COMP 125 Principles of Computing

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COMP 125Principles of Computing

• The Building Blocks Binary Numbers, Boolean
  Logic Gates

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Agenda

• Patricks Evaluation
• Attendance
• Collect Assignment 3
• Patrick will take up 2 3 on June 20th
• Final Exam
• Interested doing the final exam in last class
  August 1?
• Review Practice Problem 104 (105)
• Questions 1-4
• Chapter 4
• Lab Experience 7

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Intro to Level 2 The Hardware World

• What is involved in designing real computers to
  solve real-life problems
• Chapter 4 Digital logic design level
• How to represent and store information
• Use symbolic (Boolean) logic to design gates
• Use gates to design circuits
• Chapter 5 Organization level
• Memory
• input/output,
• Arithmetic/logic unit
• Control unit

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Chapter 4 to 4.2

• Topics
• Binary Representation
• Reliability of Binary Representation
• Binary Storage Devices
• Learning Objectives
• Understand why binary is used, to translate
  between decimal and binary and to do simple
  binary calculations
• Know the conditions of a bistable environment and
  the way light switches, magnetic cores and
  transistors meet them.

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Data Representation

• Notational conventions for representing
  information
• Decimal representation for numerical values
• 0,1,2,3,4,5,6,7,8,9
• The 26 letters A,B,C,X,Y,Z for text
• Plus lowercase and punctuation
• Signed notation ,-
• Decimal Notation for real numbers
• Decimal point separates whole number part from
  fractional part

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Data Representation

• External
• Internal

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Decimal Representation

• In decimal, the following symbols are used to
  represent different values 0, 1, 2, 3, 4, 5, 6,
  7, 8, 9.
• For example, let us write four thousand, seven
  hundred and sixty-three. 1000s  100s   10s    1s
  103    102    101    100   4    
  7     6      3 or else it can be written
  this way (4 x 1000) (7 x 100) (6 x 10) (3
  x 1) 4763

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Binary Representation

• In the binary system, just two symbols are used
  0 and 1.
• Therefore any number can be represented by 0s and
  1s only
• Lets examine a 6 digit (bit) binary number
• 1 1 1 0 0 1
• Position value 25 24 23 22 21 20
• 32 16 8 0 0 1
• 57
• This table converts binary to decimal

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Converting Decimal to Binary

• A popular method for converting a Decimal number
  into a Binary number is by dividing the number by
  two, repeatedly.
• 26 is written as 11010 in binary.
• Check it out, using the repeated "division by 2"
  method.
• 26 divided by 2 13, remainder 0
• 13 divided by 2 6, remainder 1
• 6 divided by 2 3, remainder 0
• 3 divided by 2 1, remainder 1
• 1 divided by 2 0, remainder 1

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Computer Maximums

• Every number that can be represented in base 10
  can be represented in base 2, although it takes
  more digits
• in base 10, 57 takes 2 digits
• In base 2, 57 takes 6 digits - 111001
• Every computer (virtual machine) has a maximum
  number of bits used to represent an integer
• 16 bit 216 65,535
• 32 bit 232 4,294,967,296
• Adding just one more (than max) causes an
  arithmetic overflow error.

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Signed Integers

• The leftmost will represent the sign negative
  if on (1), positive if off (0)
• Our previous 6 digit binary now needs an extra
  bit to make it negative
• Here it is in full 1 byte ( 8 bit) representation

1 0 1 1 1 0 0 1
neg. 26 25 24 23 22 21 20 - 0 32 16
8 0 0 1 - 57
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Encoding Schemes

• Weve just looked at two such schemes for numbers
• Another called twos complement is generally used
  for signed integers
• Real numbers have yet another scheme
• So far weve looked at schemes that involve
  performing some conversion
• The next uses a look-up table

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Binary Representation of Characters

• Each character is represented by a unique
  (binary) number
• The ASCII table is the most common scheme page
  141
• Uses 8 bits 28 or 256 different characters
• Only the numbers 32-126 have been assigned
  printable characters

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Binary Representation of Sound

