AI3391 Artificial Intelligence Session 25 Horn clause.pptx

AI3391 ARTIFICAL INTELLIGENCE
(II YEAR (III Sem))
Department of Artificial Intelligence and Data Science
Session 25
by
Asst.Prof.M.Gokilavani
NIET
1/24/2024 Department of AI & DS 1

TEXTBOOK:
• Artificial Intelligence A modern Approach, Third Edition, Stuart
Russell and Peter Norvig, Pearson Education.
REFERENCES:
• Artificial Intelligence, 3rd Edn, E. Rich and K.Knight (TMH).
• Artificial Intelligence, 3rd Edn, Patrick Henny Winston, Pearson
Education.
• Artificial Intelligence, Shivani Goel, Pearson Education.
• Artificial Intelligence and Expert Systems- Patterson, Pearson
Education.

Topics covered in session 25
• Propositional Logic: Knowledge-Based Agents, The Wumpus World, Logic,
Propositional Logic, Propositional Theorem Proving: Inference and proofs, Proof
by resolution, Horn clauses and definite clauses, Forward and backward chaining,
Effective Propositional Model Checking, Agents Based on Propositional Logic.

RECAP
• Knowledge: Taking decisions and reasoning to act efficiently.
• sentences are expressed in a language which is called a knowledge
representation language.
• Knowledge-base of KBA stores fact about the world.
• Types of knowledge
• Technique of knowledge representation
• Inference system: Inference system applies logical rules to the KB to deduce new
information.
• Inference Knowledge: Inferential knowledge approach represents knowledge in
the form of formal logics.
Example: Let's suppose there are two statements:
a. Marcus is a man
b. All men are mortal
Then it can be represented as:
a. man(Marcus)
b. ∀x = men (x) ----------> mortal (x)s

Basic of Logical Representation
• Logic : It consists of precisely defined syntax and semantics which
supports the sound inference.
• Each sentence can be translated into logics using syntax and
semantics.
• Syntax: Syntaxes are the rules which decide how we can construct
legal sentences in the logic. (Symbols)
• Semantics: Semantics are the rules by which we can interpret the
sentence in the logic. ( meaning of sentence).
• Pros: Logical representation enables us to do logical reasoning.
• Cons: Logical representation technique may not be very natural, and
inference may not be so efficient.

Propositional Logic
Logical representation can be categorized into mainly two logics:
• Propositional Logic
• Predicate Logic
• Propositional Logic: 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
can not be both.
• Example:
a) It is Sunday.
b) The Sun rises from West (False proposition)
c) 3+3= 7(False proposition)
d) 5 is a prime number.

Propositional Logic
• Propositional logic consists of an object, relations or function, and
logical connectives.
• These connectives are also called logical operators.

Types of Propositions
There are two types of Propositions:
a. Atomic Propositions
b. Compound propositions
a. Atomic Proposition: Atomic propositions are the simple propositions. It consists
of a single proposition symbol. These are the sentences which must be either true or
false.
Example:
a) 2+2 is 4, it is an atomic proposition as it is a true fact.
b) "The Sun is cold" is also a proposition as it is a false fact.
b. Compound proposition: Compound propositions are constructed by combining
simpler or atomic propositions, using parenthesis and logical connectives.
Example:
a) "It is raining today, and street is wet."
b) "Ankit is a doctor, and his clinic is in Mumbai."

Semantic Knowledge Representation
(predicate logic)
• Semantic networks are alternative of predicate logic for knowledge
representation.
• If-Then rule or IS-A relation (Inheritance).
• Pros: These networks are simple and easily understandable.
• Cons: Human-like memory (Which has 1015 neurons and links) to store the
information.
• Example:
Statements: a. Jerry is a cat.
b. Jerry is a mammal
c. Jerry is owned by Priya.
d. Jerry is brown colored.
e. All Mammals are animal.

Rule of Inference
• What is Inference? we need intelligent computers which can create
new logic from old logic or by evidence, so generating the conclusions
from evidence and facts is termed as Inference.
• Inference rules are the templates for generating valid arguments.
(formal sentence).
• Terminologies:
• Implication: It is one of the logical connectives which can be
represented as P → Q. It is a Boolean expression.
• Converse: The converse of implication, which means the right-hand
side proposition goes to the left-hand side and vice-versa. It can be
written as Q → P.

Rule of Inference
Contrapositive: The negation of converse is termed as contrapositive, and
it can be represented as ¬ Q → ¬ P.
Inverse: The negation of implication is called inverse. It can be represented
as ¬ P → ¬ Q.

Resolution
• The Resolution rule state that if P∨Q and ¬ P∧R is true, then Q∨R will
also be true. It can be represented as:

Proof of Resolution
• Resolution is a single inference rule which can efficiently operate on
the conjunctive normal form or clausal form.
• Unification is a key concept in proofs by resolutions.
• Clause: Disjunction of literals (an atomic sentence) is called a clause.
It is also known as a unit clause.
• Conjunctive Normal Form: A sentence represented as a conjunction
of clauses is said to be conjunctive normal form or CNF.
• Rule is also called the binary resolution rule because it only resolves
exactly two literals.

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).

Example: John likes all kind of food
Step 1: Conversion of Facts into FOL.
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.
Step 3: Negate the statement to be proved.
• In this statement, we will apply negation to the conclusion statements, which will
be written as
Step 4: Draw Resolution Graph

Horn clauses and definite clauses
• Definite clause: A clause which is a disjunction of literals
with exactly one positive literal is known as a definite clause or strict
horn clause.
• Horn clause: A clause which is a disjunction of literals with at most
one positive literal is known as horn clause. Hence all the definite
clauses are horn clauses.
Example:
(¬ p V ¬ q V k)
It has only one positive literal k.
It is equivalent to p ∧ q → k.

Topics to be covered in next session 26
• First Order Logic (FOL)
Thank you!!!

AI3391 Artificial Intelligence Session 25 Horn clause.pptx

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AI3391 Artificial Intelligence Session 25 Horn clause.pptx