Discourse Representation Theory: An overview

Discourse Representation Theory:An overview

Part I: The linguistic problem

DRT: An overview - Part I: The linguistic problem

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Outline

1. A meaning representation language

2. Compositionality

3. Lambda Calculus

4. Moving to Discourse

5. Shortcomings with FOL


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1. A meaning representation language


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A semantic analysis of NL should answer at least the following questions:

What does a given sentence mean? How is the meaning representation of a sentence

built? How do we infer some piece of information out of

another? How is the meaning representation of a discourse

built?

This boils down to have meaning representation of the linguistic utterances.


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Why First Order Logic?Apparently all natural languages show a predicate argument

structure:

Is there anything shared by these two sentences?

1. “John loves Mary”2. “Mary loves John”

They have a not so overlapping linear order, but they seem strictly related all the same.

A philosophical argument: Assume a speaker S has a meaning representation for both proper names: would it

be possible for S to understand 1. and not 2.? If so, how does it come that a speaker could grasp the meaning of infinite sentences

(he never heard before) out of finite resources (his lexicon)?

Loves(john, mary)Loves(mary, john)

Loves(x,y)


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What does a given sentence mean?

A theoretical assumption: “The meaning of a sentence is its truth value”:

(better, its truth conditions, i.e. the state of affairs of the world which make the sentence true)

FOL supports a reliable concept of satisfaction and truth on models

if we can translate a NL sentence S in a FOL formula Φ (that is called the “proposition” corresponding to S), then we have description of its meaning in terms of truth in a model M=<D,I>.

“John loves Mary” loves(john,mary) M ╞ Φ iff <I(john), I(mary)> I(loves)

“A student loves Mary” x(student(x) loves(x,Mary))M ╞ Φ iff for any variable assignment g, M,g╞ x(student(x) loves(x,Mary)) iff for some x-variant g’ of g M,g’╞ student(x) loves(x,Mary)


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We can draw inferences from the meaning of the sentences we understand

We are told that:“Rudy’s is a vegetarian restaurant”And that:“vegetarian restaurants don’t serve meat”In FOL we have a reliable way to conclude“Rudy’s doesn’t serve meat” using an inference rule

instantiated by: VegRest(rudys)

x(VegRest(x) ¬Serve(x,meat))____________________________

¬Serve(rudys,meat)


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So, does FOL capture the what, the content of Natural Language semantics?

It’s quite controversial.

Adverbs semantics:

“Milly swims slowly”. MODIFICATION OF VERBS HIGHER ORDER PREDICATION

Adjective semantics:

There is a red apple” x(apple(x) red(x)) INTERSECTION OF CLASSES

“There is a small elephant in the zoo” x(elephant(x) small(x) in_the_zoo(x)) NO!

“John is a skillful violinist” skillful(john) violinist(john) NO!NON-INTERSECTIVE ADJECTIVES

Indexical expressions semantics:

“I heard a noise coming from behind me”HOW DO WE MAP “BEHIND” ON A (SUBJECT-INDEPENDENT) MODEL?

Etc.


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Even though FOL could capture the what of meaning, in itself it doesn’t say anything about how to build FOL formulas starting from sentences, and how to do it in a systematic way:

Take for example: “John loves Mary” loves(john,mary)

It seems that: “John” contributes the constant john “Mary” contributes the constant mary “loves” contribute the binary relation loves(x,y)

So, sentence meaning flows from lexicon, from words. But how it is precisely built? Why couldn’t we derive loves(mary,john) as well?

The missing point is syntactic structure!


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Here there is a hierarchy, not just a linear order of words, and it constrains the way we can fill the argument slots in the meaning representation of the relation LOVE(x,y). But notice that there is not specific mechanism defined on meaning representations that allows this unification.

So, we need a syntax-driven semantics for Natural Language, such that:

• the lexical items in a sentence give the basic ingredients for meaning representations;

• syntactic structure tells us how the semantic contributions of the parts are to be joined together.


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2. Compositionality


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Principle of Compositionality“The meaning of a complex expression is a function of the meanings of its parts and of the syntactic rules by which they are combined”.

In order to instantiate it we need to define:

Semantic primitives for lexical entries An independent syntactic theory contributing a notion of

“parts” and “rules” of combination we’ll assume a suitable CFG and parser… a description of the “function” or mapping from syntax to

semantics.


