1. COMP541 Combinational Logic - II
Montek Singh
Sep 11, 2017

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

3. Digital Circuits
Digital Circuit = network that processes binary variables
one or more binary inputs
one or more binary outputs
inputs and outputs are called “terminals”
a functional specification
relationship between inputs and outputs
a timing specification
describes delay from inputs changing to outputs responding

4. Digital Circuits Inside the black box Example:
subcircuits or components or elements
connected by wires
wires and terminals often called “nodes”
each node has a binary value
each node is an input, an output, or “internal”
Example:
E1, E2, E3 are elements
A, B, C are input nodes
Y, Z are output nodes
n1 is an internal node

5. Types of circuits Two types: with memory and without
Combinational Circuit
output depends only on the current values of the inputs
provided enough time is given for output to respond
output does not depend on past inputs or outputs
called “memory-less”
example: AND gate
Sequential Circuit
anything not combinational is sequential
output depends on not only current inputs, but also past behavior
previous inputs and/or outputs affect behavior
has “memory”, or is “stateful”
example: counter

6. Combinational Circuits: ExamplesOR
Adder
Multi-output example
Slash notation

7. Combinational Circuits
Theorem: A circuit is combinational if:
every element is itself combinational
every node is either designated as an input, or connects to exactly one output terminal of an element
outputs of two elements are never “shorted together”
ensures that each node has a unique/unambiguous value
contains no cyclic paths
every path through the circuit visits each node at most once
no “feedback”
Conditions above ensure that output is only a function of inputs
Proof: By induction
Let’s do it interactively

8. Combinational Circuits: Examples
Which meet the conditions for combinational logic?

9. Identities in Boolean Algebra
Use identities to manipulate functions
Often used to simply a Boolean expression
e.g., so can be implemented using fewer gates

10. Table of Identities

11. Duals
Left and right columns are duals
Replace ANDs and ORs, 0s and 1s

12. Single Variable Identities

13. Commutativity
Operation is independent of order of variables

14. Associativity Independent of order in which we group
So can also be simply written as:
X+Y+Z, and
XYZ

15. Distributivity

16. Substitution
Can substitute arbitrarily large algebraic expressions for the variables
Distribute an operation over the entire expression
Example:
X + YZ = (X+Y)(X+Z)
Substitute ABC for X
ABC + YZ = (ABC + Y)(ABC + Z)

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

18. Truth Tables for DeMorgan’s

19. DeMorgan’s Thm.: “Bubble Pushing”
imagine the bubble at the output is being pushed towards the inputs
it becomes a bubble at every input, and
the shape of the gate changes from AND to OR, and vice versa

20. Algebraic/Boolean Manipulation
Apply algebraic and Boolean identities to simplify expression
example:

21. Simplification Example
Apply
Apply
Apply

22. Fewer Gates

23. Consensus Theorem The third term is redundant
Can just drop third term (consensus term)
Proof summary (for first version):
For third term to be true, Y & Z both must be 1
Then one of the first two terms is already 1!
Exercise: Provide a similar proof for the 2nd version

24. Next Lecture Next Class: More on combinational logic
Commonly-used combinational building blocks