Underspecified Representations April 2010 Underspecified Representations 1 The

Underspecified Representations April 2010 Underspecified Representations 1

The Issue • Every boxer loves a woman 1. Ax(BOXER(X) => Ey(WOMAN(Y) & LOVE(X, Y)) 2. 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 April 2010 Underspecified Representations 2

4 Approaches • • • Do nothing Montague’s original method Robin Cooper’s stores Keller Storage Hole semantics April 2010 Underspecified Representations 3

Do Nothing • • Is it really such a problem? Given 1. 2. 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? April 2010 Underspecified Representations 4

The Problem • • Every owner of a hash bar gives every criminal a big kahuna burger There are 18 readings 1. 2. 3. 18. • • • 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)))) 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 April 2010 Underspecified Representations 5

Groups of Readings {8, 9} {13, 14, 15, 16} {12} {15} {10. 11} {6, 7} {5} April 2010 {4} {18} {3} {1, 2} NB arrows represent logical implication Underspecified Representations 6

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 April 2010 Underspecified Representations 7

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. April 2010 Underspecified Representations 8

Parse Tree with Logical Forms Every boxer loves her-3 (S) Ax(BOXER(x) => LOVE(x, z 3) Every boxer (NP) u. Ax(BOXER(x) => u@x) loves her-3 (VP) y. LOVE(y, z 3) loves (TV) v. y. (v@ x. LOVE(y, x)) April 2010 Underspecified Representations her-3 NP w. (w@z 3) a woman 9

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. (w@z 3) • [vincent] = w. (w@vincent) April 2010 Underspecified Representations 10

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 April 2010 Underspecified Representations 11

Extending the Tree Every boxer loves a woman (S) a woman (NP) u. Ey(WOMAN(y)& u@y) Every boxer loves her-3 (S, 3) Ax(BOXER(x) => LOVE(x, z 3) previous tree April 2010 Underspecified Representations 12

Getting the Semantics to Work (1) u. Ey(WOMAN(y)& u@y) @ Ax(BOXER(x) => LOVE(x, z 3)) Ey(WOMAN(y)& Ax(BOXER(x) => LOVE(x, z 3)) @y) [stop] • The problem is that if we apply a woman to every boxer loves her 3 directly, no further reduction is possible. • We need to perform lambda abstraction over every boxer loves her 3, i. e. from – Ax(BOXER(x) => LOVE(x, z 3)) to – z 3. Ax(BOXER(x) => LOVE(x, z 3)) to April 2010 Underspecified Representations 13

Getting the Semantics to Work (2) u. Ey(WOMAN(y)& u@y) @ z 3. Ax(BOXER(x) => LOVE(x, z 3)) Ey(WOMAN(y)& z 3. Ax(BOXER(x) => LOVE(x, z 3)) @y) Ey(WOMAN(y)& Ax(BOXER(x) => LOVE(x, y))) [stop - success] April 2010 Underspecified Representations 14

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. April 2010 Underspecified Representations 15

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 April 2010 Underspecified Representations 16

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)> April 2010 Underspecified Representations 17

Using Cooper Storage • If <F, (B, j), . . . , (B’, k)> is a semantic representation for an NP, then the store < u. (u@zi), (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. (u@zi) • When we encounter an NP, we are faced with a choice. April 2010 Underspecified Representations 18

Using Cooper Storage • When we encounter a quantified NP, we can either pass on <F, . . other pairs. . > • or else we can pass on < u. (u@zi), (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. April 2010 Underspecified Representations 19

Parse Tree with Logical Forms Every boxer loves a woman (S) <LOVE(z 6, z 7), ( u. Ax(BOXER(x)=>u@x), 6), ( u. Ey(WOMAN(y)& u@y), 7)> Every boxer (NP) < w. (w@z 6), ( u. Ax(BOXER(x) => u@x, 6)> loves (TV) < z. u. (z@ v. LOVE(u, v))> April 2010 loves a woman (VP) < u. LOVE(u, z 7), ( u. Ey(WOMAN(Y)&u@y), 7)> a woman NP < w. (w@z 7), ( u. Ey(WOMAN(y)& u@y), 7)> Underspecified Representations 20

