Mc Quine Multiple Output

8/4/2019 Mc Quine Multiple Output

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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


Heuristically Choose aPI Reduce to CC3


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

Mc Quine Multiple Output

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