Boolean and Sequential Logic

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Boolean and Sequential Logic

• Last week
• Basic Gates
• AND
• OR
• NOT
• NOR
• XOR
• NAND

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This week

• Boolean Algebra
• Combining gates
• Using truth tables
• Sequential logic

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Boolean Algebra
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Combining gates
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Truth table for previous slide
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Truth table to logic diagram
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• Looking at the truth table on the previous slide
• Output G is only true when the inputs A is false
  and B is true, or A is true and B is false.
• The output for an AND gate is only true when both
  the inputs are true, so if we build a circuit
  that when the combinations of inputs A is false
  and B is true, or A is true and B is false we
  get an true output we have built a circuit to do
  this logic operation.

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A is false and B is true

• So if we can find a way to make the output from
  AND be true for this combination part of the
  answer.
• There is no problem with B this is true.
• A is false so we need to pass it through a device
  that we A is false the output is true NOT gate.

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• We can do a similar operation for when A is true
  and B is true
• We also need a way of combining these two parts
  together so if either combination occurs we get
  an true (1) output.
• OR gate

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Combining gates
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• So far we have looked at combinational logic, put
  these gates together with a certain set of
  inputs, you always get a known output.
• Now we are going to consider combination of logic
  gates where what the previous output is
  important. This is sequential logic.

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• This ability to remember a previous state
  extends what can be done with logic gates.
• Basis of simple memory

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R-S Flip-Flop/Latch
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• For a R-S flip-flop based around the NOR gate.
• R S Q(time1unit of time)
• 0 0 0 - stays same (e.g. if 1 to starts then
  stays as 1)
• 0 1 1 -Q is set to 1
• 1 0 0 -Q is reset
• 1 1 X -indeterminate (do not do this !)

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D-type
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• Data (D) only appears at the output Q on a clock
  pulse.
• So if D1 on a clock pulse, R0,S1 and Q1.
• So if D0 on a clock pulse R1,S0 and Q0.
• Otherwise Q stays the same.

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Summary

• Combining gate
• Using truth tables
• Producing them
• Using them to get a Boolean expression and logic
  diagram

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Summary

• Introduced the concept that an output can be
  feedback as an input, so the result is dependent
  on the previous state of the outputs as well as
  the inputs.
• Simple memory (especially D-Type)

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Sources for further reading (Boolean)

• Burrel (2004) Chapter 3 - pages 49 57
• Tanenbaum (2005) page 138-145
• Dick(2003) page 98-101

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Background reading (Sequential)

• Chalk et al (2004) pages 31-38
• Tanenbaum (2005) pages 159-164