Announcements HW5 due today Upcoming Midterm on Tuesday 1028 Will post list of topics and review problems Agenda Last time QuineMcClusky 48 Table Reductions 410 This time Multiple Output ID: 661981
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Slide1
Digital Logic Design
Lecture 15Slide2
Announcements
HW5 due today
Upcoming: Midterm on Tuesday, 10/28.
Will post list of topics and review problems.Slide3
Agenda
Last time:
Quine-McClusky
(4.8)
Table Reductions (4.10)
This time:
Multiple Output
Simplification Problem (4.12, 4.13)Slide4
The Multiple-Output Simplification Problem
General combinational networks can have several output terminals.
The output behavior of the network is described by a set of functions
, one for each output terminal, each involving the same input variables,
.
The set of functions is represented by a truth table with
columns.
Objective is to design a multiple-output network of minimal cost.Formally: A set of normal expressions that has associated with it a minimal cost as given by some cost criteria.Cost criteria: number of gates or number of gate inputs in the realization.
Slide5
Pitfalls of Naïve Approach
Multiple-output minimization problem is normally more difficult than sharing common terms in independently obtained minimal expressions.
Consider:
Slide6
Pitfalls of Naïve Approach
Naïve Approach:
Better Approach:
Slide7
Multiple Output Prime Implicants
A multiple-output prime
implicant
for a set of Boolean functions
, is a product term that:
Is a prime
implicant
of one of the individual functions Is a prime implicant of one of the product functions
Slide8
Examples of Multiple Output Prime
Implicants
0
0
0
1
0
0
0
1
01010110110010
0
0
1
1
0
1
1
1
1
1
0
1
0
1
1
1
0
1
0
0
0
1
0
001
0101
01101
100
100011
01111
10101
110
1
is a prime
implicant
of
Slide9
Examples of Multiple Output Prime
Implicants
0
0
0
1
0
0
0
0
10100101
1
1
0
1
1
0
0
0
1
0
0
0
1
0
1
0
1
1
1
1
110
11011
1010
0
0
0
10
00010
100101
11011
000
100
010
10
111
111
011
01
110
10
is a prime
implicant
of
Slide10
Multiple Output Prime Implicants
Theorem: Formulas that achieve the multiple-output minimal sum consist only of sums of multiple-output prime
implicants
such that all the terms in the expression for
or of a product function involving
Slide11
Tagged Product Terms
Term consists of two parts: a
kernel
and a
tag
.
Kernel: product term involving the variables of the function
Tag: Appended to denote which functions are implied by its kernel.Slide12
Tagged Product Term Example
0
0
0
1
0
0
0
1
1
10100101100100
0
0
1
0
1
1
--
1
1
0
0
0
1
1
1
1
1
0
0
0
1
0
0
01
1101
001011
001
00001
011--11
00011
111
Algebraic Form
Binary Form
Algebraic Form
Binary FormSlide13
Quine-McClusky
for tagged multiple-output prime
implicants
0
0
0
0
1001
2
0
103
0
1
1
5
1
0
1
7
1
1
1
0
0
0
0
1
0
0
1
2
0
1
0
3
0
1
1
5
1
0
1
7
1
1
1
The tag of a generated term has
iff
appears in both the tags of the generating terms.
A generating term is checked only if its tag is identical to the tag of the generated term.
(0,1)
0
0
(0,2)
0
0
(1,3)
0
1
(1,5)
0
1
(2,3)
0
1
(3,7)
1
1
(5,7)
1
1
(0,1)
0
0
(0,2)
0
0
(1,3)
0
1
(1,5)
0
1
(2,3)
0
1
(3,7)
1
1
(5,7)
1
1
(1,3,5,7)
1
(1,5,3,7)
1
(1,3,5,7)
1
(1,5,3,7)
1
Why doesn’t (0,1,2,3) appear?Slide14
Minimal Sums Using
Petrick’s
MethodSlide15
Multiple Outputs Prime
Implicant
Tables
B
X
X
C
X
X
A
X
X
X
E
X
X
D
X
X
X
F
X
X
X
G
X
X
B
X
X
C
X
X
A
X
X
X
E
X
X
D
X
X
X
F
X
X
X
G
X
XSlide16
Multiple Outputs Prime
Implicant
Tables
B
X
X
C
X
X
A
X
X
X
E
X
X
D
X
X
X
F
X
X
X
G
X
X
B
X
X
C
X
X
A
X
X
X
E
X
X
D
X
X
X
F
X
X
X
G
X
X
When writing down p-expression, must make a distinction between primes associated with different functions.Slide17
Multiple Outputs Prime
Implicant
Tables
B
X
X
C
X
X
A
X
X
X
E
X
X
D
X
X
X
F
X
X
X
G
X
X
B
X
X
C
X
X
A
X
X
X
E
X
X
D
X
X
X
F
X
X
X
G
X
X
P-expression:
Slide18
Manipulating P-expression into sum of product form
When calculating cost of a product term, we can disregard subscripts.
i.e.
