I know for example finding integers where their modulus n is k maps nicely to finite state machines and push down automata's work well for parsing deterministic grammars. I was wondering if there are any problems that are like that for turing machines.
Are there any problems that are easier to implement on a turing machine than a conventional computer?
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As mentioned by tia, these kinds of questions are better for cs.stackexchange.com. I would say that the question is not well stated, since conventional computers are basically a type of Turing machine. It depends on what kind of Turing machine you are working with. For example, many problems can be solved much faster on a non-deterministic Turing machine than on a classical computer (which is deterministic).