Halting Problem

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Halting Problem
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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?

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Example

• Infinite loop - we get into a loop but for some
  reason we never get out of it.

while x lt 3 x x - 1
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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
P
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

• Assume that what we want to prove is false.
• Demonstrate that a logical contradiction results.
• We can conclude that the the thing we were trying
  to prove must be true.
• Example There is no largest integer.

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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
P
Halts? Yes or No
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Lets Build Another Program

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

B
H
P
P
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What does P do?

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

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What does B do?
B

B loops
H
Yes
P
P
No
B Halts
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What Can We Do With P?

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

B
B loops
H
Yes
B
B
No
B Halts
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What if B Halts?
B
B loops
H
Yes
B
B
No
B Halts
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What if B Loops Forever?
B
B loops
H
Yes
B
B
No
B Halts
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The Paradox

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

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What Went Wrong?

• Theres nothing wrong with B.
• 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

• 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

• Ive claimed that the traveling salesperson
  problem is intractable.
• The best algorithm weve seen so far is O(N!).
• How can we know that there isnt a better
  algorithm? Maybe O(N)?
• The short answer is that we cant.
• There is a longer answer that leads us to
  conclude that TSP is very likely intractable

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P and NP

• P the set of problems that can be solved in
  polynomial time by a Turing machine.
• NP the set of problems that can be solved in
  polynomial time by a non-deterministic Turing
  machine.
• Non-deterministic Turing Machine????

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Non-Deterministic Turing Machines

• Easiest way to describe them is to say that they
  have the ability to instantly guess an answer,
  but they need to do the work of verifying that it
  is correct.
• Example Traveling salesperson decision problem.
• Deterministic TM.
• Non-deterministic TM.

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Whats the Difference?

• Everyone who is anyone believes that P and NP are
  different sets.
• In other words, there are some problems that can
  be solved in polynomial time using
  non-determinism, but cant be solved in
  polynomial time deterministically.

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NP-Completeness

• It has been proven that the problem SAT is
  NP-complete.
• (The SAT problem is to determine whether or not a
  given boolean expression can ever be true.)
• If we could solve SAT in polynomial time, then we
  could solve any other problem in NP in polynomial
  time. (No time for the proof of this.)
• It is NP-complete At least as hard as every
  other problem in NP.

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So What?

• This means that it is very very likely that SAT
  does not have a polynomial time algorithm.
• If it does, then PNP, which no one believes.
• What does any of this have to do with TSP?

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Reductions

• We would like to show a polynomial time reduction
  from SAT to TSP.
• This involves showing how to convert any instance
  of the SAT into an instance of the TSP problem
  that has the same solution.
• This would mean that any solution to the TSP
  problem would immediately give us a solution to
  SAT.
• We could conclude that the TSP solution is at
  least as hard as SAT, and by extension, at least
  has hard as every other problem in NP.

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TSP

• How do we reduce TSP to SAT?
• It can be done, but it is beyond the scope of
  this lecture )
• It turns out that many problems are NP-Complete.