1. Minimisation ENEL111

2. Minimisation Last Lecture  Sum of products  Boolean algebra This Lecture  Karnaugh maps  Some more examples of algebra and truth tables

3. Karnaugh Maps K-Maps are a convenient way to simplify Boolean Expressions. They can be used for up to 4 or 5 variables. They are a visual representation of a truth table. Expression are most commonly expressed in sum of products form.

4. Truth table to K-Map ABP 001 011 100 111 B A01 011 11 minterms are represented by a 1 in the corresponding location in the K map. The expression is: A.B + A.B + A.B

5. K-Maps Adjacent 1’s can be “paired off” Any variable which is both a 1 and a zero in this pairing can be eliminated Pairs may be adjacent horizontally or vertically B A 01 011 11 a pair another pair B is eliminated, leaving A as the term A is eliminated, leaving B as the term The expression becomes A + B

6. Returning to our car example Two Variable K-Map ABCP 0000 0010 0101 0110 1001 1010 1101 1110 A.B.C + A.B.C + A.B.C BC A00011110 01 111 One square filled in for each minterm. Notice the code sequence: 00 01 11 10 – a Gray code.

7. Grouping the Pairs BC A 00011110 01 111 equates to B.C as A is eliminated. Here, we can “wrap around” and this pair equates to A.C as B is eliminated. Our truth table simplifies to A.C + B.C as before.

8. Groups of 4 BC A00011110 011 111 Groups of 4 in a block can be used to eliminate two variables: The solution is B because it is a 1 over the whole block (vertical pairs) = BC + BC = B(C + C) = B.

9. Karnaugh Maps Three Variable K-Map  Extreme ends of same row considered adjacent A BC 00011110 0 1 00 10

10. Karnaugh Maps Three Variable K-Map example A BC 00011110 0 1 X =

11. The Block of 4, again A BC 00011110 0 11 1 11 X = C

12. Returning to our car example, once more Two Variable K-Map ABCP 0000 0010 0101 0110 1001 1010 1101 1110 A.B.C + A.B.C + A.B.C AB C00011110 0111 1 There is more than one way to label the axes of the K-Map, some views lead to groupings which are easier to see.

13. Don’t Care States Sometimes in a truth table it does not matter if the output is a zero or a one Traditionally marked with an x. We can use these as 1’s if it helps. AB C 00011110 011 1xx1 ABCP 0000 001x 0101 011x 1001 1010 1101 1110

14. Karnaugh Maps Four Variable K-Map  Four corners adjacent AB CD 00011110 00 01 11 10

15. Karnaugh Maps Four Variable K-Map example AB CD 00011110 00 01 11 10 F =

16. AB CD 00011110 00 11 01 11 11 10 11 Karnaugh Maps Four Variable K-Map solution F = B.D + A.C 1

17. Product-of-Sums We have populated the maps with 1’s using sum-of-products extracted from the truth table. We can equally well work with the 0’s AB C 00011110 0111 11 ABCP 0000 0010 0101 0111 1001 1010 1101 1110 AB C 00011110 00 1000 P = (A + B).(A + C) P = A.B + A.C equivalent

18. Inverted K Maps In some cases a better simplification can be obtained if the inverse of the output is considered  i.e. group the zeros instead of the ones  particularly when the number and patterns of zeros is simpler than the ones

19. Karnaugh Maps Example: Z5 of the Seven Segment Display 0000100001 0001000010 0011000110 0100001000 0101001010 0110101101 0111001110 1000110001 X1X2X3X4Z5 1001010010 1010X1010X 1011X1011X 1100X1101X1110X1111X1100X1101X1110X1111X 01234567890123456789 0010100101 X 1 X 2 X3 X4 00011110 00 01 11 10 Z5 = Better to group 1’s or 0’s?

20. 7-segment display If there are less 1’s than 0’s it is an easier option: X X 1 X 2 X3 X4 00011110 00 1001 01 0001 11 XXX 10 10XX Changing this to 1 gives us the corner group.

21. Tutorial - Friday Print out the CS1 tutorial questions from the website. Come to see the answers worked through.