Propositional Logic - Duke Computer Science · propositional variables. •Determine appropriate...

Lecture 03

Propositional Logic

Tuesday, January 15, 2013 Chittu Tripathy

Lecture 03

Today’s Menu


• Translating English to Propositional Logic

• Logic Puzzles

• Combinatorial Logic Circuits

• Boolean Information Retrieval

Some Applications of Propositional Logic

Propositional Equivalence • Equivalences

• Key Equivalences

• CNF and DNF

Satisfiability

Lecture 03

Translating English Sentences


Steps • Identify atomic propositions and represent using

propositional variables. • Determine appropriate logical connectives.

Example: Translate the following sentence into propositional logic:

“You can access the Internet from campus only if you are a computer science major or you are not a freshman.”

Solution: Let the variables p, q, and r represent:

p : You can access the internet from campus.

q : You are a computer science major.

r : You are a freshman.

p → (q ∨ ¬r )

Lecture 03

Logic Puzzles


• An island has two kinds of inhabitants, knights, who always tell the truth, and knaves, who always lie.

• You go to the island and meet A and B.

– A says “B is a knight.”

– B says “The two of us are of opposite types.”

Example: What are the types of A and B?

Solution: Let p and q be the statements that A is a knight and B is a knight,

respectively. Then p represents the proposition that A is a knave and q that B is a knave.

– If A is a knight, then p is true. Since knights tell the truth, q must also be true. Then (p ∧ q)∨ ( p ∧ q) would have to be true, but it is not. So, A is not a knight, and therefore, p must be true.

– If A is a knave, then B must not be a knight since knaves always lie. So, then both p and q hold since both are knaves.

Lecture 03

Combinatorial Logic Circuits


input/output signal: – 0 represents F

– 1 represents T

Logic Circuit

p1 p2 p3

pm

q1 q2 q3

qn

Lecture 03




– 1 represents T

Logic Circuit

p1 p2 p3

pm

q1 q2 q3

qn

NOT (inverter)

AND OR XOR

Logic Gates: ¬p p

p

q p ∧ q p

q p ∨q p

q p ⊕q

Lecture 03




– 1 represents T

Logic Circuit

p1 p2 p3

pm

q1 q2 q3

qn

NOT (inverter)

AND OR XOR

Logic Gates:

Two Special Logic Gates:

NAND = AND followed by NOT NOR = OR followed by NOT

¬p p p

q p ∧ q p

q p ∨q p

q p ⊕q

p

q ¬ (p ∧ q) p

q ¬ (p ∨q)

Complex Combinatorial Circuits can be constructed from these gates.

Lecture 03

Example of Combinatorial Logic Circuit ¬((p ∧ ¬q) ∨ (¬ p ∧ q)) ∧ (p ∨ q)


p

q

¬p

¬q

¬ p ∧ q

¬((p ∧ ¬q) ∨ (¬ p ∧ q))

p ∧ ¬q

p ∨ q

¬((p ∧ ¬q) ∨ (¬ p ∧ q)) ∧ (p ∨ q)

How can we show this to be equivalent to a simple AND gate, that is p ∧ q ?

Lecture 03

Document 1: The earth is perfectly spherical. It it? No. It is an

oblate spheroid. That is, the earth is slightly flatter at the poles and slightly more bulging at the equator.

Boolean Information Retrieval


Document 2: Is the cricket ball perfectly spherical? Well, almost,

but has a slightly raised sewn seam. The seam prevents it from being perfectly spherical. On the other hand, the seam helps with the swinging of the ball or to produce sideways deflection after pitching.

Query:

Document 3: Is the soccer ball perfectly spherical? Well, almost.

Actually, it is a spherical polyhedron.

(ball AND spherical) AND (bulging OR seam)

Document 1:

Document 2:

Document 3:

1 0 1 0

1 1

1 1

0 0

1 0

Result

0 1 0

Lecture 03

Propositional Equivalence


Lecture 03

Tautology, Contradiction, Contingency


• A tautology is a proposition which is always TRUE. – Example: p ∨¬p

• A contradiction is a proposition which is always FALSE. – Example: p ∧¬p

• A contingency is a proposition which is neither a tautology nor a contradiction, such as most previous propositions p we have seen

p ¬p p ∨¬p p ∧¬p

T F T F

F T T F

For any contingency p p ∨¬p is a tautology p ∧¬p is a contradiction

Lecture 03

• Two compound propositions p and q are logically equivalent if p↔q is a tautology.

• In other words, p and q always evaluate to the same truth value for any truth assignment to the propositional variables that constitute the compound propositions p and q.

• We write this as p ⇔ q or p ≡ q .

