





Algorithms are supposed to be executed by a computer
They describe the logical structure of a solution of a given problem
They are problem-oriented
We do not yet (till chapter 3) know how the computer executes them!
The goal of this chapter is to get to know the smallest building blocks of a computer
Next chapter will deal with execution of programs
1




We will deal with the:



Representation of numbers
Boolean Logic
Gates
Comparison with biology:


This chapter will have microscopic view and will handle tiny objects like cells, DNA, tissues, and genes in biology.
The big picture, including heart, lungs and other organs as well as their interconnections will be deferred to the next chapter.
This field of computer science is called
2

Binary Numbers




Numerical values: 10 digits (0..9)



Textual information: 26 letters (A..Z)
Sign-magnitude notation: + and
(left to digits)
Decimal notation: using decimal point separating 2 whole numbers where the right one is positive e.g
12.56
3

Binary Numbers





Computers?

External representation

Internal representation
External:

Looks like human representation
Important is the internal representation:

The way information is stored in memory
Internally computers use the binary numbering system
Example:



Input via keyboard: ABC
Memory representation: e.g. 0100101001010101101010100
Output to screen: ABC
4

Binary Numbers

What are binary numbers?


Base-2 positional number system
In everyday life we use base-10, where the digit itself as well as its position in a number determine the value of each digit



Moving from the right to the left the positions represent




Ones: 10 0
Tens: 10 1
Hundreds 10 2
Thousands 10 3
Example: 3049 = 3*10 3 + 0*10 2 + 4*10 1 + 9*10 0
Similarly: Binary numbers use the same idea, except that we have 2 digits only  0 and 1
5

Binary Numbers

Binary numbers:

2 digits: with symbols 0 and 1


Value is based on powers of 2 (not of 10)
The two digits 0 and 1 are called bits (binary digits)




Example: 111001 = 1*2 5 + 1*2 4 + 1*2 3 + 0*2 2 + 0*2 1 + 1*2 0
= 32 + 16 + 8 + 0 + 0 + 1
= 57
Computation is easier than base-10: simply sum up any non-zero term
Theoretically, any number in the decimal system can be represented in the decimal system and vice versa
This means: We do not loose anything if we use binary notation.
Humans use base-10 because it is more compact.
6

Binary Numbers





Computers always have a maximum number of digits (to represent numbers)
Examples: 16, 32, 64 bits
Example: maximum value is 16 bits long:
Max = 1111111111111111 (16 ones)
(compare: 999999 is the maximum mileage value if you had 6 positions in your decimal odometer)
Max = 2 15 + … .+2 0 = 65535
If an operation in a computer produces a value larger than the maximum value, an error called arithmetic overflow occurs.
Attention: in computer science there are always upper limits for numbers unlike mathematics.
7

Binary Numbers

Representing negative binary numbers:





Sign-magnitude notation (and others)
Sign-magnitude: the first bit on the left side is used to indicate the sign of the number:

First bit = 1  negative number

First bit = 0  positive number
This means that no extra symbols ‘ + ’ and ‘–’ are used internally!
Example: representing – 49 (if we had 7 bits only)
1 1 1 0 0 0 1 =
- (2
5
+ 2
4
+ 2
0
) =
- (32 +16 + 1) = -49
Hence: -49 is represented as 1110001
8

Binary Numbers




How the computer knows that 1110001 is – 49 and not the 8-bit number 1110001 =
2
6
+ 2
5
+ 2
4
+ 2
0
=
64 + 32 + 16 + 1 =
113
In other words how to differentiate between 113 and – 49?
In fact, the computer does NOT know whether or not this sequence of
8 bits is a – 49 or 113
Only the context makes it possible to make a difference:
Example: “ ball ” may mean the object used to play or a formal party.
Only the context lets us make a difference.
9


Binary Numbers
Representing Text

Binary digits are also used for text

Any letter of the alphabet is assigned a unique number
 code mapping





The binary representation of that number is stored in memory instead
Internationally used mapping: ASCII (American Standard for
Information Interchange)
ASCII uses 8 bits for each printable symbol (including letters)
Examples:

A  65  01000001


 a  97  01100001
?  63  00111111
Z  122  01111010
Again: How to know whether e.g. 65 is an ‘ A ’ or the number 65?
 Context
10








Binary Numbers
Why binary?
Why don ’ t we use decimals also internally?
Binary representation is more reliable
Information is stored using electronic devices depending on electronic quantities like current and voltage.
Using binary means we only need two stable energy states
Using decimals would mean to provide 10 stable energy states, which could be a very tricky and time-consuming engineering task.
Suppose we can store electrical charges between 0 and +45 volts
Decimal-to-Voltage Mapping Binary-to-Voltage Mapping
0  [0, 3] 0  [0, 22]
1  [3, 6]
2  [6, 9]
1  [22, 45]
… 11






