Kmap Slideshare
- 1. By Ms. Nita Arora, PGT Computer Science Kulachi Hansraj Model School e -Lesson
- 2. Subject : Computer Science (083) Unit : Boolean Algebra Topic : Minimization of Boolean Expressions Using Karnaugh Maps (K-Maps) Category : Senior Secondary Class : XII
- 3. L earning O bjectives : After successfully completing this module students should be able to: Understand the Need to simplify (minimize) expressions List Different Methods for Minimization Karnaugh Maps Algebraic method Use Karnaugh Map method to minimize the Boolean expression
- 4. P revious K nowledge : The students should be familiar with the following terms in Boolean Algebra before going through this module on K-MAPS Boolean variable, Constants and Operators Postulates of Boolean Algebra Theorems of Boolean Algebra Logic Gates- AND, OR, NOT, NAND, NOR Boolean Expressions and related terms MINTERM (Product Term) MAXTERM (Sum Term) Canonical Form of Expressions x y x+y
- 5. Minimization Of Boolean Expressions Who Developed it NEED For Minimization Different Methods What is K-Map Drawing a K-Map Minimization Steps Important Links Recap. K-Map Rules (SOP Exp.) Karnaugh Maps INDEX
- 6. Who Developed K-Maps… Name : Maurice Karnaugh, a telecommunications engineer at Bell Labs. While designing digital logic based telephone switching circuits he developed a method for Boolean expression minimization. Year : 1950 same year that Charles M. Schulz published his first Peanuts comic.
- 7. Boolean expressions are practically implemented in the form of GATES (Circuits). A minimized Boolean expression means less number of gates which means Simplified Circuit M INIMIZATION OF B OOLEAN E XPRESSION WHY we Need to simplify (minimize) expressions?
- 8. M INIMIZATION OF B OOLEAN E XPRESSION Different methods Karnaugh Maps Algebraic Method
- 9. K arnaugh M aps WHAT is Karnaugh Map (K-Map)? A special version of a truth table Karnaugh Map (K-Map) is a GRAPHICAL display of fundamental terms in a Truth Table . Don’t require the use of Boolean Algebra theorems and equation Works with 2,3,4 (even more) input variables (gets more and more difficult with more variables) NEXT
- 10. K-maps provide an alternate way of simplifying logic circuits. One can transfer logic values from a Truth Table into a K-Map. The arrangement of 0’s and 1’s within a map helps in visualizing, leading directly to Simplified Boolean Expression K arnaugh M aps……… (Contd.) NEXT
- 11. Correspondence between the Karnaugh Map and the Truth Table for the general case of a two Variable Problem Truth Table 2 Variable K-Map K arnaugh M aps……… (Contd.) A B 0 0 0 1 1 0 1 1 F a b c d A \ B 0 1 0 a b 1 c d
- 12. D rawing a K arnaugh M ap (K-Map) K-map is a rectangle made up of certain number of SQUARES For a given Boolean function there are 2 N squares where N is the number of variables (inputs) In a K-Map for a Boolean Function with 2 Variables f(a,b) there will be 2 2 =4 squares Each square is different from its neighbour by ONE Literal Each SQUARE represents a MAXTERM or MINTERM NEXT
- 13. Karnaugh maps consist of a set of 2 2 squares where 2 is the number of variables in the Boolean expression being minimized. Truth Table 2 Variable K-Map K arnaugh M aps……… (Contd.) 1 A \ B 0 1 0 0 1 1 1 1 A B F 0 0 0 0 1 1 1 0 1 1 1 1 Minterm A’B’ A’B A B’ A B Maxterm A + B A + B’ A’ + B A’ + B’ NEXT
