Introduction to Semantics
Session 4
Syntax and Semantic Interpretation
Cornelia Endriss
Cognitive Science Program
University of Osnabrück
[email protected]
Outline for Today
1.
More on Functions and Sets
•
•
•
2.
3.
Recall: λ-notation
Characteristic Functions and Sets
Currying/Schönfinkelization
Deriving Meanings Compositionally: Fragment 2
Modularity and Type Driven Interpretaton
•
•
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X-Bar Theory
Theta-Roles
Slides based on semantics textbook from I. Heim & A. Kratzer
and lecture notes from M. Krifka
2
Functions
More on
Functions and Sets
09.06.2008
Slides based on semantics textbook from I. Heim & A. Kratzer
and lecture notes from M. Krifka
3
Recall: λ-notation
• allows us to form functor expressions by abstracting from arbitrary
arguments:
• General form: λα : ϕ . γ
α is an argument variable
ϕ is the domain condition, and
γ is the value description.
• read as: "the function that maps every α such that ϕ to γ" or
"the function that maps every α such that ϕ to 1 iff γ"
• for instance:
(1)
’parent÷:= λx : x ∈ Person . parent of x
’parent÷ = the function that maps every x that is a person to the
parent of x.
(2)
’smoke÷:= λx : x ∈ D . x smokes
’smoke÷ = the function that maps every x in D onto 1 iff x smokes.
09.06.2008
Slides based on semantics textbook from I. Heim & A. Kratzer
and lecture notes from M. Krifka
4
Characteristic Functions
• Functions can be used to develop a new notation for sets.
• Sets are collections of elements (of a given universe).
• To know a set means, to be able to identify the elements of
that set.
• This information can be given as a function, namely, as a
function that maps every object in the universe to one of
two values:
–to the value True (1), if the element is in the set;
–to the value False (0), if the element is not in the set.
• Such functions are called characteristic functions of a set, as
they “characterize” the set.
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Slides based on semantics textbook from I. Heim & A. Kratzer
and lecture notes from M. Krifka
5
Characteristic Functions
• We write charA or χA for the characteristic function of set A.
• Given a universe U, and a set A, A ⊆ U, we have the
following definition of the characteristic function charA or
χA of A:
(3) charA: U →{0,1},
x → 1 if x ∈ A,
x → 0 if x ∉ A.
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Slides based on semantics textbook from I. Heim & A. Kratzer
and lecture notes from M. Krifka
6
Characteristic Functions: Example
(4) Let U, the universe, be the set of small letters {a, b, c, ... z}.
χ {b,c}:
U → {0,1}
a → 0, b → 1, c → 1, d → 0, ... z → 0
(5) {x | x is a woman}
χ{x | x is a woman}: U → {0, 1}
x → 1 if x ∈ {x | x is a woman},
i.e. if x is a woman
x → 0 if x ∉ {x | x is a woman},
i.e. if x isn’t a woman.
09.06.2008
Slides based on semantics textbook from I. Heim & A. Kratzer
and lecture notes from M. Krifka
7
Characteristic Functions
Function definition:
’works÷ = f : D → {0,1}
for all x∈D, f(x) = 1 iff x works
Set definition:
’works÷ = {x ∈ D : x works}
The two views are interchangeable:
1. Let A be a set. Then charA is the characteristic function of A,
i.e., that function f such that for any x∈A, f(x)=1
and for any x∉A, f(x)=0
2. Let f be a function with range {0,1}. Then charf, the set
characterized by f, is {x∈D : f(x)=1}
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Slides based on semantics textbook from I. Heim & A. Kratzer
and lecture notes from M. Krifka
8
Characteristic Functions as λ-Terms
• Striking similarities:
(6) {x | x is a woman}
λx∈D[x is a woman]
• Hence:
(7) Mary ∈ {x | x is a woman} iff
λx∈U[x is a woman](Mary) = 1,
i.e. if Mary is a woman
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Slides based on semantics textbook from I. Heim & A. Kratzer
and lecture notes from M. Krifka
9
Characteristic Functions as λ-Terms
•
Other example:
(8)
(9)
{x | x∈N and x≥7}
(the set of natural numbers greater or equal 7)
λx∈U[x∈N and x≥7]
(maps everything to 1 if it is a natural number greater or
equal 7, and to 0 else.)
•
Lambda notation is actually more expressive:
(10) λx∈N[x≥7]
(Partial Function: maps every natural number to 1 if it is greater
or equal 7, and to 0 else. Applied to a non-natural number, it
does not give us anything.)
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Slides based on semantics textbook from I. Heim & A. Kratzer
and lecture notes from M. Krifka
10
Set Denotations
•
We will switch back and forth between the set and the function
denotation e.g. for intransitive verbs like sleep.
