Artificial Intelligence Dr. Eng. Ahmed Moustafa Elmahalawy Computer Science and Engineering Department

Artificial IntelligenceArtificial Intelligence

Dr. Eng. Ahmed Moustafa Elmahalawy

Computer Science and Engineering Department

Artificial Intelligence Chapter 4: Knowledge Representation

4.5 Propositional Logic

Propositional Logic is a formal system in

which knowledge is represented as propositions.

Further, these propositions can be joined in

various ways using logical operators.

These expressions can then be interpreted

as truth-preserving inference rules that can be

used to derive new knowledge from the old, or test

the existing knowledge.

First, let’s introduce the proposition.

A proposition is a statement, or a simple

declarative sentence.

For example, “the book is expensive” is a

proposition.

Note that a definition of truth is not assigned

to this proposition; it can be either true or false.



In terms of binary logic, this

proposition could be false in Cairo, but

true in England. But a proposition

always has a truth value.

So, for any proposition, we can

define the true-value based on a truth

table (see the following figure).

This simply says that for any

given proposition, it can be either true

or false.

We can also negate our proposition to

transform it into the opposite truth value.

For example, if P (our proposition) is “the

book is expensive,” then ~P is “the book is not

expensive.”



Propositions can also be combined to create

compound propositions.

The first, called a conjunction, is true only if

both of the conjuncts are true (P and Q).

The second called a disjunction, is true if at

least one of the disjuncts are true (P or Q).

The truth tables for these are shown in the

following figure. These are obviously the AND and

OR truth tables from Boolean logic.



The power of propositional logic comes into

play using the conditional forms.

The two most basic forms are called

1- Modus Ponens.

2- Modus Tollens.

1- Modus Ponens is defined as:

P, (P->Q), infer Q

which simply means that given two

propositions (P and Q), if P is true then Q is true.

In English, let’s say that P is the proposition

“the light is on” and Q is the proposition “the switch

is on.” The conditional here can be defined as:

if “the light is on” then “the switch is on”


The truth table for Modus Ponens is shown

in the following Figure.



Modus Tollens takes the contradictory

approach of Modus Ponens.

With Modus Tollens, we assume that Q is

false and then infer that the P must be false.

Modus Tollens is defined as:

P, (P->Q), not Q, therefore not P.

Returning to our switch and light example,

we can say “the switch is not on,” therefore “the

light is not on.” The formal name for this method is

proof by contra positive. The truth table for Modus

Tollens is provided in the following figure.



A famous inference rule from propositional

logic is the hypothetical syllogism. This has the

form:

((P->Q) ^ (Q->R), therefore (P->R)

In this example, P is the major premise, Q is

the minor premise. Both P and Q have one

common term with the conclusion, P->R.



Propositional logic includes a number of

additional inference rules (beyond Modus Ponens

and Modus Tollens).

These inferences rules can be used to infer

knowledge from existing knowledge (or deduce

conclusions from an existing set of true premises).

4.5.1 Deductive Reasoning with

Propositional Logic

In deductive reasoning, the conclusion is

reached from a previously known set of premises.

If the premises are true, then the conclusion

must also be true.



Let’s now explore a couple of examples of

deductive reasoning using propositional logic. As

deductive reasoning is dependent on the set of

premises, let’s investigate these first.

1) If it’s raining, the ground is wet.

2) If the ground is wet, the ground is slippery.

The two facts (knowledge about the

environment) are Premise 1 and Premise 2. These

are also inference rules that will be used in

deduction.

Now we introduce another premise that it is

raining.

3) It’s raining.



Now, let’s prove that it’s slippery. First, using

Modus Ponens with Premise 1 and Premise 3, we

can deduce that the ground is wet:

4) The ground is wet. (Modus Ponens: Premise 1,

Premise 3)

Again, using Modus Ponens with Premise 3

and 4, we can prove that it’s slippery:

5) The ground is slippery. (Modus Ponens:

Premise 3, Premise 4)

4.5.2 Limitations of Propositional Logic

While propositional logic is useful, it cannot

represent general-purpose logic in a compact and

summary way.

For example, a formula with N variables has

2^N different interpretations. It also doesn’t support

changes in the knowledge base easily.



Truth values of propositions can also be

problematic, for example; consider the compound

proposition below.

This is considered true (using Modus Ponens

where P -> Q is true when P is false and Q is false,

see the truth table of Modus Ponens).

If dogs can fly, then cats can fly.

Both statements are obviously false, and

further, there’s no connection between the two. But

from the standpoint of propositional logic, they are

syntactically correct. A major problem with

propositional logic is that entire propositions are

represented as a single symbol.



