Turing Machine Deciding Language L

The core assertion is that a Turing machine $M$ decides a specific language $L$, where $L$ is a subset of all possible binary strings ($L \subseteq \{0,1\}^*$). Understanding the definition of "decides" is key.

Definition of Deciding a Language

A Turing machine $M$ decides a language $L$ if it performs two specific actions for every possible input string $w \in \{0,1\}^*$:

• Acceptance: If the input string $w$ is in the language $L$ (i.e., $w \in L$), $M$ must halt and accept $w$.
• Rejection: If the input string $w$ is NOT in the language $L$ (i.e., $w \notin L$), $M$ must halt and reject $w$.

Crucially, a Turing machine that decides a language must always halt, regardless of the input. It provides a definite yes/no answer.

Analyzing the Options

Let's examine each option based on this definition:

• Option 1 describes a machine that always halts, but doesn't specify which language it recognizes. It could halt and reject everything.
• Option 2 only covers the case for strings within $L$. It doesn't define the machine's behavior for strings outside $L$.
• Option 3 only covers the case for strings outside $L$. It doesn't define the machine's behavior for strings within $L$.
• Option 4 correctly states that $M$ accepts strings in $L$ AND rejects strings not in $L$. This fulfills both required conditions for deciding $L$.

Conclusion

The statement that accurately and completely describes a Turing machine $M$ deciding a language $L$ is the one that encompasses both the acceptance of strings belonging to $L$ and the rejection of strings not belonging to $L$. This corresponds to Option 4.