Lab 04 - Turing Machines

Key concepts

1. what is a Turing-reduction?

1.1 Prove that $ R \subsetneq RE.$

1.2 Show that the following problem is not in $ R$ : $ f(M) = 1$ iff exists $ w$ such that $ M$ accepts $ w.$

1.3 Show that the problem in 1.2 is in $ RE.$

1.4 Show that the following problem is in $ RE$ but not in $ R$ : $ f(M1, M2, w) = 1$ iff $ M1$ and $ M2$ both accept $ w.$

1.5 Prove that $ RE$ is countable.

2.1 Answer the following questions and justify your answers:

• Is $ \leq$ _{$ T$} an equivalence relation over $ 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 $ f$ $ \leq$ _{$ T$} $ f$ _{$ h$} . What does that say about $ f$ ?

2.3 Show that $ k$ -$ Vertex$ -$ Cover$ $ \leq$ _{$ T$} $ f$ _{$ h$} .

3.1 Show that the following problem is not in $ R$ : $ f(M1, M2) = 1$ iff $ M1$ and $ M2$ accept the same problem $ g.$

3.2 Show that the following problem is not in $ R$ : $ f(M) = 1$ iff $ M$ accepts all words in $ \Sigma^*.$

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

3.4 Show that the problem in 3.2 is not in $ RE.$

3.5 Show that the complement to the problem in 3.2 is not in $ RE.$