Download Artificial Intelligence Chapter 4: Knowledge Representation

Artificial
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.
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Chapter 4: Knowledge Representation
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.
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Chapter 4: Knowledge Representation
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.
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Chapter 4: Knowledge Representation
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.”
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Chapter 4: Knowledge Representation
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).
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Chapter 4: Knowledge Representation
The truth tables for these are shown in the
following figure. These are obviously the AND and
OR truth tables from Boolean logic.
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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.
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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”
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Chapter 4: Knowledge Representation
The truth table for Modus Ponens is shown
in the following Figure.
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Modus
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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.
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Chapter 4: Knowledge Representation
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.
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Chapter 4: Knowledge Representation
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Chapter 4: Knowledge Representation
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.
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Chapter 4: Knowledge Representation
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).
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4.5.1
Chapter 4: Knowledge Representation
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.
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Chapter 4: Knowledge Representation
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.
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The
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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.
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Chapter 4: Knowledge Representation
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)
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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.
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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.
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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.
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4.6
First-Order
Chapter 4: Knowledge Representation
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.
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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.
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Chapter 4: Knowledge Representation
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).
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Chapter 4: Knowledge Representation
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
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A
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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, twowheeled
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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)
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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.
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Chapter 4: Knowledge Representation
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 )
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Chapter 4: Knowledge Representation
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.
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Chapter 4: Knowledge Representation
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.
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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:
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AND
OR
NOT
Logical Conditional (then)
Logical Biconditional
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Examples
Chapter 4: Knowledge Representation
of
compound
sentences
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)
are
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We can also build conditionals using the
logical conditional connective, for example:
- If ali knows mona, then mona knows ali.
Knows( ali, mona )
Knows( mona, ali )
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This can also be written as a biconditional,
which
changes
biconditional, a
the
meaning
slightly.
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.
Knows( ali, mona )
The
Knows( mona, ali )
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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.
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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.
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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)).
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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 (
)
)
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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.
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- 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 )