Discussion
Which one of the following statements is equivalent to the following assertion?
Turing machine \(M\) decides the language \(L \subseteq \{0,1\}^{*}\)
Answer: D
The equivalent statement is M accepts all input strings in L and rejects all input strings in {0,1}* - L.
A Turing machine decides L when it halts on every input, accepting exactly the strings of L and rejecting all the others. Both halves are needed.
'Halts on all input strings' says nothing about which strings are accepted, so M could decide a different language. 'Accepts all input strings in L' is only recognition: M might accept other strings too, or loop forever on strings outside L. 'Rejects all strings outside L' is the mirror image and says nothing about the strings in L.
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