Karnaugh Maps. 1 Introduction. 2 Karnaugh maps for 2 variables.

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1 Karnaugh Maps Introduction Consider sum of products expressions. Given two variables: x, y, as inputs. An expression for the output involves some or all of the products: xy, x y, xy, x y Karnaugh maps are used to simplify the expression (but not necessarily completely). 2 Karnaugh maps for 2 variables. A Karnaugh map is matrix whose cells represent terms (products) of a sum of products. Consider a Boolean expression representing a circuit with two inputs, x and y. The map has the following format. x x y y x y x y xy xy Alternate form y x Each cell represents one term and has a value of or. We will be interested in those cells with inside. For example, consider: = x y + x y y x Note how column has all ones, these cells correspond to the case where x=. Now look at the algebra involved: = x y + x y = x ( y + y ) = x () = x Hence the expression reduces to the form represented by s in the map, where x= (or x ).

2 2 Exercise : = xy + xy + x y. simplify the expression on the right. y x In this exercise, there are three terms, thus the map contains three cells with s. Circle adjacent s: The second column represents x, the second row represents y. Thus the formula reduces to x+y (x OR y). Thus the formula is equivalent to = x+y. 3 Karnaugh maps of 3 variables x y x y xy xy Alternate form: xy Note: each row/col. differs from the previous/next by one position (e.g. changes to or in neighboring columns). The maps are considered rolled or wrapped from one end to the other, thus the last column differs from the first in one position ( to differs in the left digit only). For all Karnaugh maps, this format must be observed (it does for maps of 2 variables and will also hold for maps of 4 variables). 4 maps of 4 variables alternate form: y y w x w x wx wx wx Note: the alternate form is shorter to create; this will be used in later examples.

3 3 5 Subcubes and coverings. A subcube is a set of exactly 2 m adjacent cells containing s, for m=,, The example given on page contains a subcube (m=) of 2 adjacent cells. For 4 adjacent cells, the cells may be arranged in one row/column or in a square configuration. Examples: ) x y +x : xy The covering or subcube made up of the two cells containing s represents the case where x= and =, or, the term x. (Note that y can be or, hence it does appear in the term.) 2) x y +xy : The ones wrap around yielding y. xy Even though you could define two subcubes (one around each ), always try to find the largest subcubes in this case, wrap the cube around the ends. 3) wxy +wx+wx y +wx. Do a 4x4 map. You find a square of ones representing w. wx Note that the maps in examples and 3 are just two different ways of displaying the variables. 4) w x y +w xy +wxy +wx y : Convince yourself that the first row of the map is s and represents y. 5) w x + w xy + w x + wx + wx. This involves intersecting subcubes. wx

4 4 w x, x, x represent the three subcubes. It s okay to intersect subcubes as done above. Thus we have: w x + w xy + w x + wx + wx = w x + x + x This is a good example of how a Karnaugh map makes simplifying an expression easy, compared to using Boolean laws. Note: After simplifying using Karnaugh maps, you may need to apply the laws of Boolean algebra to simplify further. 6 EXAMPLE: Design a majority logic function 6. majority logic function: initial formula Given three inputs, we begin by writing a formula in which each term on the right side contains at least two s. M = A BC + AB C + ABC + ABC 6.2 Using a Karnaugh map A B A B AB AB C C There are three subcubes (note that we cannot circle the three s in the second row a subcube always contains 2 n s (or a power of 2). Thus we need to use two subcubes in that row. The result is: M = AB + AC + BC Note: You use fewer gates by choosing BC (for example) instead of a single cube (A BC) 7 Remember: When setting up a map, each column and row must represent a term that differs in one position from its neighbors, including wrapping around the ends. When circling subcubes, try to make them as large as possible (this results in an expression using fewer gates). Overlapping subcubes is valid. This happens often when using the largest subcubes possible.

5 5 8 Exercises to do on your own we ll go over them in class at some point. ) For each of the following, draw a Karnaugh map, and, using the best subcubes, write an expression derived from the map that is equivalent to the original expression (below). You do not need to simplify. a) x'y'' + xy'' + x'' b) w x + w x y + wx y + wx c) w x y + w x y + w x + w x + wxy + wx d) w x y + w xy ' + wxy'' + wx y' + wxy + wx'y' + wx' + w'x'' + wx'' 2) Simplify the following using any of the methods covered in class so that the corresponding equivalent circuit uses as few gates as possible. A = x + ( + y )x 9 A note on format Consider the layout: y y w x w x wx wx If you are working with terms of the form: wx (or w x, etc.), then it may be visually easier to identify terms with cells by writing the map using the following variation. w x w x wx wx y y or wx

6 Note how you can read the inputs, wx, from left to right in this layout. It doesn t really matter what the order is for the listing of input variables. The maps may look different, but the resulting expressions will be equivalent (but not necessarily equal!). 6