Question
Show that the language {(M1,M2): M1 and M2 are Turing machines with L(M1) is subset of...
Show that the language {(M1,M2): M1 and M2 are Turing machines with L(M1) is subset of L(M2) is undecidable
Answers
According to Rice's Theorm, if you have a nontrivial set of languages, then the set of (codes of) TMs that accept these languages is not decidable. In this problem, however, we have pairs of languages. One trick is to specialize the problem to make it fit Rice's theorem by fixing M2. For example, fix some M2 such that L(M2) is empty. In this case, if SUBSETTM is decidable, then its subset with fixed M2 {<M1, M2> : M1 is a Turing machine and L(M1) is empty} is decidable as well. This, in turn, means that {<M> : M is a Turing machine and L(M1) is empty} is decidable.
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