Computability and Complexity

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Existence of Undecidable Problems
Computability and Complexity Andrei Bulatov
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Math Prerequisites
We can make a list of natural numbers
1,2,3,4,5,
integers 0,1,-1,2,-2,
even rationals
These sets are countable
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Math Prerequisites
However, we cannot make a list of reals
Every real number can be thought to have an
infinite decimal representation, say,
?3.14159
Then the number where
(modulo 10) is not in the list.
The set of real numbers is uncountable
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Question
Is the set ? countable? uncountable?
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Coding up a TM
Any TM may be described by a finite string of
0s and 1s
Here is one way to do it
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• Now we can code the elements of the transition
  function

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Universal TM
Turing showed in his 1936 paper that UTMs exist
One form of UTM uses 3 tapes. To simulate the
operation of T on input x
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Universal TM description

• Find the first symbol of the coded input on
  Tape 2
• Search the list of transitions on Tape 1 for a
  transition from
• that applies to this symbol
• Simulate the effect of this transition on the
  coded input and the stored
• state (Tapes 2 and 3)
• Search the list of transitions for one that
  applies in the new situation
• Continue until a final state is reached.

(Marvin Minsky designed a UTM using only 7 states
and 4 symbols in 1962. No one has yet designed a
smaller one )
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Unsolvable problem
Problems functions from 0,1 to 0,1
(that is problems of recognizing 01-strings)
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Lemma 1 There are countably many Turing
Machines
Proof Each TM can be represented as a binary
string. Therefore the set set all TMs can be
thought as a subset of 0,1 Since 0,1
is countable, the set of all TMs is also
countable
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Lemma 2 The set of all problems is
uncountable.
Proof Each function 0,1 ? 0,1 can be
represented as a binary string f(0)
f(1) f(00) f(01) f(10) f(11)
Suppose this set is countable. Then we are able
to create a list of all problems
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Lemmas 1 and 2 implies that there are a lot more
problems than Turing Machines. Therefore at
least one of the problems cannot be solved by a
TM

QED
Note that this is an existence argument. We
cannot point out any particular undecidable
problem This is what we shall do in the next
lecture