Discourse Representation Theory: An overview

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Discourse Representation TheoryAn overview

• Part I The linguistic problem

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Outline

• A meaning representation language
• Compositionality
• Lambda Calculus
• Moving to Discourse
• Shortcomings with FOL

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• 1. A meaning representation language

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• A semantic analysis of NL should answer at least
  the following questions
• What does a given sentence mean?
• How is the meaning representation of a sentence
  built?
• How do we infer some piece of information out of
  another?
• How is the meaning representation of a discourse
  built?
• This boils down to have meaning representation of
  the linguistic utterances.

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• Why First Order Logic?
• Apparently all natural languages show a predicate
  argument structure
• Is there anything shared by these two sentences?
• 1. John loves Mary
• 2. Mary loves John
• They have a not so overlapping linear order, but
  they seem strictly related all the same.
• A philosophical argument
• Assume a speaker S has a meaning
  representation for both proper names would it be
  possible for S to understand 1. and not 2.?
• If so, how does it come that a speaker
  could grasp the meaning of infinite sentences (he
  never heard before) out of finite resources (his
  lexicon)?
• Loves(john, mary)
• Loves(mary, john)
• Loves(x,y)

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• What does a given sentence mean?
• A theoretical assumption
• The meaning of a sentence is its truth value
• (better, its truth conditions, i.e. the state of
  affairs of the world which make the sentence
  true)
• FOL supports a reliable concept of satisfaction
  and truth on models
• if we can translate a NL sentence S in a FOL
  formula F (that is called the proposition
  corresponding to S), then we have description of
  its meaning in terms of truth in a model MltD,Igt.
• John loves Mary ? loves(john,mary)
• M F iff ltI(john), I(mary)gt? I(loves)
• A student loves Mary ? ?x(student(x) ?
  loves(x,Mary))
• M F iff for any variable assignment g, M,g
  ?x(student(x) ? loves(x,Mary))
• iff for some x-variant g of g M,g student(x) ?
  loves(x,Mary)

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• We can draw inferences from the meaning of the
  sentences we understand
• We are told that
• Rudys is a vegetarian restaurant
• And that
• vegetarian restaurants dont serve meat
• In FOL we have a reliable way to conclude
• Rudys doesnt serve meat using an inference
  rule instantiated by
• VegRest(rudys)
• ?x(VegRest(x) ? Serve(x,meat))
• ____________________________
• Serve(rudys,meat)

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• So, does FOL capture the what, the content of
  Natural Language semantics?
• Its quite controversial.
• Adverbs semantics
• Milly swims slowly.
• MODIFICATION OF VERBS ? HIGHER ORDER PREDICATION
• Adjective semantics
• There is a red apple ? ?x(apple(x) ?
  red(x))
• INTERSECTION OF CLASSES
• There is a small elephant in the zoo ?
  ?x(elephant(x) ? small(x) ? in_the_zoo(x)) NO!
• John is a skillful violinist ? skillful(john)
  ? violinist(john) NO!
• NON-INTERSECTIVE ADJECTIVES
• Indexical expressions semantics

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• Even though FOL could capture the what of
  meaning, in itself it doesnt say anything about
  how to build FOL formulas starting from
  sentences, and how to do it in a systematic way
• Take for example
• John loves Mary ? loves(john,mary)
• It seems that
• John contributes the constant john
• Mary contributes the constant mary
• loves contribute the binary relation loves(x,y)
• So, sentence meaning flows from lexicon, from
  words.
• But how it is precisely built?
• Why couldnt we derive loves(mary,john) as well?
• The missing point is syntactic structure!

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Here there is a hierarchy, not just a linear
order of words, and it constrains the way we can
fill the argument slots in the meaning
representation of the relation LOVE(x,y). But
notice that there is not specific mechanism
defined on meaning representations that allows
this unification.

• So, we need a syntax-driven semantics for Natural
  Language, such that
• the lexical items in a sentence give the basic
  ingredients for meaning representations
• syntactic structure tells us how the semantic
  contributions of the parts are to be joined
  together.

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• 2. Compositionality

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• Principle of Compositionality
• The meaning of a complex expression is a
  function of the meanings of its parts and of the
  syntactic rules by which they are combined.
• In order to instantiate it we need to define
• Semantic primitives for lexical entries
• An independent syntactic theory contributing a
  notion of parts and rules of combination ?
• well assume a suitable CFG and parser
• a description of the function or mapping from
  syntax to semantics.

