Could an analog computer model a non-computable function?

Question

By an analog computer (Wikipedia), I mean in particular a physical device that models some function by physical analogy (so, in this sense, an analog computer does not have to be fully continuous). For example: the differential analyzer (Wikipedia). Obviously, a Turing machine cannot be constructed to compute a non-computable function, but:

Question: Is it possible that for there to exist an analog computer that models a non-computable function, similar to the way the differential analyzer can be used to solve differential equations by integration? What would be an argument to the contrary?

Some reasons this might be not possible / the question is ill formed:

• If we assume that the input and output of a physical device are finite, then any function that such a device models is computable. (But if we can allow for countable inputs and outputs, then non-computability is possible)
• The type of error accumulation in analog computers might make the question ill formed?
• The "analog modelling" part is too vague / matter of (non-mathematical) interpretation? Likewise, possibility here can refer to physical or mathematical possibility...

I am asking out of curiosity, not because I need an answer one way or the other for any particular reason.

Maybe this question is more suitable to the computer science or physics SE websites.

Answer 1

This is, as far as I know, an open question in physics, although I believe the answer is expected to be "no." It's a form of the Church-Turing thesis and the answer is sensitive to questions like whether time travel is possible. For example, Aaronson, Bavarian, and Gueltrini show in Computability Theory of Closed Timelike Curves that closed timelike curves allow you to solve the halting problem (if you adopt a specific model due to Deutsch of how CTCs work based on fixed points).

Answer 2

The answer is no, unless some as of yet undiscovered principle in physics allows us to physically construct infinities or non-computability into our engineered systems. Robin Gandy (Alan Turing's Ph.D. student) wrote an article about this impossibility for analog computers specifically in 1993: https://arxiv.org/abs/2311.09239. The article itself was originally handwritten and never published before he passed away a couple years later (though circulated amongst a few logicians from whom I received a copy), so I typeset it and put my copy of it online.

For an extended discussion on more modern hypercomputer proposals and why they can't be physically realized, see for example my 2014 article published in Minds & Machines: https://link.springer.com/article/10.1007/s11023-013-9317-3, which is also on arXiv in case you are behind a pay wall: https://arxiv.org/abs/1210.3304.