====== Lab 04 - Turing Machines =======

**Key concepts**

  - what is a **Turing-reduction**?


==== 1. Semi-decidable problems ====

**1.1** Prove that $math[R \subsetneq RE.]

**1.2** Show that the following problem is not in $math[R]: $math[f(M) = 1] iff exists $math[w] such that $math[M] accepts $math[w.]
  * This is actually the problem complement of establishing if $math[M] accepts the non-trivial problem $math[f(x) = 0.]

**1.3** Show that the problem in **1.2** is in $math[RE.]

**1.4** Show that the following problem is in $math[RE] but not in $math[R]: $math[f(M_1, M_2, w) = 1] iff $math[M_1] and $math[M_2] both accept $math[w.]

**1.5** Prove that $math[RE] is countable.

==== 2. Properties of Turing reductions ====

**2.1** Is $\leq_T$ an equivalence relation over $math[RE]?
  * $\leq_T$ is reflexive?
  * $\leq_T$ is transitive?
  * $\leq_T$ is symmetric? If $\leq_T$ would be symmetric, what would it signify?

**2.2** Suppose $math[f] $\leq_T$ $math[f_h.] What does that say about $math[f]?

**2.3** Show that $math[k]-$math[Vertex]-$math[Cover] $\leq_T$ $math[f_h.]

==== 3. Problems outside RE ====

**3.1** Show that the following problem is not in $math[R]: $math[f(M_1, M_2) = 1] iff $math[M_1] and $math[M_2] accept the same problem $math[g] (solve the same algorithm).

**3.2** Show that the following problem is not in $math[R]: $math[f(M) = 1] iff $math[M] accepts all words in $math[\Sigma^*] (does $math[M] accept the problem $math[g(x) = 1]?).

**3.3** Show that the problem in **3.1** is not in $math[RE.] Use the same strategy, but now choose a problem which is not in $math[RE] for the reduction. One candidate is the $math[looping] $math[problem] from the lecture.

**3.4** Show that the problem in **3.2** is not in $math[RE.]

**3.5** Show that the complement to the problem in **3.2** is not in $math[RE.]