[TelmoMenezes]

> Any system that guarantees productivity isn't Turing complete. All useful algorithms are productive.

But it does not follow that all useful algorithms can be implemented on a language that "guarantees productivity". Turing proved that the halting problem cannot be solved on a Turing machine. This opens the possibility for the existence of programs that might be "productive" in your sense, but might also never stop, and it is not possible to know in advance.

> The only thing you can do with a non-productive algorithm is convert electricity to heat.

I am not convinced of this at all. Perhaps true if you are talking about banking systems or web applications, but probably not true if you are talking about AI. Intermediary states of an endless computation might be interesting. Maybe this is a way to obtain unbounded creativity. Maybe this is the way to build minds. We don't know enough.

A more general observation: I find that we live in an era that is too obsessed with productivity at the cost of fundamental research, dreaming and imagination. I am convinced that the latter mindset is the only one that can bring qualitative changes to our culture and civilisation, and I think that our long-term survival depends on such qualitative jumps.

Of course, I also understand that someone has to take care of the plumbing...

[lmm]

> Turing proved that the halting problem cannot be solved on a Turing machine. This opens the possibility for the existence of programs that might be "productive" in your sense, but might also never stop, and it is not possible to know in advance.

Maybe. I struggle to imagine a practical case where we want to run a program that we didn't and couldn't know whether it worked though.

> I am not convinced of this at all. Perhaps true if you are talking about banking systems or web applications, but probably not true if you are talking about AI. Intermediary states of an endless computation might be interesting. Maybe this is a way to obtain unbounded creativity. Maybe this is the way to build minds. We don't know enough.

This is ridiculous reasoning. "We don't understand X, we don't understand Y, therefore X might be related to Y."

> A more general observation: I find that we live in an era that is too obsessed with productivity at the cost of fundamental research, dreaming and imagination. I am convinced that the latter mindset is the only one that can bring qualitative changes to our culture and civilisation, and I think that our long-term survival depends on such qualitative jumps.

> Of course, I also understand that someone has to take care of the plumbing...

Choosing to look at non-halting programs rather than halting programs is like choosing to look at crystal energy instead of nuclear fusion. A certain amount of willingness to question baseline assumptions is valuable, but I think our long-term survival depends far more on being willing to acknowledge the fundamental results of the field and put in the hard engineering work necessary to achieve things under the constraints of reality, rather than trying to wave them away.

[TelmoMenezes]

> Maybe. I struggle to imagine a practical case where we want to run a program that we didn't and couldn't know whether it worked though.

The vast majority of the programs used in real life are not formally proven and are written in Turing complete languages. That is already the world you live in.

> This is ridiculous reasoning. "We don't understand X, we don't understand Y, therefore X might be related to Y."

I didn't say that.

Notice that a (very simplified) model of the human brain, the recurrent neural network, is already Turing complete. Notice also that humans (and Darwinian evolution, for that matter) display a capacity for creativity that has not been successfully replicated by AI efforts yet. Notice further that non-halting computations (e.g. infinitely zooming a Mandelbrot set) are the closest thing we have to unbounded creativity.

Maybe I'm wrong, of course.

[lmm]

> The vast majority of the programs used in real life are not formally proven and are written in Turing complete languages. That is already the world you live in.

They're unproven, not believed to be unprovable. (Indeed, almost all practical programs make at least some effort to offer evidence and informal arguments for their correctness, via tests, comments on unsafe constructs, and so on).

> Notice also that humans (and Darwinian evolution, for that matter) display a capacity for creativity that has not been successfully replicated by AI efforts yet. Notice further that non-halting computations (e.g. infinitely zooming a Mandelbrot set) are the closest thing we have to unbounded creativity.

This is meaningless woo.

[TelmoMenezes]

> They're unproven, not believed to be unprovable.

A decidable language (let's say Ld) is less powerful than a Turing-complete language (Lt). This means that there are some computations that can be expressed in Lt but not in Ld, and that there are some problems that can be solved in Lt but not in Ld. These are theorems of theoretical computer science. I don't believe anyone has a clear idea of the practical implications of this limitation, and how many of the algorithms in current use are only possible in Lt.