• Multimedia data is sampled to store in digital
  form, with or without detectable differences
• Representing sound data
• Sound data must be digitized for storage in a
  computer
• Digitizing means periodic sampling of amplitude
  values

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Sound (continued)

• From samples, original sound may be approximated
• To improve the approximation
• Sample more frequently
• Use more bits for each sample value

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Digitization of an Analog Signal

• (a) Sampling the Original
• Signal
• (b) Recreating the
• Signal from the Sampled
• Values

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Representing Image Data

• Images are sampled by reading colour and
  intensity values at even intervals (a grid)
  across the image
• The size of the grid is called resolution
• 1024 x 768 or 1600 x 1200
• Each sampled point is a pixel
• Image quality depends on number of bits at each
  pixel
• True-Color (24bit) uses 1 byte for each of red,
  green blue to represent over 16 million colours
  at each pixel
• What is the storage requirement (size) of a
  True-Color image with a resolution of 1600 x 1200?

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What do the bits mean?

• The computer does not know what a specific bit
  pattern in memory means
• The meaning is imposed only as a result of how
  some process has been designed to interpret the
  pattern
• In writing programs we declare variables as a
  certain type
• integer a number
• character letter, number or character
• string group of numbers, letters or characters

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Reliability

• Electrical systems work best in a bistable
  environment, in which there are only two stable
  states
• The problem with base 10 representation is that
  it needs to store 10 unique symbols and therefore
  needs a device that has 10 stable states.
• As electrical component ages, it may drift
  (change their energy level) and therefore not
  reliably represent a state
• If it only needs to deal with voltage or no
  voltage circuit on/circuit off, nearly a 50
  drift in voltage is still tolerable when
  interpreting a stored value.

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Binary Storage Devices

• Binary Hardware Criteria
• 2 stable energy states
• A large energy barrier between them
• Ability to read the state without destroying it
• Ability to change the state as needed
• Two approaches
• Magnetic Cores historic
• Transistors current

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Magnetic Cores

• An Iron oxide-coated donut can be magnetized by
  running current through a wire strung through its
  hole
• The direction of the resulting magnetic field is
  dependant upon the direction of the current
• Who can say how big ½ a gigabyte of memory would
  be?

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Transistor

• Just a switch its turned on/off electrically
• Current technology (at texts printing)
• Switch state 1 billionth of a sec
• Density 30 to 50 million per cm2
• Made from semiconductors (silicon or gallium
  arsenide)
• Transistors are printed photographically onto a
  wafer of silicon using a mask which is used to
  produce many copies of the circuit (chip)
• VLSI Very large scale integration 3 to 10
  million transistors per cm2
• ULSI Ultra large scale integration possibly
  billions transistors per cm2

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Relationship of Transistors, Chips
Individual transistors (typically 30 50
million per chip)
DIPs are not often used any longer. Most boards
have chips soldered directly onto the board.
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Transistor Model
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Links

• Binary Numbers
• http//courses.cs.vt.edu/csonline/NumberSystems/L
  essons/BinaryNumbers/index.html
• Core Memory
• http//www.psych.usyd.edu.au/pdp-11/core.html

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Lets Practice

• Practice problems page 142
• Number 1, 2, 3 5
• Convert the following to binary using the
  standard notation
• 25, 78, 255
• Convert several bit patterns to decimal
• Start with 0110 1011

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IMPORTANT

• Bring a disc so that you can save (and take with
  you) the files that you create in LE 7
• You will need the disc for future labs as well

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Chp 4 theory cont.

