Halting Problem and TSP

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Halting Problem and TSP

• Wednesday, Week 8

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Background - Halting Problem

• Common error Program goes into an infinite
  loop.
• Wouldnt it be nice to have a tool that would
  warn us that our program has this bug?
• (We could make a lot of money if we could develop
  such a tool!)

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Example

• Infinite loop - we get into a loop but for some
  reason we never get out of it.
• for (i 0 i lt 10 i--)
• document.writeln(Help, Im in an infinite
  loop!ltBRgt)

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Example Output

• Help, Im in an infinite loop!
• Help, Im in an infinite loop!
• Help, Im in an infinite loop!
• Help, Im in an infinite loop!
• Help, Im in an infinite loop!
• Help, Im in an infinite loop!

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Definition

• Design a program, H, that will do the following
• Take as its input a description of a program P
  (in binary, say).
• Halt eventually and answer "yes" if P will
  eventually halt, or halt and answer "no" if P
  will run forever.

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Halting Problem

• In other words,
• Program H will always halt and give the correct
  answer no matter what program has been given as
  input.

H
program
Halts? Yes or No
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Proving the Halting Problem

• We want to prove that the halting problem is
  non-computable.
• In other words, there is no algorithm that we can
  use to tell whether or not a program will halt.
• We will use proof by contradiction.

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Proof by Contradiction

• Prove if xy gt 2, then xgt1 or ygt1.
• Assume that xygt2 and that xlt1 and ylt1.
• Then, xy lt 11 2.
• This is a contradiction since xy gt2!
• Thus our assumption that xlt1 and ylt1 is false.
• Note NOT(xgt1 or ygt1) (xlt1 and ylt1) by
  DeMorgans Laws.

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Proof by Contradiction

• Well assume that the Halting Problem CAN be
  computed.
• Well develop another program that uses the
  Halting Problem function.
• Well find ourselves caught in a paradox (the
  contradiction).
• Well have proven that our original assumption is
  false.

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Assume Halting Problem OK

• Let H be a program (or sub-program) that
  determines whether a program will halt.

H
program
Halts? Yes or No
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Lets Build Another Program

• Let P be a program that uses H.
• For any given program, P will call H and pass it
  the given program.

P
H
program
program
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What does P do?

• Now, P acts as follows
• P takes a program as input and feeds the program
  to H as input.
• If H answer yes, then P will enter an infinite
  loop and run forever.
• If H answer no, then P will stop.

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What does P do?
P

P loops
H
Yes
program
program
No
P halts
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The program P

• function P (program)
• var halts H(program)
• if (halts True)
• infinite loop
• else
• stop

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What Can We Do With P?

• Lets give P a copy of itself as its input.

P
P loops
H
Yes
P
P
program
No
P halts
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What If P Halts?

P
P loops
H
Yes
P
P
program
No
P halts
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What If P Loops Indefinitely?

P
P loops
H
Yes
P
P
program
No
P halts
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The Paradox

• If P is a program that halts when given itself as
  its input,
• then, when given itself as input,
• P will go into an infinite loop.
• If P is a program that loops indefinitely when
  given itself as its input,
• then , when given itself as input,
• P will halt immediately.

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What Went Wrong?

• Theres nothing wrong with P, itself.
• The problem must be with the assumption that we
  could write H.

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So, Proof by Contradiction

• We assumed that the Halting Problem COULD be
  computed.
• We developed another program that used the
  Halting Problem function.
• We found ourselves caught in a paradox (the
  contradiction).
• We proved that the Halting Problem is not
  computable.

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Conclusions

• Self-referentiality is a real problem with
  programs
• Rices theorem says that Any nontrivial property
  of programs is undecidable.
• Thus, we cant write programs to answer questions
  about programs.

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Intractability

• The Traveling Salesperson Problem is intractable.
• Proving it is beyond the scope of this class.
• We will just understand the problem and how hard
  it is.

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Traveling Sales Problem

• A salesperson has a group of cities that he/she
  needs to visit.
• There are a bunch of distances between the
  cities.
• Our salesperson has to visit all the cities by
  following a path with the least distance (or
  cost).

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TSP Example 1

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TSP Answer 1

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TSP Example 2

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TSP Answer 2

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Traveling Sales Problem

• Conclusions
• The only correct algorithm we could come up with
  had to examine all possible paths.
• The only correct algorithm that anyone has come
  up with has to examine all possible paths.
• Thus this algorithm is O(n!) and intractable.