Working with Discourse Representation Theory Patrick Blackburn

1
Working with Discourse Representation
TheoryPatrick Blackburn Johan Bos Lecture
2Building Discourse Representation Structures
2
Recap from yesterday

• Discourse representation theory DRT
• Discourse representation structure DRS
• Discourse referent
• DRS conditions
• Accessibility
• Subordination

3
More about DRS

• DRS can be viewed as a firstorder language
  without explicit quantifiers

4
More about DRS

• DRS can be viewed as a firstorder language
  without explicit quantifiers
• ?x man(x) smoke(x)

5
More about DRS

• DRS can be viewed as a firstorder language
  without explicit quantifiers

?
6
More about DRS

• DRS can be viewed as a firstorder language
  without explicit quantifiers

?

• ?x man(x) ? smoke(x)

7
More about discourse referents

• All noun phrases NPs introduce discourse
  referents
• Indefinite NPs a book
• Definite NPs the book
• Proper name Harry
• Pronoun she

8
More about discourse referents

• Verbs introduce event discourse referents
• Intransitive verbs to sleep
• Transitive verbs to read

9
Accessibility 1
?
?
?
X
10
Accessibility 1
O
-
?
-
-
?
?
X
11
Accessibility 2
?
X
?
?
12
Accessibility 2
O
O
?
X
-
-
?
-
?
13
Accessibility 3
?
?
X
?
14
Accessibility 3
O
-
?
-
O
?
X
-
?
15
Accessibility 4
?
?
X
?
16
Accessibility 4
O
-
?
-
O
?
X
-
?
17
Accessibility 5
?
?
?
X
18
Accessibility 5
O
-
?
-
O
?
O
?
X
19
Accessibility 6
?
?
?
X
20
Accessibility 6
O
-
?
-
O
?
O
O
?
X
21
Subordination
?
?
?
22
Subordination
A
?
B
C
?
D
?
E
F
23
Subordination
A subordinates B A subordinates C A subordinates
D D subordinates E E subordinates F
A
?
B
C
?
D
?
E
F
24
Subordination
A subordinates B A subordinates C A subordinates
D D subordinates E E subordinates F A
subordinates E A subordinates F .. Etc.
A
?
B
C
?
D
?
E
F
25
DRT and negation

• DRT predicts differences between the following
  DRSs wrt to the interpretation of the pronoun she
• Vincent did not dance with the woman.She was
  pretty.
• Vincent did not dance with Mia.She was pretty.
• Vincent did not dance with a woman.X She was
  pretty.

26
Negation and indefinites

• Vincent did not dance with a woman.She

?
27
Negation and definites

• Vincent did not dance with the woman.She

?
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Negation and proper names

• Vincent did not dance with Mia.She

?
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More about accessibility

• DRT predicts differences between the following
  DRSs wrt to the interpretation of the pronoun she
• Vincent danced with some woman.She was pretty.
• Vincent danced with every woman.X She was
  pretty.
• Vincent danced with no woman.X She was pretty.

30
More about accessibility

• Vincent did with some woman. She

31
More about accessibility

• Vincent did with every woman. She

?
32
More about accessibility

• Vincent did with no woman. She

?
33
Today

• We know now what DRT is, and we know what
  semantic representation is central to DRT
• But how can we construct DRSs for English
  discourses in a systematic and automatic way?
• There are various ways to do this we will
  explore the lambda-based method

34
Composing meaning

• Freges principleThe meaning of a compound
  expression is a function of the meaning of its
  parts.

35
Composing DRSs roughly

• Mia does not have a car

• Mia

• does not

• have

• a car

?
36
Composing DRSs roughly

• Mia does not have a car

?

• Mia

• does not

• have

• a car

?
37
What we need to do

• We need a mechanism to combine two smaller DRSs
  into one larger DRS
• Introduce Merge operator
• Merge reduction
• We need a mechanism to keep track of the naming
  of discourse referents
• Introduce lambda operator and application
• Beta conversion

38
What we also need

• In addition, we need something that tells us how
  and which DRSs combine
• In other words, we need syntactic structure
• In this course, we will look at two formalisms of
  syntactic theory
• Phrase Structure Grammar
• Categorial Grammar

39
Outline

• Theory
• DRS-Merging
• The lambda calculus as a glue language for
  constructing DRSs
• Practice
• A simple fragment without events
• A simple fragment with events
• Implementation example

40
The Merge

• We will introduce a new operator
• The indicates a merge between two DRSs

(
)
41
The Merge

• We will introduce a new operator
• The indicates a merge between two DRSs
• The merge is used to combine two DRSs into one
  larger DRS
• If B1 and B2 are DRSs, then so is (B1B2)

(
)
42
A merge example

• A boxer lost.
• He died.

