Wwe assume that all languages are over input alphabet {0,1}. Also, we assume tha

Question

Wwe assume that all languages are over input alphabet {0,1}. Also, we assume that a Turing machine can have any fixed number of tapes.

Sometimes restricting what a Turing machine can do does not affect the class of languages that can be recognized --- the restricted Turing machines can still be designed to accept any recursively enumerable language. Other restrictions limit what languages the Turing machine can accept. For example, it might limit the languages to some subset of the recursive languages, which we know is smaller than the recursively enumerable languages. Here are some of the possible restrictions:

1. Limit the number of states the TM may have.

2. Limit the number of tape symbols the TM may have.

3. Limit the number of times any tape cell may change.

4. Limit the amount of tape the TM may use.

5. Limit the number of moves the TM may make.

6. Limit the way the tape heads may move.

Consider the effect of limitations of these types, perhaps in pairs. Then, from the list below, identify the combination of restrictions that allows the restricted form of Turing machine to accept all recursively enumerable languages.

a) Allow the TM to run for only n2 moves when the input is of length n.

b) Allow the TM to use only 2n tape cells when the input is of length n.

c) Allow a tape cell to change its symbol only once.

d) Allow the TM to run for only 2n moves when the input is of length n.

Can you please explain to me stepwise to learn tharougly. appreciate for the help

Explanation / Answer

All the problems are partitioned into 3 classes:

Turing machine can represent 1 and 2 from above.

Now take a look at the options:

At most n^2 moves: There are many problems which can be solved at a complexity higher than O(n^2). So this is clearly less powerful than a TM.

At most 2^n cells:The way I see this is that with there will be a predictable number of states possible in such a turing machine.If you run the halting problem on this turing machine, then once all the possible states are seen, we have successfully decided the halting problem. But halting problem is undecidable. Only its "YES" instances can be found out. So, unless there is infinite length of tape, the machine will never be as powerful as the TM.

Allow tape to change symbol only once: We can circumvent this restriction by copying the input symbols on the infinitely available tape and working on it. Just make two copies at start and when one copy is finised, make a new copy of the remaining copy. So, effectively we get same power as TM. So, this will accept any recursively enumerable language.

Allow only 2^n moves with the turing machine: There are problems whose "YES" instances can be found out (RE problems) which can't be represented by this machine. We can also apply the concept of halting problem on this machine. This machine will definitely halt in future because of limited number of moves. So, it will decide halting problem which it can't do if it is real TM.

So, answer is (c)

If you have any queries or have any other ideas about this. Let me know in comments. Thank you.

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