Recursive & Recursively Enumerable Languages in TOC

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Let us understand the concept of recursive language before learning about the recursively enumerable language in the theory of computation (TOC).

Recursive Language
A language L is recursive (decidable) if L is the set of strings accepted by some Turing Machine (TM) that halts on every input.

Example
When a Turing machine reaches a final state, it halts. We can also say that a Turing machine M halts when M reaches a state q and a current symbol ‘a’ to be scanned so that δ(q, a) is undefined.

There are TMs that never halt on some inputs in any one of these ways, So we make a distinction between the languages accepted by a TM that halts on all input strings and a TM that never halts on some input strings.

Recursive Enumerable Language
A language L is recursively enumerable if L is the set of strings accepted by some TM.
If L is a recursive enumerable language then −
If w ∈ L then a TM halts in a final state,
If w ∉ L then a TM halts in a non-final state or loops forever.
If L is a recursive language then −
If w ∈ L then a TM halts in a final state,
If w ∉ L then TM halts in a non-final state.

Recursive Languages are also recursive enumerable
Proof − If L is a recursive then there is TM which decides a member in language then −
M accepts x if x is in language L.
M rejects on x if x is not in language L.
According to the definition, M can recognize the strings in language that are accepted on those strings.

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