Edit this page Backlinks This page is read only. You can view the source, but not change it. Ask your administrator if you think this is wrong. ====== 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(M1, M2, w) = 1] iff $math[M1] and $math[M2] both accept $math[w.] **1.5** Prove that $math[RE] is countable. ==== 2. Properties of Turing reductions ==== **2.1** Answer the following questions and justify your answers: * Is $math[\leq]<sub>$math[T]</sub> an equivalence relation over $math[RE]? * $math[\leq]<sub>$math[T]</sub> is reflexive? * $math[\leq]<sub>$math[T]</sub> is transitive? * $math[\leq]<sub>$math[T]</sub> is symmetric? If $math[\leq]<sub>$math[T]</sub> would be symmetric, what would it signify? **2.2** Suppose $math[f] $math[\leq]<sub>$math[T]</sub> $math[f]<sub>$math[h]</sub>. What does that say about $math[f]? **2.3** Show that $math[k]-$math[Vertex]-$math[Cover] $math[\leq]<sub>$math[T]</sub> $math[f]<sub>$math[h]</sub>. ==== 3. Problems outside RE ==== **3.1** Show that the following problem is not in $math[R]: $math[f(M1, M2) = 1] iff $math[M1] and $math[M2] 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.]