Formal Semantics

1
Formal Semantics

• Slides by Lucas Champollion
• Based on Heim Kratzer (1998)

2
The MURI pipeline
3
The big picture, more generally
Somebody sleeps
Tokenization
POS Tagging
Syntax
Semantics
(Pragmatics)
?x sleeps(x)
4
From trees to logic
compositional semantics
?xthink(bill,likes(harry,x))
or some other suitable representation
5
Examples of representation languages
Other candidates Modal logics, higher-order
predicate logics, description logics etc.
6
First-order predicate logic

• ?x (man(x))
• Everybody is a man
• ?x (man(x))
• Somebody is a man

7
First-order predicate logic

• ?x (man(x))
• Everybody is a man
• ?x (man(x))
• Somebody is a man
• ?x (rich(x) ? popular(x))
• Everybody is rich or popular
• ?x (rich(x) ? popular(x))
• Everybody is rich and popular
• ?x (rich(x) ? popular(x))
• Everybody who is rich is popular

8
First-order predicate logic

• ?x (man(x))
• Everybody is a man
• ?x (man(x))
• Somebody is a man
• ?x (rich(x) ? popular(x))
• ?x (rich(x) ? popular(x))
• ?x (rich(x) ? popular(x))

9
First-order predicate logic

• ?x (man(x) ? mortal(x))
• man(socrates)
• ____________________
• mortal(socrates)

?x ?y loves(x,y) ________________ ??y ?x
loves(x,y)
10
First-order predicate logic with lambdas

• ?x (man(x))
• Everybody is a man
• ?x (man(x))
• Somebody is a man
• ?x (man(x))
• The function that maps men to true and non-men
  to false

11
First-order predicate logic with lambdas

• ?x (man(x))
• Everybody is a man
• ?x (man(x))
• Somebody is a man
• ?x (man(x))
• The function that maps men to true and non-men
  to false
• ?x (man(x)) (john)
• The function that maps men to true and non-men
  to false applied to john
• Reduces to man(john) in one step
• This step is called function application or beta
  reduction or lambda conversion

12
The Algorithm (simplest case)

• Assume all nodes are at most binary branching
• Every word is mapped to a formula/fragment
• This information is looked up in the lexicon
• Ambiguous words have multiple entries -- use WSD
• Non-branching nodes inherit meaning from their
  daughter nodes
• Branching nodes Functional Application
• If one child denotes a function and the other
  child denotes an argument, apply function to
  argument
• Apply recursively until root is reached

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Simple Lexical Semantics

• Each proper noun is mapped to a constant
• John john
• America america

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Simple Lexical Semantics

• Each proper noun is mapped to a constant
• John john
• America america
• Adjectives, nouns, and intransitive verbs are
  mapped to one-place predicates
• A predicate is a function that returns a boolean
  value (true, false)
• sleeps ?x if x sleeps then return true
  else return false
• man ?x if x is a man then return true
  else return false
• red ?x if x is red then return true else
  return false

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Simple Lexical Semantics

• Each proper noun is mapped to a constant
• John john
• America america
• Adjectives, nouns, and intransitive verbs are
  mapped to one-place predicates
• A predicate is a function that returns a boolean
  value (true, false)
• sleeps ?x if x sleeps then return true
  else return false
• man ?x if x is a man then return true
  else return false
• red ?x if x is red then return true else
  return false

Can be simplified
16
Simple Lexical Semantics

• Each proper noun is mapped to a constant
• John john
• America america
• Adjectives, nouns, and intransitive verbs are
  mapped to one-place predicates
• A predicate is a function that returns a boolean
  value (true, false)
• sleeps ?x sleeps(x)
• man ?x man(x)
• red ?x red(x)

17
Simple Lexical Semantics

• Each proper noun is mapped to a constant
• John john
• America america
• Adjectives, nouns, and intransitive verbs are
  mapped to one-place predicates
• A predicate is a function that returns a boolean
  value (true, false)
• sleeps ?x sleeps(x)
• man ?x man(x)
• red ?x red(x)
• Transitive verbs are mapped to two-place
  predicates
• loves ?y ?x loves(x,y)

