AI subject

1. Artificial Intelligence
(CSE3002)
Winter Semester 2022-2023
Knowledge base and Inference Rules

2. The agent starts visiting from first square [1, 1], and we
already know that this room is safe for the agent. To build
a knowledge base for wumpus world, we will use some
rules and atomic propositions.
We need symbol [i, j] for each location in the wumpus
world, where i is for the location of rows, and j for
column location.
Atomic proposition variable for Wumpus world:
Knowledge-base for Wumpus world
Proves for the Wumpus-world using propositional logic:
For a 4 * 4 square board, there will be
7*4*4= 122 propositional variables.
Propositional variables = True/False
(Declarative statement)

3. Some Propositional Rules for the wumpus world:
Representation of Knowledgebase for Wumpus world:
Simple KB for Wumpus world when an agent moves from room
[1, 1], to room [2,1]:

4. • The sentences we write will suffice to derive ¬P1,2  there is no pit in [1,2]
• We label each sentence Ri so that we can refer to them:
Proportional logic to denote no pit in [1,2]
A Simple Knowledge-Base

5. Proportional Logic to denote the presence of Wumpus in 1,3

6. Finding the Wumpus using Rules of Inference

7. Inference rules are the templates for generating valid arguments. Inference rules are
applied to derive proofs in artificial intelligence, and the proof is a sequence of the
conclusion that leads to the desired goal.
Inference rules
Rules of Inference in Artificial intelligence
B

8. We can prove that wumpus is in the room (1, 3) using propositional rules which we have derived for
the wumpus world and using inference rule.
1. Apply Modus Ponens with ¬S11 and R1:
We will firstly apply MP rule with R1 which is ¬S11 → ¬ W11 ^ ¬ W12 ^ ¬ W21, and ¬S11 which will
give the output ¬ W11 ^ W12 ^ W12.
Prove that Wumpus is in the room (1, 3) using rules of inference

9. 2. Apply And-Elimination Rule:
After applying And-elimination rule to ¬ W11 ∧ ¬ W12 ∧ ¬ W21, we will get three statements:
¬ W11, ¬ W12, and ¬W21.
3. Apply Modus Ponens to ¬S21, and R2:
Now we will apply Modus Ponens to ¬S21 and R2 which is ¬S21 → ¬ W21 ∧¬ W22 ∧ ¬ W31, which
will give the Output as ¬ W21 ∧ ¬ W22 ∧¬ W31

10. 4. Apply And -Elimination rule:
Now again apply And-elimination rule to ¬ W21 ∧ ¬ W22 ∧¬ W31, We will get three statements:
¬ W21, ¬ W22, and ¬ W31.
5. Apply MP to S12 and R4:
Apply Modus Ponens to S12 and R4 which is S12 → W13 ∨. W12 ∨. W22 ∨.W11, we will get the output as W13∨
W12 ∨ W22 ∨.W11.

11. 6. Apply Unit resolution on W13 ∨ W12 ∨ W22 ∨W11 and ¬ W11 :
After applying Unit resolution formula on W13 ∨ W12 ∨ W22 ∨W11 and ¬ W11 we will get W13 ∨
W12 ∨ W22.
7. Apply Unit resolution on W13 ∨ W12 ∨ W22 and ¬ W22 :
After applying Unit resolution on W13 ∨ W12 ∨ W22, and ¬W22, we will get W13 ∨ W12 as output.

12. 8. Apply Unit Resolution on W13 ∨ W12 and ¬ W12 :
After Applying Unit resolution on W13 ∨ W12 and ¬ W12, we will get W13 as an output, hence it is
proved that the Wumpus is in the room [1, 3].

