Not Turing Equivalent

From NonTuringComputing:

If you can accurately model something on a TuringMachine, it cannot be more complex than TuringComplete. An analog neural net is a superset of TuringComplete, for instance. Although you cannot represent the weights with infinite precision on a Turing machine, you can represent a Turing machine with an analog neural net. -- SunirShah

How tied together are the phrases "not on a Turing Machine" and "not computable"?

Using the definition of computable as "That which halts on a TuringMachine" is arguing in circles of course.

[ The "Church-Turing thesis" is that anything that is computable can be computed on a Turing machine (or mechanisms established to be of comparable power, such as Lambda Calculus or combinatorial logic). This is a "thesis", not a theorem, because in fact we do not have firm idea of what "computable" means apart from mechanisms like Turing machines! This is part of the millennia-old quest for making our intuitions about math more rigorous. We have intuitions about what "compute" and "calculate" mean. So we try to construct formal systems that satisfy those intuitions. Turing machines are one of those successful ventures. Then we turn around and ask, "but does that leave out anything?", and of course the answer is somewhat circular. ]

[ However it is possible that we might someday discover a better answer to the question, "what do we really mean by "computable"? In which case we might be able to turn the Church-Turing thesis into a theorem. But this isn't a burning concern because it's entirely a question of the definition of the word "computable", and of course word definitions are extraneous to mathematical proof; modern recursive function theory tells us a lot about the mathematics of the subject. -- DougMerritt ]

A NondeterministicTuringMachine? is more powerful than a TuringMachine, I believe I've read. And an I/O Machine is more powerful still, according to Wegner. So Turing is not the tops. This leads to NonTuringComputing.

Uh, no; a NondeterministicTuringMachine? is not more powerful than a TuringMachine. It's faster--it can solve NpComplete problems in polynomial time (duh!), but it can't compute anything in finite time which requires unbounded time on a TuringMachine

On the other hand, a NonDeterministicPushdownAutomaton? is more powerful than a DeterministicPushdownAutomaton?. A NonDeterministicFiniteAutomaton? is not more powerful than a DeterministicFiniteAutomaton?.

[ There is a hierarchy that is equivalent to Chomsky's hierarchy of grammars. Finite automata are equivalent to regular expressions. They can't do all that much. Add one or two pushdown lists (stacks) or even a counter or two, and they are formally more powerful. The full power of Turing machines is attainable only if infinite loops are possible, as is obvious if you consider Goedel's theorems. The less powerful machines are incapable of simulating the more powerful machines, and that's what allows strict levels of the hierarchy to be drawn. ]

A Turing Machine can emulate an analog neural net to any desired level of accuracy. -- DaveHarris

Is that not just only in the same way that we can forecast the weather to any desired level of accuracy? -- BillWeston