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Computability and Complexity

Computability and Complexity. 4-1. Existence of Undecidable Problems. Computability and Complexity Andrei Bulatov. Computability and Complexity. 4-2. Math Prerequisites. We can make a list of natural numbers: 1,2,3,4,5 ,…. integers : 0,1,-1,2,-2 ,…. even rationals :.

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Computability and Complexity

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  1. Computability and Complexity 4-1 Existence of Undecidable Problems Computability and Complexity Andrei Bulatov

  2. Computability and Complexity 4-2 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

  3. Computability and Complexity Suppose we get a list of all real numbers: 4-3 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

  4. Computability and Complexity 4-4 Question Is the set * countable? uncountable?

  5. Computability and Complexity • First we code the states: • Then we code S as 0, L as 00, R as 000 • Then we code the alphabet: • ( is coded as an empty string of 0’s) 4-5 Coding up a TM Any TM may be described by a finite string of 0’s and 1’s Here is one way to do it:

  6. Computability and Complexity • Now we can code the whole machine by giving the whole • transition function: 4-6 • Now we can code the elements of the transition function:

  7. Computability and Complexity Definition A “Universal Turing Machine” (UTM) is a TM, U, such that, for any TM T and any input x • U(T,x) is finite iff T(x) is finite; and • the output of U(T,x) encodes the output of T(x) • Write the code for T on Tape 1 and the code for x on Tape 2 • Write the code for on Tape 3 4-7 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:

  8. Computability and Complexity 4-8 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 … )

  9. Computability and Complexity Theorem There exists a problem that cannot be solved by any Turing Machine 4-9 Unsolvable problem Problems: functions from {0,1}* to {0,1} (that is problems of recognizing 01-strings)

  10. Computability and Complexity 4-10 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

  11. Computability and Complexity (This time ) The string , where 4-11 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

  12. Computability and Complexity 4-12 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

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