Concepts in Computer Science

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Concepts in Computer Science

• COP 2500, Spring 2006
• http//www.cs.ucf.edu/courses/cop2500
• Keith Garfield
• garfield_at_cs.ucf.edu

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The World of Computer Science
Theory of Algorithms
Algorithms
Theoretical Model of Computers
Theory of Languages
Languages
Architecture
Real World
World of Theory
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What is Computer Science all About?
What problems can a computer solve?
What is the computational cost of the
solution?

• The answer to the first question tells us
  whether a problem should even be attempted using
  a computer.
• Some problems are perfectly suited for computers,
  others are not.
• The answer to the second question tells us if it
  is worth doing.
• The cost is in terms of some resource like time
  or memory usage.
• We attempt to relate cost to problem size.
• Problems are grouped into classes depending on
  their cost (Chptr 11)

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Purpose of the Course

• Computer Capabilities Inherent capabilities of
  any autonomous computing machine.
• Basic CS concepts Allows you to communicate with
  and work with CS people, and validate (or
  disprove), their ideas.
• Programming? Not trying to teach you to be
  programmers, but will require some programming to
  develop topics.
• Program design How does one go about planning a
  computer program?
• Program implementation How does my design get
  turned into working code?

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Course Outline

• Throughout the course we will use searching and
  sorting problems to illustrate points.
• Topics (roughly in the same order as they are
  introduced)
• Binary math
• Basic computer architecture
• Our model of a computer
• Introduction to Algorithms and Functions (Chapter
  2)
• The cost of computation (Chapter 9)
• Transforming algorithms into code (Chapter 3,
  supplemental text)
• Computer languages data types and data
  structures. (Chapter 3)
• Computer languages program control mechanisms
  (Chapter 4)
• Computer Languages Types of languages (Chapter
  7)
• More about functions recursion, iteration,
  modularity. (Chapter 4)
• Complex Data Structures. (Chapter 3, 5)
• The hierarchy of problems (Chapter 11)

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Some First Thoughts

• Computers are amazingly stupid. They have no
  sense of self, situation, or history.
• Computers will only do what you tell them, and
  you must give them their directions in minute
  detail.
• Computers store information in very limited ways.
• Because of the above, computers are very good at
  some types of tasks and very bad at others.
• Good Mindless repetition with accuracy.
• Bad Creativity, intuition, common sense, and
  sensing.
• We will use the above to discuss what computers
  can and cannot do Can it be computed?

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CS Concepts Computational Power

• Computational power refers to the types of
  problems a computer can solve.
• It has nothing to do with how fast a computer is
  nor how nice the graphics are.
• It turns out very simple computers are just as
  powerful than very expensive ones (although
  slower).
• So in terms of computational power, all computers
  are equal.
• This allows generalized solutions - solve a
  problem for one and we solve it for all.
• Since all computers are equal, we tend to focus
  on types of problems rather than types of
  computers.
• Is the problem computable?

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CS Concepts Computable Problems

• Since all computers are equal, we tend to focus
  on types of problems rather than types of
  computers,
• Is the problem computable? If yes, well find a
  fast computer.
• We are implying some problems are not (currently)
  computable.
• Weather prediction is the most well known
  example.
• Route planning is another.
• Natural Language Processing is another.
• This class focuses on computable problems
• Sorting and searching.
• We will look at sorting and searching methods in
  order to discuss basic CS ideas, terminology, and
  good practices.
• This corresponds to chapters 2, 3, 4, 5, and 6 of
  text.

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CS Concepts What is computable?
We will identify a small set of problems that
Are computable, and a small set that are not
Computable during this class. New problems
can be mapped (correlated, Compared, translated)
to the known problems And then deemed computable
or not.
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Patterns of 2s in Computing

• Computers operate on a binary system
• Binary systems allow only two values
• Zero or One
• On or Off
• True or False
• Early computers were composed of groups of
  mechanical switches, each of which was On or
  Off at any time.
• Modern computers store bits of data. Each bit
  can have the value of Zero or One.
• The word BIT is a shortening of BINARY DIGIT

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Computers Store Everything in Binary Form

• So computer memory is nothing more than long
  sequences of bits (0s and 1s).
• The most common unit of computer memory you see
  is the BYTE, which is simply a group of 8 bits.
• How much memory does your PC at home have? It is
  always given in bytes, not bits. The same is
  true of file sizes.
• These bits must be interpreted in special ways to
  be meaningful to humans
• All data is represented as lists of 0s and 1s
• Different types of data require different methods
  of interpretation to be meaningful. Numbers are
  different than letters, for example.
• We will talk much more about data types
  throughout the semester.