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) & u@y. • In the storage era this would be < u. Ey(WOMAN(y) & u@y> • But now we have the choice of using < w. (w@z 7), ( u. Ey(WOMAN(y) & u@y, 7)> April 2010 Underspecified Representations 21

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: <F@G, (B, j), . . . , (B, k) , (C, m), . . . , (C, n)> • It may be possible to do beta reduction on F@G April 2010 Underspecified Representations 22

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. April 2010 Underspecified Representations 23

Cooper Retrieval Rule • Let s 1 and s 2 be (possibly empty) sequences of binding operators. • If the store <F, s 1, (B, i), s 2> is associated with an expression of category S, then the store <B@ zi. F, s 1, s 2> is also associated with this expression April 2010 Underspecified Representations 24

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 April 2010 Underspecified Representations 25

< KNOW(MIA, z 2), ( u. Ay(OWNER(y) & OF(y, z 1) => u@y), 2), ( w. Ex(HASHBAR(x) & w@x), 1) > • Now we have a choice as to which item in the store to use • Suppose we choose to take the Universal quantifier first April 2010 Underspecified Representations 26

Taking the Universal first … < KNOW(MIA, z 2), ( u. Ay(OWNER(y) & OF(y, z 1) => u@y), 2), ( w. Ex(HASHBAR(x) & w@x), 1) > < u. Ay(OWNER(y) & OF(y, z 1) => u@y)@ z 2. KNOW(MIA, z 2), ( w. Ex(HASHBAR(x) & w@x), 1) > April 2010 Underspecified Representations 27

< KNOW(MIA, z 2), ( u. Ay(OWNER(y) & OF(y, z 1) => u@y), 2), ( w. Ex(HASHBAR(x) & w@x), 1) > <Ay(OWNER(y) & OF(y, z 1) => KNOW(MIA, y), ( w. Ex(HASHBAR(x) & w@x), 1) > April 2010 Underspecified Representations 28

…. . It works <Ay(OWNER(y) & OF(y, z 1) => KNOW(MIA, y), ( w. Ex(HASHBAR(x) & w@x), 1) > < w. Ex(HASHBAR(x) & w@x) @ z 1. Ay(OWNER(y) & OF(y, z 1) => KNOW(MIA, y) Ex(HASHBAR(x) & z 1…. . OF(y, z 1) … @ x Ex(HASHBAR(x) & Ay(OWNER(y) & OF(y, x) => KNOW(MIA, y) April 2010 Underspecified Representations 29

Taking the Existential first … < KNOW(MIA, z 2), ( u. Ay(OWNER(y) & OF(y, z 1) => u@y), 2), ( w. Ex(HASHBAR(x) & w@x), 1) > < w. Ex(HASHBAR(x) & w@x)@ z 1. KNOW(MIA, z 2), ( u. Ay(OWNER(y) & OF(y, z 1) => u@y), 2), > April 2010 Underspecified Representations 30

Taking the Existential first … < w. Ex(HASHBAR(x) & KNOW(MIA, z 2)), ( u. Ay(OWNER(y) & OF(y, z 1) => u@y), 2), > […] Ay(OWNER(y) & OF(Y, z 1) => Ex(HASHBAR(X) & KNOW(MIA, y))) • This is not what we wanted • The result is a formula with a free variable April 2010 Underspecified Representations 31

What went wrong • The Cooper storage mechanism ignores the hierarchical structure of the NP • a hash bar contributes the free varable z 1, but z 1 has been moved out of the core representation and is put in the store. • Hence lambda abstracting the core representation wrt z 1 is not guaranteed to take into account z 1’s contribution – which is made indirecty through the stored universal quantifier every owner. • Everything is ok if we restore UQ first since that restores z 1 to the core representation. April 2010 Underspecified Representations 32

