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  • 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
  • 3 3 Unary operators require no arguments because they automatically refer to the object that calls them. For ageClass, the ++ and -- operators would intuitively increment and decrement the age data member. LEAD IN: Return Values
  • 3 3 Unary operators require no arguments because they automatically refer to the object that calls them. For ageClass, the ++ and -- operators would intuitively increment and decrement the age data member. LEAD IN: Return Values

KMAP KMAP Presentation Transcript

  • Subject : Computer Science (083) Unit : Boolean Algebra Topic : Minimization of Boolean Expressions Using Karnaugh Maps (K-Maps) 11/23/10 Karnaugh Maps
  • L earning O bjectives : 11/23/10 Karnaugh Maps
    • 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
  • P revious K nowledge : 11/23/10 Karnaugh Maps
      • 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
  • Minimization Of Boolean Expressions 11/23/10 Karnaugh Maps
    • 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.)
    • K-Map Quiz
    EXIT Karnaugh Maps INDEX
  • 11/23/10 Karnaugh Maps 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
  • 11/23/10 Karnaugh Maps The End
    • Boolean expressions are practically implemented in the form of GATES (Circuits).
    • A minimized Boolean expression means less number of gates which means
    • Simplified Circuit
    11/23/10 Karnaugh Maps M INIMIZATION OF B OOLEAN E XPRESSION WHY we Need to simplify (minimize) expressions?
  • 11/23/10 Karnaugh Maps M INIMIZATION OF B OOLEAN E XPRESSION Different methods Karnaugh Maps Algebraic Method
  • K arnaugh M aps 11/23/10 Karnaugh Maps 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
  • 11/23/10 Karnaugh Maps 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
  • Correspondence between the Karnaugh Map and the Truth Table for the general case of a two Variable Problem 11/23/10 Karnaugh Maps 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 The diagram below illustrates the correspondence between the Karnaugh map and the truth table for the general case of a two variable problem. Truth Table Karnaugh Map A B 0 1 0 a b 1 c d
  • 11/23/10 Karnaugh Maps 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
  • Karnaugh maps consist of a set of 2 2 squares where 2 is the number of variables in the Boolean expression being minimized. 11/23/10 Karnaugh Maps Truth Table 2 Variable K-Map K arnaugh M aps……… (Contd.) 1 The diagram below illustrates the correspondence between the Karnaugh map and the truth table for the general case of a two variable problem. Truth Table Karnaugh Map 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
    • 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
    11/23/10 Karnaugh Maps 4 Variable K-Map 2 4 = 16 Cells K arnaugh M aps……… (Contd.) 3 Variable K-Map 2 3 = 8 Cells The diagram below illustrates the correspondence between the Karnaugh map and the truth table for the general case of a two variable problem. Truth Table Karnaugh Map BC A 00 01 11 10 0 1 A B C D 00 01 11 10 00 01 11 10
  • 11/23/10 Karnaugh Maps 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
  • 11/23/10 Karnaugh Maps 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
  • 11/23/10 Karnaugh Maps 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
  • Example: Reduce f(wxyz)=Σ(1,3,4,5,7,10,11,12,14,15) 11/23/10 Karnaugh Maps 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
  • OCTET REDUCTION ( Group of 8:) 11/23/10 Karnaugh Maps 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
  • OCTET REDUCTION ( Group of 8:) 11/23/10 Karnaugh Maps 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
  • OCTET REDUCTION ( Group of 8:) 11/23/10 Karnaugh Maps 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
  • OCTET REDUCTION ( Group of 8:) 11/23/10 Karnaugh Maps 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
  • OCTET REDUCTION ( Group of 8:) 11/23/10 Karnaugh Maps 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
  • QUAD REDUCTION ( Group of 4) 11/23/10 Karnaugh Maps 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
  • QUAD REDUCTION ( Group of 4) 11/23/10 Karnaugh Maps 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
  • QUAD REDUCTION ( Group of 4) 11/23/10 Karnaugh Maps 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
  • SINGLE CELL REDUCTION 11/23/10 Karnaugh Maps 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
  • PAIR REDUCTION ( Group of 2) 11/23/10 Karnaugh Maps YZ MAP ROLLING PAIR (m0,m2)
    • The term gets reduced by 1 literals i.e. 1 variables change within the group of 2( PAIR )
    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
  • 11/23/10 Karnaugh Maps
      • Groups may not include any cell containing a zero                                                             
    Karnaugh Maps - Rules of Simplification (SOP Expression) NEXT
  • 11/23/10 Karnaugh Maps
      • Groups may be horizontal or vertical, but not diagonal.                                    
    Karnaugh Maps - Rules of Simplification (SOP Expression) NEXT
  • 11/23/10 Karnaugh Maps
      • 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
  • 11/23/10 Karnaugh Maps
      • Each group should be as large as possible.                                                                          
    Karnaugh Maps - Rules of Simplification (SOP Expression) NEXT
  • 11/23/10 Karnaugh Maps
      • Each cell containing a 1 must be in at least one group.                                                                           
    Karnaugh Maps - Rules of Simplification (SOP Expression) NEXT
  • 11/23/10 Karnaugh Maps
      • Groups may overlap.                                                                    
    Karnaugh Maps - Rules of Simplification (SOP Expression) NEXT
  • 11/23/10 Karnaugh Maps
    • 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
  • 11/23/10 Karnaugh Maps
      • 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
  • 11/23/10 Karnaugh Maps
      • 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)
  • 11/23/10 Karnaugh Maps
    • Minimalization logic function with 3-10inputs.
    • Draw karnaugh map
    • Draw shema
    • Cońvert to NOR and NANDS.
    Karnaugh map minimalization software is freeware. Important Links… K-Min Karnaugh Minimizer is a tool for developers of small digital devices and radio amateurs, also for those who is familiar with Boolean algebra, mostly for electrical engineering students.
  • 11/23/10 Karnaugh Maps 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.