Working with Discourse Representation Theory Patrick Blackburn

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Working with Discourse Representation
TheoryPatrick Blackburn Johan Bos Lecture
3DRT and Inference
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This lecture

• Now that we know how to build DRSs for English
  sentences, what do we do with them?
• Well, we can use DRSs to draw inferences.
• In this lecture we show how to do that, both in
  theory and in practice.

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Overview

• Inference tasks
• Why FOL?
• From model theory to proof theory
• Inference engines
• From DRT to FOL
• Adding world knowledge
• Doing it locally

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The inference tasks

• The consistency checking task
• The informativity checking task

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Why First-Order Logic?

• Why not use higher-order logic?
• Better match with formal semantics
• But Undecidable/no fast provers available
• Why not use weaker logics?
• Modal/description logics (decidable fragments)
• But Cant cope with all of natural language
• Why use first-order logic?
• Undecidable, but good inference tools available
• DRS translation to first-order logic
• Easy to add world knowledge

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Axioms encode world knowledge

• We can write down axioms about the information
  that we find fundamental
• For example, lexical knowledge, world knowledge,
  information about the structure of time, events,
  etc.
• By the Deduction Theorem ?1 ?n ?
  iff ?1 ?n ? ?
• That is, inference reduces to validity of
  formulas.

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From model theory to proof theory

• The inference tasks were defined semantically
• For computational purposes, we need symbolic
  definitions
• We need to move from the concept of to
  --
• In other words, from validity to provability

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Soundness

• If provable then valid If -- ?
  then ?
• Soundness is a no garbage condition

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Completeness

• If valid then provable If ?
  then -- ?
• Completeness means that proof theory has captured
  model theory

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Decidability

• A problem is decidable, if a computer is
  guaranteed to halt in finite time on any input
  and give you a correct answer
• A problem that is not decidable, is undecidable

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First-order logic is undecidable

• What does this mean? It is not possible, to
  write a program that is guaranteed to halt when
  given any first-order formula and correctly tell
  you whether or not that formula is valid.
• Sounds pretty bad!

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Good news

• FOL is semi-decidable
• What does that mean?
• If in fact a formula is valid, it is always
  possible, to symbolically verify this fact in
  finite time
• That is, things are only going wrong for FOL when
  it is asked to tackle something that is not valid
• On some non-valid input, any algorithm is bound
  not to terminate

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Put differently

• Half the task, namely determining validity, is
  fairly reasonable.
• The other half of the task, showing non-validity,
  or equivalenty, satisfiability, is harder.
• This duality is reflected in the fact that there
  are two fundamental computational inference tools
  for FOL
• theorem provers
• and model builders

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Theorem provers

• Basic thing they do is show that a formula is
  provable/valid.
• There are many efficient off-the-shelf provers
  available for FOL
• Theorem proving technology is now nearly 40 years
  old and extremely sophisticated
• Examples Vampire, Spass, Bliksem, Otter

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Theorem provers and informativity

• Given a formula ?, a theorem prover will try to
  prove ?, that is, to show that it is
  valid/uninformative
• If ? is valid/uninformative, in theory, the
  theorem prover will always succeedSo theorem
  provers are a negative test for informativity
• If the formula ? is not valid/uninformative, all
  bets are off.

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Theorem provers and consistency

• Given a formula ?, a theorem prover will try to
  prove ??, that is, to show that ? is inconsistent
• If ? is inconsistent, in theory, the theorem
  prover will always succeedSo theorem provers
  are also a negative test for consistency
• If the formula ? is not inconsistent, all bets
  are off.

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Model builders

• Basic thing that model builders do is try to
  generate a usually finite model for a formula.
  They do so by iteration over model size.
• Model building for FOL is a rather new field, and
  there are not many model builders available.
• It is also an intrinsically hard task harder
  than theorem proving.
• Examples Mace, Paradox, Sem.

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Model builders and consistency

• Given a formula ?, a model builder will try to
  build a model for ?, that is, to show that ? is
  consistent
• If ? is consistent, and satisfiable on a finite
  model, then, in theory, the model builder will
  always succeedSo model builders are a partial
  positive test for consistency
• If the formula ? is not consistent, or it is not
  satisfiable on a finite model, all bets are off.

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Finite model property

• A logic has the finite model property, if every
  satisfiable formula is satisfiable on a finite
  model.
• Many decidable logics have this property.
• But it is easy to see that FOL lacks this
  property.