18
Simple Lexical Semantics

• Each proper noun is mapped to a constant
• John john
• America america
• Adjectives, nouns, and intransitive verbs are
  mapped to one-place predicates
• A predicate is a function that returns a boolean
  value (true, false)
• sleeps ?x sleeps(x)
• man ?x man(x)
• red ?x red(x)
• Transitive verbs are mapped to two-place
  predicates
• loves ?y ?x loves(x,y)
• Auxiliaries are mapped to the identity function
• is does ?x x

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Example
S
NP
VP
NP
N
V
N
John
loves
Mary
John loves Mary
20
Example
S
NP
VP
NP
N
V
N
john
?y ?x loves(x,y)
mary
John loves Mary
21
Example
S
NP
VP
NP
john
V
N
?y ?x loves(x,y)
mary
John loves Mary
22
Example
S
john
VP
NP
V
N
?y ?x loves(x,y)
mary
John loves Mary
23
Example
S
john
VP
NP
?y ?x loves(x,y)
N
mary
John loves Mary
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Example
S
john
VP
NP
?y ?x loves(x,y)
mary
John loves Mary
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Example
S
john
VP
mary
?y ?x loves(x,y)
John loves Mary
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Example
S
john
?y ?x loves(x,y) (mary)
John loves Mary
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Example
S
john
?x loves(x,mary)
John loves Mary
28
Example
?x loves(x,mary) (john)
John loves Mary
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Example
loves(john,mary)
John loves Mary
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A more involved example
S
NP
VP
NP
N
V
N
Somebody
loves
Mary
Somebody loves Mary
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The meaning of quantifier NPs

• A first try
• somebody somebody
• everybody everybody
• Somebody loves Mary. loves(somebody, mary)
• ???

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Quantifier noun phrases have a special status

• They are functions from VP meanings to booleans
• So, they are functions from one-place predicates
  to booleans
• Higher-order function a function that takes
  another function as an argument
• Examples
• everybody ?P ?z P(z)
• nobody ?P ??z P(z)
• Building quantifier NPs from quantifiers
• every ?P ?Q ?z P(z) ? Q(z)
• Question words as quantifiers
• who ?P ?z P(z)
• where ? is a query operator

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A more involved example
S
NP
VP
NP
N
V
N
?P ?z P(z)
?y ?x loves(x,y)
mary
Somebody loves Mary
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A more involved example
S
?P ?z P(z)
VP
NP
V
N
?y ?x loves(x,y)
mary
Somebody loves Mary
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A more involved example
S
?P ?z P(z)
VP
NP
?y ?x loves(x,y)
N
mary
Somebody loves Mary
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A more involved example
S
?P ?z P(z)
VP
NP
?y ?x loves(x,y)
mary
Somebody loves Mary
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A more involved example
S
?P ?z P(z)
VP
mary
?y ?x loves(x,y)
Somebody loves Mary
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A more involved example
S
?P ?z P(z)
?y ?x loves(x,y) (mary)
Somebody loves Mary
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A more involved example
S
?P ?z P(z)
?x loves(x,mary)
Somebody loves Mary
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A more involved example
?P ?z P(z) ?x loves(x,mary)
Somebody loves Mary
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A more involved example
?P ?z P(z) ?x loves(x,mary)
?P ?z P (z)
?x loves(x,mary)
Somebody loves Mary
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A more involved example
?P ?z P(z) ?x loves(x,mary)
?P ?z P (z)
?x loves(x,mary)
?z ?x loves(x,mary)(z)
Somebody loves Mary
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A more involved example
?P ?z P(z) ?x loves(x,mary)
?P ?z P (z)
?x loves(x,mary)
?z ?x loves(x,mary)(z)
?z loves(z,mary)
Somebody loves Mary
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Pronouns are mapped to variables

• Can be bound by any noun phrase (both proper
  nouns and quantifier NPs)
• Are sometimes pronounced as reflexives (himself)
• Binder-bindee relation indicated by an integer
  index
• John1 likes himself1.
• Every man1 likes a woman who hates him1.
• John1 thinks that he1 likes Mary and that Mary
  likes him1.
• Often, several ways of binding them are possible
  (anaphora resolution)
• John1 likes Bill2 because he2 pays him1.
• John1 likes Bill2 because he1 pays him2.
• John1 likes Bill2 because he1 pays himself1.
• John1 likes Bill2 because he2 pays himself2.
• Can be bound across sentences
• John1 sleeps. He1 is tired.
• though some quantifiers cause problems there
• Every man1 sleeps. He1 is tired.

