CSA4050: Advanced Topics in NLP

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CSA4050Advanced Topics in NLP

• Semantics III
• Quantified Sentences

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Outline

• Language
• Sentences
• Determiners
• Noun Phrases
• Syntactic Structure

• Logic
• Generalised Quantifiers
• Higher order functions
• Translation into Prolog

Syntax-Semantics Interface
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Determiners and Quantifiers in Language and Logic
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Syntactic Shape vs.Semantic Shape

• John walkssemanticswalk(suzie).
• Every man talkssemanticsall(X, man(X) ?
  talk(X))

• S
• NP VP
• Suzie walks
• S
• NP VP
• Det N talks
• Every man

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Problem

• Similar syntactic shape
• Dissimilar semantic shape
• How is this possible if the syntax drives the
  combination of semantic fragments as per
  rule-to-rule hypothesis?
• Answer be creative about logical forms and
  semantic combination rules

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Montague Solution

• Reorganising the semantic combination rules
  operating between VP and NP in rules such ass(S)
  --gt np(NP), vp(VP).
• We will be considering NP(VP) versus
  VP(NP).
• NPs as higher order functions
• Analyse LF of quantified sentences

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LF of Quantified Sentences

• LF of quantified sentences has a general shape
  involving
• a restrictor predicate R
• a scope predicate S
• R restricts the set of things we are talking
  about
• S says something further about set element(s)
• a logical quantifier Q
• a bound variable V
• a logical operator O connecting R and S

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Examples

• All lecturers are lazy
• ?x lecturer(x) ? lazy(x)
• Restrictor lecturers
• Scope lazy
• Quantifier All
• Operator implies
• Bound Variable x

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Examples

• There is a lazy lecturer
• ?x lecturer(x) lazy(x)
• Restrictor lecturers
• Scope lazy
• Quantifier exist
• Operator and
• Bound Variable x

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Anatomy of Quantified Sentences
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Generalized Quantifiers

• We adopt the following generalized quantifier
  representation for LF in which quantifier is a
  3-place predicateQ(ltvariablegt,ltrestrictorgt,ltscop
  egt)
• Operator is omitted.
• Examplesall(X,man(X),walk(X))exist(X,man(X),walk
  (X))the(X,man(X),climbed(X,everest))most(X,lectu
  rer(X),poor(X))

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NP as higher order function
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Encoding in Prolog

• The VP remains as before, ieXwalks(X)
• The quantified NP every man will be of the
  formQall(X,man(X),Q)
• The semantic rule for S now ensures that the NP
  function is applied to the VP function.s(S)--gt
  np(NP),vp(VP), reduce(NP,VP,S)

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DCG with QuantificationProgram 1

• grammar
• s(S) --gt np(NP), vp(VP), reduce(NP,VP,S)
• vp(VP) --gt v(V).
• lexicon
• v(Xwalk(X)) --gt walks.
• np(Qall(X,man(X),Q)) --gt every,man.

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Result

• ?- s(X,every,man,walks,).
• X all(_G397, man(_G397), _G405walk(_G405))
• all(x, man(x), ywalk(y))
• What is wrong with this?
• How can we fix it?
• We need to force the variables to be identical
  using reduce

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Result

• ?- s(X,every,man,walks,).
• X all(_G397, man(_G397), _G405walk(_G405))
• all(x, man(x), ywalk(y))
• What is wrong with this?
• The variables _G397 and _G405 are distinct. They
  should be identical.
• The consequent of the implication is a ?
  expression
• How can we fix it?
• We need to force the variables to be identical
  using reduce

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DCG with QuantificationProgram 2

• grammar
• s(S) --gt np(NP), vp(VP), reduce(NP,VP,S)
• vp(VP) --gt v(V).
• lexicon
• v(Xwalk(X)) --gt walks.
• np(Qall(X,man(X),P)) --gt every,man,
• reduce(Q,X,P).

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Result

• ?- s(X,every,man,walks,).
• X all(_G397, man(_G397),walk(_G397))
• The effect of the reduce clause is
• to identify the appropriate variables
• to remove the ? variable

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Handling Quantified NPs

• Before we cheated by having every man as a
  lexical item.np(Qall(X,man(X),P))--gt
  every,man, reduce(Q,X,P).
• Now we see what is involved in analysing the NP
  from its parts.
• Step 1 is to write a new syntactic rulenp(NP)
  --gt d(D), n(N).
• How does the semantics work?

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LF of determiners

• Key idea is determiner has LF of a 2-argument
  function corresponding to R and S which become
  bound during processing.
• ?R.?S.Q(V,R,S)
• where Q is associated with the particular
  determiner
• When we apply this function to the adjacent noun,
  we obtain the LF of the NP.

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How NP is created
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Fitting the Semantics Together

• Handle the quantified NPnp(NP) --gt d(D), n(N),
  reduce(D,N,NP).
• Add lexical entry for every
• d(RLSLall(X,R,S)) --gtevery,
  reduce(RL,X,R), reduce(SL,X,S) .

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DCG with QuantificationProgram 3

• grammar
• s(S) --gt np(NP), vp(VP), reduce(NP,VP,S).
• np(NP) --gt d(D), n(N), reduce(D,N,NP) .
• vp(VP) --gt v(VP).
• lexicon
• v(Xwalk(X)) --gt walks.
• n(Xman(X)) --gt man.
• d(RLSLall(X,R,S) --gt every,
  reduce(RL,X,R),
  reduce(SL,X,S) .

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Trace

• gt (7) s(_G510, every, man, walks, )
• gt (8) np(_L183, every, man, walks, _L184)
• gt (9) d(_L205, every, man, walks, _L206)
• lt (9) d((XR) (XS)all(X, R, S), every, man,
  walks, man, walks)
• gt (9) n(_L207, man, walks, _L208)
• lt (9) n(Zman(Z), man, walks, walks)
• gt (9) reduce((XR) (XS)all(X, R, S),
  Zman(Z), _L183)
• lt (9) reduce((Xman(X)) (XS)all(X, man(X),
  S), Xman(X), (XS)all(X, man(X), S))
• lt (8) np((XS)all(X, man(X), S), every, man,
  walks, walks)
• gt (8) vp(_L185, walks, _L186)
• gt (9) v(_L185, walks, _L186)
• lt (9) v(Ywalk(Y), walks, )
• lt (8) vp(Ywalk(Y), walks, )
• gt (8) reduce((XS)all(X, man(X), S), Ywalk(Y),
  _G510)
• lt (8) reduce((Xwalk(X))all(X, man(X),
  walk(X)), Xwalk(X), all(X, man(X), walk(X)))
• lt (7) s(all(X, man(X), walk(X)), every, man,
  walks, )

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Summary

• Quantification crops up a lot in NL
• To handle quantified sentences, there is a
  mismatch in shape between syntax and semantics
• Need to reorganise the semantic rules (NP applies
  to VP not vice versa).
• Representation of quantifier as a higher-order
  function.