[pron]

> Depends on the area the engineer works in.

Sure, but let's say that it's a conservative estimate that 99% don't need proofs. I acknowledged in the beginning that deductive proofs have a place in some niches.

> You can perfectly formalize this model using dependent types and use it there, don't you?

The logic is already a formalization of the model. Yet finding a counterexample in the model directly is very often easier than finding a deductive proof in the logic. Some people got a Turing award for that discovery.

> Doesn't mean we need to willfully embrace blind faith in incomplete induction.

Formal methods is not "blind faith". Second, we haven't willfully given up on deductive proofs; it's just that for decades we've tried to scale them and couldn't, so out of necessity we found methods that have worked better in practice. I don't know if 200 years from now people won't be able to make deductive proofs feasible, but why not use formal methods to make our software better and cheaper now?

In any event, all I want is for people to know (what formal methods research and practice already does) that software correctness is an extremely complex topic, with severe challenges that are both theoretical and practical, and anyone who suggests that there is a known specific solution to the problem or that one of the many imperfect avenues we're exploring is "the future", doesn't know what they're talking about.

[permeakra]

>Sure, but let's say that it's a conservative estimate that 99% don't need proofs

99% don't need formal verification, they are OK with unit tests. If one is actually that concerned with an effort to perform actual verification, they most likely want an actual proof.

>The logic is already a formalization of the model.

So?

>it's just that for decades we've tried to scale them and couldn't,

For decades people didn't care for formal verification. Absolute majority of 'engineers' don't see any use in it. You, on the other hand, see little use for formal verification with dependent types. Well, forgive me for drawing a parallel here.

I want some specific guarantees that are fairly easy to express and keep track with dependent types, not some fancy offshot of Hoare logic. Why would I care for a method clearly inferior for my use cases?

==

Let's return a bit and examine one specific case again. Consider situation: one writes a generic merge-join function over sorted arrays. There, input arrays must sorted with same predicate over same, possibly virtual, key.

To express this in specification on the input data one need to

- quantify over the types of both arrays

- quantify over the type of the key

- and all the functions able to produce a key of this type from elements of array 1

- and all the functions able to produce a key of this type from elements of array 2

- quantify over the ordering predicate over values of the key type

- express that ordering holds for each pair of elements of the array -- i.e. quantify over indicies of arrays 1 and 2.

- For which one needs information on index bonds attached to arrays

I can, with some effort, express it in the language of type theory. I'm very interested how one can express it using, say, first order logic that, you know, doesn't allow for quantification over predicates.

[pron]

> 99% don't need formal verification, they are OK with unit tests.

Formal verification can be cheaper than tests. It's not just about a higher confidence, but also about achieving similar confidence more cheaply. That's one of the reasons why deductive proofs are not so popular in formal methods research -- they're too inflexible.

> If one is actually that concerned with an effort to perform actual verification, they most likely want an actual proof.

But we don't need to hypothesize -- they don't. Almost all formal verification in industry uses model checkers and sound static analysis. If you are interested in addressing the needs of certain small niches, that's one thing, but the thrust of formal methods research is about helping as much software as possible, and so deductive proofs are not the focus, because they're well behind other methods.

> So?

So, if you have a model checker, you don't need to write a proof. The model-checker could automatically provide a model-theoretical proof for you, at the press of a button. That's how, say, avionics software is verified. That's how Intel's chips are verified. People can't afford to write proofs for that volume of software. People write proofs only when model checking fails, or to tie together model checking results. It's no one's first choice (unless they're researchers or hobbyists).

> For decades people didn't care for formal verification.

It doesn't seem like you're familiar with the field. There was a huge boost in formal methods in the 70s followed by a research winter when people over promised and underdelivered. Then, about 20 years ago, there was a resurgence with different goals. In general, research now focuses on managing cost vs. benefit, reducing certain kinds of bugs, and finding ways to make verifiation more flexible, as we've so far been unable to make end-to-end verification scale.

> Why would I care for a method clearly inferior for my use cases?

