KNOWLEDGE REPRESENTATION First Order Predicate Logic – Prolog Programming – Unification –

Forward Chaining-Backward Chaining – Resolution – Knowledge Representation – Ontological

Engineering-Categories and Objects – Events – Mental Events and Mental Objects – Reasoning
Systems for Categories – Reasoning with Default Information

Propositional logic in Artificial intelligence:

Propositional logic (PL) is the simplest form of logic where all the statements are made by propositions.
A proposition is a declarative statement which is either true or false. It is a technique of knowledge
representation in logical and mathematical form.

Example:

1. It is Sunday.
2. The Sun rises from West (False proposition)
3. 3+3= 7(False proposition)
4. 5 is a prime number.

Tautology: A proposition formula which is always true is called tautology, and it is also called a valid
sentence.
Contradiction: A proposition formula which is always false is called Contradiction.
Statements which are questions, commands, or opinions are not propositions such as "Where is Rohini",
"How are you", "What is your name", are not propositions.

Propositional Logic Connectives:

Truth Table:

Combine all the possible combination with logical connectives, and the representation of these
combinations in a tabular format is called Truth table. Following are the truth table for all logical
connectives:
Precedence of connectives:

Just like arithmetic operators, there is a precedence order for propositional connectors or logical
operators. This order should be followed while evaluating a propositional problem. Following is the list
of the precedence order for operators:
 Precedence Operators
First Precedence Parenthesis

Second Precedence Negation

Third Precedence Conjunction(AND)

Fourth Precedence Disjunction(OR)

Fifth Precedence Implication

Six Precedence Biconditional

Properties of Operators:
o Commutativity:
o P∧ Q= Q ∧ P, or
o P ∨ Q = Q ∨ P.
o Associativity:
o (P ∧ Q) ∧ R= P ∧ (Q ∧ R),
o (P ∨ Q) ∨ R= P ∨ (Q ∨ R)
o Identity element:
o P ∧ True = P,
o P ∨ True= True.
o Distributive:
o P∧ (Q ∨ R) = (P ∧ Q) ∨ (P ∧ R).
o P ∨ (Q ∧ R) = (P ∨ Q) ∧ (P ∨ R).
o DE Morgan's Law:
o ¬ (P ∧ Q) = (¬P) ∨ (¬Q)
o ¬ (P ∨ Q) = (¬ P) ∧ (¬Q).
o Double-negation elimination:
o ¬ (¬P) = P.

Limitations of Propositional logic:

o We cannot represent relations like ALL, some, or none with propositional logic. Example:

a. All the girls are intelligent.

b. Some apples are sweet.
Propositional logic has limited expressive power.
In propositional logic, we cannot describe statements in terms of their properties or logical
relationships.
PL logic is not sufficient, so we required some more powerful logic, such as first-order logic.

First-Order logic:
o First-order logic is another way of knowledge representation in artificial intelligence. It is an
extension to propositional logic.
o FOL is sufficiently expressive to represent the natural language statements in a concise way.

o First-order logic is also known as Predicate logic or First-order predicate logic. 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.

o As a natural language, first-order logic also has two main parts:

a. Syntax
b. Semantics

o 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.

Basic Elements of First-order logic:

Following are the basic elements of FOL syntax:

Constant 1, 2, A, John, Mumbai, cat,....

Variables x, y, z, a, b,....

Predicates Brother, Father, >,....

Function sqrt, LeftLegOf, ....

Connectives ∧, ∨, ¬, ⇒, ⇔

Equality ==

Quantifier ∀, ∃

Atomic sentences:
o 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.
o We can represent atomic sentences as Predicate (term1, term2, ......, term n).

Example: Ravi and Ajay are brothers: => Brothers(Ravi, Ajay).

Complex Sentences:
o Complex sentences are made by combining atomic sentences using connectives.
First-order logic statements can be divided into two parts:

o Subject: Subject is the main part of the statement.

o 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.