43
A merge example

• A boxer lost.
• He died.
• A boxer lost.He died.

)
(
44
Merge and accessibility

• If (B1B2) is a DRS, then
• B1 subordinates B2
• I.e., discourse referents introduced in B1 are
  accessible from B2

)
(
45
Merge and variable binding

• Which variables are bound, and which are free?

)

)
((
46
Merge and variable binding

• Which variables are bound, and which are free?

)

)
((
? free
? free
? free
47
Merge is associative

• These two DRSs do not differ in meaning

((

))
(
((

)

)
48
Merge is non-commutative

• These two DRSs differ in meaning

(
)

(
)
49
Merge Reduction

• Given a DRS with a merge, we can reduce it to a
  DRS without a merge
• This is called merge reduction
• Merge reduction is performed by taking the union
  of the universes and conditions
• Merge reduction is subject to certain conditions

50
Merge Reduction Example

(
)
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Merge Reduction Example
Mergereduction -----

(
)
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Merge Reduction Example
Mergereduction -----

(
)
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Merge Reduction Problem

• Consider the exampleA woman walks. A man talks.

(

)
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Merge Reduction Problem

• Consider the exampleA woman walks. A man talks.

Mergereduction -----
(

)
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Merge Reduction Problem

• Consider the exampleA woman walks. A man talks.

Mergereduction -----
(

)
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Constraints on merge reduction

• Given a DRS (B1B2), merge reduction can only be
  applied if
• None of the discourse referents in B2 occur as
  free variables in any of the conditions of B1

57
Constraints on merge reduction

• Given a DRS (B1B2), merge reduction can only be
  applied if
• None of the discourse referents in B2 occur as
  free variables in any of the conditions of B1
• If this criterion is not met, we can do two
  things
• Do not apply merge reduction to B1B2
• Rename B2 alpha-conversion, we will come back
  to this later

58
Today

• Theory
• DRS-Merging
• The lambda calculus as a glue language for
  constructing DRSs
• Practice
• A simple fragment
• A fragment with events
• Implementation

59
DRSs with lambdas

• We will use the lambda-calculus as a tool to
  build DRSs for sentences
• We will use ? to mark missing information in the
  DRS
• We will use _at_ to denote function application
• We call this combination ?-DRT
• Muskens
• Kuschert, Kohlhase, Pinkal

60
The ?-operator

• We will use ? to bind variables
• View variables bound by ? as placeholders for
  missing semantic information
• Examples

?u.(
u_at_x)
?x.
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The _at_ operator

• We use the _at_ operator to combine lambda-DRSs
• The expression F_at_A tells us that we want to
  substitute the argument A in the placeholders of
  function F
• This is called functional application

62
Beta-Conversion

• Performing this substitution is called
  beta-conversion
• How does this work?

?x.
_at_z
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Beta-Conversion

• Performing this substitution is called
  beta-conversion
• How does this work?
• Remove ?-prefix from functor

?x.
_at_z
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Beta-Conversion

• Performing this substitution is called
  beta-conversion
• How does this work?
• Remove ?-prefix from functor
• Substitute the argument for all bound occurrences
  of the ?

?x.
_at_z
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Beta-Conversion

• Performing this substitution is called
  beta-conversion
• How does this work?
• Remove ?-prefix from functor
• Substitute the argument for all bound occurrences
  of the ?

66
Another example

• This is functional application
• What is the functor?
• What is the argument?

?u.(
_at_
?z.
u_at_x)
?( u_at_y)
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Another example

• The functor lambda-binds u
• How many substitutions do we make?

?u.(
_at_
?z.
u_at_x)
?( u_at_y)
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Another example

• The functor lambda-binds u
• How many substitutions do we make?