Binary Numbers
Since voltage may oscillate, a decimal value representing 2 may drop to 6 volts

 how to interpret it, as 1 or as 2?
In the case of binary representation, there are only two large ranges (instead of several small ones) and it is therefore very improbable that the voltage drops or rises in a confusing way.
Therefore, we speak of the reliability or stability of the binary representation
Devices that operate on two stable states are called bi-stable ones.
However, there is (theoretically) no argument against using decimal numbers for computers if we were able to construct devices having 10 stable states.
12

Binary Numbers



Example of bi-stable states:

Full on – full off



Fully charged – fully discharged
Charged positively – charged negatively
Magnetized – non-magnetized

Magnetized clockwise - magnetized counterclockwise
Example for binary:


0 = 0 volts = full off
1 = +45 volts = full on
Only large drifts of voltages may influence data
13

Binary Numbers


Requirements for any hardware device of a binary computer:

2 stable energy states



States are separated by a large energy barrier
It is possible to sense the state of the device without destroying the stored value
It is possible to switch the state from 0 to 1 and vice versa
Examples:

On-off light switch (clearly not used to build computers)

Magnetic cores (no longer used)


1955-1975 in use
“ core memory ”
14

Binary Numbers

Cores

The 2 states for 0 and 1 are based on the direction of the magnetic field


Current left-to-right  counterclockwise magnetization (0)
Current right-to-left  clockwise magnetization (1)
Direction of magnetic field
Binary 0 current
Direction of magnetic field
Binary 1
15


Binary Numbers
Transistors



Typical density for cores: 500 bits/in 2
Today ’ s computers: millions of bits and more
Transistors are more compact, cheaper, require less power

Transistor is like a (light) switch

Off state: no electricity can flow

On state: electricity can flow

Unlike light switch: no mechanics

Switching is done electronically







Fast switching: 10-20 billionth of sec
Density: several millions of transistors / cm 2
Are based on semiconductors: silicon or gallium arsenide
Huge number of transistors on a wafer form so-called integrated circuits (IC)
ICs are also called chips
Chips are mounted inside a ceramic housing called dual in-line package
(DIP)
Input/output connectors of DIPs are called PINs 16

Binary Numbers





DIP are typically mounted on a circuit board (connecting different chips)
Production not mechanically but rather photographically
(using light)


More effective (use of masks and simple reuse)
Higher density

More accurate
Very high densities: 5-25 millions transistors/cm 2 possible
(very large scale integration or VLSI)
Ultra large scale integration or ULSI in progress
All these advantages together shifted computers from being expensive huge devices (filling rooms) to tiny (0.25 cm 2 ) and cheap (ca. $100) processors with millions of transistors, which are far more powerful than the old machines.
17

Binary Numbers
Bus
Individual transistors
IC
DIP
PINs
Circuit board
18

Binary Numbers


Theoretical concepts behind transistors: semiconductors

 Electrical engineering

 Physics
Here we use a model of a switch for a transistor:
Switch


Base = high voltage (1)

 switch closes transistor is in “on” state
Base = low (0)


Switch opens
Transistor is in “off” state
Collector Emitter
Base (control)
19

Binary Numbers

The two states of a transistor:
Switch closed
Power supply
(+5V)
1
Switch is open
+5V Power supply
Measuring
(+5V) device
0V
Measuring device
20

Boolean Logic





Computer circuits construction is based on mathematics:  Boolean logic
Relationship:

Truth value “ true ” represents binary 1

Truth value “ false ” represents binary 0
Anything stored as a sequence of 0 and 1 can be viewed as a sequence of
“ true ” and “ false ” .
Hence: These values can be manipulated using Boolean operators
Bool George:




English mathematician of mid 19 th century
1854 his book: Introduction to the Laws of Thought
Combined principles of logic with algebra
At his time his book got no special attention but 100 years later his ideas were used for computer design, now there is a branch of mathematics called Boolean algebra.
21

Boolean Logic

Boolean expressions





Any expression that evaluates to true or false
Examples:



X = 1
A < B
It is exactly 2 pm now
Operations for arithmetic are: + * / -
Operations for Boolean logic are: AND, OR, and NOT
Rules: a and b are Boolean expressions


 a AND b is true iff both a and b are true a OR b is true if either a is true, or b is true, or both are true
NOT a is true iff a is false
22



