KR-IST: Lecture 7b First-order Logic

From reasoning to knowledge representation

The combination of rulebase and inference method can be viewed as a representation of knowledge for the domain, i.e., a knowledge base (KB).

A system which packages up knowledge represented this way is a knowledge-based or expert system.

From knowledge representation to logic

With a history stretching back 2000 years, the study of logic and formal reasoning was a kind of pre-computation AI.

AI methods of knowledge representation (KR) are generally based on adapted systems of formal logic.

What is a logic?

• Sentences in the language are constructed according to formal rules and relationships.

• A semantics identifies the formal meaning of any sentence.

• An inference method allows new sentences to be generated from existing sentences.

Propositional logic

Allows facts about the world to be represented as sentences formed from:

• propositional symbols: P, Q, R, S...

• And:

• Or:

• Not:

• Implies:

• Therefore:

• wrapping parentheses: (...)

• logical constants: true, false, unknown

Propositional logic examples

• `It is humid': Q

• `If it is humid, then it is hot': Q P

• `If it is hot and humid, then it is raining': (P Q) R

Truth tables

A truth table shows how truth values combine under the relevant relationship.

Basic rules of inference

• Modus ponens

P Q
P
Q

• Modus Tolens

P Q
Q
P

Soundness and completeness

This is normally a truth value, i.e., true or false.

The associated inference method is then said to be

• sound if it does not generate false facts (i.e., contradictions), and

• complete if it is able to produce every sentence that is logically entailed by any existing set of sentences.

Semantics

The way one fact follows another should be mirrored by the way one sentence is entailed by another.

Problems with propositional logic

We cannot represent property-based generalisations.

For example, it is impossible to represent this categorical syllogism in Propositional logic:

Every person is mortal
Tony Blair is a person
Therefore Tony Blair is mortal

First-order logic

• predicates which can represent properties, e.g., mortal(person), or relationships, e.g., likes(fred, sausages),

• existentially quantified variables, e.g., at least one X such that...

• universallly quantified variables, e.g., X it is the case that...

Using predicates to represent relationships

• `Alison likes Richard and chocolate'

• likes(alison, richard) likes(alison, chocolate)

Using predicates within a rule

• `If Richard is a friend of Alison then Alison likes Richard'

• friends(alison, richard) likes(alison, richard)

Using variables with predicates to capture generalisations

• `Every elephant is grey'

• X: elephant(X) grey(X)

Using quantifiers and variables

• `There is a white aligator': X: alligator(X) white(X)

• `Alison eats everything that she likes': X: likes(alison, X) eats(alison, X)

• `There exists some bird that doesn't fly': X: bird(X) flies(X).

• `Every person has something that they love': X: person(X) Y: loves(X,Y)

More examples

Consider these

• `Every gardener likes the sun.' x: gardener(x) likes(x,Sun)

• `You can fool some of the people all of the time.' x: (person(x) t: (time(t) => can-fool(x,t)))

• `You can fool all of the people some of the time.' x: (person(x) t: (time(t) can-fool(x,t)))

• `All purple mushrooms are poisonous.' x: (mushroom(x) purple(x)) poisonous(x)

• `No purple mushroom is poisonous.' x: purple(x) mushroom(x) poisonous(x) or, equivalently, x: (mushroom(x) purple(x)) poisonous(x)

Examples cont.

• `There are exactly two purple mushrooms.' x y: mushroom(x) purple(x) mushroom(y) purple(y) (x=y) z: (mushroom(z) purple(z)) ( (x=z) (y=z) )

• `Deb is not tall.' tall(Deb)

• `X is above Y if X is on top of Y or else there is a pile of one or more other objects directly on top of one another starting with X and ending with Y.' x y: above(x,y) (on(x,y) z (on(x,z) above(z,y)))

Problems with FOL

But its expressiveness complicates the derivation of inferences. (It gets easier if we exliminate existential quantification and assume `negation by failure'.)

Also, in FOL you cannot construct sentences which make assertions about other sentences. For example, you cannot say things like `there exists a property such that...'

For this task, you need a higher-order logic.

