# Differentiate between recognizable and decidable in the Turing machine?

When we talk about Turing machines (TM) it could accept the input, reject it or keep computing which is called loop.

Now a language is recognizable if and only if a Turing machine accepts the string, when the provided input lies in the language.

Also, a language can be recognizable if the TM either terminates and rejects the string or doesn't terminate at all. This means that the TM continues with the computing when the provided input doesn't lie in the language.

Whereas, the language is decidable if and only if there is a machine which accepts the string when the provided input lies in that language and rejects the string when provided input doesn't lie in that language.

Example

• A = {hM, wi | M is a DFA and w ∈ L(M)} is decidable.

• A = {hM, wi | M is a TM and w ∈ L(M)} is recognizable.

The major differences between a recognizable and a decidable in turning machine are as follows −

Sr. NoTuring RecognizableTuring decidable
1A language which is Turing Recognizable if there is a Machine that will halt and accept only the strings in that language and not in that language, then that TM either rejects, or does not halt at all.A language is said to be Decidable if there is a Machine that will accept strings in the language and reject strings not in the language.
2A Language is called Turing Recognizable if some Turing Machine recognizes it.A Language is called Turing Decidable if some Turing Machine decides it.
3If there exists a Turing Machine such that when encountering a string in that language, the machine terminates and accepts that string then we can say that type of language is a Turing recognizable.If there exists a Turing Machine such that when encountering a string in that language, the machine terminates and accepts that string then we say that type of language is Turing decidable.
4If there exists a Turing Machine such that when encountering a string not in that language, the machine either terminates and rejects that string or doesn’t terminate at all then we can say it is Turing-Recognizable.If there exists a Turing Machine such that when encountering a string not in that language, the machine terminates and rejects that string then we can say it is Turing-Decidable.
5It is not stronger condition than Turing DecidableIt is a stronger condition than Turing-Recognizable.

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