- 14. For three and four variable expressions Maps with 2 3 = 8 and 2 4 = 16 cells are used. Each cell represents a MINTERM or a MAXTERM 4 Variable K-Map 2 4 = 16 Cells K arnaugh M aps……… (Contd.) 3 Variable K-Map 2 3 = 8 Cells BC A 00 01 11 10 0 1 A B \ C D 00 01 11 10 00 01 11 10
- 15. M inimization S teps (SOP Expression with 4 var.) The process has following steps: Draw the K-Map for given function as shown Enter the function values into the K-Map by placing 1's and 0's into the appropriate Cells 1 1 1 1 A B \ C D 00 01 11 10 00 0 0 0 1 0 3 0 2 01 0 0 0 0 11 1 1 0 0 10 1 1 0 0 0 5 0 4 0 7 0 6 0 0 12 13 15 14 8 9 11 10 NEXT
- 16. M inimization S teps (SOP Expression) Form groups of adjacent 1's . Make groups as large as possible. Group size must be a power of two. i.e. Group of 8 (OCTET), 4 (QUAD), 2 (PAIR) or 1 (Single) NEXT A B \ C D 00 01 11 10 00 0 0 0 1 0 3 0 2 01 0 0 0 0 11 1 1 0 0 10 1 1 0 0 0 5 0 4 0 7 0 6 0 0 12 13 15 14 8 9 11 10
- 17. OCTET REDUCTION ( Group of 8:) OCTET (m0,m1,m4,m5,m8, m9, m12,m13) The term gets reduced by 3 literals i.e. 3 variables change within the group of 8 ( Octets ) 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 W X YZ 0 0 0 1 1 1 1 0 Y.Z Y.Z Y. Z Y. Z 0 0 W.X 0 1 W.X 1 1 W.X 1 0 W.X NEXT
- 18. OCTET REDUCTION ( Group of 8:) OCTET (m1,m3,m5,m7,m9, m11, m13,m15) 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 W X YZ 0 0 0 1 1 1 1 0 Y.Z Y.Z Y. Z Y. Z 0 0 W.X 0 1 W.X 1 1 W.X 1 0 W.X NEXT
- 19. OCTET REDUCTION ( Group of 8:) MAP ROLLING OCTET (m0,m2,m4,m6, m8, m10, m12,m14) 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 W X YZ 0 0 0 1 1 1 1 0 Y.Z Y.Z Y. Z Y. Z 0 0 W.X 0 1 W.X 1 1 W.X 1 0 W.X 0 1 3 2 4 5 7 6 12 13 15 14 8 9 11 10 NEXT
- 20. OCTET REDUCTION ( Group of 8:) OCTET (m4,m5,m6,m7,m12, m13, m14,m15) 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 W X YZ 0 0 0 1 1 1 1 0 Y.Z Y.Z Y. Z Y. Z 0 0 W.X 0 1 W.X 1 1 W.X 1 0 W.X 0 1 3 2 4 5 7 6 12 13 15 14 8 9 11 10 NEXT
- 21. OCTET REDUCTION ( Group of 8:) MAP ROLLING OCTET (m0,m1,m2,m3 M8,m9,m10,m11) 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 W X YZ 0 0 0 1 1 1 1 0 Y.Z Y.Z Y. Z Y. Z 0 0 W.X 0 1 W.X 1 1 W.X 1 0 W.X 0 1 3 2 4 5 7 6 12 13 15 14 8 9 11 10
- 22. QUAD REDUCTION ( Group of 4) 1 1 0 0 1 1 1 1 0 1 1 1 0 1 1 0 0 WX YZ 3 2 4 5 7 6 1 12 13 15 14 8 9 11 10 QUAD (m1,m3,m5,m7) QUAD (m10,m11,m14,m15) QUAD (m4,m5,m12,m13) 0 0 0 1 1 1 1 0 Y.Z Y.Z Y. Z Y. Z 0 0 W.X 0 1 W.X 1 1 W.X 1 0 W.X The term gets reduced by 2 literals i.e. 2 variables change within the group of 4( QUAD ) NEXT
- 23. QUAD REDUCTION ( Group of 4) MAP ROLLING QUAD (m1,m3,m9,m11) QUAD (m4,m6,m12,m14) 1 1 1 0 1 1 1 1 1 1 1 1 0 1 1 0 0 WX YZ 3 2 4 5 7 6 1 12 13 15 14 8 9 11 10 0 0 0 1 1 1 1 0 Y.Z Y.Z Y. Z Y. Z 0 0 W.X 0 1 W.X 1 1 W.X 1 0 W.X NEXT
- 24. QUAD REDUCTION ( Group of 4) QUAD (m0,m2,m8,m10) CORNER ROLLING 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 WX YZ 3 2 4 5 7 6 1 12 13 15 14 8 9 11 10 0 0 0 1 1 1 1 0 Y.Z Y.Z Y. Z Y. Z 0 0 W.X 0 1 W.X 1 1 W.X 1 0 W.X
