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ELEC 2200-002 Digital Logic Circuits Fall 2008 Logic Minimization (Chapter 3)

ELEC 2200-002 Digital Logic Circuits Fall 2008 Logic Minimization (Chapter 3). Vishwani D. Agrawal James J. Danaher Professor Department of Electrical and Computer Engineering Auburn University, Auburn, AL 36849 http://www.eng.auburn.edu/~vagrawal vagrawal@eng.auburn.edu.

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ELEC 2200-002 Digital Logic Circuits Fall 2008 Logic Minimization (Chapter 3)

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  1. ELEC 2200-002Digital Logic CircuitsFall 2008Logic Minimization (Chapter 3) Vishwani D. Agrawal James J. Danaher Professor Department of Electrical and Computer Engineering Auburn University, Auburn, AL 36849 http://www.eng.auburn.edu/~vagrawal vagrawal@eng.auburn.edu ELEC2200-002 Lecture 5

  2. Understanding Minimization • Logic function: a AND b NOT AND OR F AND c AND d ELEC2200-002 Lecture 5

  3. Understanding Minimization • Reducing products: Distributivity Complementation Identity Complementation Distribitivity Consensus theorem Distributivity Complement, identity ELEC2200-002 Lecture 5

  4. Understanding Minimization • Reduced SOP: b NOT OR F AND d ELEC2200-002 Lecture 5

  5. Understanding Minimization b d • Reducing literals: bd Absorption theorem b NOT OR F d ELEC2200-002 Lecture 5

  6. Logic Minimization • Generally means • In SOP form: • Minimize number of products (reduce gates) and • Minimize literals (reduce gate inputs) • In POS form: • Minimize number of sums (reduce gates) and • Minimize literals (reduce gate inputs) ELEC2200-002 Lecture 5

  7. Product or Implicant or Cube • Any set of literals ANDed together. • Minterm is a special case where all variables are present. It is the largest product. • A minterm is also called a 0-implicant of 0-cube. • A 1-implicant or 1-cube is a product with one variable eliminated: • Obtained by combining two adjacent 0-cubes • ABCD + ABCD = ABC(D+D) = ABC ELEC2200-002 Lecture 5

  8. Cubes (Implicants) of 4 Variables A Minterm or 0-implicant or 0-cube AB CD 1 1 What is this? 1 C 1 D 1 1-implicant or 1-cube ABC B ELEC2200-002 Lecture 5

  9. Growing Cubes, Reducing Products 1-implicant or 1-cube AB C A 2-implicant or 2-cube BC 1 1 C 1 1 D 1-implicant or 1-cube ABC B What is this? ELEC2200-002 Lecture 5

  10. Largest Cubes or Smallest Products A 1 1 1 1 What is this? C 1 1 D 1 1 3-implicant or 3-cube B B ELEC2200-002 Lecture 5

  11. Implication and Covering • A larger cube covers a smaller cube if all minterms of the smaller cube are included in the larger cube. • A smaller cube implies (or subsumes) a larger cube if all minterms of the smaller cube are included in the larger cube. ELEC2200-002 Lecture 5

  12. Implicants of a Function • Minterms, products, cubes that imply the function. A 1 1 1 1 C 1 1 1 D 1 B ELEC2200-002 Lecture 5

  13. Prime Implicant (PI) • A cube or implicant of a function that cannot grow larger by expanding into other cubes. A A PI 1 1 1 1 1 1 1 1 1 1 1 1 C C 1 1 1 1 1 1 1 1 D D 1 1 1 1 B B ELEC2200-002 Lecture 5

  14. Essential Prime Implicant (EPI) • If among the minterms subsuming a prime implicant (PI), there is at least one minterm that is covered by this and only this PI, then the PI is called an essential prime implicant (EPI). • Also called essential prime cube (EPC). A EPI C Why not this? B ELEC2200-002 Lecture 5

  15. Redundant Prime Implicant (RPI) • If each minterm subsuming a prime implicant (PI) is also covered by other essential prime implicants, then that PI is called a redundant prime implicant (RPI). • Also called redundant prime cube (RPC). A EPI C RPI B ELEC2200-002 Lecture 5