Propositional Equivalence


p q ¬p ¬q ¬p ∨ q p→ q ¬q → ¬p

T T F F T T T

T F F T F F F

F T T F T T T

F F T T T T T

Example: Show that ¬p ∨ q ≡ p → q ≡ ¬q → ¬p .

Lecture 03

De Morgan’s Laws


¬(p ∧ q) ≡ ¬p ∨ ¬q ¬(p ∨ q) ≡ ¬p ∧ ¬q

p q ¬p ¬q p ∧ q ¬(p ∧q)

¬p ∨¬q p ∨q ¬(p ∨q) ¬p∧¬q

T T F F T F F T F F

T F F T F T T T F F

F T T F F T T T F F

F F T T F T T F T T

Lecture 03

Key Logical Equivalences


Identity Laws Domination Laws Idempotent laws Double Negation Law Negation Laws Commutative Laws Associative Laws Distributive Laws Absorption Laws

Exercise: Prove these laws using Truth Table.

Lecture 03

The following logical equivalences (Rosen 1.3) are often useful for solving problems. They can be proved using Truth Tables. They can use used to prove more logical equivalences!

More Logical Equivalences


Logical Equivalences Involving Conditional Statements

Logical Equivalences Involving Biconditional Statements

Lecture 03

Example of Equivalence Proof


Example: Show that

is logically equivalent to

Solution:

¬((p ∧ ¬q) ∨ (¬ p ∧ q)) ∧ (p ∨ q) ≡ p ∧ q . Exercise: Show that

Lecture 03

Disjunctive Normal Form (DNF)


A propositional formula is in disjunctive normal form (DNF) if it consists of a disjunction of (1, … ,n) disjuncts where each disjunct consists of a conjunction of (1, …, m) atomic formulas or the negation of an atomic formula. In other words, a DNF is an OR of ANDs.

(p ∧ ¬q) ∨ (¬ p ∧ q) ∨ (p ∨ q) (p ∧ q ∧ r) ∨ (¬p ∧ q ∨ ¬r) (p ∧ (q ∨ r)) ∨ (¬p ∧ q ∨ ¬r) ¬(p ∨ q)

Example:

Not DNF

DNF

Lecture 03

Conjunctive Normal Form


A compound proposition is in Conjunctive Normal Form (CNF) if it is a conjunction of disjunctions. In other words, a CNF is an AND of ORs.

(p ∨ ¬q) ∧ (¬ p ∨ q) ∧ (p ∨ q) (p ∨ q ∨ r) ∧ (¬p ∨ q ∨ ¬r) (p ∨ (q ∧ r)) ∧ (¬p ∨ q ∨ ¬r) ¬(p ∧ q)

Example:

Not CNF

CNF

Lecture 03

Proposition to CNF and DNF


⇔

Proposition

CNF DNF

Every compound proposition can be rewritten in CNF or DNF.

Lecture 03

•Construct the Truth Table for the proposition • Pick each row that evaluates to T –If a variable r in this row is T then write it as it; otherwise,

write the negation of it, i.e., ¬r –OR these written literals (literal = variable or its complement)

Proposition ⇔ DNF


p q r r p q p q → r

T T T F T F

T T F T T T

T F T F T F

T F F T T T

F T T F T F

F T F T T T

F F T F F T

F F F T F T

Truth Table for p q → r Example:

(pqr)

(pqr)

(pqr) (pqr)

(pqr)

p q → r ≡

Lecture 03

Example: Convert the following formula to CNF:

Solution:

1. Eliminate implication signs

2. Move negation inwards; eliminate double negation

3. Convert to CNF using associative/distributive laws


Proposition ⇔ CNF

Express the formula using AND, OR and NOT. To convert it to CNF move negation inwards and use the distributive and associative laws.

(p → q ) (r → p )

(p q) (r p)

(p ∧ q) (r p)

(p r p) ∧ ( q r p)

Lecture 03

Propositional Satisfiability


• A compound proposition is satisfiable if there exists a truth assignment to its variables that make it TRUE. When no such assignments exist, the compound proposition is unsatisfiable.

• A compound proposition is unsatisfiable iff its negation is a tautology.

Examples: Test if the following propositions are satisfiable.

Solution: Satisfiable. Assign T to p, q, and r.

Solution: Satisfiable. Assign T to p and F to q.

Solution: Not satisfiable. We cannot find a possible truth assignment to the variables!

1

2

3

Lecture 03

How Hard is Satisfiablity?


• Satisfiability can be checked using a truth table • Size of the truth table doubles with each variable • Therefore, size (#rows) of truth table is exponential in the

number of variables • In general, no better way is known to test satisfiability efficiently

The P=NP? Problem is all about finding an efficient (polynomial time algorithm) to test satisfiability.

Million-Dollar Problem: P = NP?

Propositional Logic - Duke Computer Science · propositional variables. •Determine appropriate...

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