Scribed Notes for 10-24-07

Class cancelled next week.

DISCOURSE REPRESENTATION THEORY

-> DRT is a kind of algebra for interpreting discourses as opposed to sentences. Sentences are no longer the scope over which an interpretation is calculated. 

-> Discourse IS NOT dialogue. Dialogue involves turn-taking and often refers to spontaneous speech. Discourse refers more to the structure of an argument. Examples include monographs written by a single author or long speeches by a single person, which have a logical train of thought. 

-> There is a whole level of linguistic structure for discourse studied by Rhetoric (or English or Lingusitics) departments. DRT is an omage to that, even though there isn't much discourse attributed to text under analysis. But we need to recognize that there is a structure that extends beyond the sentence.

DRT is concerned with anaphora:

Anaphora is the ability of referring expressions with-in all natural languages to refer to the same entities as earlier referring expressions. 

Pronominal anaphora: "John knows Mary. He loves her."

There are many other kinds of anaphora. You have anaphora of verb phrases, such as "does so" or "so do I". Some languages even have anaphoric tenses (you can only use a certain form of the past tense if you are referring to an earlier time that has been set by another sentence). 

Anaphora includes expressions that may not be anaphoric in every context:

"John knows Mary. The thrice-married actress despises him, however."

We know that "thrice-married actress" refers to "Mary" in this context.

So DRT focuses in on a number of aspects of discourse, like anaphora, that are not satisfactorilly studied in Montagovian analysis (what we've been doing)!

Now, you might think that you could resolve all these problems by stringing sentences together with conjunctions (i.e. "John knows Mary AND he loves her.")

But consider: "A man knows Mary. He loves her."
We need: exist(x) . (man(x) & knows(M)(x) & loves(M)(x))

Notice that x is bounded to the scope of the exist quantifier above. Thus, it is not suffient to just combine the sentences with "and".

So you might think that you could take all the quantifiers and pull them out to wide scope to solve the problem, but that doesn't work either.

Consider: "Exactly one boy knows Mary. He loves her."
We need: exist(x) . (every(y) . (boy(y) & knows(M)(y) <=> x=y) & boy(x) & knows(M)(x) & loves(M)(x))

The problem with widening the scope of every(y) is that while we are asserting that only one boy knows Mary, we are not asserting that only one boy loves her. Uniqueness of the boy does not extend from the first sentence to the second.

This tells us that introducing a referent is NOT the same as predicating on a referent.

DONKEY SENTENCES:

"If John owns a donkey, he beats it."

"Every farmer who owns a donkey beats it."

The challenge is to see that "a donkey" must have a universal interpretation.

For the first sentence, we need: 
every(x) . ((donkey(x) & own(x)(J)) -> beat(x)(J))

However, Montagovian analysis would typically give us:

exist(x) . ((donkey(x) & own(x)(J)) -> beat(x)(J))
Or: (exist(x) . (donkey(x) & own(x)(J))) -> beat(x)(J)

This is because under Montagovian semantics, "a donkey" would be translated to:
lambda(P) . exist(x) . (donkey(x) & P(x))

DRT is going to help us get around this problem. That's why Bos used it in the Pascal RTE Challenge.

DISCOURSE REPRESENTATION STRUCTURES:

The language of DRS will have...

1. Individual constants
2. Discourse reference markers (variables)
3. Predicate constants (n-ary)
4. Logical constants (e.g. not, and, or)
5. Equality

The combination of 1+2 are called "terms".

Each DRS is of form <V,C> where V is a finite set of reference markers and C is a finite set of constraints (i.e. conditions). 

The constraints are going to be the basic building blocks used to build the meanings from sentences, and can in fact recursively contain other DRS.

Constraints are of the forms:

1. P_n(t_1,...,t_n)
2. t_1 = t_2
3. Given X and Y as DRS: not(X), X -> Y, X or Y

So, a DRS will look like <{x_1,...,x_n}, {theta_1,...,theta_m}> where n,m >= 0.

You can think of a DRS as being like a quantifier, because the x's are bound in a scope and take scope over all the theta's. So that means that the scope of the x's extends beyond a single condition, and even beyond a single DRS.

Examples:

"John loves a girl who admires him. She loves him."
<{x, y}, {J=x, loves(y)(x), girl(y), admires(x)(y), loves(x)(y)}>

The conditions are interpreted conjunctively, and you can think of the DRS as being implicitly existentially quantified.

NOTE: The DRS is usually depicted with a box diagram (this does not appear in these notes).

"Every farmer who owns a donkey beats it."
<{}, {A->B}> 
where A = <{x,y}, {farmer(x), owns(y)(x), donkey(y)}> and B = <{}, {beat(y)(x)}>

The model theory:

We need the following new notations....

1. |= _(M,g) C: This means that condition C holds true in a model M with domain D, under the assignment g: R -> D, where R are discourse referents.

2. h |= _(M,g) C: This means that the assignment h is a "verifying embedding" in a model M with respect to g for condition C.

By saying that h is a verifying embedding, we mean that h is going to assign elements in the domain to C and that h disagrees with g on at most x_1,...,x_n where C = <{x_1,...,x_n}, ...>.

Defining these notions recursively we have...

1. |= _(M,g) P(t_1,...t_n): <[[t_1]]_(M,g),...,[[t_n]]_(M,g)> is in [[P]]_(M,g)
2. |= _(M,g) t_1 = t_2: [[t_1]]_(M,g) = [[t_2]]_(M,g)
3. |= _(M,g) not(X): There is no h such that h |= _(M,g) X
4. |= _(M,g) X -> Y: For all h, if h |= _(M,g) X then there if K such that k |= _(M,h) Y