Interactive Karnaugh Map Evaluator

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Interactive Karnaugh Map Evaluator
www.digital-logic.org
00 10 11 01
00

• By Nayda Caasi, Jay De Guzman, Joseph Shaw, and
  Robert Kost
• Professor Dr. Rasha Morsi

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Project Overview

• Project Description
• Background of a Karnaugh map
• Design Specifications
• Design Considerations
• Design Procedure
• Final Design

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Project Description

• An interactive web-based Karnaugh map evaluator.
• Used as a teaching tool.
• Not to be used as a tool to solve homework.
• Helps the student understand the concepts of
  Karnaugh maps.
• Students solve random Karnaugh map problems.

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Background of a Karnaugh Map

• Definitions
• Karnaugh Map - A visual representation of
  switching functions that helps in the creation of
  minimized expressions through the grouping of
  adjacent Boolean values on the map.
• Minterm Has a value of 1 for exactly one term
  in a truth table.
• Maxterm - Has a value of 0 for exactly one term
  in a truth table.
• SOP Sum-of-products. Each term contains a
  group of inputs anded and each term is ored
  to make the expression.
• Example ABCACBCD
• POS Product-of-sums. Each term contains a
  group of inputs ored and each term is anded
  to make the expression.
• Example (AC)(BC)(ABC)(ACD)
• Dont Care - In a truth table or Karnaugh map,
  the result can either be a 1 or 0.

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Background of a Karnaugh Map

• Can be formed from truth tables.
• Every entry in a truth table relates to a unique
  cell on a Karnaugh map.
• The top and sides of the Karnaugh map are lined
  with 1's and 0's in gray code, so that only one
  variable changes between adjacent cells along
  every dimension.
• Each grouping must have a number of cells which
  correspond to a power of 2 (groups of 1, 2, 4, or
  8 etc).

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Example 4-variable Karnaugh Map

• Each value of the cell corresponds to value in
  the truth table

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4-Variable Karnaugh Map
f(A,B,C,D) ABCDABCDABCD
ABCDABCDABCDABCDABCD
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4-Variable Karnaugh Map

• 2 Modes
• Sum-of-Products (SOP)
• Group adjacent minterms
• Product-of-Sums (POS)
• Group adjacent maxterms
• Grouping Methods
• SOP
• Include the largest possible ungrouped minterms
  in a group.
• POS
• Include the largest possible ungrouped maxterms
  in a group.

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4-Variable SOP Mode

• Grouping
• Largest Group First
• BD
• Next Largest
• CD
• Determine Final Expression
• BD CD

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Design Specifications

• A portable GUI (Graphical User Interface).
• Be accessible from anywhere.
• User solves random Karnaugh maps.
• Allow user to select group of minterms/maxterms
  and guess at an equivalent expression.
• Add expression to final expression if correct.
• Allow user to guess at a minimized expression for
  the Karnaugh map.

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Design Considerations

• What language to design it in?
• C, Java, Perl, or
• Javascript
• Why javascript?
• Powerful language.
• Familiarity.
• Accessible over the Internet.
• GUI easily created.
• Client-Side rather than Server-Side like Perl.

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Design Procedure

• What must the program do?
• Two Steps
• Step 1 Determine if an expression entered is
  equal to the Karnaugh map cells selected.
• Step 2 Determine if an expression entered is
  equal to the Karnaugh map and is fully minimized.
• How?
• Truth table comparison in step 1 and 2.
• Minimization verification in step 2.

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Truth Table Comparison

• Step 1
• Each input combination returns output equal to
  selected cells.
• Step 2
• Each input combination returns output equal to
  Karnaugh map

BC
BCCD
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Step 1 Truth Table Compare
f(A,B,C,D) BC
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Step 2 Truth Table Compare
f(A,B,C,D) BCCD
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Minimization Verification

• How to verify expression is minimized?
• The program cannot predict what the user will
  input.
• Theory 1
• Apply the fundamental Boolean algebra postulates
  and theorems to the expression.

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Boolean Theorems

• AA A
• AA A
• AA' 1
• AA 0
• A1 1
• A 0 0
• A(AB) A
• A(AB) A

• (AB)(AB') A
• (AB)(AB') A
• (AB)(AB'C) (AB)(AC)
• (AB)(AB'C) (AB)(AC)
• (AB)(A'C)(BC) (AB)(A'C)
• (AB)(A'C)(BC) (AB)(A'C).
• A(A'B) AB
• A(A'B) AB

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Errors

• Theory 1 presents an error.
• Example BCDABCABBD.
• Can not be minimized by any postulate or theorem.
• ACDABBD is equivalent and minimized more!
• Theory 2
• Remove groups of one or two terms and replace
  with a smaller term.

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Theory 2

• Example

ABC

BCD

AB
BD

New Term
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Theory 2

• What is New Term?
• A combination of the variables from the terms
  removed.
• Every reasonable combination tested.
• Example BCDABC removed. Possible new
  terms-gt

• null
• A
• B
• AB
• B
• AB
• C
• AC
• BC
• ABC
• BC

• ABC
• D
• AD
• BD
• ABD
• BD
• ABD
• CD
• ACD
• BCD
• BCD

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Theory 2

• What is a reasonable combination?
• Any new term that contains a variable and is
  negated value is unreasonable.
• Any new term that contains more variables than
  either of the terms removed is unreasonable.

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Example

• Included in the possible list of new terms
• ACD
• (BCDABCABBD) (BCDABC) (ACD)
  ACDABBD
• A more minimized expression proved to exist!

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Final Design

• An interactive website that users can access to
  evaluate random k-map examples.
• www.digital-logic.org

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User Interface
Users choose from a selection of different size
k-maps
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User Interface
User selects term
Enters minimized expression
Added to final expression
Final answer is checked
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Demonstration

• We would like to provide a full demonstration of
  the website design.