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• 5. Lambda Calculus

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• We extend FOL with a variable binding operator ?.
• Basic expressions of the kind
• ?xP(x)
• where x is a formal parameter variable and P(x)
  is a FOL expression containing x.
• Semantics Occurrences of variables buond by ?
  are place-holders for missing information. We
  abstract over the bound variables to mark the
  slots for substitution.
• ?-reduction (also called ?-conversion) is a basic
  mechanism that applies ?-expressions to terms
  yielding new expressions with all occurrences of
  formal parameters bound to the specified terms
• ?x.P(x)(a) ? P(a)
• ?x.left(x)(john) ? left(john)
• Semantics Substitute the argument term for all
  the occurrences of the bound variables in the
  functor

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• Lexical representations
• PN John ? j
• IV walks ? ?x.walks(x)
• Det a ? ? X.?Y.?z.X(z)?Y(z)
• TV loves ? ? x.?y.loves(x,y)
• We have now suitable representations to augment
  CFG with semantic attachments such that, for each
  CF rule, we have a function F to build the
  meaning of the symbols on the left out of the
  meanings of the symbols on the right
• A ? a1,,an S(A)
  F(S(a1),,S(an))
• NP ? pn S(NP) S(pn)
• NP ? det NP S(NP) S(det)S(NP)
• e.g. a man (? X.?Y.?z.X(z)?Y(z))(?x.man(x)) ?
  ?Y.?z.Man(z)?Y(z)

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John loves Mary (S) loves(j,m)
John (NP) j
loves Mary (VP) ?x.loves(x,m)
Mary(NP) m
John (PN) j
loves (TV) ?y.?x.loves(x,y)
Mary (PN) m

• We needed to know
• exactly which variables in the TVs meaning
  representation have to be replaced by the
  semantics of the TVs arguments ? we did
  it changing the semantic attachment of TV in a
  ?-expression which make variables externally
  available for binding
• how to replace them ? ?-conversion

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• 6. Moving to Discourse

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• Up to now, we addressed the meaning of linguistic
  unities up to sentences
• A discourse def a set of collocated, related
  of sentences
• NOTICE Well focus only on semantics of
  Monologue (Dialog might imply a much more
  complex interaction...)
• John went to Bills car dealership to check out
  an Acura Integra. He looked at it for about an
  hour.
• The semantics of the second sentence is not
  independent on the one of the first
• We have to look back to the semantics of
  preceding sentences to assign a meaning to the
  two pronouns
• We find expressions back in the discourse which
  have been made contextually relevant

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• We are focusing on a subset of discourse
  semantics, i.e. co-reference
• Some terminology
• Reference def The process by which expressions
  (noun phrases, proper names, pronouns, etc., also
  called referring expressions) denote, i.e.
  point to an entity (e.g. a person), called the
  referent.
• Notice we are assuming a suitable theory of
  reference and will be interested in modeling the
  phaenomenon of
• co-reference in discourse
• Anaphora def reference to an entity (called
  an antecedent) which has been previously
  introduced into the discourse. E.g.
• Mia is a woman. She loves Vincent.

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FUNCTIONAL REQUIREMENTS FOR CO-REFENCE
John went to Bills car dealership to check out
an Acura Integra. He looked at it for about an
hour.

1. A method for building a Discourse Model that
   evolves with the dynamically-changing discourse
   it represents
2. A method for mapping between referring expression
   in discourse and entities in the Discourse Model

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• 7. Shortcomings with FOL

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• Succeding co-reference
• Mia is a woman. She loves Vincent.
• A woman snorts. She collapses.
• Failing coreference
• 3. Every woman snorts. ?She collapses.
• 4. Mia didnt order a two dollar sandwich.
  ?Vincent tasted it.

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• 1. Mia is a woman. She loves Vincent.
• FOL Representations
• First attempt WOMAN(Mia) ? LOVE(x, Vincent)
• Post-processing WOMAN(Mia) ? LOVE(Mia, Vincent)

WHATS THE RULE?
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• 2. A woman snorts. She collapses.
• FOL representations
• ?z(WOMAN(z) ? SNORT(z)) ? COLLAPSE(X).
  
• ?z(WOMAN(z) ? SNORT(z) ? COLLAPSE(z)).

AND WHATS THE RULE HERE?
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• 3. Every woman snorts. She collapses.
• FOL Representations
• ?z(WOMAN(z) ? SNORT(z)) ? COLLAPSE(x)
• ?z((WOMAN(z) ? SNORT(z)) ? COLLAPSE(z))

Why doesnt it work?
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It seems that we should move to representations
which

• keep the truth-conditional dimension of meaning
  typical of FOL ? we want to be able to evaluate
  discourses on standard models as we did for
  sentences
• resolve anaphora in a systematic way
• mirror in some way the process of discourse
  understanding with the functional requirements we
  described