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Working with Discourse Representation Theory

Patrick Blackburn & Johan Bos

Lecture 3DRT and Inference

This lecture

Now that we know how to build DRSs for English sentences, what do we do with them?

Well, we can use DRSs to draw inferences.

In this lecture we show how to do that, both in theory and in practice.

Overview

Inference tasksWhy FOL?From model theory to proof theoryInference enginesFrom DRT to FOLAdding world knowledgeDoing it locally

The inference tasks

The consistency checking taskThe informativity checking task

Why First-Order Logic?

Why not use higher-order logic? Better match with formal semantics But: Undecidable/no fast provers available

Why not use weaker logics? Modal/description logics (decidable fragments) But: Can’t cope with all of natural language

Why use first-order logic? Undecidable, but good inference tools available DRS translation to first-order logic Easy to add world knowledge

Axioms encode world knowledge

We can write down axioms about the information that we find fundamental

For example, lexical knowledge, world knowledge, information about the structure of time, events, etc.

By the Deduction Theorem 1 … n |= iff |= 1& … & n

That is, inference reduces to validity of formulas.

From model theory to proof theory

The inference tasks were defined semantically

For computational purposes, we need symbolic definitions

We need to move from the concept of |= to |--

In other words, from validity to provability

Soundness

If provable then valid:

If |-- then |=

Soundness is a `no garbage` condition

Completeness

If valid then provable

If |= then |--

Completeness means that proof theory has captured model theory

Decidability

A problem is decidable, if a computer is guaranteed to halt in finite time on any input and give you a correct answer

A problem that is not decidable, is undecidable

First-order logic is undecidable

What does this mean?

It is not possible, to write a program that is guaranteed to halt when given any first-order formula and correctly tell you whether or not that formula is valid.

Sounds pretty bad!

Good news

FOL is semi-decidableWhat does that mean?

If in fact a formula is valid, it is always possible, to symbolically verify this fact in finite time

That is, things are only going wrong for FOL when it is asked to tackle something that is not valid

On some non-valid input, any algorithm is bound not to terminate

Put differently

Half the task, namely determining validity, is fairly reasonable.

The other half of the task, showing non-validity, or equivalenty, satisfiability, is harder.

This duality is reflected in the fact that there are two fundamental computational inference tools for FOL: theorem provers and model builders

Theorem provers

Basic thing they do is show that a formula is provable/valid.

There are many efficient off-the-shelf provers available for FOL

Theorem proving technology is now nearly 40 years old and extremely sophisticated

Examples: Vampire, Spass, Bliksem, Otter

Theorem provers and informativity

Given a formula , a theorem prover will try to prove , that is, to show that it is valid/uninformative

If is valid/uninformative, in theory, the theorem prover will always succeed

So theorem provers are a negative test for informativity

If the formula is not valid/uninformative, all bets are off.

Theorem provers and consistency

Given a formula , a theorem prover will try to prove , that is, to show that is inconsistent

If is inconsistent, in theory, the theorem prover will always succeed

So theorem provers are also a negative test for consistency

If the formula is not inconsistent, all bets are off.

Model builders

Basic thing that model builders do is try to generate a [usually] finite model for a formula. They do so by iteration over model size.

Model building for FOL is a rather new field, and there are not many model builders available.

It is also an intrinsically hard task; harder than theorem proving.

Examples: Mace, Paradox, Sem.

Model builders and consistency

Given a formula , a model builder will try to build a model for , that is, to show that is consistent

If is consistent, and satisfiable on a finite model, then, in theory, the model builder will always succeed

So model builders are a partial positive test for consistency

If the formula is not consistent, or it is not satisfiable on a finite model, all bets are off.

Finite model property

A logic has the finite model property, if every satisfiable formula is satisfiable on a finite model.

Many decidable logics have this property.

But it is easy to see that FOL lacks this property.

Model builders and informativity

Given a formula , a model builder will try to build a model for , that is, to show that is informative

If is satisfiable on a finite model, then, in theory, the model builder will always succeed

So model builders are a partial positive test for informativity

If the formula is not satisfiable on a finite model all bets are off.

Yin and Yang of Inference

Theorem Proving and Model Building function as opposite forces

Doing it in parallel

We have general negative tests [theorem provers], and partial positive tests [model builders]

Why not try to get of both worlds, by running these tests in parallel?

That is, given a formula we wish to test for informativity/consistency, we hand it to both a theorem prover and model builder at once

When one succeeds, we halt the other

Parallel Consistency Checking

Suppose we want to test [representing the latest sentence] for consistency wrto the previous discourse

Then: If a theorem prover succeeds in finding a proof

for PREV , then it is inconsistent If a model builder succeeds to construct a model

for PREV & , then it is consistent

Why is this relevant to natural language?

Testing a discourse for consistency

Discourse Theorem prover Model builder




Vincent is a man. ?? Model





Mia loves every man. ?? Model






Mia does not love Vincent. Proof ??

