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 (as binary) 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?

• Hilbert undecidable
• Wang Tile
• k-color
• Linear Integer Progra

mming

2.1 What is the complement of the previous problem?

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 decided?

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.

Pseudocode(M): <- input
- divide the tape on three sections:
- [word][value i][value k in binary][enc(M)]
- set the w=[word] section to "0", set the [value i] section to 0 in binary
- simulate w on M. After each transition, increment i and perform following checks
    - if M accepts (crt state of M is final), go to final state
    - if i == k:
        "increment" the current word w. E.g. "0010" may be incremented as "0011" this is the "next" binary word
        set i = 0
        repeat the same process all over

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