karnaugh map covering paolo prinetto politecnico di torino (italy) university of illinois at...
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Karnaugh map Karnaugh map coveringcovering
Karnaugh map Karnaugh map coveringcovering
Paolo PRINETTOPolitecnico di Torino (Italy)
University of Illinois at Chicago, IL (USA)
[email protected] [email protected]
www.testgroup.polito.it
Lecture
6.2
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2 6.2
Goal
This lecture presents the basic fundamentals of purely manual synthesis of Combinational networks, based on Karnaugh map covering
Students are then guided through the solution of simple problems
Algorithmic approaches are briefly introduced
The lecture eventually focuses on peculiar aspects, such as covering of chessboard-like maps and of multiple-output functions.
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3 6.2
Prerequisites
Lectures 6.1 and 3.3
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4 6.2
Homework
Students are warmly encouraged to solve the proposed exercises
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5 6.2
Further readings
Students interested in making a reference to text books on the arguments covered in this lecture can refer, for instance, to:
M. Morris Mano, C.R.Kime: “Logic and Computer Design Fundamentals,” 2nd edition updatedPrentice Hall, Upple Saddle River, NJ (USA), 2001, (pp. 45-75)
or to
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6 6.2
Further readings (cont’d)
E.J.McCluskey: “Logic design principles with emphasis on testable semicustom circuits”Prentice-Hall, Englewood Cliffs, NJ, USA, 1986, (pp. 191-250)
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7 6.2
Further readings (cont’d)
In particular, students interested in logic minimization algoritms are recommended to refer to
G. De Micheli: “Synthesis and Optimization of digital circuits,”McGraw-Hill, Inc, New York, NJ, USA, 1994, (chapters 7-8, pp. 269-440)
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8 6.2
Implicants
Covers
Algorithmic approaches
Chessboard map covering
Map composition
Multiple-output functions.
Outline
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9 6.2
An implicant of a function f is a k-cube not including any vertex of the off-set of f.
ons dcs ofs
implicantimplicant
Implicants
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10 6.2
An implicant of a function f is a k-cube not including any vertex of the off-set of f.
ons dcs ofs
implicantimplicant
Implicants
They are usually represented by:They are usually represented by:• cubic notation cubic notation • algebraic notation by product algebraic notation by product
termsterms
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11 6.2
Prime implicants
An implicant is prime iff
is not included in any other implicant of f and
is not completely included in the don’t-care set of f.
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12 6.2
Note
Since prime implicants only will be of interest in manual designing,, for sake of simplicity, the adjective “prime” will be mostly omitted hereinafter.
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13 6.2
Essential implicant
A prime implicant of a function f is essential iff it includes at least one vertex of the on-set of f which is not included in any other prime implicant of f.
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14 6.2
Implicants on Karnaugh maps
On the Karnaugh map of a function f, any k-cube whose 2k cells all correspond to vertices not belonging to the off-set of f is an implicant.
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15 6.2
Complete sum
A complete sum is a representation of a function as the sum of all its prime implicant.
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16 6.2
abcd 00 01 11 10
00 1 0 101 0 0 0 11 0 1 1 110 0 1 1
Example
Find all the prime implicants and the essential implicants.
