Question

Let us consider the following three statements:

I. Recursively enumerable languages are those that can be accepted by a Turing machine;

II. Recursive languages are those that can be decided by a Turing machine;

III. A recursively enumerable language accepted by a Turing machine that halts is recursive.

Which of the following holds?

a.Only I;
b.Only II;
c.Only I and II;
d.Only II and III;
e. All I, II, and III.

e. All I, II, and III.

Explanation:

Recursive Enumerable (RE)

RE languages or type-0 languages are generated by type-0 grammars. An RE language can be accepted or recognized by Turing machine which means it will enter into final state for the strings of language and may or may not enter into rejecting state for the strings which are not part of the language. It means TM can loop forever for the strings which are not a part of the language. RE languages are also called as Turing recognizable languages.

A recursively enumerable language is a formal language for which there exists a Turing machine (or other computable function) that will halt and accept when presented with any string in the language as input but may either halt and reject or loop forever when presented with a string not in the language. Contrast this to recursive languages, which require that the Turing machine halts in all cases.

Recursive Language (REC)

A recursive language (subset of RE) can be decided by Turing machine which means it will enter into final state for the strings of language and rejecting state for the strings which are not part of the language. e.g.; L= {anbncn | n>=1} is recursive because we can construct a turing machine which will move to final state if the string is of the form anbncn else move to non-final state. So the TM will always halt in this case. REC languages are also called as Turing decidable languages.

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