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The pronoun mechanism

• Pronoun interpretation is delegated to a special
  function g
• Any noun phrase that carries a binder index
  extends g (writes information into g)
• A pronouns meaning is given by g, applied to the
  pronouns index. (The pronoun reads information
  from g)
• he1 g(1)
• he2 g(2)
• him2 g(2) etc.
• The following is a simplified illustration

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Example Proper noun binds pronoun
S
NP
VP
NP
N
V
N
John1
loves
himself1
John loves himself
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Example Proper noun binds pronoun
S
NP
VP
NP
N
V
N
john1
?y ?x loves(x,y)
g(1)
John loves himself
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Example Proper noun binds pronoun
S
john1
VP
NP
V
N
?y ?x loves(x,y)
g(1)
John loves himself
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Example Proper noun binds pronoun
S
john1
VP
NP
?y ?x loves(x,y)
N
g(1)
John loves himself
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Example Proper noun binds pronoun
S
john1
VP
g(1)
?y ?x loves(x,y)
John loves himself
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Example Proper noun binds pronoun
S
john1
?y ?x loves(x,y) g(1)
John loves himself
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Example Proper noun binds pronoun
S
john1
?x loves(x,g(1))
John loves himself
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Example Proper noun binds pronoun
S
john1
?x loves(x,g(1))
g(1) picks up its meaning as the coindexed NP
(john) at the moment of combining
John loves himself
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Example Proper noun binds pronoun
S
john1
?x loves(x, g(1))
g(1) picks up its meaning as the coindexed NP
(john) at the moment of combining
John loves himself
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Example Proper noun binds pronoun
?x loves(x, g(1)) john
John loves himself
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Example Proper noun binds pronoun
loves(g(1), john)
John loves himself
57
Example Proper noun binds pronoun
loves(g(1), john)
We now resolve g(1), which yields john
John loves himself
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Example Proper noun binds pronoun
loves(john, john)
We now resolve g(1), which yields john
John loves himself
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Example Quantifier NP binds pronoun
S
NP
VP
NP
N
V
N
Everybody1
loves
himself1
Everybody loves himself
60
Example Quantifier NP binds pronoun
S
NP
VP
NP
N
V
N
?P ?z P(z)1
?y ?x loves(x,y)
g(1)
Everybody loves himself
61
Example Quantifier NP binds pronoun
S
?P ?z P(z)1
VP
NP
V
N
?y ?x loves(x,y)
g(1)
Everybody loves himself
62
Example Quantifier NP binds pronoun
S
?P ?z P(z)1
VP
NP
?y ?x loves(x,y)
N
g(1)
Everybody loves himself
63
Example Quantifier NP binds pronoun
S
?P ?z P(z)1
VP
g(1)
?y ?x loves(x,y)
Everybody loves himself
64
Example Quantifier NP binds pronoun
S
?P ?z P(z)1
?y ?x loves(x,y) g(1)
Everybody loves himself
65
Example Quantifier NP binds pronoun
S
?P ?z P(z)1
?x loves(x,g(1))
Everybody loves himself
66
Example Quantifier NP binds pronoun
S
?P ?z P(z)1
?x loves(x,g(1))
Replace pronoun by QNP variable at the moment of
combining
Everybody loves himself
67
Example Quantifier NP binds pronoun
S
?P ?z P(z)1
?x loves(x,g(1))
g(1) picks up its meaning as the QNP variable
(z) at the moment of combining
Everybody loves himself
68
Example Quantifier NP binds pronoun
?z P(z) ?x loves(x,g(1))
Everybody loves himself
69
Example Quantifier NP binds pronoun
?z ?x loves(x,g(1)) (z)
Everybody loves himself
70
Example Quantifier NP binds pronoun
?z loves(z,g(1))
Everybody loves himself
71
Example Quantifier NP binds pronoun
?z loves(z,g(1))
We now resolve g(1), which yields z
Everybody loves himself
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Example Quantifier NP binds pronoun
?z loves(z,z)
We now resolve g(1), which yields z
Everybody loves himself
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Traces

