1. Consider the following language L2.
L2 = {< M,w,q > | M is a TM, q is a state, and M enters
state q during the computation on w}.
Use reduction to show that L2 is undecidable.
2. Consider the problem of determining if a Turing machine rejects at least two strings, i.e., rejects two or more strings. Formulate this problem as a language and show that it is undecidable.
3. (Exercise 5.16) Consider the problem of testing whether a two-tape Turing machine ever writes a nonblank symbol on its second tape. Formulate this problem as a language, and show that it is undecidable.
4. Consider the problem of determining whether ε ∈ L(M) where M is a Turing machine. Note that we are not trying to determine whether L(M) is empty, but rather whether M accepts the empty string.
a. Formulate this problem as a language.
b. Show that this problem is undecidable.
5. Suppose that we prove a language L to be undecidable.
a. Can L be both Turing recognizable and co-Turing-recognizable? Why?
b. Can L be both not Turing recognizable and not co-Turing-recognizable? Why?