1 - Computer Science

advertisement
CPSC 171 Introduction to
Computer Science
Boolean Logic, Gates, & Circuits
Announcements
Read Chapter 4
Exam, Oct 2nd in class
Boolean Logic
A Boolean variable, A, is either true or
false
A Boolean expression, (A AND B),
evaluates to either true or false
Boolean operators include:



AND (&  • )
OR ( + )
NOT (a bar ' ¬ ~)
Boolean Operators
a AND b
true only when A and B are both true
a OR b
true when A is true, B is true, or both are true
NOT a
true when A is false
Truth Tables
Truth tables can be
used to capture
when an expression
is true, given its
inputs
a
b
a AND b
0
0
1
0
1
1
1
1
0
1
1
1
You make truth tables for AND and NOT
Example Boolean Expressions
(a AND b) OR (NOT a AND c)
a·b + ~a·c
ab+āc
Truth tables can be made for complex
expressions as well
Boolean Logic (continued)
Example:
(a AND b) OR ((NOT b) and (NOT a))
a
b
Value
0
0
1
1
0
1
0
1
1
0
0
1
Gates
Gates


Hardware devices built from transistors to
mimic Boolean logic
An electronic device that operates on a
collection of binary inputs to produce a
single binary output
AND gate (page 161 in text)


Two input lines, one output line
Outputs a 1 when both inputs are 1
Gates (continued)
OR gate (page 163 in text)


Two input lines, one output line
Outputs a 1 when either input is 1
NOT gate (page 161 in text


One input line, one output line
Outputs a 1 when input is 0 and vice versa
Figure 4.15
The Three Basic Gates and Their Symbols
Circuits
A collection of logic gates that
transforms a set of binary inputs into a
set of binary outputs
Wire gates together keeping constraints
for the number of inputs to any gate
Example Circuit
a 1
b
1
0
1
output
0
c 1
d
1
1
If a, b, c, and d are all true the output
can be determined by tracing through
the circuit
Designing Circuits
A circuit construction algorithm
1. Truth Table Construction
Determine outputs for every possible input
2. Sub-expression Construction (using AND and NOT
gates)
For each output find the rows that are 1 and build a subexpression that is true for the exact input
3. Sub-expression combination (using OR gates)
Take each subexpression and combine them, 2 at a time, using
OR gates
4. Circuit Diagram Production
Construct final circuit by converting Boolean operators into
gates
Example Circuit Design
Design a 3-input circuit that is true if
exactly two inputs are true, and false
otherwise
You Try it: Design a 2-input circuit that is
true if the inputs are the same, and
false otherwise
Examples of Circuit Design
and Construction
Compare-for-equality circuit
Addition circuit
Both circuits can be built using the circuit
design algorithm
A Compare-for-Equality Circuit
CE compares two unsigned binary
integers for equality
Built by combining together 1-bit
comparison circuits (1-CE)
Integers are equal if corresponding bits
are equal (AND together 1-CD circuits
for each pair of bits)
A Compare-for-Equality Circuit
(continued)
1-CE circuit truth table
a
b
Output
0
0
1
0
1
0
1
0
0
1
1
1
A Compare-for-Equality Circuit
(continued)
1-CE Boolean expression

First case: (NOT a) AND (NOT b)

Second case: a AND b

Combined:
((NOT a) AND (NOT b)) OR (a AND b)
Figure 4.22
One-Bit Compare-for-Equality Circuit
N-Bit Compare for Equality Circuit
AND together the 1-CE circuits, two at a
time
An Addition Circuit
Adds two unsigned binary integers,
setting output bits and an overflow
Built from 1-bit adders (1-ADD)
Starting with rightmost bits, each pair
produces

A value for that order

A carry bit for next place to the left
An Addition Circuit (continued)
1-ADD truth table

Input
 One bit from each input integer
 One carry bit (always zero for rightmost bit)

Output
 One bit for output place value
 One carry bit
Figure 4.24
The 1-ADD Circuit and Truth Table
An Addition Circuit (continued)
Building the full adder




Put rightmost bits into 1-ADD, with zero for
the input carry
Send 1-ADD’s output value to output, and
put its carry value as input to 1-ADD for
next bits to left
Repeat process for all bits
See pg 174, 175, 176
Download
Related flashcards

Logic

29 cards

Philosophical logic

16 cards

Hypotheses

16 cards

Strength athletics

14 cards

Super Rugby teams

19 cards

Create Flashcards