AI-09 Logic in AI

ARTIFICIAL INTELLIGENCE
PANKAJ DEBBARMA
Deptt. of CSE, TIT, Narsingarh
Logic in AI

CONTENTS
Logic
• Logic
• Propositional Logic
• Predicate Logic
• Well-Formed Formula

Logic
Environment
Inference
Engine
Knowledge Base
Learning
(Updating KB)
Output
Input from
Environment
Operations performed by KBA
1. TELL the KB what it perceives from the environment.
2. ASK the KB what action it should perform.
3. PERFORM the selected action.

Logic
We consider three modern roles
for logic in AI:
1) logic as a basis for
computation,
2) logic for learning from a
combination of data and
knowledge, and
3) reasoning about the
behavior of machine learning
systems.

Propositional Logic
• In propositional logic, the most fundamental
propositions are called primitive propositions.
• Primitive propositions cannot be decomposed.
• Propositions that can be decomposed are compound
propositions.
• Primitive propositions can be denoted by some
symbols, and these symbols are called atomic
formulas.
• From atomic formulas we can construct various logic
formulas corresponding to various compound
propositions.

Propositional Logic
• In propositional logic, atomic formulas are propositions,
i.e., assertions such as
A := “Aryabhata is dead.”
B := “Nagpur is on the banks of Nag river.”
C := “Sudoku is fun.”
• Atomic formulas are denoted by capital letters A, B, C, etc.
• Each atomic formula is assigned a truth value: true (1) or
false (0).
• “Propositional logic is not the study of truth, but of the
relationship between the truth of one statement and that
of another.” (Hedman 2004)

Propositional Logic: Symbols
Word Symbol Example Meaning of Example Terminus Technicus
not ¬ ¬A not A negation
and ∧ A ∧ B A and B conjunction
or ∨ A ∨ B A or B disjunction
implies → A → B A implies B implication
if and only if ↔ A ↔ B A if and only if B biconditional
Relationships between propositions

Propositional Logic
The natural language words may have slightly different
meanings.
Example:
A ∧ B and B ∧ A should always have the same meaning.
But the sentences
She became sick and she went to the doctor.
and
She went to the doctor and she became sick.
have totally different meanings.

Propositional Logic: Syntax
formula := atomicFormula | complexFormula
atomicFormula := True | False | symbol
symbol := P | Q | R | . . .
complexFormula := ( ¬ formula ) [negation]
( formula ∧ formula ) [conjunction]
[positive literal] the same as atomicFormula.
[negative literal] ¬ P where P is an atomic formula.
Formulas are also called sentences or propositions.

Propositional Logic: Laws
Implication PQ=￢P∨Q
Equivalence P↔Q=(PQ)∧(QP)
Absorption laws P∧T=P, P∧F=F, P∨T=T, P∨F=P
Involution law ￢(￢P)=P
Idempotent laws P∧P=P, P∨P=P
Complement laws P∧￢P=F , P∨￢P=T
Commutative laws P∧Q=Q∧P, P∨Q=Q∨P
Associative laws (P∧Q)∧R= P∧(Q∧R), (P∨Q)∨R= P ∨(Q ∨ R)
Distributive laws P∧(Q∨R)=(P∧Q)∨(P∧R), P∨(Q∧R)=(P∨Q)∧(P∨R)
De Morgan’s laws ￢(P∧Q)=￢P∨￢Q, ￢(P∨Q)=￢P∧￢Q

Propositional Logic: Disadvantages
Example:
• All students are present in the class.
• Some students are absent in the class.
All ( ∀ ) and Some ( ∃ ) can be represented only
by quantifiers.

Predicate Logic | First Order Logic
Think about expressing these phrases in propositional logic:
A := “Socrates is human.”
B := “All humans are mortal.”
C := “Thus, Socrates is mortal.”
How can we see that A, B, C are related?
First-order logic is richer than propositional logic:
H(a)
∀x H(x)  M(x)
M(a)
where A stands for “Socrates”, H for “is human”, and M for “is mortal”.

Predicate Logic | First Order Logic
H(a)
∀x H(x)  M(x)
M(a)
So what do we have here?
– x is a variable. Variables denote arbitrary elements (objects) of an underlying
set.
– a is a constant. Constants denote specific elements of an underlying set.
– H and M are unary relations.
– ∀ is the all quantifier. It is read “for all”.
– We can also use the connectives we already know from propositional logic.
In first-order logic, there are also relations with other arities, as well as n-ary
functions. In addition to the all quantifier, there is the existential quantifier, read
“there exists”.

Predicate Logic: Symbols
• – Let {f, g, h, . . . , f1, f2, . . .} be the set of function
symbols. Every function symbol has a given arity.
Sometimes we write fn to denote that f has arity n.
• – Let {a, b, c, . . . , a1, a2, . . .} be the set of constant
symbols. Constant symbols can be seen as 0-ary function
symbols.
• – {P, R, S, . . . , P1, P2, . . .} be the set of relation symbols.
Every relation symbol (predicate) has a given arity.
Sometimes we write Pn to denote that P has arity n.
• – {x, y, z, x1, x2, . . .} be the set of variable symbols.

Well-Formed Formula (WFF)
• Propositional logic uses symbolic language
(rules) to represent logical structure
(syntactic).
• A well-formed formula (WFF) is a string of
symbols that is grammatically (syntactically)
correct by virtue of belonging to some
language of interest.
• Which symbolic representations are WFF ?

RULES
1. A any capital letter
2. ~ can be prefixed with NOT (e.g. ~A)
3. A V B letters with connectors between two WFF
Not WFF
1. A~ NOT cannot be suffix
2. ( A ) parenthesis can be used only to connect two
WFF, single is not valid
3. (A · ) connector cannot be suffix

• The capital of India is Naypyitaw.
• Orange is gray.
• x is blue. blue(x)
• x is round. round(x)
∀x blue(x) round(x) Is it WFF ?
• taller(A,B)
• taller(A,B) ∧ taller(B,C)
WFF ?
WFF ?

• If x is a variable representing people in the
world, then
taller(x, Ravi)
∀x taller(x, Ravi)
∃x taller(x, Ravi)
• taller(∃x, Ravi)
• taller(A ∧ B, C)
Is it WFF ?
Is it WFF ?

AI-09 Logic in AI

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