1. Logic operations

2. Logic operations
We have already seen keywords or, and, not used in Python
Had a specific purpose – Boolean expressions. For example: if x >= 0 and x < 10: print("x is a single digit")
Python has a set of operators for bitwise computations:
& : bitwise AND
| : bitwise OR
~ : bitwise NOT
>>: shift to the right
<<: shift to the left

3. A bit of intuition… 21 << 1 5 | 6 5 & 6 21 >> 1 42 ? ? 10
& bit-wise AND
| bit-wise OR
^ bit-wise XOR
21 >> 1 10

4. A bit of intuition… 21 << 1 5 | 6 5 & 6 ~0 21 >> 1
Sometimes the bits are almost visible:
10101
101
110
101
110
21 << 1
5 & 6
5 | 6
000
10101
& bit-wise and
| bit-wise or
^ bit-wise exclusive or
~ bit-wise not
NOTE ~0 is -1 because it flips all the bits including sign bit ~000 is 111
21 >> 1 ~0

5. A bit of intuition… 21 << 1 5 | 6 5 & 6 ~0 21 >> 1
Sometimes the bits are almost visible:
10101
101
110
101
110
21 << 1
5 & 6
5 | 6
101010
100
111
000
10101
& bit-wise and
| bit-wise or
^ bit-wise exclusive or
~ bit-wise not
NOTE ~0 is -1 because it flips all the bits including sign bit ~000 is 111
21 >> 1 ~0
111
1010

6. AND OR NOT Truth tables x y AND(x,y) x y OR(x,y) x NOT(x) 1 1 1 1 1 1
input
output
input
output
input
output x y
AND(x,y) x y
OR(x,y) x
NOT(x) 1 1 1 1 1 1 1 1 1 1 1 1 1 1
AND outputs 1 only if ALL inputs are 1
OR outputs 1 if ANY input is 1
NOT reverses its input
Let’s construct truth tables for the operations that we have just described.
(They should have it in the handout, so we can just look over the handout)
AND OR
NOT

7. All computation x y fn(x,y) 1 1 1 1 1 1 Truth table
… consists of functions of bits, or boolean values
Boolean inputs x,y,… can only be 0 or 1 (False or True).
Boolean functions can output only 0 or 1 (False, True).
inputs
output x y
fn(x,y) 1 1 1 1 1 1
Truth table

8. Lots of bits! x y z fn(x,y,z) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
output
inputs x y z
fn(x,y,z) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
Truth table

9. Logic gates and circuits

10. Reviewing…
We have:
explored a wide range of data types
learned how different encodings are used for different types
learned that, at the core of all data stored in the computer are bits
observed different operations that can be performed on these bits (AND, OR, NOT)
We have one BIG QUESTION remaining… HOW IS COMPUTATION ACTUALLY CARRIED OUT?

11. In a computer, each bit is represented as a voltage (1 is +5v and 0 is 0v)
Computation is simply the deliberate combination of those voltages!
Feynman: Computation is just a physics experiment that always works!
101010 42
110011
ADDER circuit 51
Fenyman – awesome physicist, amazing work in quantum mechanics, Nobel prize in 1965 9
001001

12. In a computer, each bit is represented as a voltage (1 is +5v and 0 is 0v)
Computation is simply the deliberate combination of those voltages!
Feynman: Computation is just a physics experiment that always works!
101010 42
(1) set input voltages
000000
ADDER circuit 9
001001

13. In a computer, each bit is represented as a voltage (1 is +5v and 0 is 0v)
Computation is simply the deliberate combination of those voltages!
101010 42
(1) set input voltages
110011
ADDER circuit 51
(2) read output voltages 9
001001

14. In a computer, each bit is represented as a voltage (1 is +5v and 0 is 0v)
Computation is simply the deliberate combination of those voltages!
HOW? The focus for this week: Learn to design circuits that can accomplish simple computations!
101010 42
110011
ADDER circuit 51
(2) read output voltages 9
001001

