mc quine multiple output
TRANSCRIPT
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2-Level Minimization
Classic Problem in Switching Theory Tabulation Method Transformed to Set
Covering Problem
Set Covering Problem is Intractable
n-input Function Can Have 3n/n Prime
Implicants
n-input Function Can Have 2n Minterms
Exponentially Complex Algorithms for Exact
Solutions
NOTE: Sometimes the Covering Problem is Easy to Solve (when
no cyclic tables result)
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Heuristic Branch and Bound
Quine-McCluskey Method is Tabulation Method Using a Branch and
Bound Algorithm with Heuristic in Branch Operation for Solution of theCyclic Cover
Branch and Bound Approaches
BRANCH STEP
Reduce, HALT if not Cyclic Cover Heuristically Choose a PI
Solve Cyclic Cover in Two Ways
1) Assume Chosen PI is in the Final Cover Set
2) Assume Chosen PI is not in the Final Cover Set
3) Select Between 1) and 2) Depending on Minimal Cost
BOUND STEP
If Current Solution is Better Than Previous, Return from this Level ofRecursion (Note: Initially Set to Entire Set of PIs in Table)
Go to Branching Step
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Branch and Bound Diagram
Cyclic
Cover 1
Cyclic
Cover 2
Cyclic
Cover 3
CC3
Solution 1CC3
Solution 2
Initial CC1 Soln is All PIs
Heuristically Choose a
PI Reduce to CC2
Initial CC2 Soln is All PIs
Heuristically Choose aPI Reduce to CC3
Initial CC3 Soln is All PIs
Heuristically Choose a
PI1 Reduce to CC3Heuristically Choose a
PI2 Reduce to CC3
Soln 1 Fully Reduced Equal to All PIs in CC3 No Bounding
Soln 2 Better CC3 Entire
Set and Fully Reduced
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Choosing Candidate PIs
Choose PI With Fewest Literals
That is, One that Covers the most Minterms
Select One that Covers a Minterm Covered byVery Few Other Minterms
Note if Minterm Covered by Single PI, it is EPI
This Technique Chooses One that is Almost an EPI
Independent Set Heuristic
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Independent Set Heuristic
Find Maximum Set of Independent Rows in Cover Matrix
Partition Matrix as Shown
1 1 1
1 1 1 1
1 11 1
0
CA
Iis Sub-Matrix of Independent Rows
I = {I1, I2, I3, ...}
1) Choose PI inIthat Covers Most Rows inA
2) Reduce Matrix Using New EPI Selection and Dominance
3) If Matrix is 00 Solution is Found Else Go To 1)
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Exact Method Petricks Method
When Cyclic Cover Table is Found use Covering
Clauses in POS Form
Each Product Corresponds to Minterm
Transform the POS to SOP
Product Terms Represent Selected Primes
Minimum Cover Identified by Product with Fewest
Literals
Finds ALL Solutions to the Cyclic Cover
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Petricks Method Example
Write Clauses as a POS Expression:
( )( )( )( )( ) 1A D A B B C C D B D+ + + + + =
We Solve this Equation
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Solving the Satisfying Clause
It is Easy to Find a Satisfying Argument for a SOP Expression Classic Petricks Method Transforms POS to SOP
( )( )( )( )( ) 1A D A B B C C D B D+ + + + + =
( )( )( ) 1A AB AD BD BC BD C CD B D+ + + + + + + =
( )( )( ) 1A BD C BD B D+ + + =( )( ) 1A BD BC CD BD+ + + =
1ABC ACD ABD BCD BD+ + + + =
{ , , , , }SOLN ABC ACD ABD BCD BD=
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Multi-Output Functions
Minimizing Each Output Separately Usually Results in PoorMinimization
Term Sharing Occurs Only by Chance
Can Use Multi-Output Prime Implicants
More Complex Version of Tabulation Method
Can Use Characteristic Function for Multi-Output
Functions
Utilizes Principles in Multiple-Valued Logic
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Product Functions
Consider1( , , )f x y z x z x y x z= + +
2 ( , , )f x y z z x y= +x
y
z
f1
f2x y z f1 f2 f
1 f
2
0 0 0 1 0 0
0 0 1 1 1 1
0 1 0 1 1 10 1 1 0 1 0
1 0 0 0 0 01 0 1 1 1 1
1 1 0 0 0 01 1 1 1 1 1
Minterms inf1f2 are Minterms for Bothf1 andf2
f1f2 is a Product Function
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Multi-Output Prime Implicant
DEFINITIONA MOPI for a set of switching functionsf1, f2, , fm is a product of
literals which is either:
1. A Prime Implicant of One of the functions,fi, fori=1,2,,m
2. A Prime Implicant of one of the Product Functions,fifjfk
where i,j,k=1,2,,m and ij k
THEOREM
The set of all MOPIs is sufficient for the determination of
at least one multi-output minimized SOP.
