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William Stallings Computer Organization and Architecture 9 th Edition

William Stallings Computer Organization and Architecture 9 th Edition. Chapter 11. Digital Logic. Boolean Algebra. Mathematical discipline used to design and analyze the behavior of the digital circuitry in digital computers and other digital systems Named after George Boole

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William Stallings Computer Organization and Architecture 9 th Edition

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  1. William Stallings Computer Organization and Architecture9th Edition

  2. Chapter 11 Digital Logic

  3. Boolean Algebra • Mathematical discipline used to design and analyze the behavior of the digital circuitry in digital computers and other digital systems • Named after George Boole • English mathematician • Proposed basic principles of the algebra in 1854 • Claude Shannon suggested Boolean algebra could be used to solve problems in relay-switching circuit design • Is a convenient tool: • Analysis • It is an economical way of describing the function of digital circuitry • Design • Given a desired function, Boolean algebra can be applied to develop a simplified implementation of that function

  4. Boolean Variables and Operations • Makes use of variables and operations • Are logical • A variable may take on the value 1 (TRUE) or 0 (FALSE) • Basic logical operations are AND, OR, and NOT • AND • Yields true (binary value 1) if and only if both of its operands are true • In the absence of parentheses the AND operation takes precedence over the OR operation • When no ambiguity will occur the AND operation is represented by simple concatenation instead of the dot operator • OR • Yields true if either or both of its operands are true • NOT • Inverts the value of its operand

  5. Table 11.1Boolean Operators (a) Boolean Operators of Two Input Variables (b) Boolean Operators Extended to More than Two Inputs (A, B, . . .)

  6. Table 11.2Basic Identities of Boolean Algebra Table 11.2 Basic Identities of Boolean Algebra

  7. Basic Logic Gates

  8. Uses of NAND Gates

  9. Uses of NOR Gates

  10. Combinational Circuit

  11. Boolean Function of Three Variables Table 11.3 A Boolean Function of Three Variables

  12. Sum-of-Products Implementation of Table 11.3

  13. Product-of-Sums Implementationof Table 11.3

  14. Algebraic Simplification • Involves the application of the identities of Table 11.2 to reduce the Boolean expression to one with fewer elements

  15. Karnaugh Map A convenient way of representing a Boolean function of a small number (up to four) of variables

  16. Example Karnaugh Maps

  17. Overlapping Groups

  18. Table 11.4Truth Table for the One-Digit Packed Decimal Incrementer Table 11.4 Truth Table for the One-Digit Packed Decimal Incrementer

  19. Figure 11.10

  20. Quine-McCluskey MethodAn Example 1. Find all the prime implicants Group the minterms according to the number of 1s in the minterm. This way we only have to compare minterms from adjacent groups. group 0 0 0000 1 0001 2 0010 8 1000 group 1 5 0101 6 0110 9 1001 10 1010 group 2 7 0111 14 1110 group 3 CMPUT 329 - Computer Organization and Architecture II

  21. group 0 0 0000 1 0001 2 0010 8 1000 group 1 5 0101 6 0110 9 1001 10 1010 group 2 7 0111 14 1110 group 3 Quine-McCluskey MethodAn Example Column I Column II Combining group 0 and group 1: CMPUT 329 - Computer Organization and Architecture II

  22. 0,1 000-  group 0 0 0000 1 0001 2 0010 8 1000 group 1 5 0101 6 0110 9 1001 10 1010 group 2 7 0111 14 1110 group 3 Quine-McCluskey MethodAn Example Column I Column II Combining group 0 and group 1: CMPUT 329 - Computer Organization and Architecture II

  23. group 0 0 0000 1 0001 2 0010 8 1000 group 1 5 0101 6 0110 9 1001 10 1010 group 2 7 0111 14 1110 group 3 Quine-McCluskey MethodAn Example Column I Column II  0,1 000- 0,2 00-0   Combining group 0 and group 1: CMPUT 329 - Computer Organization and Architecture II

  24. group 0 0 0000 1 0001 2 0010 8 1000 group 1 5 0101 6 0110 9 1001 10 1010 group 2 7 0111 14 1110 group 3 Quine-McCluskey MethodAn Example Column I Column II Does it make sense to no combine group 0 with group 2 or 3?  0,1 000- 0,2 00-0 0,8 -000    No, there are at least two bits that are different. CMPUT 329 - Computer Organization and Architecture II

