Okay, to everybody who's watching the recording, I forgot to record the first part, which was

a recap anyway, so you can just listen to Tuesday's recording.

So we were talking about the representational layer. The representational layer allows us

to make some operations like adding another sentence, and meaningful and somewhat compositional

at the cost of having to retranslate into first-order logic for inferencing all the

time. The cost of the retranslation becomes higher and higher and higher with every sentence.

The second thing we did for inference here doesn't necessarily have any value there,

even if it could have.

Any questions so far?

There's a different semantics, a semantics we call a direct semantics, one where we directly go into

the first-order logic, and for that we give ourselves a first-order model,

i.e. a domain and an interpretation constants, and we're doing most of the things as we've always done it,

but for DRSs we have a different semantics than in first-order logic. Remember in first-order logic, sentences are true or false.

We can't do this anymore.

Sentences can be followed by other sentences, and what we do in the first one matters to the second one.

So we have to have a much better, bigger structure instead of true and false.

And the idea here is that we talk about states. States are essentially the same as variable assignments,

namely variables go to something in the domain, only that we're here applying them to discourse reference.

We're assigning values to discourse reference.

Rather than kind of like in first-order semantics, we treat the variable assignments kind of stack-like,

grow and shrink together with the three variables.

We kind of do more interesting things with the states here.

And a technical notion that we're going to need is that we have two states, phi and psi,

and a subset of discourse reference. We define that phi agrees with psi except on x,

if the obvious thing, phi of u, is the same as psi of u for all the discourse reference that are not in x.

That's going to be used later.

So to write down the meaning, we're basically doing exactly the same thing as in PLN theory,

for atomic propositions, for relations applied to a bunch of individuals.

But for the kind of really non-standard conditions, negation, disjunction, and implication, we do something interesting.

So technically, the value function for DRSs is a set of states.

And for technical reasons, we also need to have kind of the domain of that.

So we have a special evaluation function, the dynamic value function for DRSs.

And we're essentially going to call a DRS to be true if the second component of the dynamic value is non-etch.

So the dynamic evaluation of a DRSd, which is just standard DRS, is just basically we record the, it's a pair, we record the discourse reference.

And we have a set of states, namely those states, right, psi, that make the conditions true.

So far so easy.

So we have, as always in the evaluations, we have an interpretation coming in, and we have a variable assignment coming in.

Or a state coming in, in this case. We don't have any variables.

So we take the set of all states that make the conditions true, and they are allowed to do their own thing on the discourse reference that we're playing with in that DRS.

Conditions, we evaluate statically with, in this case, the extended psi.

And that's what essentially the value, the dynamic value of a DRS is just the sets of states that make the conditions true.

Everything else is to get the induction going.

The merge operator is relatively simple. The first component of the pair is just collect up the discourse reference, which is actually what the merge operations do syntactically, and take the intersection of the second components, which is exactly, since conditions are conjunctively exactly what you would expect.

Now,

the merge operator

on the discourse reference, on a discourse representation structure, just looks up whether the dynamics value is false.

The value of not D is true, i.e. non empty, true, if D is false, which in this case just means the second component is empty.

Disjunction, just as you would expect it, namely D is true or E is true, again, non empty.

And D implies E, that's the place where things become interesting.

After this, this is essentially, this says that if any state in the value of D can be extended to make E true.