1. April 2010Underspecified Representations1

2. April 2010Underspecified Representations2 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

3. April 2010Underspecified Representations3 4 Approaches Do nothing Montague’s original method Robin Cooper’s stores Keller Storage Hole semantics

4. April 2010Underspecified Representations4 Do Nothing Is it really such a problem? Given 1.Ax(BOXER(X) => Ey(WOMAN(Y) & LOVE(X,Y)) 2.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?

5. April 2010Underspecified Representations5 The Problem Every owner of a hash bar gives every criminal a big kahuna burger There are 18 readings 1.Ax((Ey(HBAR(y) & OF(x,y)) & OWNER(x)) => Az(CRIM(x) => Eu(BKB(u) & GIVE(x,z,u)))) 2.Ax(CRIM(x) => Ay((Ex (HBAR(z) & OF(y,z)) & OWNER(y)) => Eu(BKB(u) & GIVE(y,x,u)))) 3.[..] 18.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

6. April 2010Underspecified Representations6 Groups of Readings {13,14,15,16} {12} {15} {18} {10.11} {6,7} {5} {8,9} {4} {3} {1,2} NB arrows represent logical implication

7. April 2010Underspecified Representations7 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

8. April 2010Underspecified Representations8 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.

9. April 2010Underspecified Representations9 Every boxer loves her-3 (S) Ax(BOXER(x) => LOVE(x,z3) Every boxer (NP) u.Ax(BOXER(x) => u@x) loves her-3 (VP) y.LOVE(y,z3) loves (TV) v. y.(v@ x.LOVE(y,x)) her-3 NP w.(w@z3) a woman Parse Tree with Logical Forms

10. April 2010Underspecified Representations10 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@z3) [vincent] = w.(w@vincent)

11. April 2010Underspecified Representations11 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

12. April 2010Underspecified Representations12 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,z3) previous tree

13. April 2010Underspecified Representations13 Getting the Semantics to Work (1) u.Ey(WOMAN(y)& u@y) @ 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

14. April 2010Underspecified Representations14 Getting the Semantics to Work (2) u.Ey(WOMAN(y)& u@y) @ 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]

15. April 2010Underspecified Representations15 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.

16. April 2010Underspecified Representations16 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

17. April 2010Underspecified Representations17 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. –

18. April 2010Underspecified Representations18 Using Cooper Storage If is a semantic representation for an NP, then the store 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.

19. April 2010Underspecified Representations19 Using Cooper Storage When we encounter a quantified NP, we can either pass on or else we can pass on 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.

20. April 2010Underspecified Representations20 Every boxer loves a woman (S) <LOVE(z6,z7), ( u.Ax(BOXER(x)=>u@x),6), ( u.Ey(WOMAN(y)& u@y),7)> Every boxer (NP) < w.(w@z6), ( u.Ax(BOXER(x) => u@x,6)> loves a woman (VP) < u.LOVE(u,z7), ( u.Ey(WOMAN(Y)&u@y),7)> loves (TV) a woman NP < w.(w@z7), ( u.Ey(WOMAN(y)& u@y),7)> Parse Tree with Logical Forms

21. April 2010Underspecified Representations21 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 But now we have the choice of using

22. April 2010Underspecified Representations22 Combining Stores If a functor node is associated with and an argument node is associated with The the store associated with the result of applying the first to the second is: It may be possible to do beta reduction on F@G

23. April 2010Underspecified Representations23 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.

24. April 2010Underspecified Representations24 Cooper Retrieval Rule Let s1 and s2 be (possibly empty) sequences of binding operators. If the store is associated with an expression of category S, then the store is also associated with this expression

25. April 2010Underspecified Representations25 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

26. April 2010Underspecified Representations26 < KNOW(MIA,z2), ( u.Ay(OWNER(y) & OF(y,z1) => 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