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Model builders and informativity

• Given a formula ?, a model builder will try to
  build a model for ??, that is, to show that ? is
  informative
• If ?? is satisfiable on a finite model, then, in
  theory, the model builder will always succeedSo
  model builders are a partial positive test for
  informativity
• If the formula ?? is not satisfiable on a finite
  model all bets are off.

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Yin and Yang of Inference

• Theorem Proving and Model Building function as
  opposite forces

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Doing it in parallel

• We have general negative tests theorem provers,
  and partial positive tests model builders
• Why not try to get of both worlds, by running
  these tests in parallel?
• That is, given a formula we wish to test for
  informativity/consistency, we hand it to both a
  theorem prover and model builder at once
• When one succeeds, we halt the other

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Parallel Consistency Checking

• Suppose we want to test ? representing the
  latest sentence for consistency wrto the
  previous discourse
• Then
• If a theorem prover succeeds in finding a proof
  for PREV ? ??, then it is inconsistent
• If a model builder succeeds to construct a model
  for PREV ?, then it is consistent

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Why is this relevant to natural language?

• Testing a discourse for consistency

Discourse Theorem prover Model builder
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Why is this relevant to natural language?

• Testing a discourse for consistency

Discourse Theorem prover Model builder
Vincent is a man. ?? Model
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Why is this relevant to natural language?

• Testing a discourse for consistency

Discourse Theorem prover Model builder
Vincent is a man. ?? Model
Mia loves every man. ?? Model
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Why is this relevant to natural language?

• Testing a discourse for consistency

Discourse Theorem prover Model builder
Vincent is a man. ?? Model
Mia loves every man. ?? Model
Mia does not love Vincent. Proof ??
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Parallel informativity checking

• Suppose we want to test the formula ?
  representing the latest sentence for
  informativity wrto the previous discourse
• Then
• If a theorem prover succeeds in finding a proof
  for PREV ? ?, then it is not informative
• If a model builder succeeds to construct a model
  for PREV ??, then it is informative

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Why is this relevant to natural language?

• Testing a discourse for informativity

Discourse Theorem prover Model builder
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Why is this relevant to natural language?

• Testing a discourse for informativity

Discourse Theorem prover Model builder
Vincent is a man. ?? Model
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Why is this relevant to natural language?

• Testing a discourse for informativity

Discourse Theorem prover Model builder
Vincent is a man. ?? Model
Mia loves every man. ?? Model
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Why is this relevant to natural language?

• Testing a discourse for informativity

Discourse Theorem prover Model builder
Vincent is a man. ?? Model
Mia loves every man. ?? Model
Mia loves Vincent. Proof ??
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Lets apply this to DRT

• Pretty clear what we need to do
• Find efficient theorem provers for DRT
• Find efficient model builders for DRT
• Run them in parallel
• And Bobs your uncle!
• Recall that theorem provers are more established
  technology than model builders
• So lets start by finding an efficient theorem
  prover for DRT

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Googling theorem provers for DRT
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Theorem proving in DRT

• Oh no!Nothing there, efficient or otherwise.
• Lets build our own one.
• One phone call to Voronkov later
• Oops --- does it take that long to build one from
  scratch?
• Oh dear.

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Googling theorem provers for FOL
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Use FOL inference technology for DRT

• There are a lot FOL provers available and they
  are extremely efficient
• There are also some interesting freely available
  model builders for FOL
• We have said several times, that DRT is FOL in
  disguise, so lets get precise about this and put
  this observation to work

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From DRT to FOL

• Compile DRS into standard FOL syntax
• Use off-the-shelf inference engines for FOL
• Okay --- how do we do this?
• Translation function ()fo

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Translating DRT to FOL DRSs
x1xn
C1 . . . Cn
(
)fo ?x1 ?xn((C1)fo(Cn)fo)
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Translating DRT to FOL Conditions
(R(x1xn))fo R(x1xn) (x1x2)fo
x1x2 (?B)fo ?(B)fo (B1?B2)fo (B1)fo ?
(B2)fo
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Translating DRT to FOLImplicative DRS-conditions
x1xm
C1 . . . Cn
(
?B)fo ?x1?xm(((C1)fo(Cn)fo)?(B)fo
)
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Two example translations

