>>12
No, in general to prove that two formal systems are equivalent, you must write simulator of one in another and vice-versa. For instance, is not enough to write a TM simulator in Lisp, you also are supposed to write a Lisp simulator in TM.
Of course these days you know that a lot of formalisms are equivalent to TM (so you have a notion of Turing-completeness in the first place), so if you are interested in equivalence of a couple of formalisms, you check both against TM, and it's usually obvious that your formalism is not more powerful than TM.
Nevertheless, the idea behind all these shortcuts is what I said: that Lisp is no more powerful than C++ because you can write a Lisp interpreter in C++ (which might internally interpret a TM which interprets Lisp, of course). Which means something not very pleasant to
>>7 ^^
Also, Turing tarpits are interesting only as long as you try to program them on their own terms, so to speak. As soon as you are, like, OK, fuck this shit, let's implement a Turing machine here, they stop being interesting.