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Working with Discourse Representation Theory Patrick Blackburn

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Now that we know how to build DRSs for English sentences, what do we do with them? ... Oops --- does it take that long to build one from scratch? Oh dear. ... – PowerPoint PPT presentation

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Title: Working with Discourse Representation Theory Patrick Blackburn


1
Working with Discourse Representation
TheoryPatrick Blackburn Johan Bos Lecture
3DRT and Inference
2
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.

3
Overview
  • Inference tasks
  • Why FOL?
  • From model theory to proof theory
  • Inference engines
  • From DRT to FOL
  • Adding world knowledge
  • Doing it locally

4
The inference tasks
  • The consistency checking task
  • The informativity checking task

5
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

6
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.

7
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

8
Soundness
  • If provable then valid If -- ?
    then ?
  • Soundness is a no garbage condition

9
Completeness
  • If valid then provable If ?
    then -- ?
  • Completeness means that proof theory has captured
    model theory

10
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

11
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!

12
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

13
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

14
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

15
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.

16
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.

17
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.

18
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.

19
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.

20
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.

21
Yin and Yang of Inference
  • Theorem Proving and Model Building function as
    opposite forces

22
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

23
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

24
Why is this relevant to natural language?
  • Testing a discourse for consistency

25
Why is this relevant to natural language?
  • Testing a discourse for consistency

26
Why is this relevant to natural language?
  • Testing a discourse for consistency

27
Why is this relevant to natural language?
  • Testing a discourse for consistency

28
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

29
Why is this relevant to natural language?
  • Testing a discourse for informativity

30
Why is this relevant to natural language?
  • Testing a discourse for informativity

31
Why is this relevant to natural language?
  • Testing a discourse for informativity

32
Why is this relevant to natural language?
  • Testing a discourse for informativity

33
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

34
Googling theorem provers for DRT
35
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.

36
Googling theorem provers for FOL
37
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

38
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

39
Translating DRT to FOL DRSs
(
)fo ?x1 ?xn((C1)fo(Cn)fo)
40
Translating DRT to FOL Conditions
(R(x1xn))fo R(x1xn) (x1x2)fo
x1x2 (?B)fo ?(B)fo (B1?B2)fo (B1)fo ?
(B2)fo
41
Translating DRT to FOLImplicative DRS-conditions
(
?B)fo ?x1?xm(((C1)fo(Cn)fo)?(B)fo
)
42
Two example translations
  • Example 1
  • Example 2

43
Example 1
44
Example 1
)fo
(
45
Example 1
?x( (man(x))fo (walk(x))fo )
46
Example 1
?x(man(x) (walk(x))fo )
47
Example 1
?x(man(x) walk(x))
48
Example 2
49
Example 2
)fo
(
50
Example 2
(woman(y))fo (
)fo
)
?y (
?
51
Example 2
woman(y) (
)fo
)
?y (
?
52
Example 2
)
?y (woman(y) ?x((man(x))fo? (
)
)fo
53
Example 2
)
?y (woman(y) ?x(man(x) ? (
)
)fo
54
Example 2
?y (woman(y) ?x(man(x) ? ?e (
(adore(e))fo (agent(e,x))fo (theme(e,y))fo )))
55
Example 2
?y (woman(y) ?x(man(x) ? ?e (adore(e)
(agent(e,x))fo (theme(e,y))fo )))
56
Example 2
?y (woman(y) ?x(man(x) ? ?e (adore(e)
agent(e,x) (theme(e,y))fo )))
57
Example 2
?y (woman(y) ?x(man(x) ? ?e (adore(e)
agent(e,x) theme(e,y))))
58
Basic setup
  • DRS

59
Basic setup
  • DRS
  • FOL ?x?y(vincent(x) mia(y) love(x,y))

60
Basic setup
  • DRS
  • FOL ?x?y(vincent(x) mia(y) love(x,y))
  • Model D d1
    F(vincent)d1 F(mia)d1
    F(love)(d1,d1)

61
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 ?
    ?

62
Basic setup
  • DRS

63
Basic setup
  • DRS
  • FOL ?x?y(vincent(x) mia(y) love(x,y))

64
Basic setup
  • 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))

65
Basic setup
  • 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)

66
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

67
Local informativity
68
Local informativity
?
69
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

70
Local consistency
71
Local consistency
?
72
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.

73
Tomorrow
  • Presupposition and Anaphora in DRT
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