•
Example:
D = {Cassandra, Hannes, Bohemia}
(11) ’limps÷ = {Cassandra, Bohemia}
(12) ’sleeps÷ = {Cassandra, Hannes}
•
We can say things like:
(13) Cassandra ∈ ’sleeps÷
(14) ’limps÷ ⊄ ’sleeps÷
(15) |’limps÷ ∩ ’sleeps÷| = 1
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Slides based on semantics textbook from I. Heim & A. Kratzer
and lecture notes from M. Krifka
11
Function Denotations
•
Same example:
D = {Bohemia, Cassandra, Hannes}
(11’) ’limps÷ = Bohemia → 1
Cassandra → 1
Hannes
→0
•
(12’) ’sleeps÷= Bohemia → 0
Cassandra → 1
Hannes → 1
Now we can say things like:
(13’) ’sleeps÷(Cassandra) = 1
(14’) {x: ’limps÷(x) = 1} ⊄ {y: ’sleeps÷(y) = 1}
(15’) | {x: ’limps÷(x) = 1} ∩ {y: ’sleeps÷(y) = 1} | = 1
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Slides based on semantics textbook from I. Heim & A. Kratzer
and lecture notes from M. Krifka
12
Relations as Functions
•
Characteristic functions allow us to render sets as functions.
•
Relations are sets, namely, sets of pairs
(16) a. ’parent÷ as set:
{〈x, y〉 | y is a parent of x}.
b. ’parent÷ as function:
λ〈x,y〉∈{〈x,y〉| x∈Person, y∈Person}[y is a parent of x]
(17) a. 〈Achim Petry, Wolle Petry〉 ∈ {〈x, y〉 | y is a parent of x}
b. λ〈x,y〉[y is a parent of x](〈Achim Petry, Wolle Petry〉) = 1
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Slides based on semantics textbook from I. Heim & A. Kratzer
and lecture notes from M. Krifka
13
Currying 2-place relations
• Schönfinkelization/Currying
(due to the logicians Moses Schönfinkel and Huskell Curry):
parent not treated as a function over pairs, but as one that first
takes one argument (say, the child), and then another.
(18) ’parent÷, reduced to 1-place functions:
λx∈Person[λy∈Person[y is a parent of x]]
(19) λx∈Person[λy∈Person[y is a parent of x]]
(Achim Petry)(Wolle Petry)
= λy∈Person[y is a parent of Achim Petry](Wolle Petry)
= 1, as Wolfgang Petry is a parent of Achim Petry
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Slides based on semantics textbook from I. Heim & A. Kratzer
and lecture notes from M. Krifka
14
Currying 3-place relations
• We can apply the same technique for three-place relations:
(20) a. ’give÷ as relation:
b. ’give÷ as function:
{〈x, y, z〉 | x gives y to z}
λ〈x,y,z〉[x gives y to z]
c. ’give÷ as reduced function: λz[λy[λx[x gives y to z]]]
09.06.2008
Slides based on semantics textbook from I. Heim & A. Kratzer
and lecture notes from M. Krifka
15
Why Currying?
• One important advantage of using 1-place functions in
semantic analysis: arguments are not symmetric.
• One argument is usually “closer” than the other.
• E.g. for the father-function, the child-argument is closer than
the father-argument. The child argument forms a syntactic
constituent with the noun, a Noun Phrase (NP).
• (21) a. Wolfgang Petry is [NP Achim Petry’s father].
b. Wolfgang Petry is [NP the father of Achim Petry].
09.06.2008
Slides based on semantics textbook from I. Heim & A. Kratzer
and lecture notes from M. Krifka
16
Evidence for Currying
• Also, the object argument of love is closer than the subject argument
— the object forms a syntactic constituent with the verb, the Verb
Phrase (VP).
(22) Hannes [VP likes Cassandra].
• Cf. canonical word order in German:
(23) a. … weil
das Pferd den Jungen gebissen
because the horse the boy
bit
‘… because the horse bit the boy.’
b. … weil
den Jungen das Pferd gebissen
because the boy
the horse bit
‘… because the boy was bit by the horse.’
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hat. [canonical]
has
hat. [marked]
has
Slides based on semantics textbook from I. Heim & A. Kratzer
and lecture notes from M. Krifka
17
Evidence for Currying
• Further evidence – possible readings:
(24) … weil
Cassandra
because Cassandra
Hannes
gebissen hat.
