1. At the bottom, all data is binary
At the bottom, all data is binary

2. Key Questions
If 0’s and 1’s are the fundamental “atoms” of information, what are the mechanisms via which they interact?
How does information “get processed”, or computation on information occur?
What are the basic operations?

3. Gates and Circuits

4. Goals Learn function of basic logic gates AND, OR, NOT, XOR, NAND, NOR
Learn how gates can be combined into simple circuits.
Given a circuit, a truth-table, or a Boolean formula, construct the other two.
Explain the concept of universality of a collection of gates.
Explain how transistors can act as logic gates.

5. “OR” gate
inputs
output A B
A+B 1
inputs
output
truth table

6. “AND” gate
inputs
output A B
A·B 1
inputs
output
“truth table”

7. “NOT” gate
input
output
input A
~A, A’, A,¬A 1
output
truth table

8. Another view: arithmetic+ 1 1 B B
OR = Addition
AND = Multiplication

9. Elwood the Electronic Watchdog

10. output w = d+ps d p s woof 1 Elwood the Electronic Watchdog1
output w = d+ps
Use boolean arithmetic to evaluate!
Show method of writing 0/1 values on circuit lines to evaluate, then have them do it.
Show how boolean formula is derived from circuit by writing on circuit lines.
Show method of evaluating boolean formula on next slide.

11. output w = d+ps example: 010 0 + 10 = 0 + 0 = 0 example: 011 ? d p s
Elwood the Electronic Watchdog
output w = d+ps
example: 010
0 + 10 =
= 0 d p s
woof 1 1
example: 011 ?

12. CHALLENGE
Draw Circuit
Give Truth Table
A’B A B
A’B 1

13. CHALLENGE: circuit to truth table
output 1
not
and or
and
not
Can also reason about circuit to simplify process. Eg, observe that except when B and C are 1, the lower input to the OR gate will be 1, so can fill in all but two rows with 1 output. Then deal with those two rows by inputting values, or note that when A is 1, the upper AND will be 0, and so...
What is truth table?
- Use the labeling method

14. CHALLENGE: circuit to formulaB C
not
and or
and
not
What is formula?
- Use the labeling method

15. CHALLENGE: formula to circuit
A(B+C’)(C+D) + BC(A’+D)
Construct circuit
TOP-DOWN APPROACH
Do on board either bottom-up or top-down method

16. CHALLENGE: formula to truth table
A(B+C’)(C+D) + BC(A’+D)
Example: Find value if A=1, B=0, C=0, D=1
Simply plug in values and simplify!
1(0+1)(0+1) + 00(0+1)
= 1(1)(1) + 00(1) =
= 1
Do on board either bottom-up or top-down method

17. w = d+ps
Formula ?
Circuit
Truth Table ?

18. Circuit Design We can now do the following:
From formula or circuit, get the other
From formula or circuit, get the truth table
Given truth table (desired behavior), how can you construct a formula?
Given truth table (desired behavior), how can you construct a circuit?
Perhaps best to just do it by example.

19. Important Question
Are there circuits and formulas for every possible Truth Table?
In other words, if I tell you what output behavior I want, for all combinations of input values, can you create a formula and circuit which realizes that behavior?

20. Sum-of-Products method
A’B’C + A’BC + ABC A B C
output 1
Expression is giant “OR” (+), with terms corresponding to each row of truth table with a “1” output.
The term for a row is a product of variables, one for each input, and either negated (if 0) or unnegated (if 1).
A’B’C
A’BC
ABC

21. Sum-of-Products method
EXAMPLE 2: (you do it) A B C
output 1

22. Universality
The sum-of-products method shows that no matter what truth table we start with, we can create a circuit from AND, OR, NOT
This means that {AND, OR, NOT} are Universal: they can be used to make a circuit for any behavior desired
That’s why all we need to build a computer are AND, OR, NOT gates.

23. Other Gates XOR NAND NOR A B Out 1 A B Out 1 A B Out 11 A B
Out 1 A B
Out 1
Amazing Fact: {NAND} is universal all by itself!!

24. More complex circuits Two-bit adder
(“XOR” is 1 when exactly one of inputs is 1)
Play with the actual circuit in SimCir

26. Boolean Algebra
Can simplify complex expressions, just like arithmetic simplification
Example: distributive: A(B+C) = AB+AC
but also A+BC = (A+B)(A+C)
Can show some expressions equivalent to others.
Entire industry… circuit simplification/minimization.

27. Transistors Transistors to gates: Sec 4.3 in text Reading and Demo:

29. 1 micron = 1/ inch .13 micron feature resolution on chips ~1 billion gates on chip. Incredible complexity - how could such a thing be built ??!! (ultimately, we’ll grow them?!)

30. How did we get from {0,1,and/or/not} to TODAY ??!!
How is it possible to build a car, or a skyscraper?
You don’t think about a bunch of steel, wires, etc.
Complexity, difficult problems, are attacked by
breaking them up into subproblems, subsystems, often in different “layers” hierarchically.
CS not only has benefited from this method, but
it in fact studies methods for such organization.

31. Summary
Simple functional units called gates are made from transistors, and manipulate 0’s and 1’s.
AND, OR, NOT, plus others
Equivalence between circuits, formulas, and truth tables
Sum-of-products method to build formula and circuit from a given truth-table specification
Set of gates is universal - can realize any truth table or formula we care to specify
Complex circuits (we’ll see more later) built out of simple gates to do interesting things