Quantifiers in First-order logic:

o A quantifier is a language element which generates quantification, and quantification specifies
the quantity of specimen in the universe of discourse.
o 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:

a. Universal Quantifier, (for all, everyone, everything)

b. Existential quantifier, (for some, at least one).

Universal Quantifier:

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:

o For all x
o For each x
o For every x.

Example:

All man drink coffee.

∀x man(x) → drink (x, coffee).

It will be read as: There are all x where x is a man who drink coffee.

o There exists a 'x.'

o For some 'x.'
o For at least one 'x.'

Example:

Some boys are intelligent.

∃x: boys(x) ∧ intelligent(x)

It will be read as: There are some x where x is a boy who is intelligent.

Points to remember:
o The main connective for universal quantifier ∀ is implication →.
o The main connective for existential quantifier ∃ is and ∧.

Properties of Quantifiers:
o In universal quantifier, ∀x∀y is similar to ∀y∀x.
o In Existential quantifier, ∃x∃y is similar to ∃y∃x.
o ∃x∀y is not similar to ∀y∃x.

Prolog Programming:

 Prolog stands for programming in logic. In the logic programming paradigm, prolog language is
most widely available.
  Prolog is a declarative language, which means that a program consists of data based on the
facts and rules (Logical relationship) rather than computing how to find a solution.
 A logical relationship describes the relationships which hold for the given application.
 To obtain the solution, the user asks a question rather than running a program. When a user
asks a question, then to determine the answer, the run time system searches through the
database of facts and rules.
 In prolog, logic is expressed as relations (called as Facts and Rules). Core heart of prolog lies
at the logic being applied. Formulation or Computation is carried out by running a query over
these relations.

Applications of Prolog:
o Specification Language
o Robot Planning
o Natural language understanding
o Machine Learning
o Problem Solving
o Intelligent Database retrieval
o Expert System
o Automated Reasoning

Syntax and Basic Fields:

 In prolog, we declare some facts. These facts constitute the Knowledge Base of the system.
We can query against the Knowledge Base. We get output as affirmative if our query is already
in the knowledge Base or it is implied by Knowledge Base, otherwise we get output as
negative.
 So, Knowledge Base can be considered similar to database, against which we can query.
Prolog facts are expressed in definite pattern.
 Facts contain entities and their relation. Entities are written within the parenthesis separated
by comma (, ).Their relation is expressed at the start and outside the parenthesis. Every
fact/rule ends with a dot (.). So, a typical prolog fact goes as follows:

Format : relation(entity1, entity2, ....k'th entity).

Example:
friends(raju, mahesh).
singer(sonu).
odd_number(5).
Explanation :
These facts can be interpreted as :
raju and mahesh are friends.
sonu is a singer.
5 is an odd number.

Key Features :
1. Unification : The basic idea is, can the given terms be made to represent the same structure.
2. Backtracking : When a task fails, prolog traces backwards and tries to satisfy previous task.
3. Recursion : Recursion is the basis for any search in program.

Advantages :
1. Easy to build database. Doesn’t need a lot of programming effort.
2. Pattern matching is easy. Search is recursion based.
3. It has built in list handling. Makes it easier to play with any algorithm involving lists.

Disadvantages :
1. LISP (another logic programming language) dominates over prolog with respect to I/O features.
2. Sometimes input and output is not easy.

Applications :
Prolog is highly used in artificial intelligence(AI). Prolog is also used for pattern matching over natural
language parse trees.

Unification:
o Unification is a process of making two different logical atomic expressions identical by finding a
substitution.
o Unification depends on the substitution process.
It takes two literals as input and makes them identical using substitution.
Let Ψ1 and Ψ2 be two atomic sentences and 𝜎 be a unifier such that, Ψ1𝜎 = Ψ2𝜎, then it can be
o
o
expressed as UNIFY(Ψ1, Ψ2).
o Example: Find the MGU for Unify{King(x), King(John)}

Let Ψ1 = King(x), Ψ2 = King(John),

Substitution θ = {John/x} is a unifier for these atoms and applying this substitution, and both
expressions will be identical.

o The UNIFY algorithm is used for unification, which takes two atomic sentences and returns a
unifier for those sentences (If any exist).
 o Unification is a key component of all first-order inference algorithms.
o It returns fail if the expressions do not match with each other.
o The substitution variables are called Most General Unifier or MGU.

