Boolean and Sequential Logic Last week – Basic Gates AND OR NOT NOR XOR NAND

This week Boolean Algebra Combining gates Using truth tables Sequential logic

Boolean Algebra

Combining gates

Truth table for previous slide AB CD EF

Truth table to logic diagram ABCDEFG

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.

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.

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

Combining gates

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.

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

R-S Flip-Flop/Latch

For a R-S flip-flop based around the NOR gate. RSQ(time+1unit of time) 000- stays same (e.g. if 1 to starts then stays as 1) 011-Q is set to Q is reset 11X-indeterminate (do not do this !)

D-type

Data (D) only appears at the output Q on a clock pulse. So if D=1 on a clock pulse, R=0,S=1 and Q=1. So if D=0 on a clock pulse R=1,S=0 and Q=0. Otherwise Q stays the same.

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

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)

Sources for further reading (Boolean) Burrel (2004) Chapter 3 - pages 49 –57 Tanenbaum (2005) page Dick(2003) page

Background reading (Sequential) Chalk et al (2004) pages Tanenbaum (2005) pages