> I think you missed the point I was making which is that you still have to know that you're proving the right thing.

Right now, it's a missing part in many theorem prover systems (but I didn't do exhaustive researching, so it's more my point of view) : a code to succinctly describe, or state, the problem you are trying to solve. For exemple, for a sorting problem, you would state the problem in English : after sorting, for any element E, the next element in the sequence should be lower.

> It's entirely possible to have a but in the proof itself, at which point your program is going to incorrect

The premise of theorem provers is that if the problem is correctly stated, then a proof of a solution passing the prover's compiler and a few human reviews is even more unlikely to have a bug.

> Reading and understanding the Idris proof is actually more work than understanding the untyped Python version, therefore it's actually harder to say whether it's correct or not in a semantic sense.

I would be curious to see a Python proof of the Python sort function. I mean an actual logical and mathematical proof, not a unit test or fuzzy test. It would imply to create a library with a DSL around math and logic.

“Saying it’s correct in the semantic sense” = proving the code.

They found a bug in Java’s binary search after 10 years of it going unnoticed, so I think you’re overestimating your own ability to prove code correct in your head.

Generative testing/spec is absolutely useful. It doesn’t subsume static typing and much less theorem proving though, and you can QuickCheck your pre and postconditions in static languages as well.

I don't find that to be a convincing argument in the slightest. The type system in Java also failed to catch the bug, and there's no guarantee that you wouldn't end up with an error in a specification using a more advanced type system that could actually encode that constraint.

The thing to realize is that theorem proving is not a business goal in most cases. What you care about is being able to deliver software that works well in a reasonable amount of time. Sure, you might end up with a binary search bug in your code that goes unnoticed for 10 years, but clearly the world didn't stop because of that, and people have successfully delivered many projects in Java despite that bug that work perfectly fine.

> The type system in Java also failed to catch the bug

Java is not a theorem prover, and your comment was about theorem provers.

> you wouldn't end up with an error in a specification

Again, completely orthogonal to your argument. If you have an error in specification tests won't help you because you'll test the wrong thing.

> The thing to realize is that theorem proving is not a business goal in most cases. What you care about is being able to deliver software that works

Again, your comment was about "saying things are correct in the semantic sense". You're saying now "I don't care about correctness in the semantic sense, I care about delivering software that "works" (whatever this means; clearly it doesn't mean software without bugs) in a reasonable amount of time" and that's fine. I do too, but again, nothing to do with your original argument.

>Java is not a theorem prover, and your comment was about theorem provers.

No, my comment was about formal methods in general. Every type system is a form of a machine assisted proof in practice. It's just that most type systems don't allow you to prove interesting things about your code.

The main problem is that of basic cost/benefit analysis. If it takes significantly more effort to write a full formal proof, and you end up sometimes catching minor errors, the effort is not justified in most scenarios.

>Again, completely orthogonal to your argument. If you have an error in specification tests won't help you because you'll test the wrong thing.

This is completely central to my argument. I'm saying that encoding a meaningful specification using a type system is a lot more difficult than doing that using runtime contracts, or even simply reasoning about the code unassisted.

I can read through the 5 lines of Python code implementing insertion code, and be reasonably sure that it's correct. I would have a much harder time verifying that 300 lines of Idris are specifying what was intended.

> You're saying now "I don't care about correctness in the semantic sense, I care about delivering software that "works" (whatever this means; clearly it doesn't mean software without bugs) in a reasonable amount of time" and that's fine. I do too, but again, nothing to do with your original argument.

I'm saying that you have diminishing returns. What you want to show is semantic correctness, and type systems are a poor tool for doing that. So, while you can use a type system to do that in principle, the effort is not justified vast majority of the time.

> I can read through the 5 lines of Python code implementing insertion code, and be reasonably sure that it's correct.

Binary search is possibly one of the simplest and most basic CS algorithms, and yet, it took people who were "reasonably sure" 10 years to find that bug.

> I would have a much harder time verifying that 300 lines of Idris are specifying what was intended.

Then don't use Idris?

> I'm saying that you have diminishing returns.

I agree. However, that was not your argument. Your argument was that tests subsume proofs, and that's obviously wrong.

Also, types vs tests is a false dichotomy. Personally, I find a strongly typed language + QuickCheck to be the most practical way of developing complex software. YMMV, and that's fine.

>Binary search is possibly one of the simplest and most basic CS algorithms, and yet, it took people who were "reasonably sure" 10 years to find that bug.

Again, saying there is a bug is not interesting. The question is how much this bug costs you, and how much time you're willing to invest in order to guarantee that you won't make that type of a bug.

>Then don't use Idris?

I think you entirely missed the point I was making here.

>I agree. However, that was not your argument. Your argument was that tests subsume proofs, and that's obviously wrong.

My point is that tests and runtime contracts provide sufficient guarantees in practice. Nowhere have I argued that they provide the same guarantees as proofs. The argument is that the cost of the proofs is much higher, and the proofs themselves can be harder to reason about making it harder to tell they're proving the right thing.

Consider the case of Fermat's last theorem as an example. It's pretty easy to state: a^n + b^n = c^n, and it's easy to test that this is the case for a given set of inputs that you might care about. However proving that to be the case for all possible inputs is a monumental task, and there are only a handful of people in the world who would even be able to follow the proof.

>Also, types vs tests is a false dichotomy. Personally, I find a strongly typed language + QuickCheck to be the most practical way of developing complex software. YMMV, and that's fine.

Again, that is not a dichotomy I was suggesting. My point was that I think runtime contracts are a more effective way to provide a semantic specification than a static type system. I also said that static typing restricts how you're able to express yourself, leading to code that's optimized for the benefit of the type checker as opposed to that of the human reader. I'm not sure how you got types vs tests from that.

