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?

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. Complement ==== 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. Turing Machine pseudocode ==== 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. <code> 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 </code> 3.2 Which of the following pseudocode is a proper Turing Machine? Explain why. <code> Algoritm(M,w){ if size(w) > 10 then if M accepts w in k steps accept. } </code> <code> 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 } </code> <code> Algorithm(M,A) { A is a finite set of words

for each w in A

     if M(w) accepts 
        then accept  

} </code>

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