Parallel informativity checking

Suppose we want to test the formula [representing the latest sentence] for informativity wrto the previous discourse

Then: If a theorem prover succeeds in finding

a proof for PREV , then it is not informative If a model builder succeeds to construct

a model for PREV & , then it is informative


Testing a discourse for informativity
















Mia loves Vincent. Proof ??

Let`s apply this to DRT

Pretty clear what we need to do: Find efficient theorem provers for DRT Find efficient model builders for DRT Run them in parallel And Bob`s your uncle!

Recall that theorem provers are more established technology than model builders

So let`s start by finding an efficient theorem prover for DRT…

Googling theorem provers for DRT

Theorem proving in DRT

Oh no!Nothing there, efficient or otherwise.

Let`s build our own one.

One phone call to Voronkov later: Oops --- does it take that long to build one

from scratch? Oh dear.

Googling theorem provers for FOL

Use FOL inference technology for DRT

There are a lot FOL provers available and they are extremely efficient

There are also some interesting freely available model builders for FOL

We have said several times, that DRT is FOL in disguise, so lets get precise about this and put this observation to work

From DRT to FOL

Compile DRS into standard FOL syntaxUse off-the-shelf inference engines for

FOLOkay --- how do we do this?Translation function (…)fo

Translating DRT to FOL: DRSs

x1…xn

C1

.

.

.Cn

( )fo = x1… xn((C1)fo&…&(Cn)fo)

Translating DRT to FOL: Conditions

(R(x1…xn))fo = R(x1…xn)

(x1=x2)fo = x1=x2

(B)fo = (B)fo

(B1B2)fo = (B1)fo (B2)fo

Translating DRT to FOL:Implicative DRS-conditions

x1…xm

C1

.

.

.Cn

( B)fo = x1…xm(((C1)fo&…&(Cn)fo)(B)fo)

Two example translations

Example 1

Example 2

xman(x)walk(x)

y

woman(y)

xman(x)

eadore(e)agent(e,x)theme(e,y)

Example 1

xman(x)walk(x)

Example 1

xman(x)walk(x)

)fo(

Example 1

x( (man(x))fo & (walk(x))fo )

Example 1

x(man(x) & (walk(x))fo )

Example 1

x(man(x) & walk(x))

Example 2

y

woman(y)

xman(x)


Example 2

y

woman(y)

xman(x)


)fo(

Example 2

xman(x)


y ( )(woman(y))fo & ( )fo

Example 2

xman(x)


y ( )woman(y) & ( )fo

Example 2


)y (woman(y) &x((man(x))fo ( )fo)

Example 2


)y (woman(y) &x(man(x) ( )fo )

Example 2

y (woman(y) &x(man(x) e ( (adore(e))fo & (agent(e,x))fo & (theme(e,y))fo )))

Example 2

y (woman(y) &x(man(x) e (adore(e) & (agent(e,x))fo & (theme(e,y))fo )))

Example 2

y (woman(y) &x(man(x) e (adore(e) & agent(e,x) & (theme(e,y))fo )))

Example 2

y (woman(y) &x(man(x) e (adore(e) & agent(e,x) & theme(e,y))))

Basic setup

DRS:x y

vincent(x)mia(y)love(x,y)

Basic setup

DRS:

FOL: xy(vincent(x) & mia(y) & love(x,y))

x y


Basic setup

DRS:


Model: D = {d1} F(vincent)={d1} F(mia)={d1} F(love)={(d1,d1)}

x y


Background Knowledge (BK)

Need to incorporate BKFormulate BK in terms of first-order

axiomsRather than just giving to the theorem

prover (or model builder), we give it: BK & or BK

Basic setup

DRS: x y


Basic setup

DRS:


x y


Basic setup

DRS:


BK: x (vincent(x) man(x)) x (mia(x) woman(x)) x (man(x) woman(x))

x y


Basic setup

DRS:


BK: x (vincent(x) man(x)) x (mia(x) woman(x)) x (man(x) woman(x))

Model: D = {d1,d2} F(vincent)={d1} F(mia)={d2} F(love)={(d1,d2)}

x y


Local informativity

Example:

Mia is the wife of Marsellus. If Mia is the wife of Marsellus, Vincent will

be disappointed.

The second sentence is informative with respect to the first. But…

x y

mia(x)marsellus(y)wife-of(x,y)

Local informativity

x y z

mia(x)marsellus(y)wife-of(x,y)vincent(z)

wife-of(x,y) disappointed(z)

Local informativity

Local consistency

Example:

Jules likes big kahuna burgers. If Jules does not like big kahuna burgers,

Vincent will order a whopper.

The second sentence is consistent with respect to the first. But…

x y

jules(x)big-kahuna-burgers(y)like(x,y)

Local consistency

x y z

jules(x)big-kahuna-burgers(y)like(x,y)vincent(z)

u

order(z,u)whopper(u)

Local consistency

like(x,y)

DRT and local inference

Because DRS groups information into contexts, we now have natural means to check not only global, but also local consistency and informativity.

Important for dealing with presupposition.

Presupposition is not about strange logic. But about using classical logic in new ways.

Tomorrow

Presupposition and Anaphora in DRT

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