What went wrong • However, if we choose to retrieve the existential quantifier first 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. April 2010 Underspecified Representations 33

Keller Storage Rule • If the nested store <F, s> • s an interpretation for an NP, then the nested store < u. (u@zi), (<F, s>, i)> for some unique index i, is also an interpretation of that NP April 2010 Underspecified Representations 34

Parse Tree with Logical Forms Every owner of a hash bar (NP) < u. u@z 2), (< u. Ay(OWNER(y)&OF(y, z 1) => u@y), (< w. Ex(HASHBAR(x) & w@x)>, 1)>, 2)> Every (DET) < w. u. Ay(w@y => u@y)> owner (N) < x. OWNER(x)> April 2010 Owner of a hash bar (VP) < u. OWNER(u)&OF(u, z 1)), (< w. Ex(HASHBAR(x)&w@x)>, 1)> of a hash bar (PP) < v. u. (v@u&OF(u, z 1)), (< w. Ex(HASHBAR(x)&w@x)>, 1)> Underspecified Representations 35

Keller Retrieval Rule • Let s, s 1 and s 2 be (possibly empty) sequences of binding operators • If the nested store • <F, s 1, (<G, s>, i), s 2> • is an interpretation for an expression of category S, then so is • <G@ zi. F, s 1, s, s 2> April 2010 Underspecified Representations 36

Keller Retrieval <F, s 1, (<G, s>, i), s 2> <G@ zi. F, s 1, s, s 2> April 2010 Underspecified Representations 37

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. April 2010 Underspecified Representations 38

Example of a Nested Store Mia knows every owner of a hash bar <KNOW(MIA, z 2), (< u. Ay(OWNER(y)&OF(y, z 1)=>u@y), (< w. Ex(HASHBAR(x) & w@x)>, 1)>, 2)> There is only one reading April 2010 Underspecified Representations 39

Keller Retrieval <F, (<G, s>, 2)> => <G@ z 2. F, s> <KNOW(MIA, z 2), (< u. Ay(OWNER(y)&OF(y, z 1)=>u@y), (< w. Ex(HASHBAR(x) & w@x) >, 1) >, 2)> => April 2010 Underspecified Representations 40

Keller Retrieval < u. Ay(OWNER(y)&OF(y, z 1)=>u@y)@ z 2. KNOW(MIA, z 2), (< w. Ex(HASHBAR(x) & w@x)>, 1)> <Ay(OWNER(y)&OF(y, z 1)=>KNOW(MIA, y), (< w. Ex(HASHBAR(x) & w@x)>, 1)> (< w. Ex(HASHBAR(x) & w@x)@ z 1. Ay(OWNER(y)&OF(y, z 1)=>KNOW(MIA, y)>, April 2010 Underspecified Representations 41

(< w. Ex(HASHBAR(x) & w@x)@ z 1. Ay(OWNER(y)&OF(y, z 1)=>KNOW(MIA, y)>, <Ex(HASHBAR(x) & Ay(OWNER(y)&OF(y, x)=>KNOW(MIA, y)> April 2010 Underspecified Representations 42

Parse Tree with Logical Forms Every owner of a hash bar (NP) < u. u@z 2), (< u. Ay(OWNER(y)&OF(y, z 1) => u@y), 2)> Every (DET) < w. u. Ay(w@y => u@y)> owner (N) < x. OWNER(x)> April 2010 Owner of a hash bar (VP) z. (OWNER(z)&Ex(HASHBAR(x)&OF(z, x)))> of a hash bar (PP) < u. z. (u@z&Ex(HASHBAR(x)&OF(z, x)))> Underspecified Representations 43

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? • Oriented to Quantifier scope ambiguities and not other constructs. – Interaction between negation and quantification – every boxer doesn't love a woman April 2010 Underspecified Representations 44

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 underspecified representation is the rationale behind Hole Semantics April 2010 Underspecified Representations 45