13. • Models are assignments of true or false to every proposition symbol.
• Wumpus world  the relevant proposition symbols are B1,1, B2,1, P1,1, P1,2, P2,1, P2,2, and P3,1
• With seven symbols, there are 27 = 128 possible models; in three of these, KB is true.
• In those three models, ¬P1,2 is true, hence there is no pit in [1,2]. On the other hand, P2,2 is true in
two of the three models and false in one, so we cannot yet tell whether there is a pit in [2,2].
A Simple Inference Procedure

15. • In the topic of Propositional logic, we have seen that how to represent statements using
propositional logic.
• In propositional logic, we can only represent the facts, which are either true or false.
• PL is not sufficient to represent the complex sentences or natural language statements.
• The propositional logic has very limited expressive power. Consider the following
sentence, which we cannot represent using PL logic.
• "Some humans are intelligent", or
• "Sachin likes cricket."
• To represent the above statements, PL logic is not sufficient, so we required some more
powerful logic, such as first-order logic.
Limitations of Proportional Logic

16. • First-order logic is another way of knowledge representation in artificial intelligence. It is an
extension to propositional logic.
• FOL is sufficiently expressive to represent the natural language statements in a concise way.
• First-order logic is a powerful language that develops information about the objects in a more easy
way and can also express the relationship between those objects.
• First-order logic (like natural language) does not only assume that the world contains facts like
propositional logic but also assumes the following things in the world:
• Objects: A, B, people, numbers, colors, wars, theories, squares, pits, wumpus, ..
• Relations: It can be unary relation such as: red, round, is adjacent, or n-any relation such as: the sister
of, brother of, has color, comes between
• Function: Father of, best friend, third inning of, end of, ......
• As a natural language, first-order logic also has two main parts:
• Syntax
• Semantics
First order Logic

17. Formal languages

18. • In Propositional logic
• Each model links the vocabulary of the logical sentences to elements of the possible world, so
that the truth of any sentence can be determined.
• Models for propositional logic link proposition symbols to predefined truth values.
• In first order logic
• They have objects in them.
• The domain of a model is the set of objects or domain elements it contains.
• The domain is required to be nonempty—every possible world must contain at least one
object.
• It doesn’t matter what these objects are—all that matters is how many there are in each
particular model
Syntax and semantics for FOL

19. Five Objects:
• Richard the Lionheart,
King of England from
1189 to 1199;
• His younger brother, the
evil King John, who
ruled from 1199 to
1215;
• The left legs of Richard
and John; and
• A crown
First Order Logic: Example

20. • Richard and John are brothers  a relation is just the set of tuples of objects that
are related.
• The crown is on King John’s head, so the “on head” relation contains just one
tuple,
• The “brother” and “on head” relations are binary relations—that is, they relate
pairs of objects
• The model also contains unary relations, or
• properties: the “person” property is true of both Richard and John; the “king”
property is true only of John (presumably because Richard is dead at this point);
and the “crown” property is true only of the crown.
First Order Logic: Example

21. • Certain kinds of relationships are best considered as functions, in that a given
object must be related to exactly one object in this way. For example, each person
has one left leg, so the model has a unary “left leg” function that includes the
following mappings
• Models in first-order logic require total functions, that is, there must be a value
for every input tuple. Thus, the crown must have a left leg and so must each of the
left legs.
First Order Logic: Example

22. • Three kind of symbols
• Constant symbols: stand for objects;
• Predicate symbols: stand for relations;
• Function symbols: stand for functions
• These symbols will begin with uppercase letters
• Constant symbols Richard and John;
• Predicate symbols Brother , OnHead, Person, King, and Crown;
• and the function symbol LeftLeg
• Each predicate and function symbol comes with an arity that fixes the number of arguments.
Symbols and Interpretation

23. • As in propositional logic, every model must provide the information required to determine if any given
sentence is true or false.
• in addition to its objects, relations, and functions, each model includes an interpretation that specifies
exactly which objects, relations and functions are referred to by the constant, predicate, and function
symbols.
Richard refers to Richard the Lionheart and John refers to the evil King John.
Brother refers to the brotherhood relation,
OnHead refers to the “on head” relation that holds between the crown and King John;
Person, King, and Crown refer to the sets of objects that are persons, kings, and crowns.
LeftLeg refers to the “left leg” function
Symbols and Interpretation

24. Other Interpretations:
• For example, one interpretation maps Richard to the crown and John to King
John’s left leg.
• There are five objects in the model, so there are 25 possible interpretations just for
the constant symbols Richard and John.
• Notice that not all the objects need have a name—for example, the intended
interpretation does not name the crown or the legs. It is also possible for an object
to have several names; there is an interpretation under which both Richard and
John refer to the crown.
Symbols and Interpretation

