> Knowing that neural networks are theoretically Turing complete does not imply that the networks we train (ie the sets of weights we in fact encounter) have created Turing complete solutions.

• A computer algorithm is anything that runs over an automaton and taking some input encoding a question, gives us the answer.
• ML are systems where we create a model and adjust it through some configuration, until it will, given some input encoding a question, give us the answer.
• ML can only solve the problems its own system can solve. A turing complete ML system can solve anything a turing machine can.
• It stands to reason that some problems can only be truly solved through an algorithm (e.j. if the possible inputs are uncountable infinite).
• If we assume that an ML model can solve these problems, we have to assume that it can encode in its configuration algorithms, including some that we know. Otherwise we assume there's a limit.

Now I wouldn't take this to say that it would learn to be optimal. Say we trained an AI to sort lists, I could see it encoding a sorting algorithm within its network eventually, but I can't say if it'd ever discover an O(NlogN) algorithm, even if pressure was put to optimize the solution as well as being correct. But something that we can say is that neural networks may be able to do Markov Chain models internally, as its own sub-algorithm, if that's the way to solve the problem. But the assumption of this is why we think so much about neural networks nowadays.

That said the problem of sufficiently good learning is not trivial at all. And we certainly could discover its impossible to do. But at the moment, AFAIK, there's no reason not to think it can't happen.

The fact that we observed this happening is good, it basically validates the assumptions and models that we've had up to know, and implies that "sufficiently good learning" is attainable. There may still be limits (like finding the optimal algorithm, vs just an algorithm). So there's a lot of value in seeing it.

But to day-to-day applied ML research I am not sure if it really has that much of an impact, this lays ground work though.

The really interesting discovery here. More than the conclusion the interesting thing is how they reach it, the ability to reach it. As ML starts being used in more areas, we'd want to be able to audit an ML model and verify that it effectively has found a useful solution, and isn't just over-fitted beyond what we understand. Being able to identify algorithms within the system, and be able to split the AI model into simpler "steps" that do all the things, we'd be able to validate that it has found a good solution.

Again not something we need to solve now, but being able to know how to do it is a good thing to start doing already.

And on a more complex theme. This sets a better understanding of how ML models work, and in the process they can give us a hint of how intelligent systems in general work themselves, and we could then revisit that. This is like a longer-vision here. Being able to deconstruct models we may start seeing patterns and start forming more interesting math to describe intelligent systems in general. Which is where mapping it to organic models could allow proving strong AI, for example.