• Can be bound by any noun phrase (both proper
  nouns and quantifier NPs)
• Are almost never pronounced (by definition, in
  some sense)
• Sometimes pronounced anyway (resumptive
  pronouns)
• This is the man1 that Mary doesnt know what he1
  is eating.
• Binder-bindee relation indicated by an integer
  index
• Who1 does John like t1?
• Who1 does Bill think Harry likes t1?
• Beans1, John likes t1.
• John knows the man1 that Mary likes t1.
• Can only be bound by a noun phrase that has moved
  
• Who1 does John2 like t2?

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The trace mechanism

• Trace interpretation is delegated to the special
  function g
• Any noun phrase that carries a binder index
  extends g (writes information into g)
• A traces meaning is given by g, applied to the
  traces index. (The trace reads information from
  g)
• t1 g(1)
• t2 g(2) etc.
• Exactly the same as pronouns

75
Example Question word binds trace
S
S
NP
S
Aux
N
NP
VP
does
Who1
N
V
NP
Who does Harry like?
like
t1
Harry
76
Example Question word binds trace
S
S
NP
S
Aux
N
NP
VP
?x x
?P ?x P(x)1
N
V
NP
Who does Harry like?
?y?xlike(x,y)
g(1)
harry
77
Example Question word binds trace
S
S
NP
S
Aux
N
NP
VP
?x x
?P ?x P(x)1
N
?y?xlike(x,y)
g(1)
Who does Harry like?
harry
78
Example Question word binds trace
S
S
NP
S
Aux
N
NP
?xlike(x,g(1))
?x x
?P ?x P(x)1
N
Who does Harry like?
harry
79
Example Question word binds trace
S
S
NP
S
Aux
N
NP
?xlike(x,g(1))
?x x
?P ?x P(x)1
harry
Who does Harry like?
80
Example Question word binds trace
S
S
NP
S
Aux
N
harry
?xlike(x,g(1))
?x x
?P ?x P(x)1
Who does Harry like?
81
Example Question word binds trace
S
S
NP
like(harry,g(1))
Aux
N
?x x
?P ?x P(x)1
Who does Harry like?
82
Example Question word binds trace
S
S
NP
like(harry,g(1))
?x x
N
?P ?x P(x)1
Who does Harry like?
83
Example Question word binds trace
S
?x x like(harry,g(1))
NP
N
?P ?x P(x)1
Who does Harry like?
84
Example Question word binds trace
S
like(harry,g(1))
NP
N
?P ?x P(x)1
Who does Harry like?
85
Example Question word binds trace
S
like(harry,g(1))
NP
?P ?x P(x)1
Who does Harry like?
86
Example Question word binds trace
S
like(harry,g(1))
?P ?x P(x)1
Who does Harry like?
87
Example Question word binds trace
?P ?x P(x) like(harry,g(1))
Who does Harry like?
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Example Question word binds trace
?x like(harry,x)
Who does Harry like?
89
Problem Quantifiers in object position
S
NP
VP
NP
N
V
N
Mary
loves
somebody
Mary loves somebody.
90
Problem Quantifiers in object position
S
NP
VP
NP
N
V
N
mary
?y ?x loves(x,y)
?P ?z P(z)
Mary loves somebody.
91
Problem Quantifiers in object position
S
NP
VP
?P ?z P(z)
N
?y ?x loves(x,y)
mary
Mary loves somebody.
92
Problem Quantifiers in object position
S
NP
?P ?z P(z) ?y ?x loves(x,y)
N
mary
Mary loves somebody.
93
Problem Quantifiers in object position
S
NP
?z ?y ?x loves(x,y) (z)
N
mary
Mary loves somebody.
94
Problem Quantifiers in object position
S
NP
?z ?y ?x loves(x,y) (z)
A two-place function is fed only one argument
N
mary
Mary loves somebody.
95
Problem Quantifiers in object position
S
NP
?z ?x loves(x,z)
A two-place function is fed only one argument
N
mary
Mary loves somebody.
96
Problem Quantifiers in object position
S
NP
?z ?x loves(x,z)
A two-place function is fed only one argument
N
Whatever this is, its not a function, so we
cant apply it to mary
mary
Mary loves somebody.
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Proposed Solutions

