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Logical Equivalence Two formulas are logically equivalent if their truth tables are identical Logically Equivalent forms can look very different ((p  (q  r))  (q  p)) vs (p  q)  (p  q  r) ((p  q))  (q  r) vs (p  q  r)  (p  q  r)

Normal Forms Disjunctive Normal Form –Sum-of-Products –Ex: (p  q)  (p  q  r) Conjunctive Normal Form –Product-of-Sums –Ex: (p  q  r)  (p  q  r)

Disjunctive Normal Form Literal: variable or its negation Term: conjunction of m literals DNF: disjunction of n terms Every formula is logically equivalent to a formula in DNF

Disjunctive Normal Form To find DNF 1.Create truth table 2.For each line that is T, construct a term 3.Create disjunction of these terms Example: ((p  (q  r))  (q  p))

Conjunctive Normal Form Literal: variable or its negation Clause: disjunction of m literals CNF: conjunction of n clauses Every formula is logically equivalent to a formula in CNF

Conjunctive Normal Form To find CNF 1.Create truth table 2.For each line that is F, construct term 3.Negate term using DeMorgan to get clause 4.Create conjunction of clauses Example: ((p  q))  (q  r)

Logic Networks Claude Shannon (1938) Switches can be wired to produces signals 1 and 0 Combine switches in the right way and you can produce circuits to represent logic formulas

Logic Gates OR gate (+,  ) AND gate ( ,  ) INV gate ( )

Circuit Design Examples Design a network for … (a  b)  c(a  b)  (a  b) Determine the function for the network. a b a c b c

More Circuit Design Examples Design a network for … (a  c)  (b  c) Determine the function for the network. a b c

Circuit Design Examples Create network for … ABC

Minimization What is minimum? –Usually involves # connections & # gates How do we find? –Equivalence rules –Algorithmic

Two-level Minimization Minimal DNF algorithm Uses the equivalence rule: (a  b)  (a  b)  a Examples: –(a  b  c)  (a  b  c)  (b  c) –(a  b  c)  (a  b  c)  (a  b) –(a  b  c)  (a  b  c)

Quine McCluskey Alg. Takes a formula written in canonical DNF and simplifies it using the equivalence rule Produces a DNF formula with minimum number of terms

Practice Problems Mathematical Structures –Section 7.2: 1(b), 3, 9, 16, 18 –Section 7.3: 20, 21