1. CSE 370 – Winter 2002 – Logic minimization - 1 Overview zLast lecture yExamples of Karnaugh maps, including Don’t cares zToday yDesign Examples yPrime implicants and implicants

2. CSE 370 – Winter 2002 – Logic minimization - 2 we'll need a 4-variable Karnaugh map for each of the 3 output functions Design example: two-bit comparator block diagram LT EQ GT A B C D A B C D N1 N2 ABCDLTEQGT 0000010 01100 10100 11100 0100001 01010 10100 11100 1000001 01001 10010 11100 1100001 01001 10001 11010 and truth table

3. CSE 370 – Winter 2002 – Logic minimization - 3 A' B' D + A' C + B' C D B C' D' + A C' + A B D' LT= EQ= GT= K-map for EQK-map for LTK-map for GT Design example: two-bit comparator (cont’d) 0 00100010 00 D A11 010001000 B C 10011001 00 D A00 10011001 B C 010001000 11 D A00 0 00100010 B C = (A xnor C) (B xnor D) LT and GT are similar (flip A/C and B/D) A' B' C' D' + A' B C' D + A B C D + A B' C D’

4. CSE 370 – Winter 2002 – Logic minimization - 4 two alternative implementations of EQ with and without XNOR XNOR is implemented with at least 3 simple gates Design example: two-bit comparator (cont’d)

5. CSE 370 – Winter 2002 – Logic minimization - 5 block diagram and truth table 4-variable K-map for each of the 4 output functions A2A1B2B1P8P4P2P1 00000000 010000 100000 110000 01000000 010001 100010 110011 10000000 010010 100100 110110 11000000 010011 100110 111001 Design example: 2x2-bit multiplier P1 P2 P4 P8 A1 A2 B1 B2

6. CSE 370 – Winter 2002 – Logic minimization - 6 K-map for P8 K-map for P4 K-map for P2K-map for P1 Design example: 2x2-bit multiplier (cont’d) 00 00 B1 A200 011101111 A1 B2 0 00010001 0 00100010 B1 A2 010001000 100010000 A1 B2 00 01 B1 A2 01010101 01100110 A1 B2 00 00 B1 A200 100010000 A1 B2 P8 = A2A1B2B1 P4= A2B2B1' + A2A1'B2 P2= A2'A1B2 + A1B2B1' + A2B2'B1 + A2A1'B1 P1= A1B1 Do we really need it?

7. CSE 370 – Winter 2002 – Logic minimization - 7 I8I4I2I1O8O4O2O1 00000001 00010010 00100011 00110100 01000101 01010110 01100111 01111000 10001001 10010000 1010XXXX 1011XXXX 1100XXXX 1101XXXX 1110XXXX 1111XXXX block diagram and truth table 4-variable K-map for each of the 4 output functions O1 O2 O4 O8 I1 I2 I4 I8 Design example: BCD increment by 1

8. CSE 370 – Winter 2002 – Logic minimization - 8 O8 = I4 I2 I1 + I8 I1' O4 = I4 I2' + I4 I1' + I4’ I2 I1 O2 = I8’ I2’ I1 + I2 I1' O1 = I1' O8 O4 O2 O1 Design example: BCD increment by 1 (cont’d) 00 X1X0X1X0 I1 I8 010001000 XX I4 I2 01 X0X0X0X0 I1 I801 XX I4 I2 01010101 X0X0X0X0 I1 I8 10011001 XX I4 I2 10 X1X0X1X0 I1 I801 XX I4 I2

9. CSE 370 – Winter 2002 – Logic minimization - 9 Definition of terms for two-level simplification zImplicant ysingle element of ON-set or DC-set or any group of these elements that can be combined to form a subcube zPrime implicant yimplicant that can't be combined with another to form a larger subcube zEssential prime implicant yprime implicant is essential if it alone covers an element of ON-set ywill participate in ALL possible covers of the ON-set yDC-set used to form prime implicants but not to make implicant essential zObjective: ygrow implicant into prime implicants (minimize literals per term) ycover the ON-set with as few prime implicants as possible (minimize number of product terms)

10. CSE 370 – Winter 2002 – Logic minimization - 10 0X110X111 10101010 D A 100010000 11 B C 5 prime implicants: BD, ABC', ACD, A'BC, A'C'D Examples to illustrate terms 01 10101010 D A 01010101 10 B C 6 prime implicants: A'B'D, BC', AC, A'C'D, AB, B'CD minimum cover: AC + BC' + A'B'D essential minimum cover: 4 essential implicants essential

11. CSE 370 – Winter 2002 – Logic minimization - 11 Algorithm for two-level simplification zAlgorithm: minimum sum-of-products expression from a Karnaugh map yStep 1: choose an element of the ON-set yStep 2: find "maximal" groupings of 1s and Xs adjacent to that element xconsider top/bottom row, left/right column, and corner adjacencies xthis forms prime implicants (number of elements always a power of 2) yRepeat Steps 1 and 2 to find all prime implicants yStep 3: revisit the 1s in the K-map xif covered by single prime implicant, it is essential, and participates in final cover x1s covered by essential prime implicant do not need to be revisited yStep 4: if there remain 1s not covered by essential prime implicants xselect the smallest number of prime implicants that cover the remaining 1s

12. CSE 370 – Winter 2002 – Logic minimization - 12 X101X101 011101111 D A 0X010X01 X001X001 B C 3 primes around AB'C'D' Algorithm for two-level simplification (example) X101X101 011101111 D A 0X010X01 X001X001 B C 2 primes around A'BC'D' X101X101 011101111 D A 0X010X01 X001X001 B C 2 primes around ABC'D X101X101 011101111 D A 0X010X01 X001X001 B C minimum cover (3 primes) X101X101 011101111 D A 0X010X01 X001X001 B C X101X101 011101111 D A 0X010X01 X001X001 B C 2 essential primes X101X101 011101111 D A 0X010X01 X001X001 B C

13. CSE 370 – Winter 2002 – Logic minimization - 13 Combinational logic summary zLogic functions, truth tables, and switches yNOT, AND, OR, NAND, NOR, XOR,..., minimal set zAxioms and theorems of Boolean algebra yproofs by re-writing and perfect induction zGate logic ynetworks of Boolean functions and their time behavior zCanonical forms ytwo-level and incompletely specified functions zSimplification ytwo-level simplification zLater yautomation of simplification ymulti-level logic ydesign case studies ytime behavior