- 25. PAIR REDUCTION ( Group of 2) YZ MAP ROLLING PAIR (m0,m2) 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 1 0 WX 3 2 4 5 7 6 1 12 13 15 14 8 9 11 10 PAIR (m5,m7) 0 0 0 1 1 1 1 0 Y.Z Y.Z Y. Z Y. Z 0 0 W.X 0 1 W.X 1 1 W.X 1 0 W.X The term gets reduced by 1 literals i.e. 1 variables change within the group of 2( PAIR )
- 26. SINGLE CELL REDUCTION SINGLE CELL (m1) SINGLE CELL (m12) QUAD (m10,m11,m14,m15) The term is not reduced in a single cell 1 1 0 0 1 1 0 1 0 0 0 0 0 0 1 0 wx yz 00 01 11 10 00 01 11 10
- 27. Karnaugh Maps - Rules of Simplification (SOP Expression) NEXT
- 28. Groups may not include any cell containing a zero Karnaugh Maps - Rules of Simplification (SOP Expression) NEXT
- 29. Groups may be horizontal or vertical, but not diagonal. Karnaugh Maps - Rules of Simplification (SOP Expression) NEXT
- 30. Groups must contain 1, 2, 4, 8, or in general 2 n cells. That is if n = 1, a group will contain two 1's since 2 1 = 2. If n = 2, a group will contain four 1's since 2 2 = 4. Karnaugh Maps - Rules of Simplification (SOP Expression) NEXT
- 31. Each group should be as large as possible. Karnaugh Maps - Rules of Simplification (SOP Expression) NEXT
- 32. Each cell containing a 1 must be in at least one group. Karnaugh Maps - Rules of Simplification (SOP Expression) NEXT
- 33. Groups may overlap. Karnaugh Maps - Rules of Simplification (SOP Expression) NEXT
- 34. Groups may wrap around the table. The leftmost cell in a row may be grouped with the rightmost cell and The top cell in a column may be grouped with the bottom cell . Karnaugh Maps - Rules of Simplification (SOP Expression) NEXT
- 35. There should be as few groups as possible , as long as this does not contradict any of the previous rules. Karnaugh Maps - Rules of Simplification (SOP Expression) NEXT
- 36. No 0’s allowed in the groups. No diagonal grouping allowed. Groups should be as large as possible. Only power of 2 number of cells in each group. Every 1 must be in at least one group. Overlapping allowed. Wrap around allowed. Fewest number of groups are considered. Redundant groups ignored Karnaugh Maps - Rules of Simplification (SOP Expression)
- 37. M inimization S teps (SOP Expression) Select the least number of groups that cover all the 1's. Ensure that every 1 is in a group. 1's can be part of more than one group. Eliminate Redundant Groups 1 1 0 0 1 1 0 1 0 1 1 1 0 1 1 0 0 wx yz 00 01 11 10 00 01 11 10 3 2 4 5 7 6 1 12 13 15 14 8 9 11 10 NEXT
- 38. Example: Reduce f(wxyz)=Σ(1,3,4,5,7,10,11,12,14,15) PAIR (m4,m5) REDUNDANTGROUP 1 1 0 0 1 1 0 1 0 1 1 1 0 1 1 0 0 wx yz 00 01 11 10 00 01 11 10 3 2 4 5 7 6 1 12 13 15 14 8 9 11 10 QUAD (m1,m3,m5,m7) QUAD (m10,m11,m14,m15) QUAD (m3,m7,m11,m15) REDUNDANT Group PAIR (m4,m12) Minimized Expression : xy’z’ + wy + w’z
- 40. References For K-Map Minimizer Download http:// karnaugh.shuriksoft.com Thomas C. Bartee, DIGITAL COMPUTER FUNDAMENTALS, McGraw Hill International. Computer Science (Class XII) By Sumita Arora http://www.ee.surrey.ac.uk/Projects/Labview/minimisation/karrules.html
- 41. The End
Editor's Notes
- #2: 1 1 The “operator” keyword Overloading Unary operators Overloading Binary operators Constructors as conversion routines Converting between basic and user-defined types LEAD IN: Overloaded Unary Operators