E.g. Let's say there are two different expressions, P(x, y), and P(a, f(z)).

In this example, we need to make both above statements identical to each other. For this, we will
perform the substitution.

P(x, y)......... (i)

P(a, f(z))......... (ii)

o Substitute x with a, and y with f(z) in the first expression, and it will be represented as a/x and
f(z)/y.
o With both the substitutions, the first expression will be identical to the second expression and
the substitution set will be: [a/x, f(z)/y].

Conditions for Unification:

Following are some basic conditions for unification:

o Predicate symbol must be same, atoms or expression with different predicate symbol can never
be unified.
o Number of Arguments in both expressions must be identical.
o Unification will fail if there are two similar variables present in the same expression.

Unification Algorithm:

Algorithm: Unify(Ψ1, Ψ2)

Step. 1: If Ψ1 or Ψ2 is a variable or constant, then:

a) If Ψ1 or Ψ2 are identical, then return NIL.
b) Else if Ψ1is a variable,
a. then if Ψ1 occurs in Ψ2, then return FAILURE
b. Else return { (Ψ2/ Ψ1)}.
c) Else if Ψ2 is a variable,
a. If Ψ2 occurs in Ψ1 then return FAILURE,
b. Else return {( Ψ1/ Ψ2)}.
d) Else return FAILURE.
Step.2: If the initial Predicate symbol in Ψ1 and Ψ2 are not same, then return FAILURE.
Step. 3: IF Ψ1 and Ψ2 have a different number of arguments, then return FAILURE.
 Step. 4: Set Substitution set(SUBST) to NIL.
Step. 5: For i=1 to the number of elements in Ψ1.
a) Call Unify function with the ith element of Ψ 1 and ith element of Ψ2, and put the result into
S.
b) If S = failure then returns Failure
c) If S ≠ NIL then do,
a. Apply S to the remainder of both L1 and L2.
b. SUBST= APPEND(S, SUBST).
Step.6: Return SUBST.

Implementation of the Algorithm

Step.1: Initialize the substitution set to be empty.

Step.2: Recursively unify atomic sentences:

a. Check for Identical expression match.

b. If one expression is a variable vi, and the other is a term ti which does not contain variable vi,
then:

a. Substitute ti / vi in the existing substitutions

b. Add ti /vi to the substitution setlist.
c. If both the expressions are functions, then function name must be similar, and the
number of arguments must be the same in both the expression.

For each pair of the following atomic sentences find the most general unifier (If exist).

EXAMPLE:
S2 => { p(b, f(g(b)), f(g(b)); p(b, f(g(b)), f(g(b))} Unified Successfully.

And Unifier = { b/Z, f(Y) /X , g(b) /Y}.

3. Find the MGU of {p (X, X), and p (Z, f(Z))}

Here, Ψ1 = {p (X, X), and Ψ2 = p (Z, f(Z))

S0 => {p (X, X), p (Z, f(Z))}
SUBST θ= {X/Z}
S1 => {p (Z, Z), p (Z, f(Z))}
SUBST θ= {f(Z) / Z}, Unification Failed.

4. Find the MGU of UNIFY(prime (11), prime(y))

Here, Ψ1 = {prime(11) , and Ψ2 = prime(y)}

S0 => {prime(11) , prime(y)}
SUBST θ= {11/y}
S1 => {prime(11) , prime(11)} , Successfully unified.
Unifier: {11/y}.
5. Find the MGU of Q(a, g(x, a), f(y)), Q(a, g(f(b), a), x)}

Here, Ψ1 = Q(a, g(x, a), f(y)), and Ψ2 = Q(a, g(f(b), a), x)

S0 => {Q(a, g(x, a), f(y)); Q(a, g(f(b), a), x)}
SUBST θ= {f(b)/x}
S1 => {Q(a, g(f(b), a), f(y)); Q(a, g(f(b), a), f(b))}
SUBST θ= {b/y}
S1 => {Q(a, g(f(b), a), f(b)); Q(a, g(f(b), a), f(b))}, Successfully Unified.
Unifier: [a/a, f(b)/x, b/y].