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Comparing Human and Computer Formats

• The human digits 0,1,2,3,4,5,6,7,8,9
• Computer digits 0,1
• Some human numbers 17 42 31
• Computer numbers 10001 101010 11111
• The human alphabet Aa, Bb, Cc, Dd, Ee, .. Xx,
  Yy, Zz
• Computer alphabet 0, 1
• Some human words dog cat
• Computer words 0110010001101111101100111
• 0110001101100010001110100

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Lets Look at Regular Math Again

• Consider two math operations taking a number to
  a power, and taking the log of a number.
• We will focus on the number 2 (binary no
  surprise)

n 2n n log2n
0 1 1 2 2 4 3 8 8 256 9 512 10 1024 1 0 2 1 4 2 8 3 256 8 512 9 1024 10
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How Much Information can we Store?

• The number of bits assigned to a piece of
  information tells us how well we can describe the
  information.
• The most abstract case n bits allows 2n
  distinct values.

Bits Values Number Of Values
1 2 0 1 00 01 10 11 2 4
Bits Values Number Of Values
3 000 001 010 011 100 101 110 111 8
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Example 1 Color Depth

• How many colors are available if each pixel uses
  n bits?
• How many bits are required for 256 colors?
• 8 bits, because log2 256 8
• Thats one byte coincidence???

Bits Values Colors
1 2 0 1 00 01 10 11 Black White Black Dark Gray Light Gray White
Bits Values Colors
3 000 001 010 011 100 101 110 111 Black Blue Green Yellow Red Purple Orange White
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Example 2 Red-Green-Blue Format

• One way to designate colors is setting values for
  the amount of red, green, and blue present in the
  mix.
• The digits allowed in this system are 0, 1, 2,
  3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F.
• Each color component can have a value from 00 to
  FF
• Examples of RGB colors might be FF 00 FF
  (fuschia), C0 C0 C0 (silver)
• How many bits are need to store the value of the
  color?

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Example 2 Red-Green-Blue Format (cont)

• How many bits are need to store the value of the
  color?
• How many distinct colors can be specified?

How many bits are required for each
digit? There are 16 distinct digits, and log 16
4 Each color component has two digits,
so Each color component requires 8 bits (or one
byte coincidence? A color is made up of three
color components, so 8 x 3 24 bits to specify
a color.
24 bits are used to specify the color, so 224
16,777,216 (but in practice this is not done)
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Example 3 Computer Memory

• How much memory is on your computer? Is it a
  power of 2?
• Lets assume a machine having 64 MB of RAM
• How about 128 MB of RAM?

How many bits are required to specify a distinct
memory location? Lets try 25 bits ? 225
33,554,432 (not enough) Lets try 26 bits ? 226
67,108,864 distinct values. 26 bits are required
to specify a memory location.
27 bits ? 227 134, 217, 728 27 bits are now
required to specify a memory location. (twice the
memory for one more bit coincidence?)
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Example 4 Computer Memory

• What if we needed to assign a unique binary
  number to each of you in this class? How many
  bits would we need for each number?
• How many bits do we need to add if we double the
  class size?

There are about 90 of you. 1 bit ? 21 2
numbers. 2 bits ? 22 4 numbers. 3 bits ? 23
8 numbers. 4 bits ? 24 16 numbers. 5 bits ? 25
32 numbers. 6 bits? 26 64 numbers. 7 bits? 27
128 numbers. (This is more than the 90
required). Answer 7 bits. This allows 90
unique values.
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Example 5 The Hi-Lo Game

• I will give you five chances to guess an integer.
  Whenever you guess wrongly I will tell you
  whether the answer is higher or lower than your
  guess.
• Under what circumstances are you guaranteed
  success (the range of numbers can be considered
  the size of the problem)?
• The numbers range from 0 3
• The numbers range from 0 7
• The numbers range from 0 15
• The numbers range from 0 31
• The numbers range from 0 63

We want the fifth guess to be perfect, so we only
have fours guesses to play with. Each guess can
correspond to a bit of information. 24 16 so I
can discern 16 distinct values with my guesses.
Answer 0 15.
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Example 5 The Hi-Lo Game (cont)

• Here is how to guess to guarantee success (note
  the dirty trick -- even though the numbers we are
  guessing are integers, we can guess non-integers)

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By the way Memorize this!
Depth Dots 0 1 20 1
2 21 2 4 22 3 8
23 4 16 24
In general, a structure like this having depth d,
has N 2d objects at the bottom row.
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Why do we Care?

• We are trying to determine what kinds of problems
  are computable (ie capable of being solved by a
  computer).
• These patterns weve seen will show up again and
  again
• Developing methods to solve problems and to
  determine the computational cost of the
  solution.
• Determining what problem sizes are computable
  (example hi-lo with 5 guesses and a range of 0
  1000 is not computable.)
• Divide and Conquer (section 6.4)
• A basic and powerful method of solving problems.
• We attempt to cut the size of the problem in half
  at each step.
• This leads to a log n number of steps for
  problems of size n (and thats good)

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A Computer Model

• Computer science is concerned with computation
• What is computable?
• How much effort is it to compute something?
• We want the results to apply to all computers,
  not specific types or brands.
• We need to develop an abstract model of a
  computer
• Develop properties for the theoretical computer.
• Apply the properties to all real world computers.