is the same cost as
Slide19
Calculating Cost of Product Terms
The term
yields
The term
yields
Slide20
Calculating Cost of multiple output combinational network
Where
is the set of distinct terms,
is equal to the number of literals in
, unless the term consists of a single literal, in which case
. Let
be the number of terms in
unless there is only a single term, in which case
Slide21
Calculating Cost of Product Terms
The term
yields
Distinct terms:
Beta costs: 2 + 2 + 3 + 0 = 7
Alpha costs = 3 + 2 = 5
Total cost: 12
The term
yields
Distinct terms:
Beta costs: 2+2+2+2 = 8
Alpha costs = 3 +3 = 6
Total cost: 14
Slide22
Minimal Sums using Table ReductionSlide23
Table Reduction
Cost
B
X
X
3
C
X
X
3
A
X
X
X
1
E
X
X
3
D
X
X
X
3,4
F
X
X
X
3,4
G
X
X
4,5
Cost
B
X
X
3
C
X
X
3
A
X
X
X
1
E
X
X
3
D
X
X
X
3,4
F
X
X
X
3,4
G
X
X
4,5Slide24
Table Reduction
Cost
B
X
X
3
C
X
X
3
A
X
X
X
1
E
X
X
3
D
X
X
X
3,4
F
X
X
X
3,4
G
X
X
4,5
Cost
B
X
X
3
C
X
X
3
A
X
X
X
1
E
X
X
3
D
X
X
X
3,4
F
X
X
X
3,4
G
X
X
4,5
Essential prime
implicant
for
Slide25
Table Reduction
Cost
B
X
X
3
C
X
X
3
A
X
X
X
1
E
X
X
3
D
X
X
3,4
*1 F
X
1
G
X
X
4,5
Cost
B
X
X
3
C
X
X
3
A
X
X
X
1
E
X
X
3
D
X
X
3,4
*1 F
X
1
G
X
X
4,5
column cannot be removed from
part since
is not essential for
.
Slide26
Table Reduction
Cost
B
X
X
3
C
X
X
3
A
X
X
X
1
E
X
X
3
D
X
X
3,4
*1 F
X
1
G
X
X
4,5
Cost
B
X
X
3
C
X
X
3
A
X
X
X
1
E
X
X
3
D
X
X
3,4
*1 F
X
1
G
X
X
4,5
Dominated RowsSlide27
Table Reduction
Cost
B
X
X
3
C
X
X
3
A
X
X
X
1
E
X
X
3
D
X
X
3,4
*1 F
X
1
G
X
X
4,5
Cost
B
X
X
3
C
X
X
3
A
X
X
X
1
E
X
X
3
D
X
X
3,4
*1 F
X
1
G
X
X
4,5
Dominated Rows
Row A dominates Row F
Cost for Row A is not greater than cost for Row F.Slide28
Table Reduction
Cost
B
X
X
3
C
X
X
3
A
X
X
X
1
E
X
X
3
D
X
X
3,4
G
X
X
4,5
Cost
B
X
X
3
C
X
X
3
A
X
X
X
1
E
X
X
3
D
X
X
3,4
G
X
X
4,5
Slide29
Table Reduction
Cost
B
X
X
3
C
X
X
3
A
X
X
X
1
E
X
X
3
D
X
X
3,4
G
X
X
4,5
Cost
B
X
X
3
C
X
X
3
A
X
X
X
1
E
X
X
3
D
X
X
3,4
G
X
X
4,5
Only row that covers
Slide30
Table Reduction
Cost
B
X
X
3
C
X
X
3
*2 A
1
E
X
3
D
X
3,4
G
X
X
4,5
Cost
B
X
X
3
C
X
X
3
*2 A
1
E
X
3
D
X
3,4
G
X
X
4,5
Delete
Slide31
Table Reduction
Cost
B
X
X
3
C
X
X
3
E
X
3
D
X
3
G
X
X
4,5
Cost
B
X
X
3
C
X
X
3
E
X
3
D
X
3
G
X
X
4,5
Delete row ASlide32
Table Reduction
Cost
B
X
X
3
C
X
X
3
E
X
3
D
X
3
G
X
X
4,5
Cost
B
X
X
3
C
X
X
3
E
X
3
D
X
3
G
X
X
4,5
Row D is dominated by Row B.Slide33
Table Reduction
Cost
B
X
X
3
C
X
X
3
E
X
3
D
X
3
G
X
X
4,5
Cost
B
X
X
3
C
X
X
3
E
X
3
D
X
3
G
X
X
4,5
Row D is dominated by Row B.Slide34
Table Reduction
Cost
B
X
X
3
C
X
X
3
E
X
3
G
X
X
4,5
Cost
B
X
X
3
C
X
X
3
E
X
3
G
X
X
4,5
Row D is dominated by Row B.Slide35
Table Reduction
Cost
B
X
X
3
C
X
X
3
E
X
3
G
X
X
4,5
Cost
B
X
X
3
C
X
X
3
E
X
3
G
X
X
4,5
Row B is the only row covering
Slide36
Table Reduction
Cost
*1 B
X
X
3
C
X
X
3
E
X
3
G
X
X
4,5
Cost
*1 B
X
X
3
C
X
X
3
E
X
3
G
X
X
4,5
Row B is the only row covering
Slide37
Table Reduction
Cost
*1 B
3
C
X
3
E
X
3
G
X
X
4,5
Cost
*1 B
3
C
X
3
E
X
3
G
X
X
4,5
Delete columns
Slide38
Table Reduction
Cost
C
X
3
E
X
3
G
X
X
4,5
Cost
C
X
3
E
X
3
G
X
X
4,5
Delete columns
Slide39
Table Reduction
Cost
C
X
3
E
X
3
G
X
X
4,5
Cost
C
X
3
E
X
3
G
X
X
4,5
Cannot delete dominated rows since their cost is lower.
**Table is cyclic**Slide40
Table Reduction
Cost
C
X
3
E
X
3
G
X
X
4,5
Cost
C
X
3
E
X
3
G
X
X
4,5
Cannot delete dominated rows since their cost is lower.
**Table is cyclic**Slide41