7' Unsolvable Problems

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7. Unsolvable Problems

• What is computable?
• If something is computable then
• it can be computed in a FlooP program,
• or equivalently by a Turing machine.
• Such a method of computation can be described by
  a finite set of symbols.
• Not all functions are computable!
• There are more possible functions than programs.

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Countable

• A set of objects, S , is countable if EVERY
  element of S can be UNIQUELY labeled with a
  natural number.
• S1 ? 1
• S2 ? 2
• etc
• The set can be infinite, like the natural
  numbers!

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• We showed earlier that BlooP and FlooP programs
  can be enumerated in order of length to create
  the Blue Programs and Green Programs.
• Turing machines can be similarly enumerated.
• Thus, the number of programs is countably
  infinite.
• Hence there are a countably infinite number of
  problems that can be solved by computational
  methods.

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Uncountable

• While the set of countable numbers is infinite,
  it is possible to consider even larger sets,
  which are called uncountable.
• Cantors diagonal argument can be used to show
  that the set of real numbers is uncountable.
  (Basically, one assumes that a complete list of
  them can be made, and then construct a number
  which is not in the list.)

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• Let us consider functions of one variable that
  take natural numbers as arguments and values
• e.g. f(3) 5 f(4) 100
• Similar diagonal arguments show that there are an
  uncountable number of such functions!
• But, only a countable number of these functions
  are going to be computable!
• Uncountably many are uncomputable!
• Lets look at a few examples of problems which
  are known to be unsolvable

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How do we prove something is unsolvable?

• Contradiction - assume its solvable and see if it
  leads to a contradiction with something else we
  believe is true (e.g., the halting problem and
  the Church Turing thesis.)
• Reduction - see if another unsolvable problem can
  be reduced to solving it. That is, if we assume
  it is solvable, then can we show that another
  problem we believe is unsolvable can be solved?

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Some unsolvable decision problems

• The termination testerbetter known as
• The Halting Problem
• Is there an algorithm that can decide whether
  the execution of an arbitrary program halts for
  an arbitrary input?
• The Total Problem
• Is there an algorithm that can decide whether
  the execution of an arbitrary program halts for
  all possible inputs?

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The Equivalence Problem

• Does there exist an algorithm that can decide
  whether two arbitrary computable functions
  produce the same output?
• A simpler version of this is
• Does there exist an algorithm that can decide
  whether an arbitrary computable function is
  equivalent to the identity function?
• (?x f(x) 1)
• This is also unsolvable!

9
Posts Correspondence Problem

• Given a finite sequence of pairs of strings
• ?s1, t1?, ?s2, t2?,., ?sn, tn?
• Is there a sequence of indices i1, i2,ik,
  allowing repetitions, such that
• si1.sik ti1..tik ?
• Any given sequence may or may not have a
  solution.
• Is there an algorithm that can decide whether an
  arbitrary instance of this problem has a solution?

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• An Example
• Supposing we have the pairs
• ?ab, a?, ?b, bb?, ?aa, b?, ?b, aab?
• Call them 1, 2, 3, 4
• The sequence 1, 2, 1, 3, 4 produces
• ab b ab aa b a bb a b aab
• But take another instance
• ?ab, a?, ?b, ab?
• This has no solution. Why?

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• Hilberts Tenth Problem
•  
• Does a general polynomial equation
  f(x1, x2, ,xn) 0 with integer
  coefficients have an integer solution?
• e.g. 2x 3y 1 0 yes
• x2 2 0 no
•  
• Proved unsolvable in 1970 by Matiyasevich!

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Partially Solvable Problems

• Many unsolvable problems are actually partially
  solvable, because we can search for a yes answer
  and know that it will be produced in a finite
  amount of time.
• That is, an algorithm exists which can solve all
  instances which are solvable, but which may run
  forever for the unsolvable instances

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• The halting problem is partially solvable because
  for any computable function fn and any input x we
  can evaluate the expression fn(x).
• If the evaluation halts with a value, then we
  output yes.
• In any other case, we dont care!

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Posts problem

• For Posts correspondence problem we can check
  for a solution by systematically looking at all
  sequences of length 1, then length 2 etc.
• If there is a sequence giving two matching
  strings we will find it and output yes.
• Otherwise we dont care.

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• The total problem
• Is there an algorithm to decide whether an
  arbitrarily computable function halts for all
  inputs?
• This is not even partially solvable. To show a
  function is total, you would have to try it on
  all possible inputs!

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Solvable problems

• Most unsolvable problems have specific instances
  that are solvable.
• For example, consider the halting problem
• Let f(x) x 1.
• Does f halt on input x 35?
• Does f halt on any input?

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Similarly for the equivalence problem

• Are the following two functions equivalent?
• f(x) x if x is odd
• x 1 otherwise
•  
• g(x) 2x x 1 if x is even
• x otherwise

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Computational complexity

• Now we shift our attention to solvable problems
• The primary concern is the degree of complexity
  of the problems how much time or memory is
  required to find a solution?
• This depends on the nature of the problems, but
  also on the algorithms and hardware which are
  used to solve them