the emptiness and equivalence problems for Turing machines

Beyond 'does this machine accept this input', there are bigger questions about a machine's entire behavior: does it accept ANY input at all (emptiness), do two machines accept exactly the same set of inputs (equivalence), is the set it accepts a regular language (regularity), and does it halt on EVERY input (totality)? Each sounds like the kind of thing a sufficiently clever analyzer ought to settle. Each is undecidable, and reduction is how we know.

Take emptiness, E_TM = {M : M accepts no string at all}. To show it is undecidable, reduce A_TM to it. Given (M, w), build a new machine M_w that ignores its own input, simulates M on the fixed string w, and accepts only if M accepts w. Then M_w accepts at least one string (in fact every string) exactly when M accepts w, and M_w accepts nothing exactly when M does not accept w. So an emptiness-decider applied to M_w would tell us whether M accepts w, deciding A_TM, which is impossible. Equivalence EQ_TM = {(M1, M2) : they accept the same language} is even easier: compare any M against a fixed machine M_reject that accepts nothing; then M is equivalent to M_reject if and only if M accepts nothing, so emptiness reduces to equivalence, and equivalence is undecidable too.

Regularity (is the recognized language regular?) and totality (does the machine halt on all inputs?) fall the same way, and Rice's theorem packages emptiness, regularity, and equivalence-to-a-fixed-machine as instances of one sweeping result: every nontrivial property of the language a Turing machine recognizes is undecidable. The honest boundary: these are properties of the LANGUAGE the machine recognizes, not of its source code. Asking whether M has exactly 7 states, or whether its tape alphabet contains a given symbol, is a syntactic property and is perfectly decidable. Undecidability bites only when you ask about behavior, the set of inputs ultimately accepted.