Algebraic Simplification of Logic Circuits

The logic form which comes from the direct application of the truth table will work, but it is often inefficient and takes an unneccessarily large number of gates. Logic expressions can often be simplified algebraicly, and although there is no fixed procedure, the following rules are often helpful.

1. Use DeMorgan's theorem to put the original expression in a form involving only a sum of products.
   使用德摩根定理将原表达式转换为仅包含和-积形式的表达式。
2. Check the form for common factors and use the single variable theorems to eliminate terms after factoring.
   检查公因式，并利用单变量定理在因式分解后消去项。

Logic Simplification Example

This truth table can be realized in the (admittedly farfetched) form:

This example is from Tocci, Digital Systems, Sec. 4-3.

Logic Simplification Example

This truth table can be realized in the straightforward combination:

This example is from Tocci, Digital Systems, Sec. 4-3.

Logic Simplification Example

Faced with this truth table, a person with a very weird brain comes up with the logical expression:

This example is from Tocci, Digital Systems, Sec. 4-3.