Show that the set of recursive languages is closed under reversal?

Consider a Language L, over an alphabet T is known as recursive enumerable if there exists a turing machine (TM) which generates a sequence of numbers T* which have precisely the members of L.

Whereas L is said to be recursive if there exists a Turing Machine enlisting all members of L and stopping on each member of L as the input.

Thus it is clear from the above statements that every recursive language is also recursively enumerable but the converse is not true.

The precise connection between families of languages is given below.

Theorem

• Step 1 − A language L is said to be recursive over an alphabet T if and only if both L and its complement L with respect to T* are recursively enumerable.

• Step 2 − The reversal of language L^(R) is set of all strings of elements of T* that we can get by writing all elements of L in reverse order.

• Step 3 − Let us consider a language L that is recursively enumerable whose all elements are enlisted by Turing machine M.

• Step 4 − We can now design another Turing machine M’ whose strings are accepted as the reverse strings of members of L.

• Step 5 − This is also in accordance with church-turing thesis that states that every effective computational procedure can also be made by some Turing machines

• Step 6 − Now from the above discussion it is clear that L is recursive enumerable than its reversal L^(R) is also recursively enumerable.

• Step 7 − If a string X of T* is in (L^(R))’ if and only if X does not L^(R) which clearly means if and only if the reversal of X^(R) does not belong to L, which further implies that X^(R) belongs to L’⇔ X belongs to l^(R).

• That is (L^(R))’ = L’^(R)

Conclusion

If L is recursive then L’(R) and its complement (L^(R))’ is also recursively enumerable which further means that the reversal of L^(R) is also recursive.