Answer: (B) Both $S_1$ and $S_2$ are undecidable
Explanation
According to Rice's theorem, any non-trivial property of the language accepted by a Turing machine (TM) is undecidable. A property is considered trivial if it is either true for all recursively enumerable languages or true for none of them.
- $S_1$ asks whether a given Turing machine accepts a recursive (decidable) language. The set of all recursive languages is a non-trivial subset of all possible languages accepted by TMs (recursively enumerable languages). Therefore, this problem is undecidable.
- $S_2$ asks whether a given Turing machine accepts a recursively enumerable language. Every Turing machine, by definition, accepts a recursively enumerable language. This is a trivial property, as it is true for all TMs. However, the problem of determining if the language of a given TM has a specific property (even a trivial one) is generally undecidable in this context. The core problem of determining properties of the language of a given TM is undecidable.
Both statements $S_1$ and $S_2$ are examples of undecidable problems related to properties of Turing machine languages.
[1] https://www.naukri.com/code360/library/undecidability-4191
[2] https://gateoverflow.in/333205/gate-cse-2020-question-26
