MIA,y)>

Underspecified Representations

Parse Tree with Logical Forms

Every owner of a hash bar (NP)

(<u.Ay(OWNER(y)&OF(y,z1) => [email protected]),2)>

Owner of a hash bar (VP)

z.(OWNER(z)&Ex(HASHBAR(x)&OF(z,x)))>

Every (DET)

<w.u.Ay([email protected] => [email protected])>

owner (N)

<x.OWNER(x)>

of a hash bar (PP)

<u.z.

([email protected]&Ex(HASHBAR(x)&OF(z,x)))>

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Hole Semantics

- Storage methods are useful but have their limitations
- Expressivity:
- allows all possible readings to be expressed, but not some subset
One criminal knows every owner of a hash bar.

- 5 readings, but suppose we want only the subset where every owner outscopes hash bar?

- allows all possible readings to be expressed, but not some subset
- Oriented to Quantifier scope ambiguities and not other constructs.
- Interaction between negation and quantification
- every boxer doesn't love a woman

Underspecified Representations

Hole Semantics

- Neither Cooper nor Keller storage can represent all the ambiguities.
- A special mechanism is necessary to handle negation.
- But we would like to have a uniform mechanism for handling all scope ambiguities and not a special mechanism for each construct.
- The quest for a more abstract kind of under-specified representation is the rationale behind Hole Semantics

Underspecified Representations

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