Lab 03 - Turing Machines

Key concepts

1. acceptance vs decision
2. complement of a problem

1.1 Can the problem $ f(w) = 0$ (for all w in $ \Sigma^*$ ) be accepted by a Turing Machine?

1.2 Can a problem be accepted by two different Turing Machines?

1.3 Can a Turing Machine accept two different problems?

1.4 Write a Turing Machine which accepts the problem $ f(x) = 1$ iff x (encoded as binary word) is odd, but does NOT decide it.

1.5 Which of the following problems you think can be accepted and which can be decided? Use pseudocode instead of writing a TM.

Diophantine equations (Hilbert's Tenth Problem)

• A diophantine equation is a polynomial equation where only integer solutions are sought.
• Examples:
  • $ x^2+y^2=1$
  • $ x^4+y^4+z^4=w^4$
  • $ 3x^2-2xy-y^2z-7=0$
• The decision problem we are interested in is: Given a diophantine equation, does it have at least one solution?

Linear Integer Programming

• You are given a set of arithmetic constraints over integers, and try to find if a solution to the constraints exists.
• Example:
  • $ y-x\leq 1$
  • $ 3x+2y\leq 12$
  • $ 2x+3y\leq 12$

Wang Tiles

• Wang tiles are squares where each side has a specific color. An example is given below.

• Wang tiles can be used to tile surfaces, but each tile must be placed such that adjacent tiles have the same color side.
• The wang tiling decision problem is: Is it possible to tile the plane (an infinite surface) with a given set of tiles?

k-color

• You are given a undirected graph and a number of k colors. Is it possible to assign a color to each node such that no adjacent (connected by an edge) nodes have the same color?

2.1 What is the complement of the problem from Exercise 1.1 ?

2.2 What is the complement of k Vertex Cover?

2.3 If a problem is decided by some TM, can its complement be decided?

2.4 If a problem is accepted by some TM, can its complement be accepted?

3.1 Write a TM pseudocode which:

• takes a TM encoding enc(M)
• accepts if there exists a word which is accepted by M, in k steps

Suppose M is encoded on binary words, and also working on binary words, for simplicity.

3.2 Which of the following pseudocode is a proper Turing Machine? Explain why.

Algoritm(M,w){
   if size(w) > 10
     then if M accepts w in k steps
          accept.
}

Algoritm(M1,M2,w){
   k = 0
   while true
       if M1 accepts w <=(iff)=> M2 accepts w , in k steps
          then accept
       else k = k + 1
}

Algorithm(M,A) {
   // A is a finite set of words
   for each w in A
       if M(w) accepts 
          then accept  
}

Algorithm(M) {
   if M accepts all words w in Sigma*
        accept
}

Algorithm(M1,M2) {
    if M1 always accepts then
       if M2 always accepts then 
          accept
}

3.4 Write a TM pseudocode which:

• takes two TMs as input
• accepts if there exists a word which is accepted by both TMs

3.5 Write a TM pseudocode which:

• takes a word as input
• accepts if there exists a TM which accepts the word

3.6 Write a TM pseudocode which:

• takes a Turing Machine M, and a finite set of words A
• checks if all words in A are accepted by M

3.7 Write a TM pseudocode which:

• takes a Turing Machine M, and a finite set of words A
• checks if some words in A are accepted by M