6. UNIFY(knows(Richard, x), knows(Richard, John))

Here, Ψ1 = knows(Richard, x), and Ψ2 = knows(Richard, John)

S0 => { knows(Richard, x); knows(Richard, John)}
SUBST θ= {John/x}
S1 => { knows(Richard, John); knows(Richard, John)}, Successfully Unified.
Unifier: {John/x}.

Forward Chaining-Backward Chaining:

Inference engine:

The inference engine is the component of the intelligent system in artificial intelligence, which applies
logical rules to the knowledge base to infer new information from known facts. The first inference
engine was part of the expert system. Inference engine commonly proceeds in two modes, which are:

a) Forward chaining
b) Backward chaining

A. Forward Chaining

Forward chaining is also known as a forward deduction or forward reasoning method when using an
inference engine. Forward chaining is a form of reasoning which start with atomic sentences in the
knowledge base and applies inference rules (Modus Ponens) in the forward direction to extract more
data until a goal is reached.

The Forward-chaining algorithm starts from known facts, triggers all rules whose premises are satisfied,
and add their conclusion to the known facts. This process repeats until the problem is solved.

Properties of Forward-Chaining:

o It is a down-up approach, as it moves from bottom to top.

o It is a process of making a conclusion based on known facts or data, by starting from the initial
state and reaches the goal state.
 o Forward-chaining approach is also called as data-driven as we reach to the goal using available
data.
o Forward -chaining approach is commonly used in the expert system, such as CLIPS, business, and
production rule systems.

Consider the following famous example which we will use in both approaches:

Example:

"As per the law, it is a crime for an American to sell weapons to hostile nations. Country A, an enemy
of America, has some missiles, and all the missiles were sold to it by Robert, who is an American
citizen."

Prove that "Robert is criminal."

To solve the above problem, first, we will convert all the above facts into first-order definite clauses, and
then we will use a forward-chaining algorithm to reach the goal.

Facts Conversion into FOL:

o It is a crime for an American to sell weapons to hostile nations. (Let's say p, q, and r are

American (p) ∧ weapon(q) ∧ sells (p, q, r) ∧ hostile(r) → Criminal(p)

variables)
...(1)
o Country A has some missiles. ∃p Owns(A, p) ∧ Missile(p). It can be written in two definite
clauses by using Existential Instantiation, introducing new Constant T1.

Owns(A, T1) ......(2)

Missile(T1) .......(3)

∀p Missiles(p) ∧ Owns (A, p) → Sells (Robert, p, A)

o All of the missiles were sold to country A by Robert.
......(4)

o Missiles are weapons.

Missile(p) → Weapons (p) .......(5)

o Enemy of America is known as hostile.

Enemy(p, America) →Hostile(p) ........(6)

o Country A is an enemy of America.

Enemy (A, America) .........(7)

o Robert is American
American(Robert). ..........(8)

Forward chaining proof:

Step-1:
In the first step we will start with the known facts and will choose the sentences which do not have
implications, such as: American(Robert), Enemy(A, America), Owns(A, T1), and Missile(T1). All these
facts will be represented as below.

Step-2:

At the second step, we will see those facts which infer from available facts and with satisfied premises.

Rule-(1) does not satisfy premises, so it will not be added in the first iteration.

Rule-(2) and (3) are already added.

Rule-(4) satisfy with the substitution {p/T1}, so Sells (Robert, T1, A) is added, which infers from the
conjunction of Rule (2) and (3).

Rule-(6) is satisfied with the substitution(p/A), so Hostile(A) is added and which infers from Rule-(7).

