1. CSE20 Lecture 15 Karnaugh Maps Professor CK Cheng CSE Dept. UC San Diego 1

2. Example Given F =  m (3, 5), D =  m (0, 4) 0 2 6 4 1 3 7 5 b c a - 0 0 - 0 1 0 1 Primes:  m (3),  m (4, 5) Essential Primes:  m (3),  m (4, 5) Min exp: f(a,b,c) = a’bc + ab’ 2

3. Boolean Expression K-Map Variable x i and its compliment x i ’  Two half planes Rx i, and Rx i ’ Product term P (  x i * e.g. b’c’)  Intersect of Rx i * for all i in P e.g. Rb’ intersect Rc’ Each minterm  One element cell Two minterms are adjacent iff they differ by one and only one variable, eg: abc’d, abc’d’  The two cells are neighbors Each minterm has n adjacent minterms  Each cell has n neighbors 3

4. ProcedureInput: Two sets of F R D 1)Draw K-map. 2)Expand all terms in F to their largest sizes (prime implicants). 3)Choose the essential prime implicants. 4)Try all combinations to find the minimal sum of products. (This is the most difficult step) 4

5. Example Given F =  m (0, 1, 2, 8, 14) D =  m (9, 10) 1. Draw K-map 0 4 12 8 1 5 13 9 3 7 15 11 2 6 14 10 b c a d 1 0 0 1 1 0 0 - 0 0 0 0 1 0 1 - 5

6. 2. Prime Implicants: Largest rectangles that intersect On Set but not Off Set that correspond to product terms.  m (0, 1, 8, 9),  m (0, 2, 8, 10),  m (10, 14) 3. Essential Primes: Prime implicants covering elements in F that are not covered by any other primes.  m (0, 1, 8, 9),  m (0, 2, 8, 10),  m (10, 14) 4. Min exp:  m (0, 1, 8, 9) +  m (0, 2, 8, 10) +  m (10, 14) f(a,b,c,d) = b’c’ + b’d’+ acd’ 6

7. Another example Given F =  m (0, 3, 4, 14, 15) D =  m (1, 11, 13) 1. Draw K-map b c a d 1 1 0 0 - 0 - 0 1 0 1 - 0 0 1 0 0 4 12 8 1 5 13 9 3 7 15 11 2 6 14 10 7

8. 2. Prime Implicants: Largest rectangles that intersect On Set but not Off Set that correspond to product terms. E.g.  m (0, 4),  m (0, 1),  m (1, 3),  m (3, 11),  m (14, 15),  m (11, 15),  m (13, 15) 3. Essential Primes: Prime implicants covering elements in F that are not covered by any other primes. E.g.  m (0, 4),  m (14, 15) 4. Min exp:  m (0, 4),  m (14, 15), (  m (3, 11) or  m (1,3) ) f(a,b,c,d) = a’c’d’+ abc+ b’cd (or a’b’d) 8

9. Five variable K-map 0 4 12 8 c d b e 1 5 13 9 3 7 15 11 2 6 14 10 16 20 28 24 c d b e a 17 21 29 25 19 23 31 27 18 22 30 26 Neighbors of m 5 are: minterms 1, 4, 7, 13, and 21 Neighbors of m 10 are: minterms 2, 8, 11, 14, and 26 9

10. Six variable K-map d e c f d e c d e c f 48 52 60 56 d e c b 49 53 61 57 51 55 63 59 50 54 62 58 a 32 36 44 40 33 37 45 41 35 39 47 43 34 38 46 42 f f 0 4 12 8 1 5 13 9 3 7 15 11 2 6 14 10 16 20 28 24 17 21 29 25 19 23 31 27 18 22 30 26 10

11. Implicant: A product term that has non-empty intersection with on-set F and does not intersect with off-set R. Prime Implicant: An implicant that is not covered by any other implicant. Essential Prime Implicant: A prime implicant that has an element in on-set F but this element is not covered by any other prime implicants. Implicate: A sum term that has non-empty intersection with off-set R and does not intersect with on-set F. Prime Implicate: An implicate that is not covered by any other implicate. Essential Prime Implicate: A prime implicate that has an element in off-set R but this element is not covered by any other prime implicates. 11

12. Min product of sums Given F =  m (3, 5), D =  m (0, 4) 0 2 6 4 1 3 7 5 b c a - 0 0 - 0 1 0 1 Prime Implicates:  M (0,1),  M (0,2,4,6),  M (6,7) Essential Primes Implicates:  M (0,1),  M (0,2,4,6),  M (6,7) Min exp: f(a,b,c) = (a+b)(c )(a’+b’) 12

13. Corresponding Circuit a b a’ b’ c f(a,b,c,d) 13

14. Quiz 14 Given F =  m (0, 6), D =  m (2, 7), 1.Fill the Karnaugh map. 2.Identify all prime implicates 3.Identify all essential primes. 4.Find a minimal expression in product of sums format.

15. Another min product of sums example Given R =  m (3, 11, 12, 13, 14) D =  m (4, 8, 10) K-map b c a d 1 - 0 - 1 1 0 1 0 1 1 0 1 1 0 - 0 4 12 8 1 5 13 9 3 7 15 11 2 6 14 10 15

16. Prime Implicates:  M (3,11),  M (12,13),  M(10,11),  M (4,12),  M (8,10,12,14) Essential Primes:  M (8,10,12,14),  M (3,11),  M(12,13) Exercise: Derive f(a,b,c,d) in minimal product of sums expression. 16

17. 17 Summary Karnaugh Maps: Two dimensional truth table which mimics an n-variable cube with imaginary adjacency. Theme: Relation between Boolean algebra and Karnaugh maps. Key words: Primes, Essential Primes Goal: Minimal expression in the format of sum-of-products or product-of-sums.