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CSC258H1 (46)


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University of Toronto St. George
Computer Science
Steve Engels

 Coverting SOM to gates  Reducing Boolean expressions o Once you have a Sum-of-Minterms expression, it is easy to convert A B C Y this to the equivalent combination of gates 0 0 0 0 o Ex. 0 0 1 0 Y = m 0 + m 1 m + 2 3 0 1 0 0 Y = A*B*C + A*B*C + A*B*C + A*B*C 0 1 1 1 1 0 0 1 1 0 1 0 1 1 0 1 1 1 1 1 o Ex. given the above logic specs, can get the following  Then combine terms, like the last two: o Different final expressions possible depending combination process  Reducing circuits o Note example of Sum-of-Minterms (above) circuit design  Ex. about SOM, combine the middle and end terms: o To minimize the number of gates, we want to reduce the Boolean o What is considered the “simplest” expression? expression as much as possible from a collection of minterms to something smaller  In this case, “simple” denotes the lowest gate cost (G) or the  Boolean algebra review lowest gate cost with NOTs (GN)  To calculate the gate cost simply o Axioms add all the gates together  For GN cost Include the  From this we can extrapolate cost of NOT gates o Karnaugh maps (K-maps)  The technique used to to find the “simplest” expressions  Karnaught maps are a 2D gird where the values are minterms  Adjacent minterms only differ by a single value  Values of the grad are the output for that minterm o Other Boolean identities  K-maps can be of any size & have any number of inputs  Commutative Law: m0 m1 m 3 m 2
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