Computational Semantics http:www'coli'unisb'declprojectsmilcaesslli

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

• Day 5 Inference
• Aljoscha Burchardt,
• Alexander Koller,
• Stephan Walter,
• Universität des Saarlandes,
• Saarbrücken, Germany
• ESSLLI 2004, Nancy, France

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Where are we by now?
So far
John loves Mary.
Sentence
Linguistic Analysis
Why???
Why meaning?
Why logic?
Formula
love(john, mary)
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Motivations

• Why meaning?
• The big question in the background of semantics
  How do linguistic expressions relate to the
  world?
• The need for inference in a broad sense is
  omnipresent in linguistic processing Getting
  some piece of information out of another. This
  process is meaning based.
• Why logic?
• Using logic helps us in answering both problems
  at once.

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Meaning based linguistic Inferences
Peter loves Mary and she doesn't love him. No one
is happy if he isn't loved by the one he loves. ?
Peter is not happy

• Answering questions
• A "Is Peter happy" B
• Discourse
• There is my car. The roof is red.
• gt The roof of this particular car.
• Pragmatics
• A Shall we watch Athens?, B Oh, I hate
  Sports
• Answer is "no."
• ...

"No"
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Logical Inferences

• Argumentation Classical field gt Answering
  questions
• Every human is mortal, Socrates is a human
• gt Socrates is mortal.
• ?x.human(x) -gt mortal(x), human(soc)
  mortal(soc)
• Discourse, Pragmatics, ... Inference problems
  during processing
• logical relations between readings (equivalence,
  implication, contradiction)
• ?y?x.love(x,y) ? ?x?y.love(x,y)
• ?x?y.love(x,y) ? ?y?x.love(x,y)
• discourse maxims utterance consistent?
  informative?
• "lexical" inference "Brussels lowers taxes"
• presuppositions

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Next

• How do linguistic expressions relate to the
  world?
• Building logical representations is a step
  towards a scientific theory of this relation!
• They're a way of replacing something we don't
  understand by something we understand (at least
  better).
• Why? Because we have a formal way of saying what
  they mean Models.

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The big question of semantics
John loves Mary and Peter doesn't.
Semantic construction
love(john,mary) ??love(peter,mary)
???
"Understanding language"
Logics
man(john), man(peter), woman(mary),
love(john,mary)
???
Cognition / Ontology
???
?? ?
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Plan for Today

• What's the advantage of FOL-formulae?
• Interpretations and models
• Doing things with semantic representations
• Logical Inference and Proof Theory
• A calculus
• Automated Theorem Proving (first steps)
• An implementation of propositional tableaux
• A sample application

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FOL-semantics

• What does a FO-formula mean?
• It may be true or false (that's all)
• Whether it is true or false is calculated given a
  model.
• So A formula is true or false in a model.
• But what is a model?

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Models
A model can be thought of as a set of basic facts
that describe a part of the world. E.g., talking
about John, Mary, Peter, love, man and woman

• John loves Mary.
• John is a man.
• Mary doesn't love John.
• Peter is a man.
• Mary isn't a man.
• Mary is a woman.

• In this listing
• Who is there?
• Which properties do (or don't) they have?

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Formally

• This intuition is formalized as follows
• A model is an ordered pair of a set and a
  Function
• M (D, F)

The interpretation function Which properties do
these things have? (and more)
The domain What is there.
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Example model

• D John, Mary, Peter
• F (John, John),
• (Mary, Mary),
• (Peter, Peter)
• (man, John, Peter),
• (woman, Mary),
• (love, (John, Mary))

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Truth in a model

• g Assignment function, assigning values from D
  to variables

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Models as Sets of Formulae

• For our purposes, models are simply sets of
  literals (i.e. positive or negative atomic
  formulae).
• Set contains all literals that are true in the
  model.
• Our example
• man(john), man(peter), woman(mary),
  love(john,mary),?love(mary,john),
• Truth of atomic formulae without variables
• R(t1,,tn) ? M

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From theory to practice

• Models define the semantics of logical languages
• and are an interesting concept for relating
  language and the world.
• But they're also of practical importance
• They're the key to a formalization of inference.
• Now some further important logical notions.

