We 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 n ² moves when the input is of length n.

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