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 : These sets are countable

Computability and Complexity 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,  = … Suppose we get a list of all real numbers: Then the number where (modulo 10) is not in the list. The set of real numbers is uncountable

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

Computability and Complexity 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: 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)

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

Computability and Complexity 4-7 Universal TM 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) 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 : Write the code for T on Tape 1 and the code for x on Tape 2 Write the code for on Tape 3

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 No one has yet designed a smaller one … )

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

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

Computability and Complexity 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 (This time ) The string, where

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