Theory: Goals

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Theory Goals

• Reason about feasibility of device without
  actually building it.
• We can reason about what is going to happen in an
  experiment, without actually doing an experiment
• Readings Chapter 11 Sections 11.1 11.5

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Theory Goals

• To give a model of computation, to be able to
  reason about the capabilities of computers
• To study the properties of computation rather
  than being lost in the details of the machine
  hardware/software
• To study what computers may be able to do, and
  what they cannot do
• To study what can be done quickly and what cannot
  be done quickly

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Theory Goals

• We will construct a model of computation.
• Simple, functional description of a computer
• --- to capture essence of computation
• --- suppress details
• --- easy to study and reason about
• Turing Machine (there are several other models)

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Turing Machine

• (a) One sided infinite tape consisting of cells
  on which letters from a fixed FINITE alphabet may
  be written
• (b) A FINITE set of states. Machine may be in any
  one of the states at any particular time.
• (c) A head which can read/write one cell at a
  time.

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Turing Machine

0
0
1
0
0
1
0
M
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Turing Machine

• In one step TM
• (a) reads the cell being scanned by the head
• Then based on its CURRENT state and symbol read
• (b) writes something on the cell being scanned
• (c) moves the head left or right (or stay where
  it is)
• (d) changes its current state
• Simple, As powerful (computation wise) as any
  computer

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Example

• Suppose initially on the tape we have
• 11111 B 111111111 BBBBBBBBBB.
• We want the TM to convert first set of 1s to 0s
  and second set of 1s to 2s and then return to
  the initial left most cell ().

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Example

• 1) Initially it is reading , and moves the head
  right, and goes to step 2.
• 2) If the symbol being read is 1, then write a 0,
  move the head right, and repeat step 2.
• If the symbol being read is B, then move
  right and go to step 3.
• 3) If the symbol begin read is 1, then write a 2,
  move the head right, and repeat step 3.
• If the symbol being read is B, then go to
  step 4.
• 4) If the symbol being read is 0, 1, 2 or B, then
  move left.
• If the symbol being read is , then stop.

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Example

• Alphabet ,0,1,2,B
• StatesQ1,Q2,Q3,Q4
• Q1 is the starting state
• Transition Table

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11
Church Turing Thesis

• We can construct a Turing Machine for any
  algorithm.
• (Partial) Functions which take strings to strings
• Algorithms/Programs we saw before can be encoded
  as string to string operations.
• Sound/Video etc. are all coded into
  numbers/strings.
• Interpret strings/numbers appropriately.
• NOT a Theorem, only a Thesis.
• Definition of computing device not clear.
• Speed Slow,but not too bad. Within polynomial
  factor of current day computers.

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Model Some points

• FINITE set of states
• FINITE vs INFINITE tape

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Computability

• Can we solve every problem?
• In Mathematics
• Godels Theorem Not every true theorem about
  natural numbers can be proven.
• In Computer Science Not every problem can be
  solved.
• Halting Problem

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Computability

• Given as input a program P, and input x for P
• Does program P stop when run on input x?
• There is no uniform way to solve the above
  problem.
• That is there is no algorithm SOLVE(P,x) such that

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Informal Proof

• SuperSolve(P,x)
• 1. Run SOLVE(P,x)
• 2. If SOLVE(P,x) never stops, then program P also
  never stops.
• 3. If SOLVE(P,x) outputs NO, then stop.
• 4. If SOLVE(P,x) outputs YES, then go to step 5.
• 5. Go to step 5.
• End

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Complexity

• Suppose problem can be solved by the computer.
• Question How much time does it take?
• Searching Linear Search, Binary Search
• Order of Growth The order of complexity is more
  important than constant factors
• 0.5 n vs 1000 log n

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Complexity Number of Operations
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Execution time

• Assuming a fast computer (1GigaHz) (1 billion
  operations per second)

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Complexity

• Slow may not be of much use
• What is feasible?
• Polynomial time Run time polynomial in the
  length of the input.
• n2, n3, n7, 20n2 n5, etc.
• P Class of problems which can be solved in
  polynomial time.
• Note that P is invariant under different types of
  current day machines/TM. So if you can solve some
  problem in polynomial time on a supercomputer,
  then you can solve it in polynomial time on TM.

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Complexity

• Exponential time 2n
• Some problems are inherently exponential or worse
  time.

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• Travelling Salesman Problem
• Want to visit some cities New York, London,
  Tokyo,
• and come back to Singapore.
• Each city visited exactly once (except first and
  last city is both Singapore)
• Cost of travel between each pair of cities is
  given.
• Aim To minimize the cost.
• Find tour of cost less than X.

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• Difficult to solve.
• No polynomial time algorithm known.
• However, if given a tour, one can verify if the
  cost is less than X.
• Easy to verify answers. Difficult to find
  answers.
• Easy ---gt polynomial time
• Difficult --gt not polynomial time.
• NP Class of problems for which it is easy to
  verify the answers.

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• Many natural problems such as travelling salesman
  are in NP but not known to be in P.
• Much of cryptography is based on these kind of
  problems.

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Recasting/Abstraction/Reduction

• Travelling Salesman minimize cost of travel.
• What if we want to minimize time of travel?
• Distance travelled?
• All are essentially the same problem.
• Reduction A?B
• One can solve A quickly using a subprogram for
  B
• If we can solve B quickly, then we can solve A
  quickly
• If one cannot solve A quickly, then we cannot
  solve B quickly

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Recasting/Abstraction/Reduction

• Thus methods of converting (or reducing in
  computer science terminology) can often be used
  for studying complexity of problems.

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Model

• How good is our model.
• Does it capture main concerns well?
• Can we have similar model for humans/brain?
• If so can we compare the two models?
• Are Humans essentially another kind of machine?