> Reading and understanding the Idris proof is actually more work than understanding the untyped Python version, therefore it's actually harder to say whether it's correct or not in a semantic sense.

That was your original, nonsensical, point (emphasis mine).

> My point is that tests and runtime contracts provide sufficient guarantees in practice.

That's your revised point (which I don't care much about discussing), after moving the goalposts sufficiently.

> The thing to realize is that theorem proving is not a business goal in most cases. What you care about is being able to deliver software that works well in a reasonable amount of time. Sure, you might end up with a binary search bug in your code that goes unnoticed for 10 years, but clearly the world didn't stop because of that, and people have successfully delivered many projects in Java despite that bug that work perfectly fine.

I agree with you on that part. However, if the whole stack of computer technologies were more reliable, we would maybe think about new business usecases that we subconsciously dissmised because of lack of trust in current technology.

It's basic cost/benefit analysis. At some point you end up with diminishing returns on your effort that are simply not worth the investment.

Also worth noting that nobody has been able to show that static typing leads to more reliable code in practice https://danluu.com/empirical-pl/

Claiming that to be the case putting the cart before the horse. A scientific way to approach this would be to start by studying real world open source projects written in different languages. If we see empirical evidence that projects written in certain types of languages consistently perform better in a particular area, such as reduction in defects, we can then make a hypothesis as to why that is.

For example, if there was statistical evidence to indicate that using Haskell reduces defects, a hypothesis could be made that the the Haskell type system plays a role here. That hypothesis could then be further tested, and that would tell us whether it's correct or not.

This is pretty much the opposite of what happens in discussions about static typing however. People state that static typing has benefits and then try to fit the evidence to fit that claim.

I agree with you, there is no much data about the effectiveness of more rigorous software development tools. It's clearly a research topic.

My intuition is that, with advances in robotics and AI, we may see the need of more robust logical systems. At some point, mathematicians and algorithmicians may use those tools to prove new concepts, which they will be able to share and valid more quickly, which then will percolate into software engineering more quickly.

Beyond software engineering, the expirements with theorem provers may lead to new ways of exchanging information, such as news, mathematics and legal documents. There are inspirations to take from the formalizations created in theorem provers for applying automated reasoning in more domains than just programming (imho)

Edit : Intuitively, it has do to with building trust in the algorithms and informations we share at light speed. With more validated building blocks, we may explore more complex systems. Accelerating trust between different entities can only lead to plus value, I guess. That's all very abstract tho

Edit 2 : too put it in even fewer words, theorem provers may be more about collaboration than just pure technical engineering

To be honest, I think that once we have advances in AI there will come a point where you won't really be doing the kind of programming that we do today by hand. You'll have an AI assistant whom you'll give queries that are close to natural language, and it will figure out how to implement them for you. I can see that coming within a few decades for many kinds of applications such as your typical CRUD apps.

In such an hypothetical world, the "typical CRUD application" may just not even exist anymore.

I was talking about advances in consumer drones, autonomous cars, and personal robots, such as SpotMini from Boston Dynamics. More autonomous embedded systems evolving around us means different needs in term of safety in software development.

AI will have to explain the reasoning of their decisions (prove they did right) in natural language. The humans who do that in our world are scientifics, politicians, lawyers and mathematicians. Those people use a specific kind of natural language, with domain specific words to communicate. Theorem provers in software engineering are a step forward that direction imho

Sure, you wouldn't really interact with a computer the way we do now once you have AIs that can understand natural language. It would be more like having a personal secretary.

I don't think theorem provers are actually the way to get there though. AI systems are probabilistic in nature, and neural nets are self-organizing. One of the biggest problems is that it's really hard to tell how such a system arrives at a decision. The human brain itself is not based on formalism, and we find formal thinking to be very challenging. It's something that needs to be trained, and doesn't come to most people naturally. Our whole cognition is rooted in heuristics.

> One of the biggest problems is that it's really hard to tell how such a system arrives at a decision. The human brain itself is not based on formalism, and we find formal thinking to be very challenging

So far, "neural networks" in AI is a fancy name for what is nothing more than a giant equation system with many parameters. It's not even close to a biological, actual self-organizing, neural networks. It's closer to a weather model prediction.

The human brain is not based on formalisms, so let's create an AI that helps the human brain's weakness. Maybe we shouldn't try to replicate the human brain capacities, but rather create a new "form of life" complementing our biological skills.

So far, theorem provers, with expert systems, are the only works I'm aware of about systematically explaining reasoning and decisions.

Neural networks are graphs that evolve at runtime by balancing their weights based on reinforcement, and as far as I know there hasn't been much success in using formal methods for AI.

I do think theorem provers can be useful in certain contexts, and I can see AI using these tools to solve problems.

> Neural networks are graphs that evolve at runtime by balancing their weights based on reinforcement, and as far as I know there hasn't been much success in using formal methods for AI.

This is not correct in the current state of tech. Neural networks are parametrized equations systems. You train the parameters on a dataset in a training phase, then freeze the result, then distribute the model to devices. Once distributed, the "neural network" can't be modified, and stop to "learn" new cases.

Edit : I mean, you are not completely wrong, you described the training phase of the neural network. That's only half of the story tho

> Reading and understanding the Idris proof is actually more work than understanding the untyped Python version, therefore it's actually harder to say whether it's correct or not in a semantic sense.

But you don't have to write a proof to sort a list in Idris. So you're not being honest with your comparison.

Sure, but then you're not getting the formal guarantees. The whole argument here is regarding whether adding more formalism improves code quality. If you agree that less formalism is better in some cases, then it's just a matter of degrees.