Question

Turing machines - Why does Rice's Theorem not apply to a TM with a useless state?

Answer 1

Theorem (Rice’s Theorem): Let L be a language of the form
L = {hMi | L(M) has some property P}, , where
1. P is non-trivial, i.e. there exist at least one machine M such that hMi ∈ L, and
at least one machine M such that hMi ∈/ L.
2. P is indeed a property of the language of TMs, i.e. whenever L(M1) = L(M2),
we have hM1i ∈ L if and only if hM2i ∈ L.

limitation of generality we may assume that a Turing machine that recognizes the empty language does not have the property P. For if it does, just take the complement of P. The undecidability of that complement would immediately imply the undecidability of P.