  16. Selective Prime Implicant (SPI) • A prime implicant (PI) that is neither EPI nor RPI is called a selective prime implicant (SPI). • Also called selective prime cube (SPC). • SPIs occur in pairs. A EPI C B SPI ELEC2200-002 Lecture 5

  17. Minimum Sum of Products (MSOP) • Identify all prime implicants (PI) by letting minterms and implicants to grow. • Construct MSOP with PI only : • Cover all minterms • Use only essential prime implicants (EPI) • Use no redundant prime implicant (RfroPI) • Use cheaper selective prime implicants (SPI) • A good heuristic – Choose EPI in ascending order, starting from 0-implicant, then 1-implicant, 2-implicant, . . . ELEC2200-002 Lecture 5

  18. Example: F=m(1,3,4,5,8,9,13,15) A MSOP: F = AB D +A BC + A B D + ABC D C B RPI ELEC2200-002 Lecture 5

  19. Example: F=m(1,3,5,7,8,10,12,13,14) SPI A MSOP: F = A D + A D + A BC D C B ELEC2200-002 Lecture 5

  20. Functions with Don’t Care Minterms • F(A,B,C) = m(0,3,7) + d(4,5) • Include don’t care minterms when beneficial. A A C C B B F = B C +ABC F = B C +BC ELEC2200-002 Lecture 5

  21. Five-Variable Functions • F(A,B,C,D,E) = m(0,1,4,5,6,13,14,15,22,24,25,28,29,30,31) A = 0 B A = 1 B E E D D C C F = ABD + A BD+ B C E + C DE ELEC2200-002 Lecture 5

  22. Multiple-Output Minimization ELEC2200-002 Lecture 5

  23. Individual Output Minimization Need five products. F1 A F2 A D D C C B B ELEC2200-002 Lecture 5

  24. Global Minimization Need four products. F1 A F2 A D D C C B B ELEC2200-002 Lecture 5

  25. Minimized SOP and POS • F(A,B,C,D) =  m(1,3,4,7,11) + d(5,12,13,14,15) =  M(0,2,6,8,9,10) D(5,12,13,14,15) A A D D C C B B F = BD + CD + AC F = (B + D)(C + D)(A + C) F = BC +AD + C D ELEC2200-002 Lecture 5

  26. SOP and POS Circuits • F(A,B,C,D) =  m(1,3,4,7,11)+d(5,12,13,14,15) =  M(0,2,6,8,9,10) D(5,12,13,14,15) F = BC +AD + C D F = (B + D)(C + D)(A + C) A C B A C F F D D B Are two circuits functionally Identical? ELEC2200-002 Lecture 5

  27. Absorption Theorem • For two Boolean variables: A, B • A + A B = A • Proof: A B AB ELEC2200-002 Lecture 5

  28. Consensus Theorem • For three Boolean variables: A, B, C • A B +A C + B C = A B +A C • Proof: BC A B AB AC C ELEC2200-002 Lecture 5

  29. Growing Implicants to PI • F = AB +CD +ABCD initial implicants = AB +ABCD + BCD+CD consensus th. = AB + BCD +CD absorption th. = AB + BCD +CD + BD consensus th. = AB +CD + BD absorption th. A A A D D D C C C B B B ELEC2200-002 Lecture 5

  30. Identifying EPI • Find all prime implicants. • From primary implicant SOP, remove a PI. • Apply consensus theorem to the remaining SOP. • If the removed PI is generated, then it is either an RPI or an SPI. • If the removed PI is not generated, then it is an EPI ELEC2200-002 Lecture 5

  31. Example • PI SOP: F = A D +AC +CD • Is AD an EPI? F – {AD} = AC +CD, no new PI can be generated Hence, AD is an EPI. A D C AD B ELEC2200-002 Lecture 5

  32. Example (Cont.) • PI SOP: F = A D +AC +CD • Is CD an EPI? F – {CD } = A D +AC = A D +AC +C D (Consensus th.) HenceC D is not an EPI (it is an RPI) A D C B CD ELEC2200-002 Lecture 5

  33. Finding MSOP • Start with minterm or cube SOP representation of Boolean function. • Find all prime implicants (PI). • Include all EPI’s in MSOP. • Find the set of uncovered minterms, {UC}. • MSOP is minimum if {UC} is empty. DONE. • For a minterm in {UC}, include the largest PI from remaining PI’s (non-EPI’s) in MSOP. • Go to step 4. ELEC2200-002 Lecture 5

