Boolean Minimization

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Boolean Minimization

• A tour conducted by George, Gus, Morrie, Van and
  Ed.

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Admin

• 104A research papers graded and returned
• 104B paper due Nov 16.
• Common problems
• weak articles (blogs, ads), wrong writing style,
  grammar and spelling, discuss positive and
  negative
• Help available 4026 JKB
• First quiz next week (ch 1, 3, 4)
• Blackboard
• Mon after class to Weds before class

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Learning Outcomes

• Convert Boolean expressions, truth tables
  circuits
• Express Boolean expressions/truth-tables in
  minterm and maxterm forms
• State and apply De Morgans 1st 2nd theorems
• Reduce systems to NAND logic or NOR logic
  equivalent forms
• Perform algebraic minimization of Boolean
  expressions
• Convert between binary and gray code
• Explain uses of Gray Code
• Express Boolean expressions in Karnaugh Maps
• 3, 4, 5 variable forms
• dont care variables
• XOR, XNOR options
• Computer algorithms for reduction
  (Quine-McClusky)

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Logic Symbols
Logic gates
Note that you may have more than 2 inputs to a
gate
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SidebarDiscrete Mathematics

• The math of Boolean algebra is a subset of the
  field of Discrete Mathematics and also formal
  logic
• Discrete math is an important part of
  understanding computational reasoning.
• DM will be discussed several other classes
  (CS235, 236 other IT)
• Discrete math uses (many) other symbols E.G.
• OR A ? B (disjunction)
• AND A ? B (conjunction)
• NEGATE ?A or A
• XOR A B or A ? B or AB
• This last symbol can obviously be confused with
  Boolean OR.
• We will use traditional Boolean logic notation
  for this course

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Circuits, logic, truth

• Construct a circuit and a truth table for
• Simplified equivalent forms
• Sum-of-products form (minterm)
• Focus on F1 terms
• Truth table is same as ??
• How do we reduce the expression to simpler form?

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Equivalent cct table
Example
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Create minterm form

• Minterm (focus on Input/Output1 terms)
• Method
• Start with truth table
• Select 1 terms
• Combine with sum-of-products
• With example cct we get ABABABF
• EG problem 5.5

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Create maxterm form

• Maxterm (focus on Input/Output0 terms)
• Method
• Truth table
• Select 0 terms
• Combine with product-of-sums
• With example circuit we get ABF
• EG problem 5.6

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Manipulate Expressions

• Can we do all this mathematically?
• Need a way to manipulate expressions. Rules of
  adding, multiplying plus associative,
  distributive laws etc.
• Rules very similar to basic algebra
• See links page, Boolean Theorems Handout
• http//www.et.byu.edu/groups/it104/Handouts/Boolea
  n20Algebra20Theorems.doc

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Note

• Note
• If you submit problems like that they will be
  marked wrong

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DeMorgans Theorems

• Augustus DeMorgan Laws of induction
• Theorem 1 2
• Demonstrate with truth tables, circuits
• Rules Invert each term, change ltgt ?,
  invert all

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Apply De Morgan

• Convert between minterm and maxterm
• Double negate bars cancel
• (why?)
• Everything should be made as simple as possible
  but no simpler
• (Occams razor?)
• EG prob 5.12

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Why De Morgan?

• Simplify expressions
• Initial problem
• Algebraic manipulation of Boolean expressions

group terms and factorize
remove redundant 1
De Morgan x2 and simplify
distribute
Remove null De M
Simplify
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Identities

• A . A A
• A A A
• A . A 0
• A A 1
• (A) A
• ((((((A)))))) n times
• A if n even A if n odd

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Equivalent forms

• Note second-last term of previous development
• IE
• Equivalent forms of gate just a De Morgan
  transform

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Implication

• Reduce any expression to simplest form
• Minimum software logic or
• minimum hardware
• Or simplest hardware?
• Use NAND or NOR gates for everything
• Last problem on HW 1
• Single transistor NAND gate or NOR gate
• ASICs, USBs, CPUs (mixed)
• Typically 6 transistors per CPU gate

