Canonical Form (Boolean Algebra) - Application Example

Application Example

The sample truth tables for minterms and maxterms above are sufficient to establish the canonical form for a single bit position in the addition of binary numbers, but are not sufficient to design the digital logic unless your inventory of gates includes AND and OR. Where performance is an issue (as in the Apollo Guidance Computer), the available parts are more likely to be NAND and NOR because of the complementing action inherent in transistor logic. The values are defined as voltage states, one near ground and one near the DC supply voltage V_cc, e.g. +5 VDC. If the higher voltage is defined as the 1 "true" value, a NOR gate is the simplest possible useful logical element.

Specifically, a 3-input NOR gate may consist of 3 bipolar junction transistors with their emitters all grounded, their collectors tied together and linked to V_cc through a load impedance. Each base is connected to an input signal, and the common collector point presents the output signal. Any input that is a 1 (high voltage) to its base shorts its transistor's emitter to its collector, causing current to flow through the load impedance, which brings the collector voltage (the output) very near to ground. That result is independent of the other inputs. Only when all 3 input signals are 0 (low voltage) do the emitter-collector impedances of all 3 transistors remain very high. Then very little current flows, and the voltage-divider effect with the load impedance imposes on the collector point a high voltage very near to V_cc.

The complementing property of these gate circuits may seem like a drawback when trying to implement a function in canonical form, but there is a compensating bonus: such a gate with only one input implements the complementing function, which is required frequently in digital logic.

This example assumes the Apollo parts inventory: 3-input NOR gates only, but the discussion is simplified by supposing that 4-input NOR gates are also available (in Apollo, those were compounded out of pairs of 3-input NORs).