FUNCTION OPTIMIZATION

1
FUNCTION OPTIMIZATION

• Switching Function Representations can be
  Classified in Terms of Levels
• Number of Levels, k, is Number of Unique Boolean
  (binary) Operators

EXAMPLES
1-Level
2-Level
multi-level
2
k-Level functions

• 22n Possible Switching Functions of n Variables
  (actually much fewer types obtained by permuting
  and/or complementing input variables)
• 1-Level Forms
• Cannot Represent all Possible Functions
• 2-Level Forms
• Can Represent all Possible Functions
• With Additional Restrictions CANONICAL
• k-Level Forms (k?2)
• Many Different Ways to Represent a Given
  Functions
• If a multi-input Gate Represents a (binary or
  greater) Boolean Operator
• Expression can Represent a Netlist
• k Indicates depth of Netlist

EXAMPLES
2-Level
Multi-Level
a
1-Level
a
b
f
a
f
c
b
f
c
d
b
e
c
3
k-Level function properties

• 1-Level Forms
• Due to Restriction of Representable Functions,
  Not as Useful
• 2-Level Forms
• Can be Canonical
• Longest PI to PO Path is Always 2 Related to
  Delay
• Minimization Usually Means
• Minimum Number of Terms
• Minimum Number of Literals in Expression
• k-Level Forms (k?2)
• Many More Possible Representations
• Can be Optimized for Area
• Delay is More Complicated
• False Path Problem
• Spatio-temporal Correlations
• Typically an Area versus Delay Tradeoff (e.g.,
  Ripple versus CL Adder)

4
Common Terms for 2-Level Minimization

• Literal A variable in complemented or
  uncomplemented form
• Product The disjunction (AND) of a set of
  literals also represents a cube
• Support Set Set of all variables that define
  the domain of a switching function
• Minterm A disjunction (AND) containing an
  instance of each literal corresponding to a
  variable in the support set that is in the
  on-set, fon,of a function
• Dont Care The absence of a supporting variable
  in a product term
• Implicant A product term that covers one or
  more minterms in the on-set, fon, of a function
• Prime Implicant An implicant in the on-set,
  fon, of a function such that it is not a
  subproduct of any other possible implicant in the
  set.
• Essential Prime Implicant A prime implicant
  that covers at least one minterm NOT covered by
  any other implicant in the on-set, fon.

5
SOP Minimization Terms Example

• Consider

• Each Element of fon is an Implicant
• Prime Implicants are

• abc is a Prime Implicant but NOT an Essential
  Prime Implicant
• bcd is an Implicant but NOT a Prime Implicant
• bd is an Essential Prime Implicant

• Product abc
• covers minterms m4abcd and m5abcd
• disjunction of literals a, b, c
• variable d is a Dont Care
• f Represented by fon has Support Set a, b, c,
  d

6
Karnaugh Map

• Simplification Using Karnaugh Map
• Use as few loops as possible
• Use as large loop as possible
• Prime Implicant Corresponds to as large loop as
  Possible
• Karnaugh Map Simplification is Finding fon such
  that
• fon contains as few prime implicants as possible
• fon must contain all essential prime implicants
• fon contains NO implicants that are NOT prime

7
Tabulation Method - Notation

• Consider Support Set of f Sx1, x2, , xn
• xici denotes xi if ci 1 xi if ci
  0 1 if ci -
• If NO ci -, then we have a minterm
• Can be Represented by Decimal Equivalent of ci
• EXAMPLE (Sx1, x2, x3, x4)
• x11x02x03x14 m9, a minterm ? c1 c2 c3 c4
  1001 9 x01x-2x-3x04 a 4-cube ? 0--0

8
Cube Merging

• Basic Operation in Tabulation Method
• 2 Cubes that Differ in a SINGLE ci can be Merged
  into a Single Cube
• EXAMPLE
• ? 1-01? 0-01
• Merge ? and ? into ?
• ? --01

9
Tabulation Method

• Input f on as a set of minterms
• Output f on as a set of
• All Essential Prime Implicants
• As Few Prime Implicants as Possible
• Finding as few Prime Implicants as Possible is an
  NP-Hard Problem!!!!!
• Reduces to the Set Covering Problem for Unate
  Functions
• Unate function a constant or is represented by
  a SOP using either uncomplemented or complemented
  literals for each variable
• Reduces to the Minimum Cost Assignment Problem
  for Binate Functions (ex. EXOR)
• This is 2-Level (SOP) Optimization (Minimization)