?u.(
_at_
?z.
u_at_x)
?( u_at_y)
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Another example

• This is the result after substitution
• Are we ready with beta-conversion?

?z.
(
_at_x)
?z.
_at_y)
?(
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Another example

• Carrying out further substitutions
• Anything left to do?

?z.
(
_at_x)


?(
)
71
Another example

• Carrying out further substitutions
• Perhaps we can perform further reductions?

(
)


?(
)
72
Another example

• Carrying out further substitutions
• Perhaps we can perform further reductions?

(
)
?
73
Another example

• And here is the final DRS
• Btw, does this DRS make sense?

?
74
Alpha-Conversion

• Beta-conversion is not always safe
• Accidental bindings can occur when the functor
  binds a variable that occurs free in the argument
• Example

?y.
_at_x)
(
75
Alpha-Conversion

• Beta-conversion is not always safe
• Accidental bindings can occur when the functor
  binds a variable that occurs free in the argument
• Example

)
(
76
Alpha-Conversion

• Before beta-conversion, we perform
  alpha-conversion on the functor
• Alpha-conversion replaces bound variables for new
  occurrences
• Example

?y.
_at_x)
(
77
Alpha-Conversion

• Before beta-conversion, we perform
  alpha-conversion on the functor
• Alpha-conversion replaces bound variables for new
  occurrences
• Example

?u.
_at_x)
(
78
Alpha-Conversion

• Before beta-conversion, we perform
  alpha-conversion on the functor
• Alpha-conversion replaces bound variables for new
  occurrences
• Example

)
(
79
Today

• Theory
• DRS-Merging
• The lambda calculus as a glue language for
  constructing DRSs
• Practice
• A simple fragment of English
• A fragment with events
• Implementation

80
The Lexicon

• Nouns boxer, man, restaurant

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The Lexicon

• Nouns boxer, man, restaurant
• Proper names Mia, Vincent

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The Lexicon

• Nouns boxer, man, restaurant
• Proper names Mia, Vincent
• Determiners a, every, the

83
The Lexicon

• Nouns boxer, man, restaurant
• Proper names Mia, Vincent
• Determiners a, every, the
• Intransitive verbs walks, dances

84
The Lexicon

• Nouns boxer, man, restaurant
• Proper names Mia, Vincent
• Determiners a, every, the
• Intransitive verbs walks, dances
• Transitive verbs loves, admires

85
The Lexicon

• Nouns boxer, man, restaurant
• Proper names Mia, Vincent
• Determiners a, every, the
• Intransitive verbs walks, dances
• Transitive verbs loves, admires
• Adjectives big, small

86
The Lexicon

• Nouns boxer, man, restaurant
• Proper names Mia, Vincent
• Determiners a, every, the
• Intransitive verbs walks, dances
• Transitive verbs loves, admires
• Adjectives big, small
• Adverbs slowly, quickly

87
The lexicon nouns

• boxer
• restaurant

?x.
?u.
88
The lexicon proper names

• Mia
• Vincent

?p.(
p_at_x)
u_at_x)
?u.(
89
The lexicon intransitive verbs

• dances
• smokes

?x.
?y.
90
The lexicon determiners

• a
• every

?p.?q.((
p_at_x)q_at_x)
?p.?q.
p_at_x) ?q_at_x
(
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The lexicon adjectives

• big
• red

?u.?x.(
u_at_x)
u_at_x)
?u.?x.(
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Syntactic Structure

• We now know what the partial DRSs in the lexicon
  look like
• But we dont know how to put things together, at
  least not the order
• We need syntax for this

93
Example
Every
man
dances
?
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Partial DRSs
Every
man
dances
?y.
?z.
p_at_x) ?q_at_x
?p.?q. (
95
Phrase Structure Grammar

• Grammar rules s ? np vp np ? det n np ? pn
  vp ? iv vp ? tv np
• Lexical rules det ? a det ? every n ? man n
  ? car

96
Adding semantics

• Grammar rule s ? np vp
• Grammar rule with semantics sX_at_Y ? npX vpY

97
Adding semantics

• Grammar rules sF_at_A ? npF vpA npF_at_A ? detF
  nA npX ? pnX vpX ? ivX vpF_at_A ? tvF
  npA
• Lexical rules detX ? aX detX ? everyX
  nX ? manX nX ? carX

