✎ aa:lab:03 [books]

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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 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 (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.
    * V Undecidable example 1
    * V Hilbert undecidable
    * V Wang Tile
    * k-color
    * Linear Integer Programming

==== 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>