  25. group 0 0 0000 1 0001 2 0010 8 1000 group 1 5 0101 6 0110 9 1001 10 1010 group 2 7 0111 14 1110 group 3 Quine-McCluskey MethodAn Example Column I Column II Does it make sense to no combine group 0 with group 2 or 3?  0,1 000- 0,2 00-0 0,8 -000    No, there are at least two bits that are different. Thus, next we combine group 1 and group 2. CMPUT 329 - Computer Organization and Architecture II

  26. group 0 0 0000 1 0001 2 0010 8 1000 group 1 5 0101 6 0110 9 1001 10 1010 group 2 7 0111 14 1110 group 3 Quine-McCluskey MethodAn Example Column I Column II  0,1 000- 0,2 00-0 0,8 -000 1,5 0-01    Combine group 1 and group 2.  CMPUT 329 - Computer Organization and Architecture II

  27. group 0 0 0000 1 0001 2 0010 8 1000 group 1 5 0101 6 0110 9 1001 10 1010 group 2 7 0111 14 1110 group 3 Quine-McCluskey MethodAn Example Column I Column II  0,1 000- 0,2 00-0 0,8 -000 1,5 0-01    Combine group 1 and group 2.  CMPUT 329 - Computer Organization and Architecture II

  28. group 0 0 0000 1 0001 2 0010 8 1000 group 1 5 0101 6 0110 9 1001 10 1010 group 2 7 0111 14 1110 group 3 Quine-McCluskey MethodAn Example Column I Column II  0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001    Combine group 1 and group 2.   CMPUT 329 - Computer Organization and Architecture II

  29. group 0 0 0000 1 0001 2 0010 8 1000 group 1 5 0101 6 0110 9 1001 10 1010 group 2 7 0111 14 1110 group 3 Quine-McCluskey MethodAn Example Column I Column II  0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001    Combine group 1 and group 2.   CMPUT 329 - Computer Organization and Architecture II

  30. group 0 0 0000 1 0001 2 0010 8 1000 group 1 5 0101 6 0110 9 1001 10 1010 group 2 7 0111 14 1110 group 3 Quine-McCluskey MethodAn Example Column I Column II  0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001    Combine group 1 and group 2.   CMPUT 329 - Computer Organization and Architecture II

  31. group 0 0 0000 1 0001 2 0010 8 1000 group 1 5 0101 6 0110 9 1001 10 1010 group 2 7 0111 14 1110 group 3 Quine-McCluskey MethodAn Example Column I Column II  0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001    Combine group 1 and group 2.   CMPUT 329 - Computer Organization and Architecture II

  32. group 0 0 0000 1 0001 2 0010 8 1000 group 1 5 0101 6 0110 9 1001 10 1010 group 2 7 0111 14 1110 group 3 Quine-McCluskey MethodAn Example Column I Column II  0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10    Combine group 1 and group 2.    CMPUT 329 - Computer Organization and Architecture II

  33. group 0 0 0000 1 0001 2 0010 8 1000 group 1 5 0101 6 0110 9 1001 10 1010 group 2 7 0111 14 1110 group 3 Quine-McCluskey MethodAn Example Column I Column II  0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10    Combine group 1 and group 2.    CMPUT 329 - Computer Organization and Architecture II

  34. group 0 0 0000 1 0001 2 0010 8 1000 group 1 5 0101 6 0110 9 1001 10 1010 group 2 7 0111 14 1110 group 3 Quine-McCluskey MethodAn Example Column I Column II  0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010    Combine group 1 and group 2.     CMPUT 329 - Computer Organization and Architecture II

  35. group 0 0 0000 1 0001 2 0010 8 1000 group 1 5 0101 6 0110 9 1001 10 1010 group 2 7 0111 14 1110 group 3 Quine-McCluskey MethodAn Example Column I Column II  0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010    Combine group 1 and group 2.     CMPUT 329 - Computer Organization and Architecture II

  36. group 0 0 0000 1 0001 2 0010 8 1000 group 1 5 0101 6 0110 9 1001 10 1010 group 2 7 0111 14 1110 group 3 Quine-McCluskey MethodAn Example Column I Column II  0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010    Combine group 1 and group 2.     CMPUT 329 - Computer Organization and Architecture II

  37. group 0 0 0000 1 0001 2 0010 8 1000 group 1 5 0101 6 0110 9 1001 10 1010 group 2 7 0111 14 1110 group 3 Quine-McCluskey MethodAn Example Column I Column II  0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100-    Combine group 1 and group 2.     CMPUT 329 - Computer Organization and Architecture II