  34. Finding Uncovered Minterms, {UC} • {UC} = ({PI} # {PMSOP}) # {DC} • Where: • {PI} is set of all prime implicants of the function. • {PMSOP} is any partial SOP. • {DC} is set of don’t care minterms. • Sharp (#) operation between Boolean expressions X and Y, X # Y, is the set of minterms covered by X that are not covered by Y. ELEC2200-002 Lecture 5

  35. Example: # Operation • AD #CD = {A B C D, AB C D} • CD # AD = {A BC D,A BC D} A D C AD B CD ELEC2200-002 Lecture 5

  36. Minterms Covered by a Product • A product from which k variables have been eliminated, covers 2kminterms. • Example: For four variables, A, B, C, D Product AC covers 22 = 4 minterms: • AB CD • AB C D • A B CD • A B C D Obtained by inserting the eliminated variables in all possible ways. A D C B ELEC2200-002 Lecture 5

  37. Quine-McCluskey Willard V. O. Quine 1908 – 2000 Edward J. McCluskey b. 1929 Agrawal: ELEC2200-002 Lecture 5

  38. Quine-McCluskey Tabular Minimization Method • W. V. Quine, “The Problem of Simplifying Truth Functions,” American Mathematical Monthly, vol. 59, no. 10, pp. 521-531, October 1952. • E. J. McCluskey, “Minimization of Boolean Functions,” Bell System Technical Journal, vol. 35, no. 11, pp. 1417-1444, November 1956. • Textbook, Section 3.9, pp. 211-225. ELEC2200-002 Lecture 5

  39. Q-M Tabular Minimization • Minimizes functions with many variables. • Begin with minterms: • Step 1: Tabulate minterms in groups of increasing number of true variables. • Step 2: Conduct linear searches to identify all prime implicants (PI). • Step 3: Tabulate PI’s vs. minterms to identify EPI’s. • Step 4: Tabulate non-essential PI’s vs. minterms not covered by EPI’s. Select minimum number of PI’s to cover all minterms. ELEC2200-002 Lecture 5

  40. F(A,B,C,D) =  m(2,4,6,8,9,10,12,13,15) • Q-M Step 1: Group minterms with 1 true variable, 2 true variables, etc. ELEC2200-002 Lecture 5

  41. Q-M Step 2 • Find all implicants by combining minterms, and then products that differ in one variable: For example, • 2 and 6, orAB CD and A B CD → A CD, written as 0 – 1 0. • Try combining a minterm (or product) with all minterms (or products) listed below. • Include resulting products in the next list. • If minterm (or product) does not combine with any other, mark it as PI. • Check the minterm (or product) and repeat for all other minterms (or products). ELEC2200-002 Lecture 5

  42. Step 2 Executed on Example ELEC2200-002 Lecture 5

  43. Step 3 Identify EPI ELEC2200-002 Lecture 5

  44. Step 4 Covering Remaining Minterms Integer linear program (ILP), available from Matlab and other sources: Define integer {0,1} variables, xk = 1, select PIk; xk = 0, do not select PIk. Minimize k xk, subject to following constraints: x2 + x3 ≥ 1 x4 + x5 ≥ 1 x2 + x4 ≥ 1 x3 + x6 ≥ 1 A solution is x3 = x4 = 1, x2 = x5 = x6 = 0 ELEC2200-002 Lecture 5

  45. Linear Programming (LP) • A mathematical optimization method for problems where some “cost” depends on a large number of input variables. • An easy to understand introduction is: • S. I. Gass, An Illustrated Guide to Linear Programming, New York: Dover Publications, 1970. • Very useful tool for a variety of engineering design problems. • Available in software packages like Matlab. • Courses available in Math, Business and Engineering departments. ELEC2200-002 Lecture 5

  46. Q-M Solution and Verification • F(A,B,C,D) = PI1 + PI3 + PI4 + PI7 = 1-0- + -010 + 01-0 + 11-1 = AC +B CD +A BD + A B D • See Karnaugh map. A D C B ELEC2200-002 Lecture 5

  47. Further Reading • Incompletely specified functions: See Example 3.25, pages 218-220. • Multiple outputs: See Example 3.26, pages 220-222. ELEC2200-002 Lecture 5

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