Step-3:

At step-3, as we can check Rule-(1) is satisfied with the substitution {p/Robert, q/T1, r/A}, so
we can add Criminal(Robert) which infers all the available facts. And hence we reached our
goal statement.
B. Backward Chaining:
Backward-chaining is also known as a backward deduction or backward reasoning method when
using an inference engine. A backward chaining algorithm is a form of reasoning, which starts
with the goal and works backward, chaining through rules to find known facts that support the
goal.

Properties of backward chaining:

o It is known as a top-down approach.

o Backward-chaining is based on modus ponens inference rule.
o In backward chaining, the goal is broken into sub-goal or sub-goals to prove the facts
true.
o It is called a goal-driven approach, as a list of goals decides which rules are selected and
used.
o Backward -chaining algorithm is used in game theory, automated theorem proving tools,
inference engines, proof assistants, and various AI applications.
o The backward-chaining method mostly used a depth-first search strategy for proof.
Example:

Backward-Chaining proof:
In Backward chaining, we will start with our goal predicate, which is Criminal(Robert), and then
infer further rules.

Step-1:

At the first step, we will take the goal fact. And from the goal fact, we will infer other facts, and at
last, we will prove those facts true. So our goal fact is "Robert is Criminal," so following is the
predicate of it.

Step-2:

At the second step, we will infer other facts form goal fact which satisfies the rules. So as we can
see in Rule-1, the goal predicate Criminal (Robert) is present with substitution {Robert/P}. So we
will add all the conjunctive facts below the first level and will replace p with Robert.

Here we can see American (Robert) is a fact, so it is proved here.

Step-3:t At step-3, we will extract further fact Missile(q) which infer from Weapon(q), as it
satisfies Rule-(5). Weapon (q) is also true with the substitution of a constant T1 at q.

Step-4:
At step-4, we can infer facts Missile(T1) and Owns(A, T1) form Sells(Robert, T1, r) which satisfies
the Rule- 4, with the substitution of A in place of r. So these two statements are proved here.

Step-5:

At step-5, we can infer the fact Enemy(A, America) from Hostile(A) which satisfies Rule- 6.
And hence all the statements are proved true using backward chaining.
 S. Forward Chaining Backward Chaining
No.
1. Forward chaining starts from known Backward chaining starts from the goal and
facts and applies inference rule to works backward through inference rules to
extract more data unit it reaches to the find the required facts that support the goal.
goal.

2. It is a bottom-up approach It is a top-down approach

3. Forward chaining is known as data- Backward chaining is known as goal-driven
driven inference technique as we reach technique as we start from the goal and
to the goal using the available data. divide into sub-goal to extract the facts.

4. Forward chaining reasoning applies a Backward chaining reasoning applies a

breadth-first search strategy. depth-first search strategy.

5. Forward chaining tests for all the Backward chaining only tests for few
available rules required rules.

6. Forward chaining is suitable for the Backward chaining is suitable for diagnostic,
planning, monitoring, control, and prescription, and debugging application.
interpretation application.

7. Forward chaining can generate an Backward chaining generates a finite

infinite number of possible conclusions. number of possible conclusions.

8. It operates in the forward direction. It operates in the backward direction.

9. Forward chaining is aimed for any Backward chaining is only aimed for the
conclusion. required data.

Resolution
Resolution is a theorem proving technique that proceeds by building refutation proofs, i.e.,
proofs by contradictions. It was invented by a Mathematician John Alan Robinson in the year
1965.

Resolution is used, if there are various statements are given, and we need to prove a conclusion
of those statements. Unification is a key concept in proofs by resolutions. Resolution is a single
inference rule which can efficiently operate on the conjunctive normal form or clausal form.

Steps for Resolution:

1. Conversion of facts into first-order logic.
2. Convert FOL statements into CNF
3. Negate the statement which needs to prove (proof by contradiction)
4. Draw resolution graph (unification).

To better understand all the above steps, we will take an example in which we will apply
resolution.
Example:
a. John likes all kind of food.
b. Apple and vegetable are food
c. Anything anyone eats and not killed is food.
d. Anil eats peanuts and still alive
e. Harry eats everything that Anil eats.
Prove by resolution that:
f. John likes peanuts.

