[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.

[lmm]

> 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.

You're begging the question - your Turing-complete algorithm is "evaluate an expression in a Turing-complete language". It's easy to make a language accidentally Turing-complete (especially when you're thinking in a Turing-complete language), but that completeness is undesirable, and in realistic use cases it's harmful rather than helpful. No-one wants to sit waiting indefinitely to see whether a web page or font is actually going to render or not (and indeed we often end up going to great lengths to make these things Turing-incomplete in practice with timeouts and the like).

> 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.

You're not contradicting me, you're saying "most algorithms are not known to work". I don't think that's true in the sense of "algorithms" published in journals/textbooks. I would agree that most code isn't known to work, but that translates into reality: most code doesn't work, most programs have cases where they just break and also just crash every so often.

> Lots of things are surprisingly Turing-complete, and it is not trivial to prevent this for a sufficiently complex system.

For a complex system written from scratch, it's easy enough with the right tools. If you build it in a non-Turing-complete language it won't be Turing-complete, and if something is hard to do in a total way it's probably a bad idea.

Porting an existing system would be much harder, I'll agree, and porting the existing code/protocol ecosystem would be a huge ask. (I do think it's necessary though; the impact of malware attacks gets worse every day, the level of bugginess we're used to seeing in software is rapidly ceasing to be good enough).

[pron]

This entire discussion about Turing completeness is completely irrelevant. The problem that's relevant to software verification isn't the halting problem in its original formulation, but a simple corollary of it, often called "bounded halting" (which serves as the core of the time hierarchy theorem, possibly the most important theorem in computer science), which states that you cannot know whether a program halts within n steps in under n steps. The implication is that there can exist no general algorithm for checking a universal property (e.g. the program never crashes on any input) that is more efficient than running the program on every possible input until termination.

But bounded halting holds not only for Turing complete languages, but for total languages, too (using the same proof). In fact, it holds even for finite state machines (with a different proof). This is why even the verification of finite state machines is generally infeasible (or very expensive under restricted conditions) both in theory and in practice.

[lmm]

To my mind all that says is that even general total languages or finite state machines allow expressing too much. Surely one can solve this by working in a language where functions are not merely total but come with explicit bounds on their runtime. This seems like a good fit for a stratified language design like Noether - we might have to resort to including a few not-provably-bounded total functions (or even not-provably-total functions) in a practical program, but hopefully would be able to minimize this and make the parts where we were making unproven assumptions very explicit.