1

Chapter 4: Main Building Blocks of Hardware

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

2

Main Building Blocks of Hardware

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 hardware design or logic design

3

Binary Numbers How is information represented in a

computer? Information vs Data First, humans also represent information:

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

4

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

5

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: 100

Tens: 101

Hundreds 102

Thousands 103

Example: 3049 = 3*103 + 0*102 + 4*101 + 9*100

Similarly: Binary numbers use the same idea, except that we have 2 digits only 0 and 1

6

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*25 + 1*24 + 1*23 + 0*22 + 0*21 + 1*20

= 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.

7

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 = 215 + ….+20 = 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.

8

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 =- (25 + 24 + 20) = - (32 +16 + 1) = -49

Hence: -49 is represented as 1110001

9

Binary Numbers How the computer knows that 1110001 is –49 and not the 8-bit

number 1110001 = 26 + 25 + 24 + 20 =

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.

10

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

11

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 voltsDecimal-to-Voltage Mapping Binary-to-Voltage Mapping

0 [0, 3] 0 [0, 22]1 [3, 6] 1 [22, 45]2 [6, 9]…

12

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.

13

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

14

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”

15

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)

current

Binary 0 Binary 1

Direction ofmagnetic field

Direction ofmagnetic field

16

Binary Numbers Transistors

Typical density for cores: 500 bits/in2

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 / cm2

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

17

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/cm2 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 cm2) and cheap (ca. $100) processors with millions of transistors, which are far more powerful than the old machines.

18

Binary NumbersIndividual transistors

IC

DIP

PINs

Circuit board

Bus

19

Binary Numbers Theoretical concepts behind transistors: semiconductors

Electrical engineering Physics

Here we use a model of a switch for a transistor:

Collector

Base (control)

Emitter

Switch Base = high voltage (1) switch closes transistor is in “on” state

Base = low (0) Switch opens Transistor is in “off” state

20

Binary Numbers The two states of a transistor:

Power supply (+5V)

1

Switch closedSwitch is open

+5V 0VPower supply (+5V)Measuring

deviceMeasuring device

21

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 19th 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.

22

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

23

Boolean Logic Truth table for AND

a b a AND b (also written a.b)

true true truetrue false falsefalse true falsefalse false false

Truth table for ORa b a OR b (also written a+b)

true true truetrue false truefalse true truefalse false false

24

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)

25

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)

26

Gates Representation of the 3 gates looks like:

a aa

bba.b

a+b a

We use the following (and similar) representation(s):

aa

a

b ba.bAND OR NOT

aa+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 andare used instead of ., +, and

-

27

Gates Construction of an AND gate

Idea: 2 transistors in series Current flows iff both transistors are closed (state 1)

Power Supply

Input 1 (a)

Output (a.b)

Input 2 (b)

28

Gates Construction of an OR gate

Idea: 2 transistors in parallel Current flows if either of the transistor is closed

(state 1)

Power Supply

Input 1 (a)

Output (a+b)

Input 2 (b)

29

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

Input (a)Output

aResistor

30

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

31

From Gates To Circuits Example:

a

b NOT

OR

NOTAND

c

d

Circuit computes: c = a OR b d = NOT ((a OR b) AND (NOT b))

InputsOutputs

32

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

33

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 2n 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

34

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 0 0 00 0 1 00 1 0 1 (case 1)0 1 1 01 0 0 01 0 1 01 1 0 1 (case 2)1 1 1 0

Step 2: not(a).b.not(c) a.b.not(c)

35

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

36

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 (carry bit)001101 + (13)001110 (14)

= 011011 (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!)

37

From Gates To Circuits Step 1: Truth table

Ai Bi Ci Si Ci-10 0 0 0 00 0 1 1 00 1 0 1 00 1 1 0 11 0 0 1 01 0 1 0 11 1 0 0 11 1 1 1 1

1-ADD

Ai

Bi

Ci

Si

Ci-1

Inputs Outputs

38

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

39

From Gates To Circuits Step 4: Circuit itself (1-ADD)

NOT AND

Ai Bi Ci

Ci-1

Si

OR

40

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 (CN = 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 = b1 b2 … bN-1 bNA= a1 a2 … aN-1 aN

S= s0 s1 s2 … sN-1 sN

CN= 0CN-1CN-2C2C1C0 1-ADD 1-ADD 1-ADD 1-ADD

41

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

42

From Gates To Circuits Control circuits

Used to control program execution in a computer Multiplexer Decoder

Multiplexer Structure

2N 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

43

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

Selector linesRegisters

R0

R1

R4

R3

Check-zero circuit

Multiplexer

44

From Gates To Circuits Decoder

Structure N input lines 2N 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

45

From Gates To Circuits Diagram for a 4-operations decoder

Operation Codes

00 = ‘+’

01 = ‘-’

10 = ‘*’

11 = ‘/’

Decoder

Add circuit

Subtract circuit

Multiply circuit

Divide circuit

2 Inputs 4 Outputs