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Two Level Networks

Two Level Networks. SOPs. A function has, in general many SOPs Functions can be simplified using Boolean algebra. Compare the implementation of the original to the final representation. Don’t Care Conditions. In practice there are combinations that will never occur

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Two Level Networks

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  1. Two Level Networks

  2. SOPs • A function has, in general many SOPs • Functions can be simplified using Boolean algebra Compare the implementation of the original to the final representation Two-Level Networks

  3. Don’t Care Conditions • In practice there are combinations that will never occur • we may pick the most convenient assignment • A BCD input is greater than 5 F = m(6,7,8,9)+ d(10,11,12,13,14,15) don’t cares = 0 F= ABC + ABC don’t cares = 1 F= BC + A Two-Level Networks

  4. Karnaugh Map Adjacent cells are at Hamming distance 1 F = C + AB Two-Level Networks

  5. Examples • F = m(1,4,5)+ d(0) • F = m(0,3,5,6) • F = m(0,2,3,4,5,7) Two-Level Networks

  6. 4-variable Karnaugh Maps • F = m(1,4,5,6,14)+ d(3,7) • F = m(0,4,5,6,9,12,13) • F = m(0,2,5,6,7,8,10,13) Two-Level Networks

  7. Prime Implicants • Given two logic functions f and g, if g(x) = 1 for all x such that f(x) = 1, then g contains f, denoted f ≤ g. • If a logic function f contains a product c, then c is an implicant of f. Furthermore, if c is a minterm, then c is a minterm of f. • A product P is a sub-product of Q, if all the literals in P also appear in Q. • Let P be an implicant of a logic function f. If no other implicant Q of f is a sub-product of P, then P is a prime implicant (PI) of f. Two-Level Networks

  8. Prime Implicants in K maps • largest blocks (loops) of 2i 1-cells (may include don’t care cells) • Show examples Two-Level Networks

  9. Minimum SOP • Consider an SOP that consists of PIs. If any SOP that is obtained by removing any of the PIs does not represent the original function f, then the original SOP is an irredundant sum-of-product expression (ISOP) of f. • Among the ISOPs for f, one with the minimum number of products is a minimum sum-of-product expression (MSOP). Among the MSOPs for f, one with the minimum number of literals is a exactminimum sum-of-product expression (exact MSOP) of f. • Let m be a minterm of a logic function f. If only one PI of f covers m, the m is a distinguished minterm. Among all the PIs of f, if there is only one that covers m, then it is an essential prime implicant. Two-Level Networks

  10. Example • F = m(2,5,6,7,10,11) F = ABD + ABC + ACD + BCD + ABC (SOP) F = ABD + ABC + BCD + ABC (ISOP) F = ABC + ACD + ABC (MSOP) Two-Level Networks

  11. Quine-McCluskey (Tabular) Minimization • Two step process utilizing tabular listings to: • Identify prime implicants (implicant tables) • Identify minimal cover (cover tables) • All work is done in tabular form • Number of variables is not a limitation • It does affect the size of the table • Basis for many computer implementations • Proper organization and term identification are key factors for correct results Two-Level Networks

  12. Quine-McCluskey Minimization • Minterms are initially listed one per line in groups • Each group contains terms with the same number of true and complemented variables • Terms are listed in numerical order within group • Terms and implicants are identified using one of three common notations • full variable form • cellular form • 1,0,- form Two-Level Networks

  13. Notation Forms • Full variable form - variables and complements in algebraic form • hard to identify when adjacency applies • very easy to make mistakes • Cellular form - terms are identified by their decimal index value • Easy to tell when adjacency applies; indexes must differ by power of two (one bit) • Implicants identified by term nos. separated by comma; differing bit pos. in () following terms Two-Level Networks

  14. Notation Forms (cont.) • 1,0,- form - terms are identified by their binary index value • Easier to translate to/from full variable form • Easy to identify when adjacency applies, one bit is different • - shows variable(s) dropped when adjacency is used • Different forms may be mixed during the minimization Two-Level Networks

  15. Example of Different Notations F(A, B, C, D) =  m(4,5,6,8,10,13) Full variable Cellular 1,0,- 1 ABCD 4 0100 ABCD 8 1000 2 ABCD 5 0101 ABCD 6 0110 ABCD 10 1010 3 ABCD 13 1101 Two-Level Networks

  16. Implication Table Column I 0000 0100 1000 0101 0110 1001 1010 0111 1101 1111 Implication Table • Quine-McCluskey Method • Tabular method to systematically find all prime implicants • ƒ(A,B,C,D) = Σ m(4,5,6,8,9, 10,13) + Σ d(0,7,15) • Part 1: Find all prime implicants • Step 1: Fill Column 1 with active-set and DC-set minterm indices. Group by number of true variables. NOTE: DCs are included in this step! Two-Level Networks

