# Karnaugh Map

**Topics:**Karnaugh map, Race condition, Don't-care

**Pages:**6 (1639 words)

**Published:**October 4, 2011

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An example Karnaugh map

The Karnaugh map (K-map for short), Maurice Karnaugh's 1953 refinement of Edward Veitch's 1952 Veitch diagram, is a method to simplify Boolean algebra expressions. The Karnaugh map reduces the need for extensive calculations by taking advantage of humans' pattern-recognition capability, permitting the rapid identification and elimination of potential race conditions. In a Karnaugh map the boolean variables are transferred (generally from a truth table) and ordered according to the principles of Gray code in which only one variable changes in between adjacent squares. Once the table is generated and the output possibilities are transcribed, the data is arranged into the largest possible groups containing 2n cells (n=0,1,2,3...)[1] and the minterm is generated through the axiom laws of boolean algebra. Contents[hide] * 1 Example * 1.1 Truth table * 1.2 Karnaugh map * 1.3 Solution * 1.4 Inverse * 1.5 Don't cares * 2 Race hazards * 2.1 Elimination * 2.2 2-variable map examples * 3 See also * 4 References * 5 Further reading * 6 External links | [edit] Example

Karnaugh maps are used to facilitate the simplification of Boolean algebra functions. The following is an unsimplified Boolean Algebra function with Boolean variables A, B, C, D, and their inverses. They can be represented in two different notations: * f(A,B,C,D,E) = ∑(6,8,9,10,11,12,13,14) Note: The values inside ∑ are the minterms to map (i.e. rows which have output 1 in the truth table).

*

[edit] Truth table

Using the defined minterms, the truth table can be created:

#| A| B| C| D| E| f(A,B,C,D,E)|

0| 0| 0| 0| 0| 0| |

1| 0| 0| 0| 1| 0| |

2| 0| 0| 1| 0| 0| |

3| 0| 0| 1| 1| 0| |

4| 0| 1| 0| 0| 0| |

5| 0| 1| 0| 1| 0| |

6| 0| 1| 1| 0| 1| |

7| 0| 1| 1| 1| 0| |

8| 1| 0| 0| 0| 1| |

9| 1| 0| 0| 1| 1| |

10| 1| 0| 1| 0| 1| |

11| 1| 0| 1| 1| 1| |

12| 1| 1| 0| 0| 1| |

13| 1| 1| 0| 1| 1| |

14| 1| 1| 1| 0| 1| |

15| 1| 1| 1| 1| 0| |

[edit] Karnaugh map

K-map showing minterms and boxes covering the desired minterms. The brown region is an overlapping of the red (square) and green regions. The input variables can be combined in 16 different ways, so the Karnaugh map has 16 positions, and therefore is arranged in a 4 × 4 grid. The binary digits in the map represent the function's output for any given combination of inputs. So 0 is written in the upper leftmost corner of the map because ƒ = 0 when A = 0, B = 0, C = 0, D = 0, E = 0. Similarly we mark the bottom right corner as 1 because A = 1, B = 0, C = 1, D = 0, E = 0 gives ƒ = 1. Note that the values are ordered in a Gray code, so that precisely one variable changes between any pair of adjacent cells. After the Karnaugh map has been constructed the next task is to find the minimal terms to use in the final expression. These terms are found by encircling groups of 1s in the map. The groups must be rectangular and must have an area that is a power of two (i.e. 1, 2, 4, 8…). The rectangles should be as large as possible without containing any 0s. The optimal groupings in this map are marked by the green, red and blue lines. Note that groups may overlap. In this example, the red and green groups overlap. The red group is a 2 × 2 square, the green group is a 4 × 1 rectangle, and the overlap area is indicated in brown. The grid is toroidally connected, which means that the rectangular groups can wrap...

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