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5. Lambda Calculus


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We extend FOL with a variable binding operator . Basic expressions of the kind:

xP(x)where x is a formal parameter variable and P(x) is a FOL expression containing x.

Semantics: Occurrences of variables buond by are place-holders for missing information. We abstract over the bound variables to mark the slots for substitution.

-reduction (also called -conversion) is a basic mechanism that applies -expressions to terms yielding new expressions with all occurrences of formal parameters bound to the specified terms:

x.P(x)(a) P(a)

x.left(x)(john) left(john)

Semantics: Substitute the argument term for all the occurrences of the bound variables in the functor


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Lexical representations: PN: “John” j IV: “walks” x.walks(x) Det: “a” X.Y.z.X(z)Y(z) TV: “loves” x.y.loves(x,y)

We have now suitable representations to augment CFG with semantic attachments such that, for each CF rule, we have a function F to build the meaning of the symbols on the left out of the meanings of the symbols on the right:

A a1,…,an S(A) = {F(S(a1),…,S(an))}

NP pn S(NP) = {S(pn)}

NP det NP S(NP) = {S(det)S(NP)}

e.g. “a man” ( X.Y.z.X(z)Y(z))(x.man(x)) Y.z.Man(z)Y(z)


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John loves Mary (S)

loves(j,m)

John (NP) j

loves Mary (VP)x.loves(x,m)

Mary (PN) m

Mary(NP) m

loves (TV)

y.x.loves(x,y)

John (PN) j

We needed to know:

• exactly which variables in the TV’s meaning representation have to be replaced by the semantics of the TV’s arguments we did it changing the semantic attachment of TV in a -expression which make variables externally available for binding

• how to replace them -conversion


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6. Moving to Discourse


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Up to now, we addressed the meaning of linguistic unities up to sentences:

A discourse = def a set of collocated, related of sentences

NOTICE: We’ll focus only on semantics of “Monologue” (Dialog might imply a much more complex interaction...)

“John went to Bill’s car dealership to check out an Acura Integra. He looked at it for about an hour”.

The semantics of the second sentence is not independent on the one of the first

We have to look back to the semantics of preceding sentences to assign a meaning to the two pronouns

We find expressions back in the discourse which have been made contextually relevant


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We are focusing on a subset of discourse semantics, i.e. co-reference

Some terminology:

Reference = def The process by which expressions (noun phrases, proper names, pronouns, etc., also called “referring expressions”) denote, i.e. point to an entity (e.g. a person), called the referent.

Notice: we are assuming a suitable theory of reference and will be interested in modeling the phaenomenon of

co-reference in discourse

Anaphora = def reference to an entity (called an “antecedent”) which has been previously introduced into the discourse. E.g.:

“Mia is a woman. She loves Vincent”.


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“John went to Bill’s car dealership to check out an Acura Integra. He looked at it for about an hour”.

1. A method for building a Discourse Model that evolves with the dynamically-changing discourse it represents

2. A method for mapping between referring expression in discourse and entities in the Discourse Model

FUNCTIONAL REQUIREMENTS FOR CO-REFENCE:


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7. Shortcomings with FOL


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Succeding co-reference:

1. Mia is a woman. She loves Vincent.

2. A woman snorts. She collapses.

Failing coreference:

3. Every woman snorts. ?She collapses.

4. Mia didn’t order a two dollar sandwich. ?Vincent tasted it.


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1. “Mia is a woman. She loves Vincent”.

FOL Representations:

First attempt: WOMAN(Mia) LOVE(x, Vincent)

Post-processing: WOMAN(Mia) LOVE(Mia, Vincent)

WHAT’S THE RULE?


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2. A woman snorts. She collapses.

FOL representations:

z(WOMAN(z) SNORT(z)) COLLAPSE(X).

z(WOMAN(z) SNORT(z) COLLAPSE(z)).

AND WHAT’S THE RULE HERE?


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3. Every woman snorts. She collapses.

FOL Representations:

z(WOMAN(z) SNORT(z)) COLLAPSE(x)

z((WOMAN(z) SNORT(z)) COLLAPSE(z))

Why doesn’t it work?


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It seems that we should move to representations which:

keep the truth-conditional dimension of meaning typical of FOL we want to be able to evaluate discourses on standard models as we did for sentences

resolve anaphora in a systematic way mirror in some way the process of discourse

understanding with the functional requirements we described

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Discourse Representation Theory: An overview