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 ====== 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