Problem 4

Give a clear description of a TM that recognizes the language A = {0^m#0ⁿ#0^m–n | m>n = 1}. In other words, a string belongs to A if and only if it consists of three #-delimited strings of 0s such that the number of 0s in third string is equal to the number of 0s in the first string minus the number of 0s in the second string. Note that the constraint m>n = 1 implies that all three strings of 0s are nonempty. Your description should be clear and detailed. You do not have to write a Turing machine program (e.g., like Figure 3.8 or Figure 3.10 of our text), but it should be possible to write such a program based on your description.

Problem 5

Recall the ‘emptiness testing’ language E_TM = {M> | M is a TM and L(M)=Æ}. Define a mapping reduction from ??_"# (the complement of E_TM) to the language B = {Q> | Q is a TM such that 011ÎL(Q)}. You can use the fact that ??_"# is Turing recognizable and can assume there exists a recognizer Rfor ??_"# in your construction (you don’t have to describe how R works).