CA Lecture 5b: Digital Logic

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CA Lecture 5b Digital Logic

• Properties of Boolean Algebra
• The Sum-of-Products (SOP) Form
• The Product-of-Sums (POS) Form Read Appendix A of
  textbook p450 p456

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Properties of Boolean Algebra
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Properties of Boolean Algebra Cont.

• The postulates are basic axioms of Boolean
  algebra and therefore need no proofs.
• The theorems can be proven from the postulates.
• Each relationship has both an AND form and an OR
  form as a result of the principle of duality.
• The dual form is obtained by replacing AND with
  OR and OR with AND, 1s with 0s, and 0s with
  1s.

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Properties of Boolean Algebra Cont.

• The commutative property states that the order
  that two variables appear in an AND or OR
  function is not significant.
• The distributive property shows how a variable is
  distributed over an expression.
• The identity property states that a variable that
  is ANDed with 1 or is ORed with 0 produces the
  original variable.
• The complement property states that a variable
  that is ANDed with its complement is logically
  false, and a variable that is ORed with its
  complement is logical true.

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Properties of Boolean Algebra Cont.

• The zero and one theorems state that a variable
  that is ANDed with 0 produces a 0, and a variable
  that is ORed with 1 produces a 1.
• The idempotence theorem states that a variable
  that is ANDed or ORed with itself produces the
  original variable.
• The associative theorem states that the order of
  ANDing or ORing is logically of no consequence.
• The involution theorem states that the complement
  of a complement leaves the original variable (or
  expression) unchanged.

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DeMorgans Theorem
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Computational Completeness

• Computational completeness means that any digital
  logic circuit can be created from these logic
  gates.
• Thus not all of the logic gates discussed so far
  are needed to achieve computational completeness.
• Three sets of logic gates that are computational
  complete are AND, OR, NOT, NAND, and NOR.

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Example of computational completeness

• A computationally complete set of logic gates can
  implement other logic gates that are not part of
  the set. E.g. implement the OR function with the
  NAND set.

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Example of computational completeness

• NAND gates alone implement all other logic gates,
  e.g. OR gate.
• Computational completeness implies functional
  equivalence among logic gates, which is important
  for many practical reasons.

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The Majority Function

• True whenever more than half of its inputs are
  true and can be thought of as a balance that tips
  to the left or right depending on whether there
  are more 0s or 1s at the input.
• No single logic gate discussed so far implements
  the majority function directly, we transform the
  function into a two-level AND-OR equation and
  then implement the function with an arrangement
  of logic gates from the set AND, OR, NOT (for
  instance).

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The Majority Function
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The Sum-of-Products Form

• One way to represent logic equations is to use
  the sum-of-products (SOP) form, in which a
  collection of ANDed variables are ORed together.
• The boolean logic equation that describes the
  majority function is in the following SOP

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AND-OR Implementation of the Majority Function
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Common Notations Used at Circuit Intersections
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The Sum-of-Products Form Cont.

• When a product term contains exactly one instance
  of every variable, either in true or complemented
  form, it is called a minterm.
• A minterm has a value of 1 for exactly one of the
  entries in the truth table.
• A new form of the equation

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The Product-of-Sums Form

• Another way to represent logic equations is to
  use the product-of-sums (POS) form, in which a
  collection of ORed variables are ANDed together.
• The boolean logic equation that describes the
  majority function is in the following POS
• A maxterm contains every variable in either true
  or false form for exactly once.
• A maxterm only has a value of 0 once in the truth
  table.

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OR-AND Implementation of Majority
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Complexity of circuit

• Complexity of circuit is affected by the
  following
• Gate count number of logic gates used in the
  circuit. The lower the gate count, the better the
  complexity of circuit.
• Gate input count number of inputs to all of the
  logic gates. The lower the gate input count, the
  better the complexity of circuit.
• Complexity of the wiring topology.

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Complexity of circuit

• For the example above (majority function), both
  SOP and POS forms have the same gate count (8)
  and gate input count (19), there is no difference
  in circuit complexity.
• However, for other functions the differences can
  be significant.

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Tutorial five questions

• Textbook appendix A A.2, A.24