Hannes bit
has
… because Cassandra bit Hannes. [only possible reading]
• Even further evidence – topicalization:
(25) a. [Den Jungen gebissen] hat das Pferd ___ nicht.
the boy
bit
has the horse
not
‘The horse did not bite the boy.’
b. *[Das Pferd gebissen] hat ___ den Jungen nicht.
the horse bit
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has
the boy
Slides based on semantics textbook from I. Heim & A. Kratzer
and lecture notes from M. Krifka
.
not
18
Binary Branching
• In other words, there is evidence for binary branching trees –
reflected by lexical entries that take their arguments one after the
other (curried functions):
’like÷(’Cassandra÷) (’Hannes÷) =
S
= [λy[y likes
NP
VP
N
V
NP
Hannes
likes
N
’Hannes÷
=
= 1 iff
’like÷
= λx[λy[y likes x]]
Cassandra
]] (
)
likes
’like÷(’Cassandra÷)
= λx[λy[y likes x]](
= [λy[y likes
)
]]
’Cassandra÷=
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Slides based on semantics textbook from I. Heim & A. Kratzer
and lecture notes from M. Krifka
19
Transitive Verbs
• Transitive verbs are functions to functions, not sets of pairs,
or functions from pairs of individuals to the truth values.
• The good reason is that this assumption allows us to model
(i) the binary branching structure of our syntax together
with the assumptions that
(ii) semantic interpretation rules operate locally (Locality)
and
(iii) that semantic interpretation is functional application
(Frege's Conjecture).
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Slides based on semantics textbook from I. Heim & A. Kratzer
and lecture notes from M. Krifka
20
Deriving Meanings
Compositionally
Fragments of English
(Fragment 2)
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Slides based on semantics textbook from I. Heim & A. Kratzer
and lecture notes from M. Krifka
21
So far: Fragment 1
(A) Ontology
- individuals in the domain D: D = the set of actual individuals
- the truth values, True and False: {0,1}
- functions from D to {0,1}
(B ) Lexicon
’Cassandra÷ = Cassandra =
’Hannes÷ = Hannes =
’Ann÷ = Ann
’Jan÷ = Jan
…
’limps÷ = f : D → {0,1} for all x∈D, f(x)=1 iff x limps
’works÷ = f : D → {0,1} for all x∈D, f(x)=1 iff x works
’smokes÷ = f : D → {0,1} for all x∈D, f(x)=1 iff x smokes
09.06.2008
Slides based on semantics textbook from I. Heim & A. Kratzer
and lecture notes from M. Krifka
22
So far: Fragment 1
(C) Rules
S
(S1)
(S2)
(S3)
(S4)
(S5)
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If α has the form
β γ
NP
If α has the form
β
VP
If α has the form
β
N
If α has the form
β
V
If α has the form
β
, then ’α÷ = ’γ÷ (’β÷)
, then ’α÷ = ’β÷
, then ’α÷ = ’β÷
, then ’α÷ = ’β÷
, then ’α÷ = ’β÷
Slides based on semantics textbook from I. Heim & A. Kratzer
and lecture notes from M. Krifka
23
Adding transitive verbs
add to the lexicon:
S
NP
VP
N
V
Hannes
likes
NP
’likes÷ =
f : D→{g:g is a function from D→{0,1}}
for all x,y∈D, f(x)(y) =1 iff y likes x.
N
add Rule (S6):
Cassandra
VP
If α has the form , then ’α÷ = ’β÷(’γ÷)
β γ
add to ontology:
functions from D to
functions from D to {0,1}
09.06.2008
Slides based on semantics textbook from I. Heim & A. Kratzer
and lecture notes from M. Krifka
24
Semantic types
Introduce semantic types:
e
individuals ("e" for "entities")
Saturated
denotations
t
truth values
<e,t>
functions from individuals to truth values
<e,<e,t>>
functions from individuals to functions
from individuals to truth values
Unsaturated
denotations
Recursive definition of semantic types:
(i)
e and t are semantic types.
(ii)
If σ and τ are semantic types the <σ, τ> is a
semantic type.
(iii) Nothing else is a semantic type.
09.06.2008
Slides based on semantics textbook from I. Heim & A. Kratzer
and lecture notes from M. Krifka
25
Denotation domains
Corresponding denotation domains:
De
domain of individuals
domain of truth values
Dt
D<e,t>
domain of functions from individuals to
truth values
D<e,<e,t>>
domain of functions from individuals to
functions from individuals to truth values
For any semantic types σ and τ, D< σ, τ > is the set of all
functions from Dσ to Dτ.