  38. group 0 0 0000 1 0001 2 0010 8 1000 group 1 5 0101 6 0110 9 1001 10 1010 group 2 7 0111 14 1110 group 3 Quine-McCluskey MethodAn Example Column I Column II  0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 Again, there is no need to try to combine group 1 with group 2.        CMPUT 329 - Computer Organization and Architecture II

  39. group 0 0 0000 1 0001 2 0010 8 1000 group 1 5 0101 6 0110 9 1001 10 1010 group 2 7 0111 14 1110 group 3 Quine-McCluskey MethodAn Example Column I Column II  0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 Again, there is no need to try to combine group 1 with group 3.      Lets try to combine group 2 with group 23.   CMPUT 329 - Computer Organization and Architecture II

  40. group 0 0 0000 1 0001 2 0010 8 1000 group 1 5 0101 6 0110 9 1001 10 1010 group 2 7 0111 14 1110 group 3 Quine-McCluskey MethodAn Example Column I Column II  0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1    Combine group 2 and group 3.      CMPUT 329 - Computer Organization and Architecture II

  41. group 0 0 0000 1 0001 2 0010 8 1000 group 1 5 0101 6 0110 9 1001 10 1010 group 2 7 0111 14 1110 group 3 Quine-McCluskey MethodAn Example Column I Column II  0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1    Combine group 2 and group 3.      CMPUT 329 - Computer Organization and Architecture II

  42. group 0 0 0000 1 0001 2 0010 8 1000 group 1 5 0101 6 0110 9 1001 10 1010 group 2 7 0111 14 1110 group 3 Quine-McCluskey MethodAn Example Column I Column II  0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011-    Combine group 2 and group 3.      CMPUT 329 - Computer Organization and Architecture II

  43. group 0 0 0000 1 0001 2 0010 8 1000 group 1 5 0101 6 0110 9 1001 10 1010 group 2 7 0111 14 1110 group 3 Quine-McCluskey MethodAn Example Column I Column II  0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -110    Combine group 2 and group 3.       CMPUT 329 - Computer Organization and Architecture II

  44. group 0 0 0000 1 0001 2 0010 8 1000 group 1 5 0101 6 0110 9 1001 10 1010 group 2 7 0111 14 1110 group 3 Quine-McCluskey MethodAn Example Column I Column II  0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -110    Combine group 2 and group 3.       CMPUT 329 - Computer Organization and Architecture II

  45. group 0 0 0000 1 0001 2 0010 8 1000 group 1 5 0101 6 0110 9 1001 10 1010 group 2 7 0111 14 1110 group 3 Quine-McCluskey MethodAn Example Column I Column II  0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -110    Combine group 2 and group 3.       CMPUT 329 - Computer Organization and Architecture II

  46. group 0 0 0000 1 0001 2 0010 8 1000 group 1 5 0101 6 0110 9 1001 10 1010 group 2 7 0111 14 1110 group 3 Quine-McCluskey MethodAn Example Column I Column II  0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -110    Combine group 2 and group 3.       CMPUT 329 - Computer Organization and Architecture II

  47.   group 0 0 0000  1 0001 2 0010 8 1000 group 1    5 0101 6 0110 9 1001 10 1010  group 2   7 0111 14 1110 group 3 Quine-McCluskey MethodAn Example Column I Column II 0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -110 10,14 1-10 We have now completed the first step. All minterms in column I were included. We can divide column II into groups. CMPUT 329 - Computer Organization and Architecture II

  48. Column I Column II  0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -110 10,14 1-10   group 0 0 0000  1 0001 2 0010 8 1000 group 1    5 0101 6 0110 9 1001 10 1010  group 2   7 0111 14 1110 group 3 Quine-McCluskey MethodAn Example CMPUT 329 - Computer Organization and Architecture II

  49. 0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -110 10,14 1-10   group 0 0 0000  1 0001 2 0010 8 1000 group 1    5 0101 6 0110 9 1001 10 1010  group 2   7 0111 14 1110 group 3 Quine-McCluskey MethodAn Example Column I Column II Column III CMPUT 329 - Computer Organization and Architecture II

  50. 0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -110 10,14 1-10   group 0 0 0000  1 0001 2 0010 8 1000 group 1    5 0101 6 0110 9 1001 10 1010  group 2   7 0111 14 1110 group 3 Quine-McCluskey MethodAn Example Column I Column II Column III CMPUT 329 - Computer Organization and Architecture II

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