1. 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)

2. Heuristic – Branch and Bound

3. Heuristic – Branch and Bound
Quine-McCluskey Method is Tabulation Method Using a Branch and Bound Algorithm with Heuristic in Branch Operation for Solution of the Cyclic Cover
Branch and Bound Approaches
BRANCH STEP
Reduce, HALT if not Cyclic Cover
Heuristically Choose a PI
Solve Cyclic Cover in Two Ways
Assume Chosen PI is in the Final Cover Set
Assume Chosen PI is not in the Final Cover Set
Select Between 1) and 2) Depending on Minimal Cost
BOUND STEP
If Current Solution is Better Than Previous, Return from this Level of Recursion (Note: Initially Set to Entire Set of PIs in Table)
Go to Branching Step

4. Branch and Bound Diagram
Cyclic
Cover 1
Initial CC1 Soln is All PIs
Heuristically Choose a
PI – Reduce to CC2
Cyclic
Cover 2
Initial CC2 Soln is All PIs
Heuristically Choose a
PI – Reduce to CC3
Cyclic
Cover 3
Initial CC3 Soln is All PIs
Heuristically Choose a
PI1 – Reduce to CC3
Heuristically Choose a
PI2 – Reduce to CC3
CC3
Solution 1
CC3
Solution 2
Soln 2 Better CC3 Entire Set and Fully Reduced
Soln 1 Fully Reduced Equal to All PIs in CC3 – No Bounding

5. Choosing Candidate PIs
Choose PI With Fewest Literals
That is, One that Covers the most Minterms
Select One that Covers a Minterm Covered by Very 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

6. Independent Set Heuristic
Find Maximum Set of Independent Rows in Cover Matrix
Partition Matrix as Shown
I is Sub-Matrix of Independent Rows I = {I1, I2, I3, ...} A C
Choose PI in I that Covers Most Rows in A
Reduce Matrix Using New EPI Selection and Dominance
If Matrix is 00 Solution is Found Else Go To 1)

7. Exact Method – Petrick’s 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

8. Petrick’s Method Example

9. Petrick’s Method Example
Write Clauses as a POS Expression:
We Solve this Equation

10. Solving the Satisfying Clause
It is Easy to Find a Satisfying Argument for a SOP Expression
Classic “Petrick’s Method” Transforms POS to SOP

11. Multi-Output Functions
Minimizing Each Output Separately Usually Results in Poor Minimization
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

12. Product Functions Considerx f1 y f2 z
Minterms in f1•f2 are Minterms for Both f1 and f2
f1•f2 is a Product Function

13. Multi-Output Prime Implicant
DEFINITION
A MOPI for a set of switching functions f1, f2, …, fm is a product of literals which is either:
A Prime Implicant of One of the functions, fi, for i=1,2,…,m
A Prime Implicant of one of the Product Functions, fi•fj•…•fk where i,j,k=1,2,…,m and ij k
THEOREM
The set of all MOPIs is sufficient for the determination of at least one multi-output minimized SOP.

14. Tagged Product Terms
Could Generate Using K-map or Tabulation Method for Each Output Separately AND all Product Functions
too lengthy
instead use Tagged Product Terms
Tagged Product Terms have Two Parts:
kernel – a product term of literals (as normal)
tag – appended entity to kernel that indicates which function outputs it applies to

15. Generating MOPIs

16. Generating MOPIs

17. Generating MOPIs 0-0f1- and 0-1-f2 DO 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 f1- and 001f1f2 Results in 000f1- Being Checked ONLY

18. MOPI Cover Table

19. MOPI Cover Table F is an essential row for f1, only
Column dominance, remove m5 under f2

20. MOPI Cover Table
Will use Petrick’s method applied to multiple outputs

21. Exact Solution

22. Exact Solution (Cont.) Minimum Product Term: {A,B,G}
f1ON = All MOPIs that have f1 in Tag
f2ON = All MOPIs that have f2 in Tag

23. Hazard-Free 2-Level Minimization

24. Hazard-Free 2-Level Minimization
Tabular Method Can Be Applied To Realize 2-Level Designs That Eliminate Certain Hazards
Considers Delay of Logic Circuits
Example:

25. Hazard Types Static Hazard – Output value the same after input change
0-Hazard 1-Hazard
Dynamic Hazard – Output value different after input change

26. Analysis of Networks with Static Hazards
SOP Expression with 1-Hazard
POS Expression with 0-Hazard

27. Elimination of Hazard
Prime Implicant added to eliminate static 1-hazard

28. 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

29. 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
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 pair
There are no 1-sets that contain exactly one pair of complementary literals

30. Tabular Approach Steps
Find prime implicants using tabular approach
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

31. Problems to consider Finding prime implicants using tabular approach
Hazard, what is it?
Use Covering to solve non-hazard realizations
Petrick Function and its uses
How to program Petrick Function?
How to program Backtracking?
How to program Branch–and-Bound?
How can we generalize the concept of Petrick Function?