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17 6.2
abcd
SolutionPrime implicants
00 01 11 10
00 1 0 101 0 0 0 11 0 1 1 110 0 1 1
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18 6.2
abcd
SolutionPrime implicants
00 01 11 10
00 1 0 101 0 0 0 11 0 1 1 110 0 1 1
a’ c’ d’a’ c’ d’
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19 6.2
abcd
SolutionPrime implicants
00 01 11 10
00 1 0 101 0 0 0 11 0 1 1 110 0 1 1
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20 6.2
abcd
SolutionPrime implicants
00 01 11 10
00 1 0 101 0 0 0 11 0 1 1 110 0 1 1
a’ b d’a’ b d’
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21 6.2
abcd
SolutionPrime implicants
00 01 11 10
00 1 0 101 0 0 0 11 0 1 1 110 0 1 1
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22 6.2
abcd
SolutionPrime implicants
00 01 11 10
00 1 0 101 0 0 0 11 0 1 1 110 0 1 1
b’ c’ d’b’ c’ d’
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23 6.2
abcd
SolutionPrime implicants
00 01 11 10
00 1 0 101 0 0 0 11 0 1 1 110 0 1 1
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24 6.2
abcd
SolutionPrime implicants
00 01 11 10
00 1 0 101 0 0 0 11 0 1 1 110 0 1 1
a b’a b’
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25 6.2
abcd
SolutionPrime implicants
00 01 11 10
00 1 0 101 0 0 0 11 0 1 1 110 0 1 1
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26 6.2
abcd
SolutionPrime implicants
00 01 11 10
00 1 0 101 0 0 0 11 0 1 1 110 0 1 1
a ca c
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27 6.2
abcd
SolutionPrime implicants
00 01 11 10
00 1 0 101 0 0 0 11 0 1 1 110 0 1 1
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28 6.2
abcd
SolutionPrime implicants
00 01 11 10
00 1 0 101 0 0 0 11 0 1 1 110 0 1 1
b cb c
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29 6.2
Solution Essential implicants
abcd 00 01 11 10
00 1 0 101 0 0 0 11 0 1 1 110 0 1 1
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30 6.2
Solution Essential implicants
abcd 00 01 11 10
00 1 0 101 0 0 0 11 0 1 1 110 0 1 1
b cb c
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31 6.2
Property
If you erase a variable from a prime implicant of a function f, the resulting product term is no longer an implicant of f.
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Exampleabcd 00 01 11 10
00 1 0 101 0 0 0 11 0 1 1 110 0 1 1
b cb c
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Exampleabcd 00 01 11 10
00 1 0 101 0 0 0 11 0 1 1 110 0 1 1
b cb c
cc
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Exampleabcd 00 01 11 10
00 1 0 101 0 0 0 11 0 1 1 110 0 1 1
b cb c
abcd 00 01 11 10
00 1 0 101 0 0 0 11 0 1 1 110 0 1 1
cc
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Exampleabcd 00 01 11 10
00 1 0 101 0 0 0 11 0 1 1 110 0 1 1
b cb c
abcd 00 01 11 10
00 1 0 101 0 0 0 11 0 1 1 110 0 1 1
cc
It’s not an implicant, since it covers some 0s
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36 6.2
Implicants
Covers
Algorithmic approaches
Chessboard map covering
Map composition
Multiple-output functions
Outline
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37 6.2
Function Covers
A cover of a function f is a function C including all the vertices of the on-set and no vertex of the off-set of f:
ons(f) C ons(f) dcs(f)
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38 6.2
Function Covers
A cover of a function f is a function C including all the vertices of the on-set and no vertex of the off-set of f:
ons(f) C ons(f) dcs(f)
dcsdcs ofsofsonsons
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39 6.2
Function Covers
A cover of a function f is a function C including all the vertices of the on-set and no vertex of the off-set of f:
ons(f) C ons(f) dcs(f)
dcsdcs ofsofs
CoverCover
onsons
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40 6.2
A cover C is irredundant iff erasing any of its terms would result in a set of cubes which is not a cover of f.
Irredundant cover
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41 6.2
Finding irredundant covers
Finding an irredundant cover of a function relies on the following theorem, proved by Quine in ’52.
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42 6.2
Theorem
A minimal sum must always consist of a sum of prime implicants if any definition of cost is used in which the addition of a single literal to any expression increases the cost of the expression.
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43 6.2
Corollary
If an s-o-p function f is minimal w.r.t. either the # of logic gates or the # gate inputs, then each of its product term corresponds to a prime implicant of f.
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44 6.2
Cover by s-o-p expressions
A possible cover of a function f is given by a set of prime implicants covering any “1” of f, at least once, and, if needed, some of the “-” of f.
ons dcs ofs
CoverCover
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45 6.2
Covering incompletely specified functions
When dealing with incompletely specified functions, the vertices of dcs can be freely covered or not, in dependence of designer’s convenience.
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46 6.2
abcd 00 01 11 10
00 1 0 101 0 0 0 11 0 1 1 110 0 1 1
Example
Find an irredundant
cover.