09.06.2008
Slides based on semantics textbook from I. Heim & A. Kratzer
and lecture notes from M. Krifka
26
Semantic types so far
Type
expressions
examples
<t>
truth values
sentences
Cassandra limps
<e>
individuals
names
Hannes
<e,t>
functions
intr.verbs
walks
<e,<e,t>>
functions
trans.verbs
likes
<t,t>
functions
sent. oper.
it is not the case that
<t,<t,t>>
functions
sent. oper.
and, or
09.06.2008
Slides based on semantics textbook from I. Heim & A. Kratzer
and lecture notes from M. Krifka
27
Recall: Semantic Rules
• So far two separate semantic rules:
S
(S1)
If α has the form
, then ’α÷ = ’γ÷ (’β÷)
β γ
(S6)
If α has the form
VP
, then ’α÷ = ’β÷(’γ÷)
Typedriven
interpretation
β γ
• Both cases: meaning of one subexpression applied to meaning of
the other. But specified explicitly which is to be applied to which.
• However, reverse application would be impossible.
(S1&6) If α has the form
09.06.2008
S
, then ’α÷ = ’γ÷ (’β÷) or ’β÷ (’γ÷)
β γ
-- whichever makes sense.
Slides based on semantics textbook from I. Heim & A. Kratzer
and lecture notes from M. Krifka
28
General Semantic Principles
Replace our semantic rules by 3 General Principles:
(TN)
If α is a terminal node, and ’α÷ is in the domain of ’ ÷,
then ’α÷ is given in the lexicon.
(NN)
If α is a non-branching node and β is its daughter and β is in
the domain of ’ ÷, then also α is in the domain of ’ ÷ and
’α÷ = ’β÷.
(FA)
If α is branching and {β, γ} are its daughters, then α is in the
domain of ’ ÷ if both β and γ are in the domain of ’ ÷ and ’β÷ is a
function with ’γ÷ in its domain. In this case: ’α÷ = ’β÷ (’γ÷).
Note that no linear order is specified!
Interpretability Principle
A tree α is interpretable iff all of its nodes are in the domain of ’ ÷.
09.06.2008
Slides based on semantics textbook from I. Heim & A. Kratzer
and lecture notes from M. Krifka
29
Fragment 2
Lexicon
’Ann÷ =
’works÷ =
’likes÷ =
’and1÷ =
’and2÷ =
Ann
λx∈De . x works
λx∈De . [λy∈De . y likes x]
λp∈Dt . [λq∈Dt . p=q=1]
λP∈D<e,t> . [λQ∈D<e,t> .[λx∈De .P(x)=Q(x)=1]]
Principles for composition: TN, NN, FA
Three general principles:
- Frege's conjecture (compositionality)
- locality
- binary branching
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Slides based on semantics textbook from I. Heim & A. Kratzer
and lecture notes from M. Krifka
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The Interplay of
Syntax and Semantics
The Interplay of
Syntax and
Semantics
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Slides based on semantics textbook from I. Heim & A. Kratzer
and lecture notes from M. Krifka
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Syntax and Semantics
• Semantics is a separate module interpreting syntactic trees.
• Semantics and syntax are independent:
– syntactically ill-formed trees may be interpretable
(26) *Limps Hannes.
(27) *Likes Hannes Cassandra.
– syntactically well-formed trees may be uninterpretable
(28) # Cassandra limps Hannes. (29) # Greeted Ann.
Question: Is this actually “# ” or “*”??
Usual Answer in the syntactic literature: “*”!
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Slides based on semantics textbook from I. Heim & A. Kratzer
and lecture notes from M. Krifka
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Type-Driven Interpretation
• We can also explain why the following sentences are illformed by way of theta-theory or semantics.
(28) # Cassandra limps Hannes.
e
<e,t> e
t
(29) # Greeted
<e,<e,t>
Ann.
e
<e,t>
?
So is it syntax ( –criterion) or semantics (type-theory)
that excludes these sentences?
09.06.2008
Slides based on semantics textbook from I. Heim & A. Kratzer
and lecture notes from M. Krifka
33
Type-Driven Interpretation
• There are good reasons to believe that it is actually semantics
and not syntax!
• Note that the –criterion makes stronger predictions than the
conjecture to type-driven interpretation.
• Suppose ’÷ is of type <e,t>
• Prediction of –theory: must be in the neighborhood of
something of type e so that it can assign its –role.
• According to type-driven interpretation: α could be the
argument of another function, say one of type <<e,t>,e>.
Cassandra is a horse.
The horse sleeps.
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Slides based on semantics textbook from I. Heim & A. Kratzer
and lecture notes from M. Krifka
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Homework
Read Chapter 3 of Heim & Kratzer’s textbook, Sections 2.4, 2.5, and 3
of Krifka’s notes, and go through the lecture notes again.
Apart from that no homework!
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Slides based on semantics textbook from I. Heim & A. Kratzer
and lecture notes from M. Krifka
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Thank you!
Thank you!
09.06.2008
Slides based on semantics textbook from I. Heim & A. Kratzer
and lecture notes from M. Krifka
36