  17. Implication Table Column I 0000  0100  1000  0101  0110  1001  1010  0111  1101  1111  Column II 0-00 -000 010- 01-0 100- 10-0 01-1 -101 011- 1-01 -111 11-1 Minimization - First Pass • Quine-McCluskey Method • Tabular method to systematically find all prime implicants • ƒ(A,B,C,D) = Σ m(4,5,6,8,9,10,13) + Σ d(0,7,15) • Part 1: Find all prime implicants • Step 2: Apply Adjacency - Compare elements of group with N 1's against those with N+1 1's. One bit difference implies adjacent. Eliminate variable and place in next column. E.g., 0000 vs. 0100 yields 0-00 0000 vs. 1000 yields -000 When used in a combination, mark with a check. If cannot be combined, mark with a star. These are the prime implicants. Repeat until nothing left. Two-Level Networks

  18. Implication Table Column I 0000  0100  1000  0101  0110  1001  1010  0111  1101  1111  Column II 0-00 * -000 * 010-  01-0  100- * 10-0 * 01-1  -101  011-  1-01 * -111  11-1  Column III 01-- * -1-1 * Minimization - Second Pass • Quine-McCluskey Method • Step 2 cont.: Apply Adjacency - Compare elements of group with N 1's against those with N+1 1's. One bit difference implies adjacent. Eliminate variable and place in next column. E.g., 0000 vs. 0100 yields 0-00 0000 vs. 1000 yields -000 When used in a combination, mark with a check. If cannot be combined, mark with a star. These are the prime implicants. Repeat until nothing left. Two-Level Networks

  19. Prime Implicants: A AB 00 01 11 10 CD 00 X 1 0 1 01 0 1 1 1 D 11 0 X X 0 C 10 0 1 0 1 B Prime Implicants Two-Level Networks

  20. Prime Implicants: A AB 00 01 11 10 CD 00 X 1 0 1 01 0 1 1 1 D 11 0 X X 0 C 10 0 1 0 1 B Prime Implicants (cont.) Stage 2: find smallest set of prime implicants that cover the active-set recall that essential prime implicants must be in final expression Two-Level Networks

  21. 5 X X 9 X X 4 X X 6 X 8 X X X 10 X 13 X X 0,4(0-00) 0,8(-000) 8,9(100-) 8,10(10-0) 9,13(1-01) 4,5,6,7(01--) 5,7,13,15(-1-1) Coverage Table Coverage Chart Note: Don’t include DCs in coverage table; they don’t have to be covered by the final logic expression! rows = prime implicants columns = ON-set elements place an "X" if ON-set element is covered by the prime implicant Two-Level Networks

  22. 5 X X 9 X X 5 X X 4 X X 6 X 8 X X X 10 X 13 X X 9 X X 4 X X 6 X 8 X X X 10 X 13 X X 0,4(0-00) 0,8(-000) 8,9(100-) 8,10(10-0) 9,13(1-01) 4,5,6,7(01--) 5,7,13,15(-1-1) 0,4(0-00) 0,8(-000) 8,9(100-) 8,10(10-0) 9,13(1-01) 4,5,6,7(01--) 5,7,13,15(-1-1) Coverage Table (cont.) Coverage Chart If column has a single X, than the implicant associated with the row is essential. It must appear in minimum cover rows = prime implicants columns = ON-set elements place an "X" if ON-set element is covered by the prime implicant Two-Level Networks

  23. 5 X X 9 X X 4 X X 6 X 8 X X X 10 X 13 X X 0,4(0-00) 0,8(-000) 8,9(100-) 8,10(10-0) 9,13(1-01) 4,5,6,7(01--) 5,7,13,15(-1-1) Coverage Table (cont.) Eliminate all columns covered by essential primes Two-Level Networks

  24. 5 X X 5 X X 9 X X 9 X X 4 X X 6 X 8 X X X 10 X 13 X X 4 X X 6 X 8 X X X 10 X 13 X X 0,4(0-00) 0,8(\000) 8,9(100-) 8,10(10-0) 9,13(1-01) 4,5,6,7(01--) 5,7,13,15(-1-1) 0,4(0-00) 0,8(-000) 8,9(100-) 8,10(10-0) 9,13(1-01) 4,5,6,7(01--) 5,7,13,15(-1-1) Coverage Table (cont.) Eliminate all columns covered by essential primes Find minimum set of rows that cover the remaining columns Two-Level Networks

  25. Cover Table Reduction • Detection of essential rows • Elimination of dominated rows • Elimination of dominated columns • Detection of Pseudo-essential rows • repeat step 2 to 4 while reductions are possible Two-Level Networks

  26. Solution Methods for Cyclic Tables • Example F = m(0,2,3,4,5,7) • naïve algorithm • Assume that a row is in the final solution, reduce the covering, and obtain a minimum cover • Assume that a row is not in the final solution, reduce the covering, and obtain a minimum cover • select the lower from (1) and (2) Two-Level Networks

  27. Branch and Bound • Branch • Choose one row from the covering table. Let it be A. • Assume A is in the final solution---find minimal cover • Assume A is not in the final solution---find minimal cover • Select smaller of (b) and (c) • Bound • if the cost of the solution is equal to the lower bound of the cost of the minimum cover, then stop the search. • if the solution of the current problems cannot be better than the best solution found, then stop the search. Two-Level Networks

  28. Algebraic Method for Cyclic Tables Let gi = 1 iff Pi is present in the SOP Two-Level Networks

  29. Algebraic Method for Cyclic Tables P1 + P5 is the MSOP Two-Level Networks

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