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Inference and Entailment

• Valid inference Truth of premises guarantees
  truth of conclusion.
• Entailment Talking about all models.
• Concept directly captures syllogistic reasoning.

P, Q, R
For all M, g such that
we have
and
and

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Validity

• A related notion Truth of a formula in all
  models Validity
• A iff for all M,g
• Validity formalizes the notion of tautology,
  e.g.
• Sylvester is either a cat or not.
• cat(s) v ? cat(s)
• Relation to entailment via the deduction theorem
• A B iff A?B

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Where are we now?

• Why meaning? ?
• Why logic? ?
• Relation to the world Models ?
• Inferences Entailment and validity ?
• How to compute with these notions?

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How to work with all models?

• Entailment and validity are both defined with
  respect to all models.
• Problem There are infinitely many models.
• How can we work with these notions then?
• Idea Tell whether a formula is valid or not just
  by looking at it!
• The answer A calculus.

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Calculi

• Calculi are rule-based systems for manipulating
  formulae according to their structure.
• Some of the resulting configurations are called
  proofs.
• Formulas with proofs are called theorems.
• A good calculus produces a proof iff its input
  formula is valid.

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"Good" Calculi

• Good Calculi are
• Sound Only valid formulae get a proof.
• Complete All valid formulae get a proof.
• In other words All and only theorems are valid.
• -
• To achieve this, one has to give the right rules.
  Let's try

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Tableaux The intuition I

• Truth conditions tell us what would have to hold
  in a model for a given formula, e.g.
• A and B hold in all models for A ? B
• For A ? B, there are two kinds of models Those
  for A and those for B.
• If we go on decomposing a formulas that way, we
  end up with sets of literals
• ? models
• Example smoke(john) ? (? love(mary,john) ? ?
  love(john,mary))
• ? smoke(john), ?love(john, mary)
• ? smoke(john), ?love(mary, john)

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Tableaux The intuition II

• We know If a formula is valid, it's always true.
• I.e. No model makes it false.
• Making formulae false
• (smoke(john) ? walk(john))F
• ? ?smoke(john), ?walk(john)
• (smoke(john) ? ?smoke(john)) F
• ? smoke(john), ?smoke(john)

"sign"
?
?
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Tableaux

• If we want to know whether a formula is valid, we
  systematically try to find a model that would
  make it false
• hoping that we find none.
• That is, all attempts should lead to
  contradictions.
• Next A look at

(
)F
?((p?q) ?(?p??q))
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A simple fragment

• Next The rules for a tableaux calculus for
  predicate logic without variables and
  quantifiers.
• Actually propositional logic
• Advantage 1 Decidable
• Advantage 2 Rules are easy
• Disadvantage Boring and restricted
• More is possible but not here and now.

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Preprocessing

• Reduce the number of connectives by translating ?
  and ? to ? and ?.
• Use logical equivalences
• A ? B ? ? (?A ? ?B) De Morgan
• A ? B ? ? (A ? ?B)

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Tableaux Inference Rules
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Mary loves Bill or John loves Mary'' John
loves Mary ???
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Summing up

• Using predicate logic as representation language
  seemed to be a design decision on Monday.
• Now we're happy we did it
• Models tell us when sentences are true.
• Models give us a concept of logical inference.
• This concept can be mechanized by calculi.
• After the break Calculi can be implemented in
  provers. And provers are useful!

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More logics - Changing the language and/or the
semantics.

• Different phenomena, different logics
• Intensional logic (John seeks a unicorn)
• Temporal logics (tense)
• Dynamic logics (anaphora)
• Higher Order (quantifiers)
• Different tasks different tools
• Decidability and complexity
• From propositional over first order to higher
  order
• In between. E.g. Description logics.