1. Working with Discourse Representation Theory Patrick Blackburn & Johan Bos Lecture 2 Building Discourse Representation Structures

2. Recap from yesterday  Discourse representation theory [DRT]  Discourse representation structure [DRS]  Discourse referent  DRS conditions  Accessibility  Subordination x man(x) smoke(x)

3. More about DRS  DRS can be viewed as a first–order language without explicit quantifiers x man(x) smoke(x)

4. More about DRS  DRS can be viewed as a first–order language without explicit quantifiers   x [man(x) & smoke(x)] x man(x) smoke(x)

5. More about DRS  DRS can be viewed as a first–order language without explicit quantifiers x man(x)smoke(x) 

6. More about DRS  DRS can be viewed as a first–order language without explicit quantifiers x man(x)smoke(x)    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 y book(y) y y y=harry y y=?

8. More about discourse referents  Verbs introduce [event] discourse referents  Intransitive verbs: to sleep  Transitive verbs: to read e sleep(e) agent(e,x) e read(e) agent(e,x) patient(e,x)

9. Accessibility 1 x man(x) u smoke(u) v snort(v)  y car(y)  z smoke(z)  X

10. Accessibility 1 x man(x) u smoke(u) v snort(v)  y car(y)  z smoke(z)  X - - - O

11. Accessibility 2 x man(x) u smoke(u) v snort(v)  y car(y)  z smoke(z)  X

12. Accessibility 2 x man(x) u smoke(u) v snort(v)  y car(y)  z smoke(z)  X O O -- -

13. Accessibility 3 x man(x) u smoke(u) v snort(v)  y car(y)  z smoke(z)  X

14. Accessibility 3 x man(x) u smoke(u) v snort(v)  y car(y)  z smoke(z)  X O O - - -

15. Accessibility 4 x man(x) u smoke(u) v snort(v)  y car(y)  z smoke(z)  X

16. Accessibility 4 x man(x) u smoke(u) v snort(v)  y car(y)  z smoke(z)  X O O - - -

17. Accessibility 5 x man(x) u smoke(u) v snort(v)  y car(y)  z smoke(z)  X

18. Accessibility 5 x man(x) u smoke(u) v snort(v)  y car(y)  z smoke(z)  X O O O - -

19. Accessibility 6 x man(x) u smoke(u) v snort(v)  y car(y)  z smoke(z)  t X

20. Accessibility 6 x man(x) u smoke(u) v snort(v)  y car(y)  z smoke(z)  t X O O OO - -

21. Subordination x man(x) u smoke(u) v snort(v)  y car(y)  z smoke(z) 

22. Subordination x man(x) u smoke(u) v snort(v)  y car(y)  z smoke(z)  A B C D EF

23. Subordination x man(x) u smoke(u) v snort(v)  y car(y)  z smoke(z)  A B C D EF A subordinates B A subordinates C A subordinates D D subordinates E E subordinates F

24. Subordination x man(x) u smoke(u) v snort(v)  y car(y)  z smoke(z)  A B C D EF A subordinates B A subordinates C A subordinates D D subordinates E E subordinates F A subordinates E A subordinates F ….. Etc.

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 … x u x=vincent u = ???  y e woman(y) dance(e) agent(e,x) patient(e,y)

27. Negation and definites  Vincent did not dance with the woman. She … x y u x=vincent woman(y) u = y  e dance(e) agent(e,x) patient(e,y)

28. Negation and proper names  Vincent did not dance with Mia. She … x y u x=vincent y=mia u = y  e dance(e) agent(e,x) patient(e,y)

29. 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 … x y e u x=vincent woman(y) dance(e) agent(e,x) patient(e,y) u = y

31. More about accessibility  Vincent did with every woman. She … x u x=vincent u = ??? y woman(y) e dance(e) agent(e,x) patient(e,y) 

32. More about accessibility  Vincent did with no woman. She … x u x=vincent u = ??? y e woman(y) dance(e) agent(e,x) patient(e,y) 

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  Frege’s principle The meaning of a compound expression is a function of the meaning of its parts.