25. • An Atomic sentence (or atom for short) is formed from a predicate symbol
optionally followed by a parenthesized list of terms, such as Brother (Richard,
John).
• Atomic sentences can have complex terms as arguments  Married(Father
(Richard), Mother (John))
• An atomic sentence is true in a given model if the relation referred to by the
predicate symbol holds among the objects referred to by the arguments.
Atomic Sentences and Complex Sentences

26. • We can use logical connectives to construct more complex sentences, with the
same syntax and semantics as in propositional calculus. Here are four sentences
that are true in the model of Figure under our intended interpretation:
Atomic Sentences and Complex Sentences

27. • The syntax of FOL determines which collection of symbols is a logical expression
in first-order logic.
• The basic syntactic elements of first-order logic are symbols. We write statements
in short-hand notation in FOL.
• Basic Elements of First-order logic:
Syntax of First-Order logic

28. • Atomic sentences are the most basic sentences of first-order logic.
• These sentences are formed from a predicate symbol followed by a parenthesis
with a sequence of terms.
• We can represent atomic sentences as Predicate (term1, term2, ......, term n).
• Example: Ravi and Ajay are brothers: => Brothers(Ravi, Ajay).
Chinky is a cat: => cat (Chinky).
Atomic sentences

29. • Complex sentences are made by combining atomic sentences using connectives.
• First-order logic statements can be divided into two parts:
• Subject: Subject is the main part of the statement.
• Predicate: A predicate can be defined as a relation, which binds two atoms together in a statement.
• Consider the statement: "x is an integer.", it consists of two parts, the first part x is the subject
of the statement and second part "is an integer," is known as a predicate.
Complex Sentences

30. • A quantifier is a language element which generates quantification, and
quantification specifies the quantity of specimen in the universe of discourse.
• These are the symbols that permit to determine or identify the range and scope of
the variable in the logical expression.
• There are two types of quantifier:
• Universal Quantifier, (for all, everyone, everything)
• Existential quantifier, (for some, at least one).
Quantifiers in First-order logic

31. • Universal quantifier is a symbol of logical representation, which specifies that the statement
within its range is true for everything or every instance of a particular thing.
• The Universal quantifier is represented by a symbol ∀, which resembles an inverted A.
• If x is a variable, then ∀x is read as:
• For all x
• For each x
• For every x.
• Example:
• All man drink coffee.
• Let a variable x which refers to a cat so all x can be represented in UOD as below:
Universal Quantifier

32. •The main connective for universal quantifier ∀ is implication →.
•The main connective for existential quantifier ∃ is and ∧.

33. • Existential quantifiers are the type of quantifiers, which express that the statement within its scope is true
for at least one instance of something.
• It is denoted by the logical operator ∃, which resembles as inverted E. When it is used with a predicate
variable then it is called as an existential quantifier.
• If x is a variable, then existential quantifier will be ∃x or ∃(x). And it will be read as:
• There exists a 'x.'
• For some 'x.'
• For at least one 'x.'
Existential Quantifier:

34. Some Examples of FOL using quantifier:
• 1. All birds fly.
In this question the predicate is "fly(bird)."
And since there are all birds who fly so it will be represented as follows.
∀x bird(x) →fly(x).
• 2. Every man respects his parent.
In this question, the predicate is "respect(x, y)," where x=man, and y= parent.
Since there is every man so will use ∀, and it will be represented as follows:
∀x man(x) → respects (x, parent).
• 3. Some boys play cricket.
In this question, the predicate is "play(x, y)," where x= boys, and y= game. Since there are some
boys so we will use ∃, and it will be represented as:
∃x boys(x) → play(x, cricket).
• 4. Not all students like both Mathematics and Science.
In this question, the predicate is "like(x, y)," where x= student, and y= subject.
Since there are not all students, so we will use ∀ with negation, so following representation for
this:
¬∀ (x) [ student(x) → like(x, Mathematics) ∧ like(x, Science)].