====== Lab 03 - Turing Machines =======

**Key concepts**

  - acceptance vs decision
  - complement of a problem


==== 1. Accepting and deciding a decision problem ====

**1.1** Can the problem $math[f(w) = 0] (for all w in $math[\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 $math[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: 
    * $math[x^2+y^2=1]
    * $math[x^4+y^4+z^4=w^4]
    * $math[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:
    * $math[y-x\leq 1]
    * $math[3x+2y\leq 12]
    * $math[2x+3y\leq 12]

**Wang Tiles**
  * Wang tiles are squares where each **side** has a specific color. An example is given below.
{{ :aa:lab:wang_11_tiles.svg.png?200 |}}
  * 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 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. 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.

/* Solution:
<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>

<code>
Algorithm(M) {
   if M accepts all words w in Sigma*
        accept
}
</code>

<code>
Algorithm(M1,M2) {
    if M1 always accepts then
       if M2 always accepts then 
          accept
}
</code>

**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