35. Composing DRSs [roughly]  Mia does not have a car x x=mia  have(…,…) y car(y)  Mia  does not  have  a car

36. Composing DRSs [roughly]  Mia does not have a car x x=mia y car(y) have(x,y)  x x=mia  have(…,…) y car(y)  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 x boxer(x) lose(x) y die(y) y=x ( ; )

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 ( B1;B2 ) x boxer(x) lose(x) y die(y) y=x ( ; )

42. A merge example  A boxer lost.  He died. y die(y) y=x x boxer(x) lose(x)

43. A merge example  A boxer lost.  He died.  A boxer lost. He died. x boxer(x) lose(x) y die(y) y=x ( ; ) x boxer(x) lose(x) y die(y) y=x

44. Merge and accessibility  If ( B1;B2 ) is a DRS, then  B1 subordinates B2  I.e., discourse referents introduced in B1 are accessible from B2 x boxer(x) lose(x) y die(y) y=x ( ;)

45. Merge and variable binding  Which variables are bound, and which are free? x ….(x) ….(y) ….(z) y ….(x) ….(y) ….(z) (( ;);) z ….(x) ….(y) ….(z)

46. Merge and variable binding  Which variables are bound, and which are free? x ….(x) ….(y) ….(z) y ….(x) ….(y) ….(z) (( ;);) z ….(x) ….(y) ….(z)  free

47. Merge is associative  These two DRSs do not differ in meaning x ….(x) ….(y) ….(z) y ….(x) ….(y) ….(z) ((; ); ) z ….(x) ….(y) ….(z) x ….(x) ….(y) ….(z) y ….(x) ….(y) ….(z) z ….(x) ….(y) ….(z) (( ; ; ))(

48. Merge is non-commutative  These two DRSs differ in meaning x boxer(x) lose(x) y die(y) y=x ( ; ) x boxer(x) lose(x) ( ; y die(y) y=x )

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 x boxer(x) lose(x) y die(y) y=x ( ; )

51. Merge Reduction Example x boxer(x) lose(x) y die(y) y=x ( ; ) Merge reduction ----->

52. Merge Reduction Example x boxer(x) lose(x) y die(y) y=x ( ; ) x y boxer(x) lose(x) die(y) y=x Merge reduction ----->

53. Merge Reduction Problem  Consider the example: A woman walks. A man talks. x woman(x) walk(x) x man(x) talk(x) (;)

54. Merge Reduction Problem  Consider the example: A woman walks. A man talks. x woman(x) walk(x) x man(x) talk(x) (;) Merge reduction ----->

55. Merge Reduction Problem  Consider the example: A woman walks. A man talks. x woman(x) walk(x) x man(x) talk(x) (;) x woman(x) man (x) walk(x) talk(x) Merge reduction ----->

56. Constraints on merge reduction  Given a DRS ( B1;B2 ), 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 ( B1;B2 ), 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 B1;B2  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 @ 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: boxer(x) x. x x=vincent ;u@x) u.(

61. The @ operator  We use the @ operator to combine lambda-DRSs  The expression F@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? boxer(x) x. @z

63. Beta-Conversion  Performing this substitution is called beta-conversion  How does this work?  Remove -prefix from functor boxer(x) x. @z

64. 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 boxer(x) @z x.

65. 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 boxer(z)

66. Another example  This is functional application  What is the functor?  What is the argument? x man(x) u.( ;u@x) y man(y)  ( ;u@y) @ z. run(z)

67. Another example  The functor lambda-binds u  How many substitutions do we make? x man(x) u.( ;u@x) y man(y) @ z. run(z)  ( ;u@y)

68. Another example  The functor lambda-binds u  How many substitutions do we make? x man(x) u.( ;u@x) y man(y) @ z. run(z)  ( ;u@y)

69. Another example  This is the result after substitution  Are we ready with beta-conversion? x man(x) (; @x) y man(y) ; @y) z. run(z) z. ((

70. Another example  Carrying out further substitutions  Anything left to do? x man(x) (; @x) y man(y) ; z. run(z) run(y) (( )

71. Another example  Carrying out further substitutions  Perhaps we can perform further reductions? x man(x) (; ) y man(y) ; run(x) run(y) (( )

72. Another example  Carrying out further substitutions  Perhaps we can perform further reductions? x man(x) (; ) y man(y) run(y) run(x) 

73. Another example  And here is the final DRS  Btw, does this DRS make sense? x man(x) run(x) y man(y) run(y) 

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: x mia(x) love(x,y) x vincent(x) ; y. @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: x mia(x) love(x,x) x vincent(x) ; ) (

76. Alpha-Conversion  Before beta-conversion, we perform alpha-conversion on the functor  Alpha-conversion replaces bound variables for new occurrences  Example: x mia(x) love(x,y) x vincent(x) ; y. @x) (

77. Alpha-Conversion  Before beta-conversion, we perform alpha-conversion on the functor  Alpha-conversion replaces bound variables for new occurrences  Example: v mia(v) love(v,u) x vincent(x) ; u. @x) (

78. Alpha-Conversion  Before beta-conversion, we perform alpha-conversion on the functor  Alpha-conversion replaces bound variables for new occurrences  Example: v mia(v) love(v,x) x vincent(x) ; ) (

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

81. The Lexicon  Nouns: boxer, man, restaurant  Proper names: Mia, Vincent

82. 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: boxer(x) x. u restaurant(u) u.