• Example 1
• Example 2

x
man(x) walk(x)
y
woman(y) ?
e
adore(e) agent(e,x) theme(e,y)
x
man(x)
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Example 1
x
man(x) walk(x)
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Example 1
x
man(x) walk(x)
)fo
(
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Example 1
?x( (man(x))fo (walk(x))fo )
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Example 1
?x(man(x) (walk(x))fo )
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Example 1
?x(man(x) walk(x))
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Example 2
y
woman(y) ?
e
adore(e) agent(e,x) theme(e,y)
x
man(x)
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Example 2
y
woman(y) ?
e
adore(e) agent(e,x) theme(e,y)
x
man(x)
)fo
(
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Example 2
e
adore(e) agent(e,x) theme(e,y)
x
man(x)
(woman(y))fo (
)fo
)
?y (
?
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Example 2
e
adore(e) agent(e,x) theme(e,y)
x
man(x)
woman(y) (
)fo
)
?y (
?
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Example 2
e
adore(e) agent(e,x) theme(e,y)
)
?y (woman(y) ?x((man(x))fo? (
)
)fo
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Example 2
e
adore(e) agent(e,x) theme(e,y)
)
?y (woman(y) ?x(man(x) ? (
)
)fo
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Example 2
?y (woman(y) ?x(man(x) ? ?e (
(adore(e))fo (agent(e,x))fo (theme(e,y))fo )))
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Example 2
?y (woman(y) ?x(man(x) ? ?e (adore(e)
(agent(e,x))fo (theme(e,y))fo )))
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Example 2
?y (woman(y) ?x(man(x) ? ?e (adore(e)
agent(e,x) (theme(e,y))fo )))
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Example 2
?y (woman(y) ?x(man(x) ? ?e (adore(e)
agent(e,x) theme(e,y))))
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Basic setup
x y
vincent(x) mia(y) love(x,y)

• DRS

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Basic setup
x y
vincent(x) mia(y) love(x,y)

• DRS
• FOL ?x?y(vincent(x) mia(y) love(x,y))

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Basic setup
x y
vincent(x) mia(y) love(x,y)

• DRS
• FOL ?x?y(vincent(x) mia(y) love(x,y))
• Model D d1
  F(vincent)d1 F(mia)d1
  F(love)(d1,d1)

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Background Knowledge (BK)

• Need to incorporate BK
• Formulate BK in terms of first-order axioms
• Rather than just giving ? to the theorem prover
  (or model builder), we give it BK ? or BK ?
  ?

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Basic setup
x y
vincent(x) mia(y) love(x,y)

• DRS

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Basic setup
x y
vincent(x) mia(y) love(x,y)

• DRS
• FOL ?x?y(vincent(x) mia(y) love(x,y))

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Basic setup
x y
vincent(x) mia(y) love(x,y)

• DRS
• FOL ?x?y(vincent(x) mia(y) love(x,y))
• BK ?x (vincent(x) ? man(x)) ?x (mia(x) ?
  woman(x)) ?x (man(x) ? ? woman(x))

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Basic setup
x y
vincent(x) mia(y) love(x,y)

• DRS
• FOL ?x?y(vincent(x) mia(y) love(x,y))
• BK ?x (vincent(x) ? man(x)) ?x (mia(x) ?
  woman(x)) ?x (man(x) ? ? woman(x))
• Model D d1,d2
  F(vincent)d1 F(mia)d2
  F(love)(d1,d2)

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Local informativity

• Example
• Mia is the wife of Marsellus.
• If Mia is the wife of Marsellus, Vincent will be
  disappointed.
• The second sentence is informative with respect
  to the first. But

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Local informativity
x y
mia(x) marsellus(y) wife-of(x,y)
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Local informativity
x y z
mia(x) marsellus(y) wife-of(x,y) vincent(z)

wife-of(x,y)

disappointed(z)
?
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Local consistency

• Example
• Jules likes big kahuna burgers.
• If Jules does not like big kahuna burgers,
  Vincent will order a whopper.
• The second sentence is consistent with respect to
  the first. But

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Local consistency
x y
jules(x) big-kahuna-burgers(y) like(x,y)
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Local consistency
x y z
jules(x) big-kahuna-burgers(y) like(x,y) vincent(z)

?
u
order(z,u) whopper(u)

like(x,y)
?
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DRT and local inference

• Because DRS groups information into contexts, we
  now have natural means to check not only global,
  but also local consistency and informativity.
• Important for dealing with presupposition.
• Presupposition is not about strange logic. But
  about using classical logic in new ways.

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Tomorrow

• Presupposition and Anaphora in DRT