1. CMSC250
Fall 2018
Circuits 1 1

2. Logic == Math?
What calculations can we do with logic? Add, subtract, multiply? George Boole – 1800’s. Boolean logic Claude Shannon – Logic == circuits == math T = 1 p v q == p+q p v ~q F = 0 p ^ q == p*q is ~ p == (1-q) p + (1-q)

3. Find Boolean formula for:
p, q & r are the variables. p q r
output 1 3
For each output row which has a 1, form a statement which is true only for the values of the inputs in that row For this example, it would be rows 1, 2, and 4, and the statements would be
(p  q  r)
(p  q  r)
(p  q  r)
Then OR them all together. For this example the result would be
(p  q  r)  (p  q  r)  (p  q  r)
This procedure always works.

4. Find Boolean Formula
For each row with output 1 obtain “mini-formula” which is 1 exactly on that row.
OR together all of the mini-formulas
111, 110, and 100 all output 1
p^q^r, p^q^~r, p^~q^~r
(p^q^r)(p^q^~r)(p^~q^~r) 4

5. Basic logic gates AND gate: OR gate: NOT gate: 5
We can recast everything we’ve done so far in terms of circuits. These circuits are what are actually inside a computer and allow it to work. AND, OR, and NOT gates really exist! 5

6. Basic logic gates AND gate: OR gate: NOT gate: 6
We can recast everything we’ve done so far in terms of circuits. These circuits are what are actually inside a computer and allow it to work. AND, OR, and NOT gates really exist! 6

7. Circuit to Boolean formula
Given this circuit, convert to logic

8. Circuit to Boolean formula
Given this circuit, convert to logic p v ~(p ^ r)

9. Draw a circuit for: p, q & r are inputs.
Simplify before building the circuit. p q r
output 1
For each output row which has a 1, form a statement which is true only for the values of the inputs in that row For this example, it would be rows 1, 2, and 4, and the statements would be
(p  q  r)
(p  q  r)
(p  q  r)
Then OR them all together. For this example the result would be
(p  q  r)  (p  q  r)  (p  q  r)
This procedure always works. 9

10. Number conversions
Different number system bases are used when convenient
some commonly-used bases are 10 (decimal), 2 (binary), 8 (octal), 16 (hexadecimal)
the base tells how many different numerals are used
the base also determines the value of each place
Conversions from anything to base 10
use the definition of the number system
Conversions from base 10 to anything
use repeated integer division
Circuits can calculate or solve arithmetic functions with base 2 numbers. So we take a detour to base 2.
On the board:
In base 10: 268 = 2 * 10^2 + 6 * 10^1 + 8 * 10^0.
This ONLY uses the digits 0 through 9
A number in base 2 only uses the digits 0 and 1.
In base 2: some number like = 1 * 2^5 + 0 * 2^4 + 1 * 2^3 + 2 * 2^2 + 1 * 2^0 10

11. Addition of binary numbers
Carry if the number would be too large for the number system- if it is greater than 1
1001 + 10
1001 + 11
1011 + 10
1101 +
111
1011
1100
1101
10100 11

12. Addition of octal and hexadecimal numbers
Carry if the number would be too large for the number system (larger than 7 or 15)
7238 +
128
2658 +
338
ABC16 +
1216
CDE16 +
ED16
7358
3208
ACE16
DCB16
Explain: Octal is base 8, therefore the octal digits are 0 through 7. Any value which is larger than 7 must result in a carry to the next digit position to the left.
Hexadecimal is base 16, so ten digit symbols 0-9 aren’t enough to represent all the hexadecimal digits. So we use letters A, B, C, D, E, and F to represent the hexadecimal digits with values 10, 11, 12, 13, 14, and 15.
Hexadecimal is commonly used in computer science, since expressing numbers in binary is tedious. However, computer memory is often organized in terms of bytes, which are 8 bits. Therefore two hexadecimal digits can be used to indicate the first four bits and the second four bits of any byte, more conveniently than 8 bits. For example, the binary number is 6D in hexadecimal. Similarly, it's 155 in octal.
Octal is less common but sometimes used as well. 12

13. Two's complement To represent negative values in binary:
Find the binary equivalent of the absolute value.
Pad on the left to completely fill the bits in the specified bit width
Switch all of the 1's to 0's and 0's to 1's.
Add 1 to the result.
Example: find the 8-bit two's complement representation of -43:
4310 =
= -4310 13

14. Using a circuit for adding two bits
input
output p q
carry
sum 1
Write as a logic expression
Translate to circuits
The key is that we’ve translated arithmetic in base 2 to truth tables, now we can do circuits!
Note order of rows; this is not a truth table. 14

15. Half adder
Sum = (x v y) ^ ~ (x ^ y) Carry = (x ^ y)

16. Full adder
Three bits in (x, y, previous carry)

17. Parallel adders
Chain these half adders and full adders together for multi-bit addition
A3A2A1A0 + B3B2B1B0 = S3S2S1S0 X3 Y3 X2 Y2
This performs addition of 2 3-bit numbers! X1 Y1

18. Topic not covered
Simplifying circuits: there are techniques that exist (which are complex). 18