• Chapter 4 (156 178)
• Boolean Logic
• Gates
• Building Computer Circuits (to 178)

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George Boole (1815 1864)

• Created a new form of logic containing the values
  true and false and the operators AND OR NOT
• Developed a set of rules to interpret and
  manipulate expressions that contain these values
• 100 years later his theories became the framework
  for designing computer systems

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Boolean Logic

• Rules for manipulating 2 logical values true
  and false
• Anything stored in a sequence of binary digits
  can also be viewed as a sequence of logical
  values, true and false
• The operators AND OR NOT map a set of True False
  values into a single (true, false) result

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Boolean Logic

• Boolean Logic deals with manipulating the two
  logical values true and false
• A Boolean Expression is defined as any expression
  that evaluates to either true or false
• Examples (x1), (a?b), (cgt5.23)
• A Boolean Expression combines arithmetic
  expressions using the boolean operators
• AND (x y 25 AND y lt 100)
• OR (x y OR x lt 10)
• NOT (NOT (x y))

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Rule for performing AND operation

• a AND b, also written as a?b, is true only if the
  value of a and the value of b is true
• Otherwise the expression a?b has the value of
  false
• To check a test score S in the range 90 to 100
  inclusive
• Sgt90 AND Slt100

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Truth Table for AND Operation
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Rule for performing OR operation

• a OR b, also written as ab, is true if the value
  of a is true, if value of b is true, or both are
  true
• Otherwise the expression ab has the value of
  false
• To check a students major
• (major math) OR (major comp science)

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Truth Table for OR Operation
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Boolean logic (OR and AND)

• What conditions must be satisfied for the
  necessary outcome (buzzer sounding, alarm
  sounding, etc.) ?
• Determine whether all conditions must be met or
  whether satisfying only one of the conditions is
  necessary.
• You have a buzzer in your car that sounds when
  your keys are in the ignition AND the door is
  open.
• You have a fire alarm installed in your house.
  This alarm will sound if it senses heat OR smoke.
  
• There is a federal election coming up. People are
  allowed to vote if they are a Canadian citizen
  AND they are 18.

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Rule for performing NOT operation

• Not a, also written as a, is true if a has the
  value of false and false if a has the value of
  true
• NOT reverses or complements the value of the
  Boolean expression
• To compare a students gpa
• (gpa gt 3,5) is true if your gpa is greater than
  3.5
• NOT (gpagt3.5) is true only if your gpa 3.5

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Truth Table for NOT Operation
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Lets Practice

• Page 159
• Problem 1 2

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Gates

• A gate operates on a collection of binary inputs
  to produce a binary output
• Consider the value 1 equivalent to true and the
  value 0 equivalent to false
• Then AND, OR, NOT gates directly implement the
  corresponding Boolean expression

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AND, OR, NOT Gates
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IMPORTANT NOTE

• The text takes a different (slightly more
  complex) approach to building gates that we will
  here.
• Be sure that youre comfortable with the texts
  approach as well as this approach.

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Gates and Transistors

• AND 2 transistors in series
• The output will be 1 only if both inputs are 1
• OR 2 transistors in parallel
• Only if both inputs are 0 will the output be 0
• NOT 1 transistor 1 resistor
• If input is 0, output is 1
• If input is 1, output is 0

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Why Gates and not transistors?

• A transistor is too elementary a device
• we dont have to deal with transistors,
  resistors and capacitors or voltages, current and
  resistance the building block of circuit
  construction (for us) will be gates
• Using gates as building blocks is another
  example of using abstraction in computer science.

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Building Computer Circuits

• A combinational circuit (just circuit to us) is
  made up of logic gates
• That transform a set of binary inputs into a set
  of binary outputs,
• Where the values of the outputs depend only on
  the values of the inputs
• A circuit is constructed from AND, OR NOT gates
• These gates can be connected in any way so long
  as the connections do not violate the proper
  number of inputs and outputs for each gate
• Each AND - two inputs, one output
• Each OR gate two inputs , one output
• Each NOT gate - one input, one output

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Building Computer Circuits

• Every Boolean expression can be represented
  pictorially as a circuit diagram
• The pictorial view is often used during the
  design stage, allowing overall visualization
• Every output value in a circuit diagram can be
  written as a Boolean expression
• An expression may be a more convenient way to
  perform verification and optimization

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Our First Circuit

• c (a OR b)
• d NOT ( (a OR b) AND (NOT b ))

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Building Computer Circuits

• Construct circuits using AND, OR, and NOT gates.
• Apply the sum of products algorithm to construct
  the circuit diagram for any truth table. (AND the
  reverse)
• Understand the purpose and construction of
  multiplexors and decoders