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47 6.2
SolutionIrredundant cover
abcd 00 01 11 10
00 1 0 101 0 0 0 11 0 1 1 110 0 1 1
f =
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48 6.2
SolutionIrredundant cover
abcd 00 01 11 10
00 1 0 101 0 0 0 11 0 1 1 110 0 1 1
a’ c’ d’a’ c’ d’
f =
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49 6.2
SolutionIrredundant cover
abcd 00 01 11 10
00 1 0 101 0 0 0 11 0 1 1 110 0 1 1
a’ c’ d’a’ c’ d’
f = a’c’d’
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50 6.2
SolutionIrredundant cover
abcd 00 01 11 10
00 1 0 101 0 0 0 11 0 1 1 110 0 1 1
f = a’c’d’
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51 6.2
SolutionIrredundant cover
abcd 00 01 11 10
00 1 0 101 0 0 0 11 0 1 1 110 0 1 1
a b’a b’
f = a’c’d’
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52 6.2
SolutionIrredundant cover
abcd 00 01 11 10
00 1 0 101 0 0 0 11 0 1 1 110 0 1 1
a’ b’a’ b’
f = a’c’d’ + ab’
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53 6.2
SolutionIrredundant cover
abcd 00 01 11 10
00 1 0 101 0 0 0 11 0 1 1 110 0 1 1
f = a’c’d’ + ab’
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54 6.2
SolutionIrredundant cover
abcd 00 01 11 10
00 1 0 101 0 0 0 11 0 1 1 110 0 1 1
b cb c
f = a’c’d’ + ab’
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55 6.2
SolutionIrredundant cover
abcd 00 01 11 10
00 1 0 101 0 0 0 11 0 1 1 110 0 1 1
b cb c
f = a’c’d’ + ab’ + bc
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56 6.2
Implicants
Covers
Algorithmic approaches
Chessboard map covering
Map composition
Multiple-output functions.
Outline
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57 6.2
Algorithmic approaches
Several algorithmic approaches have been proposed to generate minimal covers.
They can be clustered as:
Exact logic minimization algorithms
Heuristic logic minimization algorithms
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58 6.2
Note
A detailed presentation of these algorithms is outside she scope of this course.
In the following we shall just introduce the basic ideas behind some of them.
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59 6.2
Exact approaches
Exact approaches mostly rely on two steps:
Generating all the prime implicants of the function ons(f) dcs(f)
Selecting a subset of these prime implicants that is an irredundant minimum cost cover of ons(f).
Each step can be performed in several ways.
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60 6.2
Minimum cover finding
Prime implicantgeneration
Quine-McCluskey Consensus
Set of prime Set of prime
iimplicantmplicantss
Quine-McCluskey Petrick
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61 6.2
Minimum cover finding
Prime implicantgeneration
Quine-McCluskey Consensus
Set of prime Set of prime
iimplicantmplicantss
Quine-McCluskey Petrick
The # of prime implicants of an n input functions is
O (3n / n)
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62 6.2
Minimum cover finding
Prime implicantgeneration
Quine-McCluskey Consensus
Set of prime Set of prime
iimplicantmplicantss
Quine-McCluskey Petrick
• Start from the set of minterms of the functions
ons(f) dcs(f)
• Recursively apply the distance one merging theorems:
x y + x’ y = y
x y + x = x
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63 6.2
0 1 0 1 1
0 1 1 1 1
0 1 - 1 1
Example of merging
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64 6.2
Minimum cover finding
Prime implicantgeneration
Quine-McCluskey Consensus
Set of prime Set of prime
iimplicantmplicantss
Quine-McCluskey Petrick
• Start from any s-o-p expression of the functions
ons(f) dcs(f)
• Recursively apply the consensus theorem:
xy + x’z = xy + x’z + yz
x + xy = x
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65 6.2
Minimum cover finding
Prime implicantgeneration
Quine-McCluskey Consensus
Set of prime Set of prime
iimplicantmplicantss
Quine-McCluskey Petrick
It’s an NP problem
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66 6.2
Minimum cover finding
Prime implicantgeneration
Quine-McCluskey Consensus
Set of prime Set of prime
iimplicantmplicantss
Quine-McCluskey Petrick
It generates all the minimum covers of the target function
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67 6.2
Minimization tools
Several minimization tools are available.
They will be covered in Lecture 6.4.
