Reference no: EM133035864

Question 1 Show that L(G₁) ∩ L(G₂) = Φ is co-CE but not computable (decidable). Here G₁ and G₂ are context-free grammars.

Question 2 For this question the alphabet is {a, b}. Suppose that the language L is CE but not computable; this means that L¯ cannot be CE. We define a new language as follows:

K= {aω|ω ∈ L} ∪ {bv|v ∈ L¯}.

1. Show that K is not computable.
2. Show that K is not CE.
3. Show that K is not co-CE.

Question 3 Suppose that M is a Turing machine and w is a word. Is the question "Does M ever use more than 330 cells on its tape while processing ω?" decidable or not. Prove your answer Hint: Rice's theorem will not help you.

Question 4

1. Here is a question on regular languages just to get you in shape for the final exam. Suppose that L is a regular language and to is any word, not necessarily in L. We define the set

L/ω = {x ∈ ∑*|xω ∈ L}.

Show that L/ω is regular.

2. Suppose that G is a context-free grammar. Show that the question "Is L(G) regular?" is undecidable. Here is a possible approach. Let N be some language that is known to be context-free but not regular (for example, {aⁿbⁿ|n ≥ 0}). Now consider the language L = (N#∑*) U (∑*#L(G)), where # is some symbol that is not in L(G) or N. Prove that L is always context-free but is regular if and only if L(G) = ∑*. This, by itself, does not complete the question, so you have to complete all the remaining steps as well as proving the claim. Also, you should think about why I put part (1) together with this question? You are free to ignore this hint, but if you do so, I will mark you just as rigourously as the people who used the hint.

Question 5 You are playing a video game under the following idealized conditions. The computer memory is unbounded and you have no time limit for finishing the game.

The game board is the set of points in the plane with integer coordinates and time moves in discrete integer steps. There is a hidden submarine: you do not not know its location, you do not know its speed and you do not know its direction of motion. The speed and direction of motion do not change throughout the game. The speed is a natural number and the direction of motion is either "up", "down", "left" or "right". For example, the submarine could start at (2,3) have speed 7 and move right. Then at step 0 it is at (2,3), at step 1 it is at (9,3), at step 2 it is at (16,3) and so on.

At every step you get to zap a point: you enter the coordinates and if the submarine is at that point, at that time step, you will destroy it. Of course, there is no point zapping a position before it gets there or after it leaves. Give a strategy or scheme that is guaranteed to get the submarine at some finite stage. I repeat, you do not know where it started, you do not not know its direction and you do not know its speed; you only know that the speed and direction do not change. You get zero points for this question if your strategy works probabilistically; this question has nothing to do with probability.

Hint: Ask yourself, "why is this on the homework for this class? It is a direct application of something I have discussed in class." Also, thinking geometrically will not help you.