Special-purpose logics

For example

• fuzzy logic: evaluation in terms of probability;

• modal logic: evaluation in terms of a propositional attitude such as belief. Good for sentences containing `should', `must' etc.

• temporal logic: evaluation in terms of truth at a particular moment in time.

The Frame problem

This affects all varieties of knowledge representation but is particularly apparent where evaluation is in terms of truth, and rules are used to define the results of actions.

Frame problem example

paint(X, C) color(X, C)
move(X, P) position(X, P)

and it is known that

paint(tony, blue).
move(tony, garden).

We should then be able to infer that

colour(tony, blue) position(tony, garden)

But the inference is, in fact, logically unsound

There is the possibility that the colour of tony gets changed by the move action.

Nothing in what we know rules this out.

Addressing the frame problem

Such rules are known as frame axioms.

For example

move(X, P) color-before-move(X, C) color(X, C).

asserts the fact that moving an object will not affect its colour.

However, this is not satisfactory.

Since most actions do not affect most properties of a situation, in a domain comprising actions and properties, we are going to need approximately frame axioms.

The Epistemological Frame Problem

Imagine being the designer of a robot that has to carry out an everyday task, such as making a cup of tea. Now, suppose the robot has to take a tea-cup from the cupboard. The present location of the cup is represented as a sentence in its database of facts alongside those representing innumerable other features of the ongoing situation, such as the ambient temperature, the configuration of its arms, the current date, the colour of the tea-pot, and so on. Having grasped the cup and withdrawn it from the cupboard, the robot needs to update this database. The location of the cup has clearly changed, so that's one fact that demands revision. But which other sentences require modification?

Summary

• From reasoning to knowledge representation

• From knowledge representation to logic

• Propositional logic

• Truth tables

• Basic rules of inference

• Soundness and completeness

• Semantics

• Problems with propositional logic

• First-order logic

• Using variables with predicates and quantifiers to capture generalisations

• Problems with FOL

• Special-purpose logics

• The Frame problem

Questions

• What does it mean to say that predicate calculus is a `first-order' logic?

• What is the difference between implication and conjunction?

Exercises

• Represent the following assertions using propositional logic: Fred enjoys swimming or Fred enjoys dancing; Fred enjoys swimming and Fred enjoys dancing; Fred doesn't dance; If Fred enjoys dancing then Fred enjoys swimming.

• Represent the same assertions using first-order logic.

• State the inference known as modus ponens. State the inference known as modus tollens.

• Write out the truth table for the exclusive-or relation.

• List the principle limitations of first-order logic and give an example of a situation in which they would be significant.

Exercises cont.

• Consider this categorical syllogism.

Every knowledge representation is formal
Propositional logic is a knowledge representation
Therefore propositional logic is formal

• Produce the most accurate propositional representation of this assertion you can think of.

• Produce the best predicate logic representation you can think of.

Exercises cont.

• Represent the first three verses of the song `House of the Rising Sun' in FOL.

There is a house in New Orleans
They call the Rising Sun
And it's been the ruin of many a poor boy
And God I know I'm one
My mother was a tailor
She sewed my new bluejeans
My father was a gamblin' man
Down in New Orleans
Now the only thing a gambler needs
Is a suitcase and trunk
And the only time he's satisfied
Is when he's on a drunk

Exercises cont.

• Using first-order logic, represent as accurately as possible the information contained in these comments on the availability of mortgages during the credit crunch.

To be considered for the best mortgage deals during the current
difficult conditions, you must borrow substantially less than the
full purchase price, have a perfect credit record and be able to
act fast.
Only people who have built up savings over
several years and have shown their ability to live on less than
their salary are able to get a mortgage.
It is first-time buyers who are hardest hit by the need to stump up
a bigger deposit in order to get the choice of the best deals.

Exercises cont.

• Use FOL to represent the information contained in the lyric to `When a man loves a woman'.

When a man loves a woman
Can't keep his mind on nothing else
He'll trade the world
For the good things he's found
If she's bad, he can't see it
She can do no wrong
Turn his back on his best friend
If he put her down
When a man loves a woman
Spend his very last time
Tryin' to hold on to what he needs
He'd give up all his comfort
Sleep out in the rain
If she says that's the way it ought to be