Step-1: Conversion of Facts into FOL

In the first step we will convert all the given statements into its first order logic.

Step-2: Conversion of FOL into CNF

In First order logic resolution, it is required to convert the FOL into CNF as CNF form makes easier
for resolution proofs.

o Eliminate all implication (→) and rewrite

a. ∀x ¬ food(x) V likes(John, x)
b. food(Apple) Λ food(vegetables)
c. ∀x ∀y ¬ [eats(x, y) Λ ¬ killed(x)] V food(y)
d. eats (Anil, Peanuts) Λ alive(Anil)
e. ∀x ¬ eats(Anil, x) V eats(Harry, x)
f. ∀x¬ [¬ killed(x) ] V alive(x)
g. ∀x ¬ alive(x) V ¬ killed(x)
h. likes(John, Peanuts).
Move negation (¬)inwards and rewrite
 a. ∀x ¬ food(x) V likes(John, x)
b. food(Apple) Λ food(vegetables)
c. ∀x ∀y ¬ eats(x, y) V killed(x) V food(y)
d. eats (Anil, Peanuts) Λ alive(Anil)
e. ∀x ¬ eats(Anil, x) V eats(Harry, x)
f. ∀x ¬killed(x) ] V alive(x)
g. ∀x ¬ alive(x) V ¬ killed(x)
h. likes(John, Peanuts).
Rename variables or standardize variables

a. ∀x ¬ food(x) V likes(John, x)
b. food(Apple) Λ food(vegetables)
c. ∀y ∀z ¬ eats(y, z) V killed(y) V food(z)
d. eats (Anil, Peanuts) Λ alive(Anil)
e. ∀w¬ eats(Anil, w) V eats(Harry, w)
f. ∀g ¬killed(g) ] V alive(g)
g. ∀k ¬ alive(k) V ¬ killed(k)
h. likes(John, Peanuts).

Eliminate existential instantiation quantifier by elimination.

In this step, we will eliminate existential quantifier ∃, and this process is known
as Skolemization. But in this example problem since there is no existential quantifier so
all the statements will remain same in this step.
Drop Universal quantifiers.
In this step we will drop all universal quantifier since all the statements are not implicitly
quantified so we don't need it.

a. ¬ food(x) V likes(John, x)
b. food(Apple)
c. food(vegetables)
d. ¬ eats(y, z) V killed(y) V food(z)
e. eats (Anil, Peanuts)
f. alive(Anil)
g. ¬ eats(Anil, w) V eats(Harry, w)
h. killed(g) V alive(g)
i. ¬ alive(k) V ¬ killed(k)
j. likes(John, Peanuts).
Step-3: Negate the statement to be proved

In this statement, we will apply negation to the conclusion statements, which will be written as
¬likes(John, Peanuts)

Step-4: Draw Resolution graph:

Now in this step, we will solve the problem by resolution tree using substitution. For the above
problem, it will be given as follows:

Hence the negation of the conclusion has been proved as a complete contradiction with the given
set of statements.

Explanation of Resolution graph:

o In the first step of resolution graph, ¬likes(John, Peanuts) , and likes(John, x) get
resolved(canceled) by substitution of {Peanuts/x}, and we are left with ¬
food(Peanuts)
o In the second step of the resolution graph, ¬ food(Peanuts) , and food(z) get resolved
(canceled) by substitution of { Peanuts/z}, and we are left with ¬ eats(y, Peanuts) V
killed(y) .
o In the third step of the resolution graph, ¬ eats(y, Peanuts) and eats (Anil,
Peanuts) get resolved by substitution {Anil/y}, and we are left with Killed(Anil) .
o In the fourth step of the resolution graph, Killed(Anil) and ¬ killed(k) get resolve by
substitution {Anil/k}, and we are left with ¬ alive(Anil) .
o In the last step of the resolution graph ¬ alive(Anil) and alive(Anil) get resolved.