10.4.3 Using Lambdas

Using lambdas.

Let's return to the sentence ``A woman walks''. According to our grammar, a determiner and a common noun can combine to form a noun phrase. Our semantic analysis couldn't be simpler: we will associate the NP node with the functional application that has the determiner representation as functor and the noun representation as argument. Structurally, this is of course the same thing that we did in Section 10.3.4. Only this time the semantic contributions of constituents are generally -expressions, and we will simply read the @-symbols as application markers. In fact it will turn out that the combination of functional application and -reduction is a method of such generality that we can even completely disregard the phrase-indices (such as and ) that were part of our ``proto''-representations in Section 10.3.4.

Building a structured application...

As you can see from the picture, we use the -expression as our representation for the indefinite determiner ``a''. We'll take a closer look at this representation soon, after we've looked at how it does its job in the semantic construction process. But there's one thing that we have to remark already now. While the -bound variables in the examples we've seen so far were placeholders for missing constant symbols, and in our determiner-representation stand for missing predicates. The version of -calculus introduced here does not distinguish variables that stand for different kinds of missing information. Nevertheless we will stick to a convention of using lower case letters for variables that stand for missing constant symbols, and capital letters otherwise.

But now let's carry on with the analysis of the sentence ``A woman walks''. We have to incorporate the intransitive verb ``walks''. We assign it the representation . The following tree shows the final representation we obtain for the complete sentence:

The S node is associated with . We obtain this representation by a procedure analogous to that performed at the NP node. We associate the S node with the application that has the NP representation just obtained as functor, and the VP representation as argument.

...and reducing it.

Now instead of hand-tailoring lots of specially dedicated post-processing rules, we will simply -reduce as often as possible the expression we find at the S node. We must follow its (bracketed) structure when we perform -reduction. So we start with reducing the application . We have to replace by , and drop the prefix. The whole representation then looks as follows:

Beta conversion movie!

Let's go on. This time we have two applications that can be reduced. We decide to get rid of the . Replacing Q by we get:

Again we have the choice where to go on -reducing, but this time it should be clear that our choice doesn't make any difference for the final result (in fact it never does. This property of -calculus is called confluence ). Thus let's -reduce twice. We have to replace both and by . Doing so finally gives us the desired:

Determiner

Finally, let's have a closer look at the determiner-representation we've been using. Remember it was . Why did we choose this expression? In a way, there isn't really an answer to this question, except simply: Because it works.

So now let's have a closer look at why it works. We know that a determiner must contribute a quantifier and the pattern of the quantification. Intuitively, indefinite determiners in natural language are used to indicate that there is something of a certain kind (epressed in the so-called restriction of the determiner), about which one is going to say that it also has some other property (expressed in the so-called scope of the determiner). In the sentence ``A woman walks'', the ``a'' indicates that there is something of a kind, which is then specified to be ``woman'', that also has a certain property, which is then specified as ``walk''.

So for the case of an indefinite determiner, we know that the quantifier in its first-order formalization has to be existential, and that the main connective within the quantification is a conjunction symbol. This is the principle behind formalizing indefinite determiners in first-order logic.

Now clever use of -bound variables in our determiner representation allows us to leave unspecified all but just these two aspects. All that is already ``filled in'' in the representation is the quantifier and a little bit about the internal structure of its scope, namely the main connective . The rest is ``left blank'', and this is indicated using variables.

The second crucial aspect of a -expression is the order of prefixes. This is where the role of syntactic structure comes in: It should be obvious from the further construction process why we had to choose and not - the reason is simply that phrases and sentences containing determiners are generally built up syntactically as they are. So when deciding about the order of -prefixes of a meaning representation, one has to think of the right generalizations over the syntactic use of its natural language counterpart.