Boolean Logic
Truth table for AND a true true false false
Truth table for OR a true true false false b true false true false b true false true false a AND b (also written a.b) true false false false a OR b (also written a+b) true true true false
23

Boolean Logic



Example for AND




Check if the weight w of an object is between 90 and 100 kg
W >= 90 does not suffice  130 does not match
W <= 100 does not suffice  45 does not match
We have to build the following Boolean expression:

(W >= 90) AND (w<= 100)
Example for OR




Looking for students majoring in either computer science or biology
Test for major = computer science does not suffice
Test for major = biology does not suffice
OR expression needed:

(major = computer science) OR (major = biology)
Example for NOT

GPA should not exceed 3.5 using “ > ” operator

NOT (GPA > 3.5)
24

Gates


Why Boolean logic at all?

 Gates are constructed using Boolean logic
Gate:



A device that transforms a set of binary inputs to one binary output
In general, a gate can do any transformation, but we will only deal with 3 gates:



AND gate
OR gate
NOT gate
Computers can be built using only these 3 gates (thanks to
Boolean algebra)
25

Gates
 a b
Representation of the 3 gates looks like: a a.b
a+b a b

We use the following (and similar) representation(s): a a
AND a.b
OR a+b a NOT b b

 a, b  {0, 1} with interpretation similar to {false, true}

Example 0.0 = 0, 0.1 = 0, 0+1=1, not(1) = 0
Often also  and  are used instead of .
, +, and
a a
26

Gates

Construction of an AND gate


Idea: 2 transistors in series
Current flows iff both transistors are closed (state 1)
Output (a.b)
Power Supply
Input 1 (a) Input 2 (b)
27

Gates

Construction of an OR gate

Idea: 2 transistors in parallel

Current flows if either of the transistor is closed (state 1)
Power Supply Output (a+b)
Input 1 (a) Input 2 (b)
28

Gates

Construction of a NOT gate

Idea: 1 transistors and a resistor

Current flows (to output) iff the transistor is open (state 0)
Power
Supply
Resistor
Output a
Input (a)
29

From Gates To Circuits




Circuit (also called combinational circuit):

Collection of logic gates that

Transforms a set of binary inputs to a set of binary outputs

And where any output only depends on the current input
Any gate is a circuit and not vice versa
We will discuss circuits based on AND, OR, and NOT gates
Internal connections must conform to used gates!!!


OR and AND gates: 2 inputs and 1 output
NOT gate: one input and one output
30

From Gates To Circuits

Example: a
Inputs b
OR
NOT AND NOT

Circuit computes:

 c = a OR b d = NOT ((a OR b) AND (NOT b)) c
Outputs d
31

From Gates To Circuits




Combinational circuits vs sequential circuits

Combinational: output depends only on current input

Sequential: feedback loops from output to input are allowed meaning that output may depend on previous input
Sequential circuits good for memory design (they have the ability to “ remember ” inputs)
In a sense when designing computer hardware we don ’ t have to think in terms of current and voltage, or transistor. The main building blocks of circuits are gates (  abstraction)
How to build circuits from gates?
 Circuit construction algorithm
32





From Gates To Circuits
Circuit construction algorithm:


Here we use: sum-of-product algorithm
It has 4 steps
Step 1: Construct Truth Table
 n input lines  2 n rows in the truth table


Simple example: OR gate  2 inputs  4 rows
Each input and each output is assigned a column in the truth table
Step 2: Construct subexpression based on AND and NOT

Scan the column of each output line

Whenever you see 1, build a subexpression for that output line based on inputs using the following 2 rules



Any 0 of the row corresponds to its negated input
Any 1 of the row corresponds to its input
Combine the inputs (of that row) using ANDs
Step 3: Combine subexpressions using OR

For each output line combine the corresponding subexpressions of step 2 using
ORs
33

From Gates To Circuits


Step :4 Circuit Diagram Production

Convert Boolean expressions of step 3 into a circuit diagram using
AND, OR, and NOT gates
Example:

Step 1: a b c d (output)
0
1
1
1
1
0
0
0
0
1
1
0
0
0
1
1
0
0
1
0
1
1
0
1
0
0
1 (case 1)
0
0
0
1 (case 2)
0

Step 2:
 not(a).b.not(c)
 a.b.not(c)
34

From Gates To Circuits




Step 3
 d = not(a).b.not(c) + a.b.not(c)
Step 4



Draw the diagram
We need here 2 NOT gates, 4 AND gates, and 1 OR gate
 7 gates are needed
Described algorithm for circuit construction is NOT always optimal:

Always correct circuit


Not always the minimum number of gates
Compare chapter 3: algorithms should be correct and if possible efficient
Our example could be solved directly:

Don ’ t use the input a


 d = b.not(c)
 only two gates (instead of 7) are needed at minimum!!!
Less gates means circuit is cheaper, compacter, and faster
35

From Gates To Circuits

Designing a full-adder circuit (for unsigned numbers)

Problem: given 2 N-bit binary numbers A and B, design a circuit that performs the binary addition and yields an N-bit binary sum S




S = A + B in binary
Example: N = 6
11
001101 +
001110
= 011011
(carry bit)
(13)
(14)
(27)
Consider each column i separately:

We need Ai, Bi, Ci as inputs, where Ci is the used carry bit.