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Minimization methods

• Intuition
• Quickly run out of steam
• Truth tables
• Minterm or maxterm
• De Morgan
• Graphically
• Karnaugh
• Computer algorithms
• Quine McClusky

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Karnaugh Maps

• Graphical minimization
• See the link
• 2, 3, 4 terms (next slide)
• Minterm or maxterm
• Wrap-around
• Map wraps horizontally and vertically
• 4 corners of a 4-var map are adjacent
• Dont Cares

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Example K-maps

• 2, 3, 4 terms
• Note counting
• Not binary
• Only one bit change
• gray code
• May not fill all cells
• Depends on design

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Minimization

• Minimization
• Omit flipping bits

ACD AD
ABD AB
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EG Dont Care

• Website customer uses Credit card or charges to
  Account.
• There is a limit (max) used for validating
  customer ID
• Boss tells you If credit card C and over limit
  L then validate ID. If not over limit do not
  validate ID
• If Account A and over limit query ID, if below
  limit do not query
• Logic Validate or query if CL or if AL. Do NOT
  validate or query if CL or if AL.
• Draw Kmap. Fill in dont cares

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K-map for ID function

• K-map shown for ID I
• Show where I is 0, 1
• Some conditions cant occur (EG CA) so we dont
  care what the output may be. (X)
• I C.L.A A.L.C
• L (minimized)

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Checkerboard

• Karnaugh Map w. checkerboard
• Truth table
• Function?
• And this one?

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Karnaugh more

• XOR, XNOR
• Look for checkerboard squares
• Depends whether XOR/XNOR gates available
• Look for overlapping regions
• More than four vars
• 5 vars
• More than 5
• Use of karnaugh.exe
• Doesnt do dont-cares, xors etc.

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Overlapping areas

• Look for overlapping areas

Without overlap Y ABD AD With Overlap Y
BD AD (minimum Y D(B A) (after
algebraic simplification one less gate)
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5-variable K-map

• 5- variable K-map
• Two 4 variable maps side-by-side
• Look for overlap between layers

Y ACDE ABCD AD
E 0
E 1
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Quine McClusky

• An algorithm for reducing expressions
• Algorithms are programmable
• Works with n variables
• One explanation of it
• http//www.embedded.com/showArticle.jhtml?articleI
  D29111968

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The people (ref Wikipedia)

• George Boole - a logical child
• Augustus De Morgan a nerd
• Maurice Karnaugh a physicist and mathematician
• Willard Van Orman Quine (Van) - a philosopher
• Edward McClusky - an engineer

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George Boole

• British mathematician and philosopher
• November 2, 1815 December 8, 1864
• died before computers invented
• son of tradesman, school teacher, professor of
  math
• Wrote about logic (pioneer)
• Inventor of Boolean algebra,
• Obscure in his lifetime
• Rediscovered by Claude Shannon 1937
• Shannons was, "possibly the most important, and
  also the most famous, master's thesis of the
  century."
• Also developed Shannons Theorem
• basis of all modern computer arithmetic

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Augustus De Morgan

• British mathematician and logician
• June 27, 1806 (India) March 18, 1871
• De Morgan crater on the Moon
• Died before computers invented
• Formulated De Morgan's laws
• introduced the term,
• rigorous mathematical induction.

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Maurice Karnaugh

• American Physicist
• Oct 4, 1924
• Physics and Mathematics
• Bell Laboratories and IBM Research
• Invented Karnaugh Maps in 1950 (Same year as
  first Peanuts cartoon)

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Van Quine

• Willard Van Orman Quine
• June 25, 1908 December 25, 2000
• Spent his entire career teaching philosophy and
  mathematics at Harvard University, his alma
  mater, where he held the Edgar Pierce Chair of
  Philosophy from 1956 to 1978.
• A computer program whose output is its source
  code is called a "quine," named after him.

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Edward McClusky

• McCluskey worked on electronic switching systems
  at the Bell Telephone Laboratories from 1955 to
  1959
• 1959 Princeton Professor of Electrical
  Engineering and Director of the University
  Computer Center.
• 1966 joined Stanford University,
• Currently Emeritus Professor of Electrical
  Engineering and Computer Science,