27. April 2010Underspecified Representations27 Taking the Universal first … < KNOW(MIA,z2), ( u.Ay(OWNER(y) & OF(y,z1) => u@y), 2), ( w.Ex(HASHBAR(x) & w@x),1) > u@y)@ z2. KNOW(MIA,z2), ( w.Ex(HASHBAR(x) & w@x),1) >

28. April 2010Underspecified Representations28 < KNOW(MIA,z2), ( u.Ay(OWNER(y) & OF(y,z1) => u@y), 2), ( w.Ex(HASHBAR(x) & w@x),1) > KNOW(MIA,y), ( w.Ex(HASHBAR(x) & w@x),1) >

29. April 2010Underspecified Representations29 ….. It works KNOW(MIA,y), ( w.Ex(HASHBAR(x) & w@x),1) > < w.Ex(HASHBAR(x) & w@x) @ 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)

30. April 2010Underspecified Representations30 Taking the Existential first … < KNOW(MIA,z2), ( u.Ay(OWNER(y) & OF(y,z1) => u@y), 2), ( w.Ex(HASHBAR(x) & w@x),1) > < w.Ex(HASHBAR(x) & w@x)@ z1. KNOW(MIA,z2), ( u.Ay(OWNER(y) & OF(y,z1) => u@y), 2),>

31. April 2010Underspecified Representations31 Taking the Existential first … < w.Ex(HASHBAR(x) & KNOW(MIA,z2)), ( u.Ay(OWNER(y) & OF(y,z1) => u@y), 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

32. April 2010Underspecified Representations32 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.

33. April 2010Underspecified Representations33 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.

34. April 2010Underspecified Representations34 Keller Storage Rule If the nested store s an interpretation for an NP, then the nested store,i)> for some unique index i, is also an interpretation of that NP

35. April 2010Underspecified Representations35 Every owner of a hash bar (NP) < u.u@z2), ( u@y), (,1)>,2)> Every (DET) u@y)> Owner of a hash bar (VP) < u.OWNER(u)&OF(u,z1)), (,1)> owner (N) of a hash bar (PP) < v. u.(v@u&OF(u,z1)), (,1)> Parse Tree with Logical Forms

36. April 2010Underspecified Representations36 Keller Retrieval Rule Let s, s1 and s2 be (possibly empty) sequences of binding operators If the nested store,i),s2> is an interpretation for an expression of category S, then so is

37. April 2010Underspecified Representations37 Keller Retrieval,i),s2>

38. April 2010Underspecified Representations38 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.

39. April 2010Underspecified Representations39 Example of a Nested Store Mia knows every owner of a hash bar <KNOW(MIA,z2), ( u@y), (,1)>,2)> There is only one reading

40. April 2010Underspecified Representations40 Keller Retrieval,2)> => <KNOW(MIA,z2), ( u@y), (< w.Ex(HASHBAR(x) & w@x) >,1) >,2)> =>

41. April 2010Underspecified Representations41 Keller Retrieval u@y)@ z2.KNOW(MIA,z2), (,1)> KNOW(MIA,y), (,1)> (< w.Ex(HASHBAR(x) & w@x)@ z1.Ay(OWNER(y)&OF(y,z1)=>KNOW(MIA,y)>,

42. April 2010Underspecified Representations42 (< w.Ex(HASHBAR(x) & w@x)@ z1.Ay(OWNER(y)&OF(y,z1)=>KNOW(MIA,y)>, KNOW(MIA,y)>

43. April 2010Underspecified Representations43 Every owner of a hash bar (NP) < u.u@z2), ( u@y),2)> Every (DET) u@y)> Owner of a hash bar (VP) z.(OWNER(z)&Ex(HASHBAR(x)&OF(z,x)))> owner (N) of a hash bar (PP) < u. z. (u@z&Ex(HASHBAR(x)&OF(z,x)))> Parse Tree with Logical Forms

44. April 2010Underspecified Representations44 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

45. April 2010Underspecified Representations45 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