My guess is that the complexity of the software in use nowadays vastly exceeds our ability to do such an analysis, but maybe you know something I don't. Otherwise, while it is true that they are not believed to be unprovable, it is also true that they are not believed to be provable.

> This is meaningless woo.

I'm not sure I should reply to this, because it is just name calling, but for other people reading this (and you, if you're still interested): people who study artificial creativity and related fields such as Artificial Life take these ideas seriously and have interesting philosophical definitions and mathematical formalisms to address them. I have been to conferences sponsored by serious universities and other organisations such as ACM and IEEE where ideas such as the generative power of the Mandelbrot set are seriously discussed. There are several attempts to quantify creativity and to connect the idea of creativity with computer science.

It is important to not have a mind so open that the brain falls off, but I suggest that you may be going too far in the opposite direction.

[lmm]

> I don't believe anyone has a clear idea of the practical implications of this limitation, and how many of the algorithms in current use are only possible in Lt.

> My guess is that the complexity of the software in use nowadays vastly exceeds our ability to do such an analysis, but maybe you know something I don't. Otherwise, while it is true that they are not believed to be unprovable, it is also true that they are not believed to be provable.

Mathematicians and computer scientists deliberately seek out problems that are not in Ld, and have only found constructed examples, mostly minor variations on the same "diagonalization" argument. Any algorithm that is known to work is necessarily in Ld, and those constitute the overwhelming majority of algorithms that are published or used, for obvious reasons. (And those that are merely believed to work are, in the overwhelming majority of cases, believed to work for reasons that translate directly into a belief that they could be proven to work).

[TelmoMenezes]

> have only found constructed examples, mostly minor variations on the same "diagonalization" argument

Anything that is Turing-complete cannot be implemented in Ld (by definition). Off the top of my head, this includes: Recurrent Neural Networks, CSS, Minecraft, TrueType fonts, x86 emulators (MOV is Turing-complete) and Conways' Game of Life.

Of course you can argue that lots of things can be implemented in an Ld language. Sure, I have nothing against it, but it's not like desiring Turing-completeness is an absurd requirement.

> Any algorithm that is known to work is necessarily in Ld

No, most algorithms are known to work correctly for the common cases that are tested for + all the edges cases the developers can think of or encounter in real life. For non-trivial software, this is a minuscule subset of the possible states. Lots of things are surprisingly Turing-complete, and it is not trivial to prevent this for a sufficiently complex system.

[tom_mellior]

> > the existence of programs that might be "productive" in your sense, but might also never stop, and it is not possible to know in advance

> I struggle to imagine a practical case where we want to run a program that we didn't and couldn't know whether it worked though.

First-order theorem proving is a very practical case. Proof in first-order logic is "recursively enumerable", which means that we can write provers that, given a formula F, produce a proof of F in finite time if such a proof exists. But in general we don't know if a proof exists or how long it will take to find it. So if we start a prover, it will just sit there and not look "productive" until it either returns a proof or we kill it because we're tired of waiting.

[lmm]

But we wouldn't want to just leave such a theorem prover running indefinitely. Far more practical to have it check all possible proofs of length up to n (trivial to do in a total language), or at most run it for a particular number of days, months or years. At some point you do end the experiment.

[tom_mellior]

Sure. You usually run the prover with a timeout. The point is, this is a practical example of running a program where you cannot predict whether it will "work" or not. Additionally, when you reach the timeout, you will not have gained any information, i.e., the process is not "productive" in the sense discussed above.

Edit: Can't reply to your reply. Interesting. Yes, true, "there is no proof of length < N" is some information, but it's not information that tells you anything about the provability of your formula. It's not productive information. You can keep trying, but you won't be able to overturn the semidecidability of first-order logic in a Hacker News comment thread.

[lmm]

If the prover simply went into an infinite loop while checking the first proof candidate and never got as far as checking the second one, would it be working correctly? I would consider such a prover not useful, and a useful prover would be one that told me something when reaching the timeout, such as that there were no proofs for the given theorem shorter than a certain length.

[thamster]

This is handled in university CS algorithms courses. In such a case, dovetailing or interleaving the execution of the TM on the input would be used. Sipser pp. 150–152 https://courses.engr.illinois.edu/cs373/sp2009/lectures/old/...