Samenvatting Taaltheorie en Taalverwerking Deeltentamen 2

Taaltheorie en Taalverwerking Samenvatting 2
Lecture 5a
Computational Semantics:
 Two important questions for computational semantics:
o Semantic representations: how can we represent the meaning of linguistic expressions
in a computational way? How can we establish a link to the world? With Predicate
Logic.
o Semantic construction (or semantic analysis): how can we systematically associate
semantic representations to linguistic expressions?

Semantic Representations:
 To be able to use semantic information computationally, we need a formal and precise way of
representing what linguistic expressions mean. Two main approaches:
o Symbolic approaches: we can represent meaning (knowledge) using logic and reason
with the meaning representations using inference procedures.
o Statistical approaches: we can represent meaning (knowledge) using frequencies or
probabilities derived from observations.

PL as Semantic Representation Formalism:
 We will use Predicate Logic (PL or FOL) as our semantic representation formalism. We make a
distinction between syntax and semantics:
o The syntax of PL tells us what are well-formed formulas: we have logical connectives,
quantifiers, predicates, variables, constants …
o The semantics of PL specifies when a formula is true in a formal model. Formal models
are abstract representations of situations in the real world.
 PL acts as an intermediate representation level that allows us to connect language to the
world




o We use PL formulas as meaning representations for sentences
o We use PL models to represent world situations
o We know how to check if a PL formula is true in a model
o Therefore, we connect language to the world
 We want to be able to establish a systematic (non-arbitrary) relation between sentences and
formulas

,Semantic Construction:
 Principle of Compositionality: the meaning of a sentence is a function of the meaning of its
parts
 How do we put the meaning of the parts together?
o We may be able to associate a representation with each word in a sentence, but how
is this information combined?
o The meaning of a sentence is not only based on the words that make it up, but also on
the ordering, grouping and structural relations among such words  its syntactic
structure
 Syntax tells us how to hierarchically decompose a sentence into sub-parts
o If we associate a semantic representation with each lexical item and …
o Describe how the semantic representation of a syntactic constituent is to be built up
from the representation of its sub-parts, then …
o We have at our disposal a compositional semantics: a systematic way of constructing
semantic representations for sentences.
 Now we have a plausible strategy for finding a way to systematically associate PL formulas
with sentences. We need to:
o Specify a reasonable syntax by means of a formal grammar such as CFG for the
fragment of natural language of interest
o Specify semantic representations for the lexical items
o Specify how the semantic representation of a syntactic constituent is constructed in
terms of the representations of its sub-parts
 We will use a notational extension of PL to do this: lambda calculus

Lambda Abstraction:
 We shall view the lambda calculus as a notational extension of PL that allows us to bind
variables with a new operator λ:
o The prefix λx binds the occurrence of x in Sleep(x)
o We often say the prefix λx abstracts over x, and call expressions with such prefixes
lambda expressions or lambda abstractions




o We can use on lambda expression as the body of another one:
 We can think of the lambda calculus as a toll dedicated to gluing together the items needed to

build semantic representations
o The purpose of abstracting over variables is to mark the slots where we want
substitutions to be made
o Lambda abstractions can be seen as functors that can be applied to arguments (we
will use the symbol @ for functional application)
o A compound expression of this sort refers to the application of the functor λx.Sleep(x)
to the argument b
 Β-conversion (or λ-reduction)
o Compound expressions F@A can be seen as instructions to
 Throw away the λx prefix of the functor F, and
 Replace any occurrence of x bound by the λ-operator with the argument A

, o This replacement or substitution processes is called Β-conversion:



 Lambda abstraction, functional application and Β-conversion are the main ingredients we
need to deal with semantic construction:
o Once we have devised lambda abstractions to represent lexical items, we only need to
use functional application and Β-conversion to combine semantic representations
compositionally
 Given a syntactic constituent R with subparts Ra and Rb, we need to specify
which subpart is to be thought as the functor F and which as the argument A
 We then construct the semantic representation of R by functional application
F@A




λ -abstractions for Lexical Items
 Representing different basic categories
o Intransitive verbs (verbs with no object) and nouns are 1-place relations which are
missing their argument:



o What about determiners such as ‘a’ and ‘every’ in NPs like ‘a boxer’?
 For instance, we’d like to represent the meaning of ‘a boxer walks’ as:


 What does each word contribute to this formula? And what is the
contribution of the determiner?
 If ‘boxer’ contributes Boxer(x) and ‘walks’ contributes Walk(x) to the formula,
then the determiner ‘a’ must contribute something like Ǝx …
 Two bits are missing:
 The contribution of the NP (the restriction)
 The contribution of the VP (the scope)
 We can use lambda abstraction to mark the missing arguments that will be
filled in during semantic construction
 This is the representation for existential determiners: