Data Manipulation

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Data Manipulation

• CSCI130
• Instructor Dr. Imad Rahal

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Overview of Computer Hardware

• necessary components of a computer
• CPU, Main memory
• components needed for convenience
• computer wont be very practical to use otherwise
• Secondary/Auxiliary storage, I/O devices
• Main memory
• Connects to the motherboard
• Divided into two major parts

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Overview of Computer Hardware

• RAM --- Random Access Memory
• memory registers which store data before/after
  CPU processing
• Available for users and programs so store data in
  and read data from
• Volatile --- does not persist when no electric
  power is supplied to its circuits
• ROM --- Read Only Memory
• Permanent
• Holds programs that are vital for the operation
  of the computer
• As the name indicates, can be read but never
  altered
• CPU
• Central Processing Unit
• Single silicon chip with circuits attached to it
• Known as microprocessor
• Sits on a circuit board known as the motherboard

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Data Manipulation

• Computing an answer to an equation
• 53 62 7
• Assume our computer cant directly multiply,
  subtract, or raise to power
• Multiplication task
• 1 Get 1st number, Number_1
• 2 Get 2nd number, Number_2
• 3 Answer 0
• 4 Add Number_1 to Answer
• 5 Subtract 1 from Number_2
• 6 if Number_2gt0, go to step 4
• 7 Stop

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Data Manipulation

• All tasks done by a general-purpose computer can
  be accomplished through the following set of
  operations
• Input/output
• Not mentioned in book but important
• Store data
• numbers (positive, negative or fractions), text,
  pictures, etc
• Compare data (numbers, pictures, sounds, letters)
• Add
• Move data from one storage (memory) location to
  another
• Editing a text document

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Data Manipulation

• Adding and comparing bit patterns is sufficient
  to achieve an operational machines
• Hard-wired vs. programmed
• This is done by circuits for adding and comparing
  bit patterns in registers
• Circuits are made up of logical gates
• Gates and Truth Tables
• Gates needed are NOT, AND, and OR
• NOT Gate
• Single input and single output
• Reverses input
• 1 ? 0 and 0 ? 1
• If there is a strong electric current
• ? shut it off
• If there is no/weak electric current
• ? turns it on
• Like a power switch

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Data Manipulation

• AND Gate
• Accepts two inputs (or more) and yields one
  output
• Output is 0 when any input is 0
• Requires power coming from both lines in order to
  give out power

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Data Manipulation

• OR Gate
• Accepts two inputs and yields one output
• Output is 1 when any input is 1
• Requires power coming from at least one of the
  input lines in order to give out power

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Data Manipulation

• These three simple gates are combined to create
  circuits that perform more complicated operations
• Circuits, in turn, might then be used (thru
  programs) to perform even more complicated tasks
• Gate combinations can be expressed in three ways
• (1) Through Expressions
• A AND B ? AB
• A OR B ? AB
• NOT A ? A
• AB AB

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• Enough rows to hold all input combinations
• 1 letter ? 21 rows
• 2 letters ? 22 rows
• 3 letters ?23 rows
• n letters ?2n rows

• (2) Through Circuit diagrams
• Given an expression
• Draw a gate after its inputs have been drawn
• Try AB AB
• (3) Through Truth Tables
• Each of the representations can be derived from
  the other
• Derive truth table given expression
• One column for each letter
• Make 1 additional column for every sub-expression
  (order parentheses, NOTs, ANDs, ORs)

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Practice

• Try A (ABBC)
• Design circuit
• Rule once you have drawn all the inputs of a
  gate, you can draw the gate itself
• Find truth table
• How many rows?

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Data Manipulation

• given a circuit diagram ? expression ? truth
  table
• Mark every output wire by its label

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Sum of Products Method

• Given a truth table, how to find expression?
• Sum-of-product method
• For each row with a 1 in the final column
• AND letters with a 1 in their column and negation
  of the letters with a 0 in their column
• Connect the resulting AND groups with ORs

• AB
• AB
• Ignore
• AB

? ABABAB ? Complicated and UGLY ? !!!
(requires space, and is costly and slow)
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Simplification

• Why simplify?
• UGLY ?
• Circuits supposed to be as simple as possible
• Save on speed (operation execution---fewest gates
  as possible because every gates slows the
  operation a bit)
• (in general) more gates ? more time
• Save space (on motherboard)
• Notebooks!
• Save money (not as critical)

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Simplification of Expressions

• Laws of Boolean Algebra
• Commutative Law
• AB BA, ABBA
• E.g. addition and multiplication
• Distributive Law
• A(BC) AB AC, E.g. multiplication over
  addition
• A(BC) (AB)(AC)
• Idempotency Law
• AA A, AAA

• Double Negation
• (A) A
• E.g. -(-5) 5
• DeMorgans Law
• (AB) AB,
• (AB) AB
• Identities
• A0 0, A0A
• A1 A, A11
• AA 0, AA 1

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Simplification of Expressions

• To simplify (reduce the number of gates)
• (1) look at two or more terms sharing one or more
  letters
• use distributive Law
• AB AC A(BC)
• ABABAB // distributive law (1)
• A(BB) AB // identities
• A(1) AB // identities
• A AB // distributive law (2)
• (AA)(AB) // identities
• (1)(AB) // identities
• AB

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Simplification of Expressions

• ABBBAB
• AB1 AB
• 1

Identities
Identities
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Simplification of Expressions
Distributive law
Idempotency Law
Idempotency Law
Distributive Law
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Circuit for Equivalence

• We need to compare the data contents of two
  registers
• Data is in binary
• compare them bit by bit
• Start right to left
• Take two inputs
• If both 0s or 1s, output 1
• Otherwise, output a zero
• ABAB
• (Sum of products method)
• Ask yourself when am I getting a 1?
• ? (AB) AB (simpler)
• Draw circuit