15. of these boolean functions…
We need only three building circuits to compute anything at all
input
output
input
output x y
AND(x,y) x y
OR(x,y)
input
output x
NOT(x) 1 1 1 1 1 1 1 1 1 1 1 1 1 1
input
input x x y x y
These are standard symbols for the gates. Reviewing our basic knowledge
of these boolean functions…
x + y x xy
output
"x or y"
output
"not x"
"x and y"

16. Circuits from logic gates... ?
What are all of the inputs that make this circuit output 1?

17. Logisim
HW 5 – Use Logisim (a free circuit simulation package) to design circuits to perform simple computations
Hard? Well, let's recall our claim – we only need AND, OR and NOT to compute anything at all…

18. Constructive Proof ! xy + xy Formula !
For each input row whose output needs to be 1, build an AND circuit that outputs 1 only for that specific input!
Specify a truth table defining any function you want i ii
input
output x y
fn(x,y)
iii 1 1
OR them all together 1 1 1 1
Formula !
Minterm Expansion Principle – algorithm for building expressions from truth tables
xy + xy
minterm

19. Minterm expansion readily converts into logic gates…

20. Constructive Proof ! xy + xy x y fn(x,y) 1 1 1 1 1 1
For each input row whose output needs to be 1, build an AND circuit that outputs 1 only for that specific input!
Specify a truth table defining any function you want i ii
input
output x y
fn(x,y)
iii
OR them all together 1 1 1 1
xy + xy 1 1
NOT
AND
Does this work for the red row? x y
Does this work for any other rows?

21. Constructive Proof ! xy + xy x y fn(x,y) 1 1 1 1 1 1
For each input row whose output needs to be 1, build an AND circuit that outputs 1 only for that specific input!
Specify a truth table defining any function you want i ii
input
output x y
fn(x,y)
iii
OR them all together 1 1
xy + xy 1 1 1 1
How did we handle the rows outputting zero?
NOT
AND OR
NOT
AND x y

22. Constructive Proof !
For each input row whose output needs to be 1, build an AND circuit that outputs 1 only for that specific input!
Specify a truth table defining any function you want i ii
input
output x y
fn(x,y)
This is a constructive proof that AND, OR, and NOT suffice to build any (boolean) function!
iii 1 1
OR them all together 1 1 1 1
How did we handle the rows outputting zero?
NOT
AND OR
NOT
AND x y

23. But... ALL functions are just boolean functions: binary!
Constructive Proof !
For each input row whose output needs to be 1, build an AND circuit that outputs 1 only for that specific input!
Specify a truth table defining any function you want i ii
But... ALL functions are just boolean functions: binary!
input
output x y
fn(x,y)
This is a constructive proof that AND, OR, and NOT suffice to build any (boolean) function!
iii 1 1
OR them all together 1 1 1 1
How did we handle the rows outputting zero?
NOT
AND OR
NOT
AND x y

24. Example

25. Try it! the less than or equal circuit (<=) Truth Table Formula
We usually know what we want to do…
We just have to determine how to build it!
Truth Table
Formula
"English"
input bits
output bit
f(x,y) should output 1 when x <= y otherwise, output 0 x y
x <= y 1 1 1 1
1,1,0,1
-x-y + -xy + xy = -x + xy
BETTER -> recognize that only one case is a 0, and use a NOT
Another circuit to do Less than or Equal 1 1 1
Circuit

26. Try it! xy + xy + xy the less than or equal circuit (<=)
We usually know what we want to do…
We just have to determine how to build it!
Truth Table
Formula
"English"
input bits
output bit
f(x,y) should output 1 when x <= y otherwise, output 0
xy + xy + xy x y
x <= y 1 1 1 1
1,1,0,1
-x-y + -xy + xy = -x + xy
BETTER -> recognize that only one case is a 0, and use a NOT
Another circuit to do Less than or Equal 1 1 1
Circuit

27. x <= y