Computational Semantics http://www.coli.uni-sb.de/cl/projects/milca/esslli

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Computational Semanticshttp//www.coli.uni-sb.de/
cl/projects/milca/esslli

• Aljoscha Burchardt,
• Alexander Koller,
• Stephan Walter,
• Universität des Saarlandes,
• Saarbrücken, Germany
• ESSLLI 2004, Nancy, France

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Computational Semantics

• How can we compute the meaning of e.g. an English
  sentence?
• What do we mean by meaning?
• What format should the result have?
• What can we do with the result?

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The Big Picture

• Sentence John smokes.
• Syntactic Analyses S
• NP VP
• John smokes
• Semantics Construction smoke(j)
• Inference ?x.smoke(x)?snore(x),smoke(j)
• gt snore(j)

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Course Schedule

• Monday - Thursday Semantics Construction
• Mon.Tue. Lambda-Calculus
• Wed.Thu. Underspecification
• Friday Inference
• (Semantic) Tableaux

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The Book

• If you want to read more about computational
  semantics, see the forthcoming book Blackburn
  Bos, Representation and Inference A first
  course in computational semantics. CSLI Press.

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Today (Monday)

• Meaning Representation in FOL
• Basic Semantics Construction
• ?-Calculus
• Semantics Construction with Prolog

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Meaning Representations

• Meaning representations of NL sentences
• First Order Logic (FOL) as formal language
• John smokes.
• gt smoke(j)
• Sylvester loves Tweety.
• gt love(s,t)
• Sylvester loves every bird.
• gt ?x.(bird(x) ? love(s,x))

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In the Background Model Theory

• ?x.(bird(x) ? love(s,x)) is a string again!
• Mathematically precise model representation,
  e.g. cat(s), bird(t), love(s,t), granny(g),
  own(g,s), own(g,t)
• Inspect formula w.r.t. to the model Is it true?
• Inferences can extract information Is anyone not
  owned by Granny?

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FOL Syntax (very briefly)

• FOL Formulae, e.g. ?x.(bird(x) ? love(s,x))
• FOL Language
• Vocabulary (constant symbols and
  predicate/relation symbols)
• Variables
• Logical Connectives
• Quantifiers
• Brackets, dots.

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What we have done so far

• Meaning Representation in FOL ?
• Basic Semantics Construction
• ?-Calculus
• Semantics Construction with Prolog

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Syntactic Analyses

• Basis Context Free Grammar (CFG)
• Grammar Rules
• S ? NP VP
• VP ? TV NP
• TV ? love
• NP ? john Lexical Rules / Lexicon
• NP ? mary
• ...

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Example Syntactic Analyses
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Example Semantic Lexicon
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Example Semantics Construction
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Example Semantics Construction
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Compositionality

• The meaning of the sentence is constructed from
• The meaning of the words john, mary, love(?,?)
  (lexicon)
• Paralleling the syntactic construction (semantic
  rules)

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Systematicity

• How do we know that e.g. the meaning of the VP
  loves Mary is constructed as
• love(?,mary) and not as
• love(mary,?) ?
• Better How can we specify in which way the bits
  and pieces combine?

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Systematicity (ctd.)

• Parts of formulae (and terms), e.g. for the VP
  love Mary?
• love(?,mary) bad not FOL
• love(x,mary) bad no control over free
  variable
• Familiar well-formed formulae (sentences)
• ?x.love(x,mary) Everyone loves Mary.
• ?x.love(mary,x) Mary loves someone.

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Using Lambdas (Abstraction)

• Add a new operator to bind free variables
• ?x.love(x,mary) to love Mary
• The new meta-logical symbol ? marks missing
  information in the object language (?-)FOL
• We abstract over x.
• How do we combine these new formulae and terms?

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Super Glue

• Glueing together formulae/terms with a special
  symbol _at_
• ?x.love(x,mary) john
• ?x.love(x,mary)_at_john
• Often written as ?x.love(x,mary)(john)
• How do we get back to the familiar
  love(john,mary)?

