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Circuit Design

Circuit Design. 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.

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Circuit Design

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  1. Circuit Design

  2. 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)

  3. 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)

  4. 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

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

  6. 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

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

  8. 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

  9. Logic Gates • OR gate (+, ) • AND gate (, ) • INV gate (  )

  10. 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

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

  12. Circuit Design Examples • Create network for …

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

  14. 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)

  15. 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

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

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