# ODEs and Zeno Machines

Disclaimer: I know nothing about differential equations, and I don't know if this belongs here...

Wikipedia states that a Zeno machine is a hypothetical machine able to compute infinite steps in finite time, by making use of supertasks. Building such a machine is physically impossible. My question is this: Is the Zeno machine an example of an differential equation and if so, is my assumption correct that we can utilize a differential equation solver to solve the Zeno machine differential.equation, like in this paper, which shows a different but fundamentally the same concept?

• Here is a differential equation: $y' = y$. A Zeno machine is some type of machine. I don't see too much similarity. – Yuval Filmus May 13 '20 at 7:24

No, they are not the same.

There are different equivalent representations for any problem (e.g. Turing machines, neural networks, differential equations, ...). But changing from one representation to another does not change whether the problem is fundamentally solvable - otherwise they would not constitute equivalent representations.

Zeno machines can be considered as Turing machines plus hypercomputation which, if it were possible, would permit solving certain previously undecidable problems.

Differential equations can be seen as extension of algebraic equations (or, conversely: an algebraic equation is a differential equation where no derivatives appear).

However, those two types of extensions are independent of each other:

1. There exist differential equations which are solvable by Turing machines.
2. There exist differential equations which are not solvable by Zeno machines.

If what made Zeno machines different from Turing machines was that only Zeno machines could represent differential equations, then there could be no Turing machines which can solve for differential equations.

But if you search for "differential equation solver" you get a lot of results: there exist programs which can solve certain differential equations. So a Zeno machine is not necessary for solving differential equations. Therefore Zeno machines and differential equations cannot be the same.

You can interpret the computations of a Zeno machine as a differential equation only insofar as you can already interpret the computations of a Turing machine as a differential equation. So neither do differential equations tell you anything about Zeno machines nor do Zeno machines tell you anything about differential equations.

• I think you might misinterpret my question. Again, I don't understand differential equations, but it seems as if you could interpret the result of the computations of a Zeno machine as a differential equation to solve..,. Correct me if I'm wrong. – ZeroMaxinumXZ May 13 '20 at 8:43
• @ZeroMaxinumXZ No, I interpret it correctly. You can interpret the computations of a Zeno machine as a differential equation only insofar as you can already interpret the computations of a Turing machine as a differential equation. So neither do differential equations tell you anything about Zeno machines nor do Zeno machines tell you anything about differential equations. – user66554 May 13 '20 at 8:58