4.6 First-Order Logic (Predicate

Logic)

One issue with propositional logic is that it’s

not very expressive. For example, when we

declare a proposition such as:

_ The ground is wet.

It’s not clear which ground we’re referring to.

Nor can we determine what liquid is making the

ground wet. Propositional logic lacks the ability to

talk about specifics.

In the other hand, we’ll explore predicate

calculus (otherwise known as First-Order Logic, or

FOL).

Using FOL, we can use both predicates and

variables to add greater expressiveness as well as

more generalization to our knowledge.


In FOL, knowledge is built up from

1- Constants (the objects of the knowledge)

2- A set of predicates (relationships between the

knowledge)

3- Some number of functions (indirect references

to other knowledge).



4.6.1 Atomic Sentences

A constant refers to a single object in our

domain.

A sample set of constants include:

ali, mona, bicycle, scooter, the-stranger, colorado

A predicate expresses a relationship

between objects, or defines properties of those

objects.

A few examples of relationships and

properties are defined below:

owns, rides, knows, person, sunny, book, two-

wheeled



With our constants and predicates defined,

we can now use the predicates to define

relationships and properties of the constants (also

called Atomic sentences).

First, we define that both Ali and Mona are

‘Persons.’ The ‘Person’ is a property for the

objects (Ali and Mona).

_Person( ali)

_Person( mona)

The above may appear as a function, with

Person as the function and Ali or Mona as the

argument.

But in this context, Person(x) is a unary

relation that simply means that Ali and Mona fall

under the category of Person.

Now we define that Ali and Mona both know

each other.



We use the knows predicate to define this

relationship.

Note that predicates have arity, which refers

to the number of arguments. The ‘Person’

predicate has an arity if one where the predicate

‘knows’ has an arity of two.

_Knows( ali, mona )

_Knows( mona, ali )

We can then extend our domain with a

number of other atomic sentences, shown and

defined below:

_Rides( ali, bicycle ) - ali rides a bicycle.

_Rides( mona, scooter ) - mona rides a scooter.

_Two-Wheeled (bicycle)- A Bicycle is two-

wheeled.

_Book( the-stranger ) - The-Stranger is a book.

_Owns( mona, Book(the-stranger) ) - mona owns a

book called The Stranger.


Finally, a function allows us to transform a

constant into another constant.

For example, the sister_of function is

demonstrated below:

_Knows( ali, sister_of( hassan ) ) -ali knows

Hassan’s sister.



4.6.2 Compound Sentences

Recall from propositional logic that we can

apply Boolean operators to build more complex

sentences.

In this way, we can take two or more atomic

sentences and with connectives, build a compound

sentence.

A sample set of connectives is shown below:


AND

OR

NOT

Logical Conditional (then)

Logical Biconditional


Examples of compound sentences are

shown below:

- ali and mona know one another.

Knows(ali, mona) Knows(mona, ali)

- ali knows mona, and mona does not know ali.

Knows( ali, mona ) Knows( mona, ali )

-ali rides a scooter or ali rides a bicycle.

Rides( ali, scooter) Rides( ali, bicycle)

We can also build conditionals using the

logical conditional connective, for example:

- If ali knows mona, then mona knows ali.




This can also be written as a biconditional,

which changes the meaning slightly. The

biconditional, a b simply means “b if a and a if

b,” or “b implies a and a implies b.”

- ali knows mona if mona knows ali.


NOTE Another way to think about the

biconditional is from the construction of two

conditionals in the form of a conjunction, or:

(a b) (b a)

This implies that both are true or both are false.


4.6.3 Variables

So far, we’ve explored sentences where all

of the information was present, but to be useful,

we need the ability to construct abstract sentences

that don’t specify specific objects.

This can be done using variables.



For example:

- If x Knows mona, then x is a Person.

Knows( x, mona ) Person( x )

If we also knew that: ‘Knows( ali, mona)’ then

we could deduce that ali is a person (Person(ali)).

4.6.4 Quantifiers

A quantifier is used to determine the quantity

of a variable.

In first-order logic, there are two quantifiers,

1- The universal quantifier ( )

2- The existential quantifier ( )



The universal quantifier is used to indicate

that a sentence should hold when anything is

substituted for the variable.

The existential quantifier indicates that there

is something that can be substituted for the

variable such that the sentence holds.

Let’s look at an example of each.

- There exists x, that is a Person.

x. Person( x )

- For all people, there exists someone that Knows

ali or mona.

x. x. Person( x ) (Knows( x, mona) Knows

(x, ali))

- For any x, if there is someone x that Knows

mona, then x is a Person.

x. x. Knows( x, mona ) Person( x )


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Artificial Intelligence Dr. Eng. Ahmed Moustafa Elmahalawy Computer Science and Engineering Department