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68 6.2
Implicants
Covers
Algorithmic approaches
Chessboard map covering
Map composition
Multiple-output functions
Outline
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69 6.2
Chessboard-like maps
When dealing with chessboard (or chessboard-like) maps, it may be convenient to resort to exor functions.
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70 6.2
Design the sum output bit Si of a 1-bit full-adder.
Example
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71 6.2
Design the sum output bit Si of a 1-bit full-adder.
Example
00 01 11 10
0 0 1 0 1
1 1 0 1 0
ai bi
ci
Si
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72 6.2
Design the sum output bit Si of a 1-bit full-adder.
Example
00 01 11 10
0 0 1 0 1
1 1 0 1 0
ai bi
ci
Si
Si = ai bi ci
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73 6.2
Design the sum output bit Si of a 1-bit full-adder.
Example
00 01 11 10
0 0 1 0 1
1 1 0 1 0
ai bi
ci
Si
ai
bici
Si
Si = ai bi ci
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74 6.2
Implicants
Covers
Algorithmic approaches
Chessboard map covering
Map composition
Multiple-output functions.
Outline
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75 6.2
Map composition
It can sometimes be convenient to get the cover of a function by resorting to a set of proper composing functions:
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76 6.2
Map composition
It can sometimes be convenient to get the cover of a function by resorting to a set of proper composing functions:
abcd 00 01 11 10
00
01
11
10
f
ab
00 01 11 10
00
01
11
10
F2
cd00 01 11 10
00
01
11
10
abcd
F1
op
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77 6.2
Map composition
It can sometimes be convenient to get the cover of a function by resorting to a set of proper composing functions:
abcd 00 01 11 10
00
01
11
10
f
ab
00 01 11 10
00
01
11
10
F2
cd00 01 11 10
00
01
11
10
abcd
F1
op
f = F1 op F2f = F1 op F2
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78 6.2
A typical example:covering chessboard-like maps
A typical example is provided by covering maps that have either some additional 0’s w.r.t. the chessboard or some additional 1’s, or both.
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79 6.2
f f EE RR00 00 --00 11 0011 00 not possiblenot possible11 11 11
Maps with additional 0’s
Implement the target function f as:
f = E · R
where:
• E is an exor gate having as many inputs as f
• R is a masking function (having as many inputs as f), in charge of masking those 1’s of E for which f must be 0.
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80 6.2
00 01 11 10
00 0 1 0 1
01 1 0 1 0
11 0 1 0 0
10 1 0 0 0
Example #1
abcd
f
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81 6.2
00 01 11 10
00 0 1 0 1
01 1 0 1 0
11 0 1 0 0
10 1 0 0 0
Example #1
abcd
f
Additional 0’s
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82 6.2
abcd 00 01 11 10
00 0 1 0 1
01 1 0 1 0
11 0 1 0 0
10 1 0 0 0
f
ab
00 01 11 10
00 - 1 - 1
01 1 - 1 -
11 - 1 - 0
10 1 - 0 -
R
cd00 01 11 10
00 0 1 0 1
01 1 0 1 0
11 0 1 0 1
10 1 0 1 0
abcd
E
Solution
abcd
E
a’c’
Rf
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83 6.2
Example #2
00 01 11 10
00 1 0 0 0
01 0 1 0 0
11 0 0 1 0
10 0 0 0 1
abcd
U
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84 6.2
Solution
U = 1 when ab=cd, i.e., when (a=c) (b=d), and thus:
U = (a c)’ (b d)’
00 01 11 10
00 1 0 0 0
01 0 1 0 0
11 0 0 1 0
10 0 0 0 1
abcd
U
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85 6.2
Solution
One can get the same result on the basis of the following reasoning:
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86 6.2
Solution
U = F G
