Computability and Complexity 4-1

Existence of UndecidableProblems

Computability and ComplexityAndrei 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:

33

23

13

32

22

12

31

21

11

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, =3.14159…

Suppose we get a list of all real numbers:

.0 .3

.0 .2

.0 .1

333231

232221

131211

aaa

aaa

aaa

Then the number where (modulo 10) is not inthe list.

321.0 bbb iii ab

The set of real numbers is uncountable

Computability and Complexity 4-4

Question

Is the set * countable? uncountable?

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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)

000000000000000

,,,,,{ 43210 qqqqqQ

000000000000000

,,,,,{ 54321 sssss

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• Now we can code the elements of the transition function:

000100001100001),,(),,(01000100100100),,(),,(

332

3120

SsqqLsqsq

• Now we can code the whole machine by giving the whole transition function:

),,(),,(),,(),,( 3323120

0001000011000011010001001001001SsqqLsqsq

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 oninput x:

• Write the code for T on Tape 1 and the code for x on Tape 2

• Write the code for on Tape 30q

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.

0q

(Marvin Minsky designed a UTM using only 7 states and 4 symbols in1962. 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)

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

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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 allproblems

.3

.2

.1

333231

232221

131211

aaa

aaa

aaa (This time )}1,0{ija

The string , where321 bbb

0 if ,1

1 if ,0

ii

ii

ia

ab

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