Question

Which one of the following statements is equivalent to the following assertion?
Turing machine 𝑀 decides the language 𝐿⊆{0,1}∗

• A. Turing machine 𝑀 halts on all input strings in {0,1}∗
• B. Turing machine 𝑀 accepts all input strings in 𝐿
• C. Turing machine 𝑀 rejects all input strings in {0,1}∗−𝐿
• D. Turing machine 𝑀 accepts all input strings in 𝐿 and rejects all input strings in {0,1}∗−𝐿

Hint:

A Turing machine 'decides' a language only if it halts on every input and its accept or reject outcome exactly matches membership in L.

Answer 1

The Correct Option is D

Step 1: By definition, a Turing machine M decides a language \(L \subseteq \{0,1\}^*\) if M halts on every input string and its halting outcome correctly reflects membership in L.
Step 2: This requires two things simultaneously: M must accept every string that belongs to L, and M must reject every string that does not belong to L, that is every string in \(\{0,1\}^* - L\).
Step 3: Option (A) only states that M halts on all strings, but halting alone says nothing about whether the classification is correct, so it is not equivalent by itself. Option (B) only talks about acceptance on L and ignores what happens on the complement. Option (C) only talks about rejection on the complement and ignores what happens on L itself.
Step 4: Only the combined statement that M accepts all strings in L and rejects all strings in \(\{0,1\}^* - L\) captures both correctness in classification and, as a consequence, guarantees halting on every input, since every string is either in L or in its complement.
Final Answer: Option (D): Turing machine M accepts all input strings in L and rejects all input strings in \(\{0,1\}^* - L\)