• Quantifier NPs are ambiguous
• somebody1 ?P ?z P(z)
• subject position only
• somebody2 ?P ?x ?z P(x,z)
• object position only
• Quantifier NPs can act on their context
• Intuition somebody ?Context ?z
  Context(z)
• What does context mean?
• The continuation of the current computation
  (Barker, Shan)
• The surrounding tree, accessible after movement
  (May, much of GB)
• Quantifier scope is resolved in a separate step
• Use a representation language that leaves
  quantifier scope underspecified (most of
  computational semantics)

98
Example Quantifier Raising

• Quantifier NPs move out of object position
• They move above the clause so they have access to
  the whole context
• Similar mechanism to wh-movement, topicalization
  etc. -- except that we dont hear it
• A separate step before interpretation starts

99
Example Quantifier Raising
S
S
NP
N
NP
VP
N
V
NP
Somebody1
loves
t1
Mary
Mary loves somebody.
100
Example Quantifier Raising
S
S
NP
N
NP
VP
N
V
NP
?P ?z P(z)1
?y?xloves(x,y)
g(1)
mary
Mary loves somebody.
101
Example Quantifier Raising
S
S
NP
N
NP
VP
?y?xloves(x,y)
g(1)
N
?P ?z P(z)1
mary
Mary loves somebody.
102
Example Quantifier Raising
S
S
NP
N
NP
?xloves(x,g(1))
N
?P ?z P(z)1
mary
Mary loves somebody.
103
Example Quantifier Raising
S
S
NP
N
mary
?xloves(x,g(1))
?P ?z P(z)1
Mary loves somebody.
104
Example Quantifier Raising
S
?xloves(x,g(1)) mary
NP
N
?P ?z P(z)1
Mary loves somebody.
105
Example Quantifier Raising
S
loves(mary,g(1))
NP
N
?P ?z P(z)1
Mary loves somebody.
106
Example Quantifier Raising
S
loves(mary,g(1))
?P ?z P(z)1
Mary loves somebody.
107
Example Quantifier Raising
?z loves(mary,g(1))
Mary loves somebody.
108
Example Quantifier Raising
?z loves(mary,z)
Mary loves somebody.
109
Quantifier scope ambiguities

• Somebody loves everybody.
• In this country a woman gives birth every 15
  min. Our job is to find that woman and stop her.
  (Groucho Marx)
• In Quantifier Raising approach
• Which quantifier raises highest?
• see blackboard
• In continuations approach
• In which order are the quantifiers evaluated?
• In underspecification approach
• Collect quantifiers, delay decision to a later
  stage

110
Scopal ambiguity is pervasive

• Most politicians can fool most voters most of the
  time, but no politician can fool all voters all
  of the time.
• A participant of every course gave two
  presentations.

111
Constraints on quantifier scope

• Similar to islands
• Some man loves every woman
• (?? and ?? are both OK)
• Some man thinks that he loves every woman
• (?? is OK, ?? is not possible)
• Polarity items are sensitive to negation
• We did not see any man
• (?? is OK, ?? is not possible)
• We did not see some man
• (?? is OK, ?? is not possible)
• We did not see a man
• (?? and ?? are both OK)

112
Appendix Type theory in a nutshell

• Types help us determine which way function
  application applies
• In more refined versions of the theory, when we
  have more operations than just function
  application, types tell us which operations to
  apply
• We start with just two basic types
• t - a truth value (boolean type)
• e - an entity (individual type)
• The set of types is the smallest set such that
• All basic types are types
• If ?and ? are types then lt?,?gt is a type
  (complex type)
• Complex types are used to specify the types of
  input and output values to functions
• A function from entities (e) to booleans (t) will
  have the type lte,tgt (one-place predicate type)
• A function from one-place predicates (lte,tgt) to
  booleans (t) will have the type