Computer Science 101 A Survey of Computer Science

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Computer Science 101A Survey of Computer Science

• Models of Computation

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Models

• Captures the essence the important properties
  of the real thing.
• Probably differs in scale from the real thing.
• Suppresses details of the real thing.
• Lacks the full functionality of the real thing.

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Modeling Cycle

• The details that are present/suppressed, etc.
  depend on the study the model is to be used for.
• Too many details/features may make the model too
  slow, too unwieldy, too expensive, infeasible to
  work with.
• Too few details (or the wrong details) may make
  the model inaccurate for the purposes at hand.
• Usually, there is a cycle of designing,
  implementing, testing, and then refining.

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Computer Models

• Many fields of research natural and social
  sciences in particular make heavy use of
  computer models for predictions and explanations.
• Professor Eason in Physics works with very
  complex and computationally intensive models of
  the heart.
• Professor Conners in Geology works with complex
  models of geological faults to explain history of
  the site.
• And many, many more such examples.

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Properties of a Computing Agent

• Can accept input
• Can store information in and retrieve it from
  memory
• Can take actions according to algorithm
  instructions, depending on current state of the
  agent and its input.
• Can produce output

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Historical Background

• Around 1900 there was a very optimistic spirit
  within the mathematical community.
• Some even believed it might be possible to have
  an algorithm that would take as input
  mathematical statements and output whether the
  statement was true or false.
• This would put mathematicians out of business
  as far as the need for proving new theorems.

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Historical Background

• In 1931, Austrian logician, Kurt Godel, showed
  that any reasonably complex mathematical systems
  (enough to do number theory) would contain true
  statements that could not be proven in the
  system.
• So, a mathematician, might devote a lifetime to
  attempting to prove a result for which no proof
  could exist.

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Historical Background

• In mid 1930s, mathematicians began to look
  closely at computational procedures (algorithms)
  themselves what does it mean to be computable
  by algorithm, etc. Could there be an algorithm
  that told whether a statement is provable or not
  (not whether it is true or not).
• British mathematician, Alan Turing, proposed a
  model of a computing agent, The Turing Machine,
  which gradually gained acceptance (equivalent to
  many other proposed models).

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Historical Background

• Church-Turing Thesis If there is an algorithm to
  do a symbol manipulation task, then there is a
  Turing machine to do that task.
• Or, contrapositively If such a task can not be
  done with a Turing machine, then there is no
  algorithm to do the task.

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Turing MachinesModels of Computing Agents

• We can view a Turing Machine as consisting of
  three main units
• A tape for holding the input and intermediate
  results
• A state unit where the system is always in one of
  a finite number of states
• A control unit that determines how the tape is
  processed and how the system changes state.

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The Tape

• The tape consists of individual cells,
  theoretically extending infinitely in either
  direction.
• Each cell can hold one symbol from the machines
  alphabet a finite set of symbols including b
  (blank).
• The tape holds the original input, a finite
  string of non-blank symbols.
• At any point in time the tape will have only a
  finite number of non-blanks.
• There is a read/write head that is pointing to a
  particular cell and is capable of reading what is
  in that cell and writing a new symbol there.

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States of the Machine

• The states of the machine refer roughly to the
  status of the machine in doing its task.
• There are only a finite number of states for a
  given Turing machine. At any point the machine is
  in exactly one of these states.
• With a computer, we could think of being in state
  of beginning the fetch phase, or having just
  loaded something into the register, etc.
• In an algorithm, maybe we are in state of having
  just gotten the current temperature from user.

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State Unit
1
2
8
7
3
6
4
5
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Control (Steps) of Turing Machine

• A Turing Machine TM operates in a step by step
  fashion think of a clock ticking and at each
  click, an action is taken.
• The action taken at a given step depends on
• The state the machine is in
• The tape symbol in the cell where the tape head
  is located
• The action consists of
• State change (what is the next state possibly
  same state)
• Tape action
• New symbol is written into the cell where tape
  head is located (possibly same symbol)
• The tape head is moved one cell to the left or
  one cell to the right.

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TMs more details

• Initialization TM begins
• In State 1
• With tape head on leftmost non-blank cell
• Leftmost tape symbol is blank
• There can not be two possible actions for a given
  state and tape symbol
• The TM halts if it there is no action specified
  for its current situation
• If a TM halts, the non-blank portion of its tape
  can be considered to be its output.
• It could be required that a TM always halt on
  appropriate input (maybe, maybe not)

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Anatomy of an Instruction

• A TM instruction is given as a 5-tuple(current
  state, current symbol, next symbol, next state,
  direction)
• For example, the instruction (2,1,0,4,L) would
  mean
• If we are in State 2 and if tape head is reading
  a 1 then
• Replace the 1 with a 0 Move to state
  4 Move tape head one cell to the left

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Turing Machine Example

• Suppose a TM has the following instructions
• (1,1,1,2,R)
• (1,0,0,1,R)
• (2,1,1,1,R)
• (2,0,0,2,R)
• (1,b,1,3,R)
• (2,b,0,3,R)
• Suppose input is 101

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Turing Machine Example

• (1,1,1,2,R)
• (1,0,0,1,R)
• (2,1,1,1,R)
• (2,0,0,2,R)
• (1,b,1,3,R)
• (2,b,0,3,R)
• Suppose input is 111

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Turing Machine Example Odd parity
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Turing Machine Example

• (1,1,1,2,R)
• (1,0,0,1,R)
• (2,1,1,1,R)
• (2,0,0,2,R)
• (1,b,1,3,R)
• (2,b,0,3,R)

• Strategy Scan from left to right keeping track
  of whether we have seen an even number of 1s or
  an odd number of 1s
• States
• State 1 is state where we have seen an even
  number of 1s
• State 2 is state where we have seen an odd number
  of 1s
• State 3 is the halt state

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Turing Machine ExampleState Transition Diagram

• (1,1,1,2,R)
• (1,0,0,1,R)
• (2,1,1,1,R)
• (2,0,0,2,R)
• (1,b,1,3,R)
• (2,b,0,3,R)