We need Si, and Ci-1 as output, where Ci-1 is the carry to be used for the next column.
Develop first a circuit 1-ADD to perform single bit addition (one column only!)
36

From Gates To Circuits

Step 1: Truth table
1
1
0
1
1
0
0
Ai
0
0
1
1
0
1
0
1
Bi
0
1
0
1
0
1
1
0
Ci
0
0
0
0
1
1
Si
0
1
1
1
1
1
0
1
0
0
Ci-1
0
Ai
Bi
Ci
Inputs
1-ADD
Outputs
Si
Ci-1
37

From Gates To Circuits


Step 2: Building subexpressions


Subexpression of Column Si are:
 not(Ai).not(Bi).Ci


 not(ai).Bi.not(Ci)
Ai.not(Bi).not(Ci)
Ai.Bi.Ci
Subexpression of Column Ci-1 are:



 not(Ai).Bi.Ci
Ai.not(Bi).Ci
Ai.Bi.not(Ci)
Ai.Bi.Ci
Step 3: Combining them


Si = not(Ai).not(Bi).Ci + not(ai).Bi.not(Ci) + Ai.not(Bi).not(Ci) + Ai.Bi.Ci
Ci-1 = not(Ai).Bi.Ci + Ai.not(Bi).Ci + Ai.Bi.not(Ci) + Ai.Bi.Ci
38


From Gates To Circuits
Step 4: Circuit itself (1-ADD)
Si
Ci-1
NOT AND
39
OR
Ai Bi Ci



From Gates To Circuits
Back to the full-adder

We have now a device (1-ADD) that performs single bit addition

We use N such 1-ADD devices for N-bit addition



The 1-ADD at the most right position uses carry = 0 (C
N
= 0)
A potential carry should be forwarded to the left 1-ADD device
The 1-ADD device at the most right position also produces a carry bit, which is not used for the addition itself
Full adder circuit
B = b
1 b
2
… b
N-1 b
N
A= a
1 a
2
… a
N-1 a
N
C
0
1-ADD
C
1
1-ADD
C
2
C
N-2
1-ADD
C
N-1
1-ADD
C
N
= 0
S= s
0 s
1 s
2
… s
N-1 s
N
40

From Gates To Circuits

Analysis of full-adder:





Number of NOT gates: 3*N
Number of AND gates: 16*N
Number of OR gates: 6*N
For N = 32: NOTs = 96, ANDs = 512, ORs = 192
Number of transistors:

1 NOT  1 Transitor, 1 AND (resp. 1 OR)  2 Transistors each




Number NOT-transistors: 96*1 = 96
Number AND-transistors: 512*2 = 1024
Number OR-transistors: 192*2 = 384
 Number of transistors is: 1504
41

From Gates To Circuits


Control circuits



Used to control program execution in a computer
Multiplexer
Decoder
Multiplexer



Structure

2 N input lines


N selector lines (also input)
1 output line
The selector lines represent the binary representation of one input line
The value (0/1) of that input line is forwarded to the output line
42

From Gates To Circuits

Use of Multiplexer



Suppose computer has to repeatedly check whether or not one of 4 memory locations (rather processor registers) has become 0
Input: 4 lines (4 registers)
2 selector lines: 00, 01, 10, 11
R0
R1
R3
R4
Registers
Multiplexer
Check-zero circuit
Selector lines
43

From Gates To Circuits


Decoder



Structure

N input lines

2 N output lines
The input lines represent the binary representation of one output line.
The value 1 is forwarded to the (selected) output line - the output line is activated.
Example for decoder usage


Suppose a computer has 4 operation +, -, *, /
In order to activate one operation, the numbers 00, 01, 10, 11 are used as input for a decoder
44

From Gates To Circuits

Diagram for a 4-operations decoder
Operation Codes
00 = ‘+’
01 = ‘-’
10 = ‘*’
11 = ‘/’
Decoder
2 Inputs 4 Outputs
Add circuit
Subtract circuit
Multiply circuit
Divide circuit
45