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Precedence

• The order of precedence rule for boolean
  operations is NOT first, AND second and OR third.
• So
• x AND NOT y OR z
• Means
• (x AND (NOT y)) OR z
• Parentheses can be used to change the order and
  are recommended where needed to enhance
  readability

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Circuit Construction Algorithm

• Sum of Products Algorithm
• Construct the truth table
• Every output of 1 denotes a sub-expression
• Create sub-expressions using AND NOT
• each sub-expression becomes a sub-circuit of the
  completed circuit
• Combine sub-expressions using OR
• sub-circuits are joined with OR gates
• Construct the circuit

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Truth Table Construction

• Determine the binary value that should appear on
  each output line of the circuit for every
  possible combination of inputs
• If a circuit had N input lines, there are 2N
  combinations of input values
• See the truth tables on pg 166

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Subexpression Construction using AND and NOT gates

• Choose an output column
• For every place you find a 1 in the output
  column, you build a subexpression that produces
  the value of 1
• If the input is a 1, use the input value
• If the input is a 0, first take the NOT of that
  input and use that complemented input value in
  your subexpression
• See the example, page 167

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Subexpression Combination using OR gates

• Take each of the subexpressions produced in step
  3 and combine them, two at a time, using OR gates
• The Boolean expression produced in step 3
  implements exactly the function of the truth table

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Circuit Diagram Production

• Convert the Boolean expression produced at the
  end of step 3 into a circuit diagram using the
  AND,OR, NOT gates to implement the AND, OR, Not
  operators appearing in the Boolean expression

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Circuit Construction

• IMPORTANT The Sum of Products Algorithm
  doesnt always produce an optimal circuit
• The circuit on the right has the same output as
  the one on the left.

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Lets Practice

• Page 164 (169)
• Problem 2

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Problem 2

• Heres a truth table
• Create the sub-expressions
• Combine them
• Construct the circuit

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Problem 2

• Create the sub-expressions
• NOT a AND b
• a AND NOT b
• Combine them
• NOT a AND b OR a AND NOT b
• Construct the circuit diagram (next slide)

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Problem 2
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Compare for Equality Circuit

• We can combine simple 1-CE circuits to create
  an N-bit CE circuit

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Addition Circuit

• Binary addition
• 11 (the carry bit)
• 001101 (binary value 13)
• 001110 (binary value 14
• 011011 (binary value 27)

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1-Add Circuit and Truth Table

• The truth table for the 2 binary digits, ab, the
  carry digit from the previous column c and the
  outputs sum and the new carry digit
• Create sub-expressions for Si
• Combine the subexpressions for Si

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Sum Output for 1-Add
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1-Add
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The Full Adder Add Circuit
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Size and Complexity of the ADD Circuit

• How many transistors are needed in a 32-Bit
  Adder?
• NOT 32 x 3 96 NOT gates _at_ 1 tran/gate
  96
• AND 32 x 16 512 AND gates _at_ 3 tran/gate
  1536
• OR 32 x 6 192 OR gates _at_ 3
  tran/gate 576
• 
  Total 2208
• Optimized 32-bit addition circuits can be
  constructed with fewer transistors (500-600)
• NOTE if vacuum tubes were used instead of
  transistors, our ADD computer would be the size
  of a refrigerator
• With todays technology our ADD circuit fits into
  an area smaller than the period at the end of
  this sentence.

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Test Your Knowledge

• Page 172 (178) Do practice problem 1 for
  ungraded homework. We will take it up next class.

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Control Circuits

• Control circuits
• determine the order in which operations are
  performed
• select the correct data values to process
• Sequencing and decision-making circuits
• Multiplexor
• 2N input, 1 output N selector lines
• Decoder
• N input 2N output lines

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Multiplexor

• N selector lines can select one output from 2N
  input lines
• Useful to select data

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Multiplexor Example
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Decoder

• N input lines can determine which one (and ONLY
  one) of 2N output lines will be turned on
• Useful to select the correct instruction

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Decoder Example