10
Tabulation Method

• STEP 1
• Convert Minterm List (specifying f on) to Prime
  Implicant List
• STEP 2
• Choose All Essential Prime Implicants
• If all minterms are covered HALTElse GO To
  STEP 3
• STEP 3
• Formulate the Reduced Cover Table Omitting the
  rows/cols of EPI
• If Cover Table can be Reduced using Dominance
  Properties, Go To Step 2
• Else Must Solve the Cyclic Cover Problem 1)
  Use Exact Method (exponentially complex) 2) Use
  Heuristic Method (possibly non-optimal result)
• NOTE Quine-McCluskey Refers to Using a Branch
  and Bound Heuristic
• NOTE Petricks Method is Exact Technique
  Generates all Solutions Allowing the Best to be
  Used

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Tabulation Method STEP 1

1. Partition Prime Implicants (or minterms)
   According to Number of 1s
2. Check Adjacent Classes for Cube Merging Building
   a New List
3. If Entry in New List Covers Entry in Current List
   Disregard Current List Entry
4. If Current List New List HALTElse Current
   List ? New List New List ? NULL Go To Step 1

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STEP 1 - EXAMPLE

• f on m0, m1, m2, m3, m5, m8, m10, m11, m13,
  m15 ? (0, 1, 2, 3, 5, 8, 10, 11, 13, 15)

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STEP 1 - EXAMPLE

• f on m0, m1, m2, m3, m5, m8, m10, m11, m13,
  m15 ? (0, 1, 2, 3, 5, 8, 10, 11, 13, 15)

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STEP 1 - EXAMPLE

• f on m0, m1, m2, m3, m5, m8, m10, m11, m13,
  m15 ? (0, 1, 2, 3, 5, 8, 10, 11, 13, 15)

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STEP 1 - EXAMPLE

• f on m0, m1, m2, m3, m5, m8, m10, m11, m13,
  m15 ? (0, 1, 2, 3, 5, 8, 10, 11, 13, 15)

f on A,B,C,D,E,F,G 00--, -01-, -0-0, 0-01,
-101, 1-11, 11-1
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STEP 2 Construct Cover Table

• PIs Along Vertical Axis (in order of of
  literals)
• Minterms Along Horizontal Axis

NOTE Table 4.2 in book is incomplete
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STEP 2 Finding the Minimum Cover

• Extract All Essential Prime Implicants, EPI
• EPIs are the PI for which a Single x Appears in a
  Column

• C is an EPI so f onC, ...
• Row C and Columns 0, 2, 8, and 10 can be
  Eliminated Giving Reduced Cover Table
• Examine Reduced Table for New EPIs

18
STEP 2 Reduced Table
Distinguished Column
Essential row

• The Row of an EPI is an Essential row
• The Column of the Single x in the Essential Row
  is a Distinguished Column

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Row and Column Dominance

• If Row P has xs Everywhere Row Q Does Then Q
  Dominates P if P has fewer xs
• If Column i has xs Everywhere j Does Then j
  Dominates i if i has fewer xs
• If Row P is equal to Row Q and Row Q does not
  cost more than Row P, eliminate Row P, or if Row
  P is dominated by Row Q and Row Q Does not cost
  more than Row P, eliminate Row P
• If Column i is equal to Column j, eliminate
  Column i or if Column i dominates Column j,
  eliminate Column i

20
STEP 3 The Reduced Cover Table

• Initially, Columns 0, 2, 8 and 10 Removed

• No EPIs are Present
• No Row Dominance Exists
• No Column Dominance Exists
• This is Cyclic Cover Table
• Must Solve Exactly OR Use a Heuristic

21
The Cyclic Cover Table

• For now, we Arbitrarily Choose a PI
• Later we will Study Exact and Heuristic Methods

• Arbitrarily Choose F so fonC, F, ... This
  Choice May Lead to a Non-Optimal Result!!!!
• Form Reduced Cover and Go To Step 2

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STEP 3 Dominance

• Initially, Reduced Table has Columns 11 and 15
  Removed

• G is Dominated by E
• B is Dominated by A
• Form Reduced Cover Table and Go To Step 2

23
STEP 2 The Reduced Cover

• Initially, Table has Rows G and B Removed

• Secondary EPIs A and E
• All Columns Covered
• Eliminate D
• fonC, F, A, E

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Result Check
cd
cd
0
0
0
1
1
1
1
0
ab
0
0
0
1
1
1
1
0
ab
1
1
1
1
0
0
1
1
0
0
1
1
1
0
1
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
0
1
1
0
1
1
Final Result f onA, C, E, F
Initial Minterm List fon m0, m1, m2, m3, m5,
m8, m10, m11, m13, m15 ? (0, 1, 2, 3, 5, 8,
10, 11, 13, 15)