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Solution Space

• Consider
• How many answers are there to 2 2 ?
• How many answers are there to Guess what color I
  am thinking of between red, green, and blue?
• How many answers are there to Whats your
  favorite color between red, green, and blue?
• How many answers are their to Is 3 gt 2 ?
• How many answers are their to Guess which
  integer I am thinking of between 0 and 1.
• How many answers are there to Guess which number
  I am thinking of between 0 and 1.

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Solution Space
Solution Space The number of possible answers
right or wrong to a question.

• Solution space is sometimes called search space
  since often the answers must be searched through
  in some fashion.
• Sometimes, solution space can give a clue as to
  the hardness of a problem
• Large search spaces with no good mechanism
  (algorithm) for extracting the correct answer are
  intractable problems.
• Small search spaces imply a problem can be solved
  even without an elegant solution.

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A Computer Model

• Computer science is concerned with computation
• What is computable?
• How much effort is it to compute something?
• We want the results to apply to all computers,
  not specific types or brands.
• We need to develop an abstract model of a
  computer
• Develop properties for the theoretical computer.
• Apply the properties to all real world computers.

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Computer Architecture (the Basics)

• The following are components of a computer system
  from a commercial standpoint
• The Central Processing Unit (CPU)
• Memory (Temporary data storage)
• Disk Space (permanent data storage)
• Monitor
• Keyboard
• Other input/output devices
• Do we need all of this for our abstract model?

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Computer Architecture (the Basics)

• Computer science is concerned with computation
• What is computable?
• How much effort is it to compute something?
• Computer science focuses on the parts of the
  computer that deal with computation
• Memory stores data and instructions
• The CPU operates on the data per the instructions
• Not input/output devices (monitors, keyboards,
  mice, etc)
• We need a model for the CPU.
• We need a model for memory

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The CPU (Processor) Model

• A real CPU has many elements, each devoted to
  specific types of operations (addition,
  multiplication, working with real numbers etc)
• Our model of a Processor is much simpler
• A processor is a black box that can perform
  arithmetic and logical operations.
• A processor performs one discrete operation at a
  time
• Discrete Add two numbers
• Discrete Compare two numbers to see which is
  larger
• Discrete Fetch a value from memory
• Not Discrete Find the average of 20 numbers
  (too many operations)

Processor Model A black box that performs
specific instructions representing discrete
arithmetic and logical operations.
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The Memory Model

• Real Computers have a hierarchy of memory
• Registers hold individual pieces of data inside
  the PCU itself
• Cache holds a limited amount of data readily
  available for the PCU
• RAM (Random Access Memory) contains programs and
  data currently in use (temporary)
• Disks contain data for permanent storage
• In the real world this has implications in terms
  of speed of accessing data. We are not concerned
  with this in computer science.

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The Memory Model

• Our model of a computer only has one level of
  memory
• Any memory location can be accessed just as
  easily as any other using its numerical address
• That is, you can pick any location at random and
  access it (Random Access Memory RAM)
• How much memory? Infinite.

Memory Model Memory is composed of a single row
of storage locations, each referenced by a
numerical address.
0 1 2 3 4 5 6 7 8 9
10 11 12
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A Computer Model
Processor Model A black box that performs
specific instructions representing discrete
arithmetic and logical operations.
Memory Model Memory is composed of a single row
of storage locations, each referenced by a
numerical address.
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Summary of Theoretical Computer Model

• This course is investigating what computers can
  and cannot do.
• We want the results of this course to apply to
  all computers, not specific types or brands.
• Therefore we use an abstract computer model that
  has the following characteristics
• Processor A black box that performs specific
  instructions representing discrete arithmetic and
  logical operations.
• Memory An infinite number of boxes or
  pigeonholes, each referred to by a numeric
  address.
• Now we can talk about what this thing can do.

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Introduction to Variables

• Remember that memory is a long series of boxes
  referred to by numerical addresses.
• We could allow any numeric memory location to be
  used in our algorithms to store data in memory
  and then retrieving the data when we needed it.
• The problem with using numeric memory addresses
  directly is twofold.
• Programs with more than a few pieces of data
  would drive you crazy trying to keep track of
  where you put it.
• Different real world machines have different
  memeory structures.

1
8
x
y
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Introduction to Variables

• We use the concept of a variable to assign a name
  to the memory location.
• Lets call our data "x" and "y".
• We use a data declaration to tell the computer
  that we are using variable names to help us store
  data.
• Declarations need to specify the variable NAME,
  TYPE, and STRUCTURE.
• The computer will assign memory locations to each
  variable
• Here, x is associated with memory location 1.
• Here, y is associated with memory location 8.

1
8
x
y
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Data Types and Data Structures

• When specifying data in a program we need to
  describe its name, type and its structure.
• Data's type impose meaning onto data (semantics)
  and data's structure impose organization (syntax)
  onto data.
• Data Type (definition) A label applied to data
  that tells the computer how to interpret and
  manipulate data.
• Type tells the computer how much space to reserve
  for variables and how to interpret operations on
  them.
• Data Structure (definition) The way data is
  organized logically.
• Describes how different pieces of data are
  organized.