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Functional Application

• Glueing is known as Functional Application
• FA has the Form Functor_at_Argument
• ?x.love(x,mary)_at_john
• FA triggers a very simple operation
• Replace the ?-bound variable by the argument.
• ?x.love(x,mary)_at_john
• gt love(john,mary)

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?-Reduction/Conversion

• Strip off the ?-prefix,
• Remove the argument (and the _at_),
• Replace all occurences of the ?-bound variable by
  the argument.
• ?x.love(x,mary)_at_john
• love(x,mary)_at_john
• love(x,mary)
• love(john,mary)

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Semantics Construction with Lambdas

• S John loves Mary
• (?y?x.love(x,y)_at_mary)_at_john

NP John john
VP loves Mary ?y?x.love(x,y)_at_mary
TV loves ?y?x.love(x,y)
NP Mary mary
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Example Beta-Reduction

• (?y?x.love(x,y)_at_mary)_at_john
• ?gt (?x.love(x,mary))_at_john
• ?gt love(john,mary)

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In the Background

• ?-Calculus
• A logical standard technique offering more than
  ?-abstraction, functional _at_pplication and
  ß-reduction.
• Other Logics
• Higher Order Logics
• Intensional Logics
• ...
• For linguistics Richard Montague (early
  seventies)

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What we have done so far

• Meaning Representation in FOL ?
• Basic Semantics Construction ?
• ?-Calculus ?
• Semantics Construction with Prolog

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Plan

• Next, we
• Introduce a Prolog represenation.
• Specify a syntax fragment with DCG.
• Add semantic information to the DCG.
• (Implement ß-reduction.)

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Prolog Representation Terms and Formulae

• (man(john)((love(mary,john))))
• (happy(john)gt(happy(mary)v(happy(fido)))
• forall(x,happy(x))
• exists(y,(man(y)((happy(y))))
• lambda(x,)

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Prolog RepresentationOperator Definitions
Binding Strength

• - op(950,yfx,_at_).  application
• - op(900,yfx,gt).  implication
• - op(850,yfx,v).    disjunction
• - op(800,yfx,).    conjunction
• - op(750, fy,).     negation
• forall(x,man(x)love(x,mary)gthate(mary,x))
• (Mary hates every man that doesnt love her.)

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Definite Clause Grammar

• Prologs built-in grammar formalism
• Example grammar
• s --gt np,vp.
• vp --gt iv.
• vp --gt tv,np.
• ...
• np --gt john.
• iv --gt smokes.
• Call s(john,smokes,).

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Adding Semantics to DCG

• Adding an argument to each DCG rule to collect
  semantic information.
• Phrase rules of our first semantic DCG
• s(VP_at_NP) --gt np(NP),vp(VP).
• vp(IV) --gt iv(IV).
• vp(TV_at_NP) --gt tv(TV),np(NP).

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Lexicon Of Our First Semantic DCG

• np(john) --gt john.
• np(mary) --gt mary.
• iv(lambda(X,smoke(X))) --gt
• smokes,vars2atoms(X).
• tv(lambda(X,lambda(Y,love(Y,X)))) --gt
• loves,vars2atoms(X),
• vars2atoms(Y).

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Running Our First Semantics Constrution

• ?- s(Sem,mary,smokes,).
• Sem lambda(v1, smoke(v1))_at_mary
• ?- , betaConvert(Sem,Result).
• Result smoke(mary)
• Note that we use some special predicates of
  freely available SWI Prolog (http//www.swi-prolog
  .org/).

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betaConvert(Formula,Result) 1/2

• betaConvert(Functor_at_Arg,Result)-betaConvert(Funct
  or,lambda(X,Formula)),!,substitute(Arg,X,Formula
  ,Substituted),betaConvert(Substituted,Result).
• The input expression is of the form Functor_at_Arg.
• The functor has (recursively) been reduced to
  lambda(X,Formula).
• Note that the code displayed in the reader is
  wrong. Corrected pages can be downloaded.

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betaConvert(Formula,Result) 2/2

• betaConvert(Formula,Result)-
• compose(Formula,Functor,Formulas),
• betaConvertList(Formulas,Converted), compose(Re
  sult,Functor,Converted).
• Formula exists(x,man(x)(lambda(z),walk(z)_at_x))
• Functor exists
• Formulas x,man(x)(lambda(z),walk(z)_at_x))
• Converted x,man(x)walk(x)
• Result exists(x,man(x)walk(x))

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Helper Predicates

• betaConvertList(,).
• betaConvertList(FR,F_ResR_Res)-        bet
  aConvert(F,F_Res),        betaConvertList(R,R_Res
  ).
• compose(Term,Symbol,Args)-
• Term .. SymbolArgs.
• substitute() (Too much for a slide.)

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Wrapping It Up

• go -
• resetVars,
• readLine(Sentence),
• s(Formula,Sentence,),
• nl, print(Formula),
• betaConvert(Formula,Converted),
• nl, print(Converted).

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Adding More Complex NPs

• NP A man gt ?x.man(x)
• S A man loves Mary
• Lets try it in a system demo!

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Tomorrow

• S A man loves Mary
• gt love(?x.man(x),mary)
• How to fix this.
• A DCG for a less trivial fragment of English.
• Real lexicon.
• Nice system architecture.