98
Example derivation
S
VP
NP
IV
N
DET
man
Every
dances
99
Example derivation
S
VP
NP
IV
?y.
DET
N
p_at_x) ?q_at_x
?z.
?p.?q (
Every
man
dances
100
Example derivation
S
_at_?y.
p_at_x) ?q_at_x
?p.?q. (
VP
NP
Application NP?DET N
N
IV
DET
?z.
man
dances
Every
101
Example derivation
S
?q. (
_at_x) ?q_at_x
?y.
VP
NP
?-conversion
N
IV
DET
?z.
man
dances
Every
102
Example derivation
S

) ?q_at_x
?q. (
VP
NP
?-conversion
N
IV
DET
?z.
man
dances
Every
103
Example derivation
S
?q.
?q_at_x
VP
NP
-reduction
N
IV
DET
?z.
man
dances
Every
104
Example derivation
S
?q.
?q_at_x
VP
?z.
NP
No operation required VP?IV
N
IV
DET
man
dances
Every
105
Example derivation
_at_?z.
S
?q.
?q_at_x
Application S?NP VP
VP
NP
N
IV
DET
man
dances
Every
106
Example derivation
S
? ?z. _at_x
?-conversion
VP
NP
N
IV
DET
man
dances
Every
107
Example derivation
S
?
VP
NP
?-conversion
N
IV
DET
man
dances
Every
108
Lexical Semantics trans. verbs

• admires

?y. ?x.
109
The lexicon trans. verbs

• admires
• admires

?y. ?x.
?u.?x.u_at_?y.
110
Today

• Theory
• DRS-Merging
• The lambda calculus as a glue language for
  constructing DRSs
• Practice
• A simple fragment
• A fragment with events
• Implementation

111
Events in DRT

• Recap of Davidsonian event analysis
• How do we introduce events systematically?
• Some more grammar rules
• Lexical entries of verbs, prepositional phrases,
  and adverbs
• Example derivation

112
First attempt
S
VP
NP
IV
?y.
DET
N
p_at_x) ?q_at_x
?z.
?p.?q (
Every
man
dances
113
First attempt
S
?
VP
NP
IV
?y.
DET
N
p_at_x) ?q_at_x
?z.
?p.?q (
Every
man
dances
114
Example with adverb
S
VP
NP
VP
IV
ADV
Every
man
dances
quickly
115
First attempt
?
S
VP
NP
?q.
VP
?q_at_x
?i.
IV
ADV
?z.
Every
man
dances
quickly
116
Second attempt
S
VP
NP
?v.?z.?m.v_at_z_at_?i. m_at_i
?q.
VP
?q_at_x
IV
ADV
?z.?m. m_at_e
Every
man
dances
quickly
117
Second attempt
S
?z.?m. m_at_e
VP
NP
?q.
VP
?q_at_x
IV
ADV
Every
man
dances
quickly
118
Second attempt
?
?
S
?m. m_at_e
VP
NP
VP
IV
ADV
Every
man
dances
quickly
119
Third attempt
S
VP
NP
?v.?n.?m.v_at_n_at_?i. m_at_i
?q.
VP
?q_at_x
IV
ADV
?n.?m.n_at_ ?z. m_at_e
Every
man
dances
quickly
120
Third attempt
S
?n.?m.n_at_?z. m_at_e
VP
NP
?q.
VP
?q_at_x
IV
ADV
Every
man
dances
quickly
121
Third attempt
m_at_e
S
?m.
?
VP
NP
VP
IV
ADV
Every
man
dances
quickly
122
Event

• We will always end up with a lambda
• Introduce a rule T ? S
• This rule elimates the last lambda

123
Third attempt
T
?m.
_at_ ?i.
?
S
m_at_e
VP
NP
VP
IV
ADV
Every
man
dances
quickly
124
Third attempt
T
?
S
VP
NP
VP
IV
ADV
Every
man
dances
quickly
125
Today

• Theory
• DRS-Merging
• The lambda calculus as a glue language for
  constructing DRSs
• Practice
• A simple fragment
• A fragment with events
• Implementation

126
Whats next?

• Tomorrow
• DRSs, what do we do with them?
• DRT and inference
• Thursday
• Resolving anaphora in DRT
• Presupposition