1. COMP541 Combinational Logic - II
Montek Singh
Jan 19, 2010

2. Today Basics of Boolean Algebra (review)
Identities and Simplification
Basics of Logic Implementation
Minterms and maxterms
Going from truth table to logic implementation

3. Identities Use identities to manipulate functions
You can use distributive law …
… to transform from to

4. Table of Identities

5. Duals
Left and right columns are duals
Replace AND and OR, 0s and 1s

6. Single Variable Identities

7. Commutativity
Operation is independent of order of variables

8. Associativity Independent of order in which we group
So can also be written as
and

9. Distributivity
Can substitute arbitrarily large algebraic expressions for the variables
Distribute an operation over the entire expression

10. DeMorgan’s Theorem Used a lot NOR  invert, then AND
NAND  invert, then OR

11. Truth Tables for DeMorgan’s

12. Algebraic Manipulation
Consider function

13. Simplify Function
Apply
Apply
Apply

14. Fewer Gates

15. Consensus Theorem The third term is redundant Proof summary:
Can just drop
Proof summary:
For third term to be true, Y & Z both must be 1
Then one of the first two terms is already 1!

16. Complement of a Function
Definition: 1s & 0s swapped in truth table
Mechanical way to derive algebraic form
Take the dual
Recall: Interchange AND and OR, and 1s & 0s
Complement each literal

17. Mechanically Go From Truth Table to Function

18. From Truth Table to Func
Consider a truth table
Can implement F by taking OR of all terms that are 1

19. Standard Forms Not necessarily simplest F
But it’s a mechanical way to go from truth table to function
Definitions:
Product terms – AND  ĀBZ
Sum terms – OR  X + Ā
This is logical product and sum, not arithmetic

20. Definition: Minterm
Product term in which all variables appear once (complemented or not)

21. Number of Minterms For n variables, there will be 2n minterms
Like binary numbers from 0 to 2n-1
Often numbered same way (with decimal conversion)

22. Maxterms
Sum term in which all variables appear once (complemented or not)

23. Minterm related to Maxterm
Minterm and maxterm with same subscripts are complements
Example

24. Sum of Minterms Like Slide 18
OR all of the minterms of truth table row with a 1
“ON-set minterms”

25. Sum of Products
Simplifying sum-of-minterms can yield a sum of products
Difference is each term need not be a minterm
i.e., terms do not need to have all variables
A bunch of ANDs and one OR

26. Two-Level Implementation
Sum of products has 2 levels of gates

27. More Levels of Gates? What’s best? Hard to answer
More gate delays (more on this later)
But maybe we only have 2-input gates
So multi-input ANDs and ORs have to be decomposed

28. Complement of a Function
Definition: 1s & 0s swapped in truth table
Mechanical way to derive algebraic form
Take the dual
Recall: Interchange AND and OR, and 1s & 0s
Complement each literal

29. Complement of F Not surprisingly, just sum of the other minterms
“OFF-set minterms”
In this case
m1 + m3 + m4 + m6

30. Product of Maxterms
Recall that maxterm is true except for its own case
So M1 is only false for 001

31. Product of Maxterms
Can express F as AND of all rows that should evaluate to 0 or

32. Product of Sums Result: another standard form ORs followed by AND
Terms do not have to be maxterms

33. Recap Working (so far) with AND, OR, and NOT Algebraic identities
Algebraic simplification
Minterms and maxterms
Can now synthesize function (and gates) from truth table

34. Next Time Lab Prep More on combinational logic Demo lab software
Talk about FPGA internals
Overview of components on board
Downloading and testing
More on combinational logic