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Simplifying Boolean Expressions

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Algebra can remove unnecessary hardware

The purpose is not merely to make an expression prettier. An equivalent expression with fewer operations may require fewer gates, fewer connections, less loading, less power, and less accumulated propagation delay.

Example 1 — absorption

Y = A + A•B

Y = A (absorption)

Example 2 — use the easy pieces first

Y = A•1 + B•0

Y = A + 0 (identity and dominance)

Y = A

Example 3 — factor before simplifying

Y = A•B + A•~B

Y = A(B + ~B) (factor A)

Y = A•1 (complement)

Y = A

Another way to read Example 3

“A AND B, OR A AND NOT B” simply says that A is required, while B may be either state. Therefore B does not actually affect the result.

Do not optimize blindly

In real hardware, the mathematically smallest expression is not automatically the best implementation. Available ICs, timing, fan-out, hazards, power, board space, cost, and testability can change the practical choice.

Try these yourself

Simplify A + A•B. A B A+B 0

Absorption gives A + A•B = A.

Simplify A•1 + B•0. 0 1 A A+B

A•1 = A and B•0 = 0, so A + 0 = A.

Why simplify a Boolean expression before building it? To make the truth values less accurate It can reduce gate count, wiring, load, power, and accumulated delay To turn digital logic into AC To eliminate all specifications

A smaller equivalent circuit can use fewer devices and interconnections and can also reduce loading and propagation delay.

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