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When a Boolean Function is realized as a canonical Normal Form (cDNF or cCNF) normally a large number of gates is necessary with a large number of gate inputs.

A simplified circuit implementation is therefore required. The following considerations indicate how such a reduction in circuit complexity can be achieved.

- Neighbouring fields of a Karnaugh Diagram differ in only one argument value, e.g. x
_{i}= 0 and x_{i}= 1. Is the function value for those fields (the content of the field) equal, then the argument x_{i}is irrelevant (i.e. the value of these two fields does not depend on x_{i}). In the description of the function these two single fields can be replaced by one combined area, that covers both fields. This new combined field will be expressed without the argument x_{i}, i.e. the parameter x_{i}was eliminated.

- Combined fields of this kind can on their part now be combined to even larger fields (2 -> 4, 4 -> 8, etc.). With each doubling of the area another argument will be eliminated. The combination of 2
^{n}fields eliminates n arguments. Obviously this means that only a number of fields can be combined that can be expressed as a power of 2 (2^{n}= 2,4,8,...).

Seen from the Boolean Algebra viewpoint this shows that the creation of larger areas means nothing else but repeatedly applying the equation

. (2.7) |

A Boolean function whose expression contains combined areas of this type does of course not present a canonical normal form anymore, but just a conjunctive (CNF) or disjunctive normal form (DNF).

**Example:
**

The following example demonstrates on the K-Map how these combined fields can be formed:

Figure 2.21: Forming combined areas. |

In this example the following 1-formations are possible:

- 4-Field m
_{0}, m_{1}, m_{4}, m_{5} - 2-Field m
_{1}, m_{3} - 2-Field m
_{4}, m_{6}

Taking into consideration these field combinations, the function will be:

(2-tier) | (2.8) |

or, introducing the function XOR:

(3-tier). | (2.9) |

Obviously this example does not allow the formation of any 0-areas, so that only the realization as a cKNF is possible:

. | (2.10) |

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