abcd 00 01 11 10
00 1 0 0 0
01 0 1 0 0
11 0 0 1 0
10 0 0 0 1
U
ab
00 01 11 10
00 1 0 - -
01 0 1 - -
11 - - 1 0
10 - - 0 1
G
cd00 01 11 10
00 1 1 0 0
01 1 1 0 0
11 0 0 1 1
10 0 0 1 1
abcd
F
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87 6.2
Solution
F = (a c)’
G = (b d)’
U = F G = (a c)’ (b d)’
abcd 00 01 11 10
00 1 0 0 0
01 0 1 0 0
11 0 0 1 0
10 0 0 0 1
U
ab
00 01 11 10
00 1 0 - -
01 0 1 - -
11 - - 1 0
10 - - 0 1
G
cd00 01 11 10
00 1 1 0 0
01 1 1 0 0
11 0 0 1 1
10 0 0 1 1
abcd
F
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88 6.2
Example #3
00 01 11 10
00 0 - 0 -
01 - 0 - 0
11 0 - 0 -
10 1 0 1 0
abcd
U
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89 6.2
Solution
00 01 11 10
00 0 - 0 -
01 - 0 - 0
11 0 - 0 -
10 1 0 1 0
abcd
U
One can get alternative solutions, according to the values assigned to “-”
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90 6.2
Solution #1
00 01 11 10
00 0 1 0 1
01 1 0 1 0
11 0 1 0 1
10 1 0 1 0
abcd
U
00 01 11 10
00 0 - 0 -
01 - 0 - 0
11 0 - 0 -
10 1 0 1 0
abcd
U
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91 6.2
Solution #1
00 01 11 10
00 0 1 0 1
01 1 0 1 0
11 0 1 0 1
10 1 0 1 0
abcd
U
U = a b c d
00 01 11 10
00 0 - 0 -
01 - 0 - 0
11 0 - 0 -
10 1 0 1 0
abcd
U
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92 6.2
Solution #2
00 01 11 10
00 0 0 0 0
01 0 0 0 0
11 0 1 0 1
10 1 0 1 0
abcd
U
00 01 11 10
00 0 - 0 -
01 - 0 - 0
11 0 - 0 -
10 1 0 1 0
abcd
U
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93 6.2
Solution #2
00 01 11 10
00 0 0 0 0
01 0 0 0 0
11 0 1 0 1
10 1 0 1 0
abcd
U
U = (a b d) c’
00 01 11 10
00 0 - 0 -
01 - 0 - 0
11 0 - 0 -
10 1 0 1 0
abcd
U
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94 6.2
Implicants
Covers
Algorithmic approaches
Chessboard map covering
Map composition
Multiple-output functions
Outline
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95 6.2
Multiple outputs minimization
Methods used so far are cannot guarantee an optimal covering when dealing with multiple outputs functions.
In most cases a minimal solution must resort to sharing some implicants among two or more functions.
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96 6.2
Example
Minimize the expression of the outputs X and Y of a given circuits, defined as follows:
ab
cd 00 01 11 10
00 0 0 0 0
01 1 0 0 1
11 1 1 1 1
10 0 0 0 0
ab
cd 00 01 11 10
00 0 1 1 0
01 0 0 0 0
11 0 1 1 0
10 0 1 1 0
X Y
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97 6.2
ab
cd 00 01 11 10
00 0 0 0 0
01 1 0 0 1
11 1 1 1 1
10 0 0 0 0
ab
cd 00 01 11 10
00 0 1 1 0
01 0 0 0 0
11 0 1 1 0
10 0 1 1 0
X Y
Independent covers
X = b’d + cd Y = bc + bd’
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98 6.2
1st implementation
The solution looks optimal, since no shareable implicants exist:
X = b’d + cd
Y = bc + bd’
Cost:
4 and functions ( 4 and gates)
2 or functions ( 2 or gates)
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99 6.2
Better implementation
The two covers share an implicant which is not prime:
X = b’d + bcdbcd
Y = bd’ + bcdbcd
ab
cd 00 01 11 10
00 0 0 0 0
01 1 0 0 1
11 1 1 1 1
10 0 0 0 0
ab
cd 00 01 11 10
00 0 1 1 0
01 0 0 0 0
11 0 1 1 0
10 0 1 1 0X Y
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100 6.2
X = b’d + bcdbcd
Y = bd’ + bcdbcd
b’dbcdbd’
X
Y
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101 6.2
X = b’d + bcdbcd
Y = bd’ + bcdbcd
Cost:
3 and functions ( 3 and gates)
2 or functions ( 2 or gates)
b’dbcdbd’
X
Y
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102 6.2
Note
A systematic approach to the manual optimization of multiple output functions is outside the scope of the present course.
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103 6.2