Circuit Design
Ultimately the point of boolean algebra is to facilitate good circuit design and ultimately to produce electronic equipment. Simplification and conversion to functionally complete sets are just a means to this end. All of the operators we have encountered can exist as logic gates in a circuit, and by combining these gates we can produce useful circuits - or "components".
When constructing a large system, we can group gates into components to abstract away a level of the overall circuit diagram - very useful when a large scale integration circuit could have hundreds of thousands of logic gates on one chip. This is shown in the example below, where the two components comprising a half-adder circuit are combined into one component (represented as a square in the circuit diagram)
A half-adder circuit, as shown above is an example of a combinatorial circuit - this means it has output values which at any point only relate to the input values and nothing else. Combinatorial circuits hold no state (or memory) unlike a sequential circuit - where the output will depend on input values, and the current state (or previous state) of the circuit. An example of sequential circuit is a flip-flop component, which is stores its current value - keeping its state until an input changes it - thus the flip-flop is essentially one bit of memory.
References:
- Janet Lavery - Durham University Computer Systems, Machine Architecture Lecture 4, 2007
- Hardware Algorithms for Arithmetic Models - http://www.aoki.ecei.tohoku.ac.jp/arith/mg/algorithm.html