CS355 The Mathematics and Theory of Computer Science

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CS355 The Mathematics and Theory of Computer
Science

• Reductions

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Undecidable Problem

• Let us look at an undecidable problem concerning
  simple manipulation of strings.
• It is called the Post correspondence problem
  (PCP).
• Can describe the problem as a type of puzzle.

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PCP

• We begin with a collection of dominos, each
  containing two strings, one on each side for
  e.g.
• And a collection of dominos looks like

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PCP

• Goal Make a list of these dominos (repeats
  allowed) so that the string we get on the top of
  the dominos is the same as the string we get on
  the bottom of them.
• This list is called a match. E.g.

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PCP

• Reading off the top and bottom strings gives us
• abcaaabc
• For some collection of dominos there may not be a
  match! E.g.
• Top strings longer than bottom / No c in bottom

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PCP

• THE PCP is to determine whether a collection of
  dominos has a match.
• Problem is unsolvable by algorithm.
• Let us state the problem precisely and then
  express it as a language.

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PCP

• An instance of the PCP is a collection P of
  dominos
• and a match is a sequence i1, i2, ,il, where
  ti1, ti2, ,til bi1, bi2, ,bil. The problem
  is to determine whether P has a match. Let
• PCP P is an instance of the PCP with a
  match

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PCP

• PCP is an undecidable problem and there is a
  proof in Sipser if anyone is interested in
  looking it up.
• Let us look at a simpler undecidable problem.