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### Existence of Undecidable Problems in Computability and Complexity ###

This lecture explores foundational concepts in computability and complexity theory, focusing on the existence of undecidable problems. It begins with an overview of countable sets like natural numbers and integers, contrasting them with the uncountable set of real numbers. The discussion includes coding Turing Machines (TMs) and Universal Turing Machines (UTMs), showcasing how every TM can be represented as a finite string. Key theorems demonstrate that there are more problems than TMs, leading to the conclusion that some problems cannot be solved by any TM. Understanding these concepts is crucial for grasping the limits of computation. ###

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### Existence of Undecidable Problems in 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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