Data Manipulation

CSCI130Instructor: Dr. Imad Rahal

Layered ArchitectureLAYER Order

Application SW: Excel & Access 2

High-order P.L.: Visual Basic 1

Low-order P.L.: Assembly 3

System SW: O.S. 3

Machine Language 4

Data Representation 5

HW: Circuit Design 6

0

clock

16-3000 MHz (~3.0 GHz)

Overview of Computer Hardware

“necessary” components of a computer CPU, Main memory

components needed for convenience computer won’t be very practical to use otherwise Secondary/Auxiliary storage, I/O devices

Main memory Connects to the motherboard Divided into two major parts

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

Data Manipulation Computing an answer to an equation:

5*3 + 62 – 7 Assume our computer can’t 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_2>0, go to step 4 7: Stop

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

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

NOT Truth Table

A NOT A (A’)

1 0

0 1

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

AND Truth Table

A B A AND B (AB)

0 0 0

0 1 0

1 0 0

1 1 1

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

OR Truth Table

A B A OR BA+B

0 0 0

0 1 1

1 0 1

1 1 1

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 A+B NOT A A’

A’B’ + AB

(2) Through Circuit diagrams Given an expression Draw a gate after its inputs have been drawn Try A’B’ + 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)

NOT AND OR

A B A’ B’ A’B’

AB A’B’+AB

0 0 1 1 1 0 1

0 1 1 0 0 0 0

1 0 0 1 0 0 0

1 1 0 0 0 1 1

Enough rows to hold all input combinations

1 letter 21 rows 2 letters 22 rows 3 letters 23 rows n letters 2n rows

Practice

Try A + (A.B’+B.C)’ How many gates?

Design circuit Parenthesis, NOT, AND, and then OR

Find truth table How many rows?

Data Manipulation

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

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

A B ?

0 0 1

0 1 1

1 0 0

1 1 1

A’B’ A’B Ignore AB

A’B’+A’B+AB Complicated and UGLY !!! (requires space, and is costly and slow)

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)

Simplification of Expressions

Laws of Boolean Algebra Commutative Law

A+B = B+A, A.B=B.A E.g. addition and

multiplication Distributive Law

A.(B+C) = A.B + A.C, E.g. multiplication over addition

A+(B.C) = (A+B).(A+C) Idempotency Law

A+A = A, A.A=A

Double Negation (A’)’ = A E.g. -(-5) = +5

DeMorgan’s Law (A+B)’ = A’.B’, (A.B)’ = A’+B’

Identities A.0 = 0, A+0=A A.1 = A, A+1=1 A.A’ = 0, A+A’ = 1

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(B+C)

A’B’+A’B+AB // distributive law (#1) A’(B’+B) + AB // identities A’(1) + AB // identities A’ + AB // distributive law (#2) (A’+A)(A’+B) // identities (1)(A’+B) // identities A’+B

Simplification of Expressions

A.B’+A’.B’+A’.BDistributive law

Idempotency Law

Idempotency Law

Distributive Law

B’. (A + A’) + A’.B B’ + A’.B (B’ + A’).(B’ + B) B’ + A’ (B.A)’

Identities

Distributive

Distributive

DeMorgans

Simplification of Expressions

AB’+B+B’+AB AB’+1 + AB

1 Identities

Identities

Simplification of Expressions

Distributive law

Idempotency Law

Idempotency Law

Distributive Law

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

A’B’+AB (Sum of products method)

Ask yourself: when am I getting a 1? (A+B)’ +AB (simpler)

Draw circuit

A B A=B

0 0 1

0 1 0

1 0 0

1 1 1

Circuit for Equivalence

Circuit for Addition