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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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

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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.

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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

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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

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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.

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