Boolean Algebra

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Boolean Algebra
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Boolean Algebra

• A Boolean algebra
• A set of operators (e.g. the binary operators ,
  , INV)
• A set of axioms or postulates

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Postulates

• Commutative xy yx xyyx
• Distributive x(yz)(xy)(xz) x(yz)xy
  xz
• Identities x0x x1x
• Complement xx1 xx0
• Closure xy xy
• Associative (xy)zx(yz) (xy)zx(yz
  )

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Properties of Boolean Algebra

• Complement of a variable is unique.
• (x) x -- involution
• xx x --idempotent
• x11
• xxyx -- absorption
• (xy)xy -- DeMorgans Law
• xyxzyz xy xz --
  consensus

• Complement of a variable is unique.
• (x) x -- involution
• xx x xxx --idempotent
• x11 x00
• xxyx x(xy)x -- absorption
• (xy)xy (xy)xy -- DeMorgans Law
• xyxzyz xy xz (xy) (xz)
  (yz) (xy) (xz) -- consensus
• Duality ? and 0 ? 1

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Proof of Consensus

• a lt b essentially implies that if a 1 then b
  1 and if a 0 then b could be anything
• Theorem In Consensus xy xz yz xy xz
  (How?)
• We Prove that xy xz yz gt xy xz and xy
  xz yz lt xy xz (This would imply equality
  and prove the theorem)
• Proof First xy xz yz gt xy xz. Let xy
  xz a and yz b
• So we want to prove that a b gt a which is
  true by definition of the inequality
• Second xy xz yz lt xy xz. This
  inequality could be split into xy xz lt xy
  xz and yz lt xy xz
• All we need to do is to prove yz lt xy xz

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Proof of Consensus

• yz lt xy xz
• We know that a lt b iff ab 0 (How?)
• So if the above inequality is true yz(xy xz)
  0 must be true. Which is always true.
• Hence Proved.

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Boolean Functions

• A Boolean function is a mapping f(x) Bn ? B.
• Constant function f(x1,,xn) b.
• Projection (to the i-th axis) f(x1,,xn) xi.
• A Boolean function is complete if f(x) is defined
  for all x?Bn. Otherwise the point x that f(x) is
  not defined is called a dont care condition.
• Operations on Boolean functions
• Sum (fg)(x) f(x) g(x)
• Product (fg)(x) f(x)g(x)
• Complement (f)(x) (f(x))

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Representations of Boolean Functions

• Algebraic expressions
• f(x,y,z) xyz
• Tabular forms
• Venn diagrams
• Cubical representations
• Binary decision diagrams (BDD)

• Algebraic expressions
• f(x,y,z) xyz

• Algebraic expressions
• f(x,y,z) xyz
• Tabular forms

• Algebraic expressions
• f(x,y,z) xyz
• Tabular forms
• Venn diagrams

• Algebraic expressions
• f(x,y,z) xyz
• Tabular forms
• Venn diagrams
• Cubical representations

x y z f
0 0 0 0
0 0 1 1
0 1 0 0
0 1 1 1
1 0 0 0
1 0 1 1
1 1 0 1
1 1 1 1
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Representations of Boolean Functions

• Algebraic expressions
• f(x,y,z) xyz
• Tabular forms
• Venn diagrams
• Cubical representations
• Binary decision diagrams (BDD)

x y z f
0 0 0 0
0 0 1 1
0 1 0 0
0 1 1 1
1 0 0 0
1 0 1 1
1 1 0 1
1 1 1 1
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Booles Expansion Theorem

• The cofactor of f(x1,,xn) w.r.t. xi (or xi) is
  a Boolean function , s.t.
• Booles Expansion Theorem

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Booles Expansion Examples

• f(x) x f(1) x f(0) for single-variable
  x
• (x f(0)) (xf(1)) (from duality)
• f(xy) f(xy) f(1) f(y)
• Define g(x,y) f(xy)f(xy)
• g(x,y) xg(1,y) xg(0,y) xf(1)f(y)
  xf(y)f(1) f(1)f(y)
• What if expand w.r.t variable y?
• f(xy) f(xy) yf(1)f(1) yf(x)f(x)
  yf(1)yf(1)f(0) f(1)(yf(1) yf(0))
• f(1)f(y)

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Booles Expansion Examples
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Complete Expansion
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Canonical Forms

• A form is called canonical if the representation
  of the function in that form is unique.
• Minterm Canonical Form
• AND-OR circuits
• Pro and Cons of Canonical Forms
• Unique up to permutation
• Inefficient

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Normal (Standard) Forms

• SOP (Disjunctive Normal Form)
• A disjunction of product terms
• A product term (1 is also considered a product
  term)
• 0
• The Primary Objective During Logic Minimization
  is to Remove the Redundancy in the
  Representation.

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Implicants

• An implicant of a function is a product term that
  is included in the function.
• An implicant is prime if it cannot be included in
  any other implicants.
• A prime implicant is essential if it is the only
  one that includes a minterm.
• Example f(x,y,z) xy yz
• xy(not I),xyz(I, not PI),
• xz(PI,not EPI), yz(EPI)

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Specification for Incompleteness

• A Boolean function is incomplete if f(x) is NOT
  defined for some x?Bn. Such point x is called a
  dont care condition.
• Tabular representation
• 1-set, 0-set, dont care set
• 1-set xy
• 0-set xy
• Dont care set xy, xy

x y f
0 0 0
0 1 -
1 0 -
1 1 1
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Dont Care Conditions

• Satisfiability dont cares of a subcircuit
  consist of all input patterns that will never
  occur.
• Observability dont cares of a subcircuit are the
  input patterns that represent situations when an
  output is not observed.
• Example

x
y
y0