1. CS1502 Formal Methods in Computer Science
Lecture Notes 5
Normal Forms
Boolean Logic

2. Normal Forms
Using logical equivalences, we can transform logical sentences into more standard forms. Important for applications of computers science.
We will use the equivalences we saw earlier (e.g., DeMorgan’s laws, …)

3. Reminder: Equivalences
Two FOL sentences P and Q mean the same thing (are logically equivalent, written P  Q) iff they have the same truth value in all situations. If two sentences are logically equivalent, you can substitute one for the other.
Identity Laws: P ^ T  P; P v F  P
Domination Laws: P v T  T; P ^ F  F
Idempotent Laws: P v P  P; P ^ P  P
Double Negation: ~~P  P
Commutative Laws: P v Q  Q v P; P ^ Q  Q ^ P
Associative Laws: (P v Q) v R P v (Q v R)
(P ^ Q) ^ R  P ^ (Q ^ R)
Distributive Laws: P v (Q ^ R)  (P v Q) ^ (P v R)
P ^ (Q v R)  (P ^ Q) v (P ^ R)
DeMorgan’s Laws: ~(P ^ Q)  ~P v ~Q
~(P v Q)  ~P ^ ~Q

4. Negation Normal Form NNF
A sentence S is in negation normal form if the  is moved as far inside S as possible. Use DeMorgan’s and double negation laws [(A   B)  C]  (A   B)   C ( A    B)   C ( A  B)   C

5. Another Example
~(~(P ^ Q) v R)
P ^ Q ^ ~R

6. Conjunctive and disjunctive normal forms
CNF: a sentence that is a conjunction of one or more disjunctions of one or more literals
DNF: a sentence that is a disjunction of one or more conjunctions of one or more literals
First put the sentence in negation normal form, then continue to transform it.
CNF example: (~A v C v D) ^ (A v ~C v E) ^ W
DNF example: (~A ^ C ^ D) v (A ^ ~C ^ E) v W

7. We will use these normal forms when we do resolution
Example
[(A   B)  C]  (A   B)   C ( A    B)   C ( A  B)   C NNF, CNF ( A   C)  (B   C) NNF, DNF
We will use these normal forms when we do resolution
theorem proving

8. CNF or DNF? Cube(a) ^ Small(a) ^ ~Red(a) Trick question!!
CNF: all outside connectives are ^
(each term contains just one literal)
DNF: all inside connectives are ^
(there is just one term)

9. Boolean Algebra
The circuits in computers have inputs (0 or 1) and produce outputs (0 or 1). In 1938, Claude Shannon showed how the rules of propositional logic could be used to design circuits (in his MS thesis at MIT). These rules form the basis for Boolean Algebra.
1 is True and 0 is False.
+ is v ∙ is ^ ¯ is ~ (sometimes ‘) is Xor
All the logical equivalences we learned apply, as do the truth-table methods
Example of hardware minimization: Not covered; we ran out of time.

10. Not covered, we ran out of time.
Gates are Building Blocks of Circuits
AND
And, Xor, Not, Or, Nor, Nand
Not covered, we ran out of time.