88. The lexicon: proper names Mia: Vincent: x mia(x) x vincent(x) u.( ;u@x) p.( ;p@x)

89. The lexicon: intransitive verbs dances: smokes: dance(x) x. smoke(y) y.

90. The lexicon: determiners a: every: x p.q.(( ;p@x);q@x) x ;p@x)  q@x p.q. (

91. The lexicon: adjectives big: red: red(x) u.x.( ;u@x) big(x) u.x.( ;u@x)

92. Syntactic Structure  We now know what the partial DRSs in the lexicon look like  But we don’t know how to put things together, at least not the order  We need syntax for this

93. Example Everyman dances dance(x) x man(x) 

94. Partial DRSs Everyman dances dance(z) z. man(y) y. ;p@x)  q@x p.q. ( x

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 s:X@Y  np:X vp:Y

97. Adding semantics  Grammar rules s:F@A  np:F vp:A np:F@A  det:F n:A np:X  pn:X vp:X  iv:X vp:F@A  tv:F np:A  Lexical rules det:X  a:X det:X  every:X n:X  man:X n:X  car:X

98. Example derivation S NP DET N VP IV Every man dances

99. Example derivation S NP DETN VP IV Everymandances dance(z) z. man(y) y. ;p@x)  q@x p.q ( x

100. Example derivation S NP DET N VP IV Every mandances dance(z) z. man(y) @y. ;p@x)  q@x p.q. ( x Application NP  DET N

101. Example derivation S NP DET N VP IV Every mandances dance(z) z. man(y) ;y. @x)  q@x q. ( x -conversion

102. Example derivation S NP DET N VP IV Every mandances dance(z) z. man(x) ; )  q@x q. ( x -conversion

103. Example derivation S NP DET N VP IV Every mandances dance(z) z. x man(x)  q@x q. ;-reduction

104. Example derivation S NP DET N VP IV Every mandances dance(z) z. x man(x)  q@x q. No operation required VP  IV

105. Example derivation S NP DET N VP IV Every mandances dance(z) @z. x man(x)  q@x q. Application S  NP VP

106. Example derivation S NP DET N VP IV Every mandances dance(z) x man(x)  z. @x -conversion

107. Example derivation S NP DET N VP IV Every mandances dance(x) x man(x)  -conversion

108. Lexical Semantics: trans. verbs admires: admire(x,y) y. x.

109. The lexicon: trans. verbs admires: admire(x,y) y. x. admire(x,y) u. x.u@ 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 NP DETN VP IV Everymandances e dance(e) agent(e,z) z. man(y) y. ;p@x)  q@x p.q ( x

113. First attempt S NP DETN VP IV Everymandances e dance(e) agent(e,z) z. man(y) y. ;p@x)  q@x p.q ( x x man(x) e dance(e) agent(e,x) 

114. Example with adverb S NP VP IV Everymandances VP quickly ADV

115. First attempt S NP VP IV Everymandances e dance(e) agent(e,z) z.  q@x q. x man(x) VP quickly ADV quick(i) i. ?

116. Second attempt S NP VP IV Everymandances e dance(e) agent(e,z) z. m. ;m@e  q@x q. x man(x) VP quickly ADV quick(i) v.z.m.v@z@i. ;m@i

117. Second attempt S NP VP IV Everymandances z. m. ;m@e  q@x q. x man(x) VP quickly ADV e dance(e) agent(e,z) quick(e)

118. Second attempt S NP VP IV Everymandances m. ;m@e  x man(x) VP quickly ADV e dance(e) agent(e,z) quick(e) ?

119. Third attempt S NP VP IV Everymandances e dance(e) agent(e,z) n. m.n@ z. ;m@e  q@x q. x man(x) VP quickly ADV quick(i) v.n.m.v@n@i. ;m@i

120. Third attempt S NP VP IV Everymandances n. m.n@ z. ;m@e  q@x q. x man(x) VP quickly ADV e dance(e) agent(e,z) quick(e)

121. Third attempt S NP VP IV Everymandances m.  x man(x) VP quickly ADV e dance(e) agent(e,z) quick(e) ;m@e

122. Event  We will always end up with a lambda  Introduce a rule T  S  This rule elimates the last lambda

123. Third attempt NP VP IV Everymandances m. @ i.  x man(x) VP quickly ADV e dance(e) agent(e,z) quick(e) ;m@e S T event(i)

124. Third attempt NP VP IV Everymandances  x man(x) VP quickly ADV e dance(e) agent(e,z) quick(e) event(e) S T

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. What`s next?  Tomorrow  DRSs, what do we do with them?  DRT and inference  Thursday  Resolving anaphora in DRT  Presupposition