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Tagged Product Terms Could Generate Using K-map or Tabulation Method for Each
Output SeparatelyAND all Product Functions
too lengthy
instead use Tagged Product Terms
Tagged Product Terms have Two Parts:
1. kernel a product term of literals (as normal)
2. tag appended entity to kernel that indicates which function outputs it
applies tox y z f1 f2 f1 f2 TPTs
0 0 0 1 0 0 000f1 -
0 0 1 1 1 1 001f1f2
0 1 0 1 1 1 010f1f2
0 1 1 0 1 0 011 -f2
1 0 0 0 0 0
1 0 1 1 1 1 101f1f2
1 1 0 0 0 0
1 1 1 1 1 1 111f1f2
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Generating MOPIs
0 0 0 f1 -0 0 1 f1 f2
0 1 0 f1 f2
0 1 1 - f2
1 0 1 f1 f2
1 1 1 f1 f2
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Generating MOPIs
0 0 0 f1 -
0 0 1 f1 f2
0 1 0 f1 f2
0 1 1 - f2
1 0 1 f1 f2
1 1 1 f1 f2
0 0 - f1 -0 - 0 f1 -
0 - 1 - f2
- 0 1 f1 f20 1 - - f2
- 1 1 - f2
1 - 1 f1 f2
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Generating MOPIs
0-0f1- and 0-1-f2DO NOT combine
No Common Tag Elements
They WOULD Combine Under Cube Merging for Single Output
Function
Only Place Check Mark on Terms with COMMON Tag Outputs
EXAMPLE 000f1- and 001f1f2 Results in 000f1- Being Checked ONLY
0 0 0 f1 -
0 0 1 f1 f2
0 1 0 f1 f2 G
0 1 1 - f2
1 0 1 f1 f2
1 1 1 f1 f2
0 0 - f1 - B0 - 0 f1 - C
0 - 1 - f2
- 0 1 f1 f2 D0 1 - - f2 E
- 1 1 - f2
1 - 1 f1 f2 F
- - 1 - f2 A
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MOPI Cover Table
f1 f2m0 m1 m2 m5 m7 m1 m2 m3 m5 m7
B X Xf1 C X X
A X X X X
f2 E X XD X X X X
*F X X X Xf1f2G X X
5
1&7
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MOPI Cover Table
Fis an essential row forf1, only
Column dominance, remove m5 underf2
f1 f2m0 m1 m2 m5 m7 m1 m2 m3 m5 m7
B X Xf1 C X X
A X X X X
f2 E X XD X X X X
*F X X X Xf1f2G X X
5
1&7
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MOPI Cover Table
Will use Petricks method applied to multiple outputs
f2 f2
m0 m1 m2 m1 m2 m3 m7
B X Xf1
C X X
A X X Xf2E X X
D X X
*F Xf1f2
G X X
B+CB+D C+G A+D E+G A+E A+F
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Exact Solution
( )( )( )( )( )( )( ) 1B C B D C G A D E G A E A F+ + + + + + + =
( )( )( )( )( ) 1B CD C G A DE E G A F+ + + + + =
( )( )( ) 1BC BG CD A DEF E G+ + + + =
( )( ) 1BC BG CD AE AG DEF+ + + + =
(
) 1
ABCE ABCG BCDEF ACDE ACDG
CDEF ABEG ABG BDEFG
+ + + +
+ + + + =
( ) 1ABCE ACDE ACDG CDEF ABEG ABG BDEFG+ + + + + + =
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Exact Solution (Cont.)
Minimum Product Term:
{A,B,G}
f1ON
=
All MOPIs that havef1in Tagf2
ON=All MOPIs that havef2in Tag
1
2
f F B G
f A G
= + +
= +
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Hazard-Free2-Level Minimization
Tabular Method Can Be Applied To Realize 2-LevelDesigns That Eliminate Certain Hazards
Considers Delay of Logic Circuits
Example:
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Hazard Types
Static Hazard Output value the sameafter input change
0-Hazard 1-Hazard
Dynamic Hazard Output value
different after input change
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Analysis of Networks with Static
Hazards
SOP Expression with 1-Hazard
POS Expression with 0-Hazard
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Elimination of Hazard
Prime Implicant added to eliminate
static 1-hazard
Oth H d Cl ifi ti f M
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Other Hazard Classifications for More
than One Input Change Logic Hazard Hazard caused by the particular
implementation. Can be eliminated by adding Pis
Function Hazard Presence of hazard due to the function
realized by the output. Present for transitions in which more
than one input changes. Cannot be eliminated
T b l A h t H d F
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Tabular Approach to Hazard-Free
Design
Uses Minimum number of prime implicants
A Network will contain no static or dynamic
hazards if its 1-sets satisfy the following two
conditions
1. For each pair of adjacent input states that both
produce a 1 output, there is at least one 1-set that
includes both input states of the pair2. There are no 1-sets that contain exactly one pair
of complementary literals
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Tabular Approach Steps
1. Find prime implicants using tabular approach
2. Create prime implicant table, however,
columns will be any essential single minterm& all pairs of adjacent states (these have
been found in the second table of 1)
See Example Handout