I don't understand why you think it's inferior. It is absolutely not, and it is superior where it matters most. Expressiveness is the same, so specification is the same, but verification is flexible. You can use deductive proofs, or verifiation methods that have shown more promise.

> I'm very interested how one can express it using, say, first order logic that, you know, doesn't allow for quantification over predicates.

What? Of course FOL allows quantification over predicates and functions if they are elements of the domain, as they are in say, set theory (in fact, there is an assumption at the base of modern logic, that any mathematical theory could be expressed in some first-order language). Or, you could use HOL, if you like. But again, the theory is not the problem here. It's the chosen verification tool (and BTW, formal methods practitioners normally prefer to weaken the specification power if it reduces the cost of verification). Your example, in particular, is easily expressed in FO set theory, or many kinds of many-sorted FOL, or in HOL. For example,

    ∀ K, V . ∀ f ∈ K → V . ...

is a trivial, perfectly normal, first-order logic over sets. In fact, dependent type theories like Lean's strive to be interpretable in set theory. But really, the choice theory is the least important issue with dependent types or software verification in general.

If you are, however, interested in theory (as I am), you can read my series on TLA+. It is a first-order logic that is strictly richer and more expressive than lambda-calculus-based dependent type theories (unless you deeply embed it in them, and describe your computations in monads): https://pron.github.io/tlaplus

[permeakra]

>Almost all formal verification in industry uses model checkers and sound static analysis.

But does it happen because it is the only option or the best option? Also, encoding semantics specs is hard, so proper verification of algorithmic side is usually not an option anyway.

>I don't understand why you think it's inferior.

Because it absolutely is. It forces me to use an additional tool when I don't really need to with good enough type system.

Ugh. Look, example: memory safety. For quite some time memory leak needed to be tested for. We have a set of tools for that. Than memory control discipline in C++ arise. And then we have linear types in Rust. With Rust we don't need a separate checker: everything needed to ensure memory safety is embedded into the language type system.

Similarly, with linear algebra basic constrains can be expressed in current GHC type system. However, promoting input values to type level is awkward at best, so Haskell needs a small advancement here. Again, I don't need separate model checker here, moderately advance GHC type system works just fine.

> Of course FOL allows quantification over predicates

Wiki disagrees with you. Oh, well.

>If you are, however, interested in theory (as I am), you can read my series on TLA+.

As I said, at best I might be interested in HOL Isabelle. And even that is doubtful, so far Agda and Coq appear more useful for what I'm interested in.

[pron]

> But does it happen because it is the only option or the best option?

I don't understand the difference. Model-checking scales better than deductive proofs and is far cheaper. If you have less than 5 years to write a 10 KLOC program or your program is larger, then deductive proofs won't work. Does that mean it's better or that it's the only choice?

> Also, encoding semantics specs is hard, so proper verification of algorithmic side is usually not an option anyway.

This is true for all formal methods, but here, too, most of them beat deductive proofs. With deductive proofs you have to annotate every unit (or work much harder). Model-checking (and concolic tests) can handle partial specifications. This is yet another reason why formal methods research is not focusing its efforts on deductive proofs.

> Again, I don't need separate model checker here, moderately advance GHC type system works just fine.

It doesn't, though. Memory safety is a compositional property, as is every kind of property that can be easily checked by deductive proofs (or almost compositional, requiring one or two inferences). Most correctness properties are not compositional. That deductive proofs (of propositions that aren't compositional) don't scale is not some hypothesis of mine. They've been tried for years, and they just don't. The biggest programs ever verified with them are ~10KLOC and they took years of expert work.

> Wiki disagrees with you. Oh, well.

No, it doesn't. But if you think it does, it means you don't yet know what signatures, theories, structures, interpretations and models are -- the very basics of all formal logics -- so explaining that might take a while (in short, if propositions and functions are part of the structure, of course FOL can quantify over them; if it couldn't, it wouldn't have been the lingua franca of mathematics and the crowning achievement of formal logic). You can read my very short introduction to formal logic here: https://pron.github.io/posts/tlaplus_part2#what-is-a-logic . It's relevant to any formal logic you may want to learn at some point, as they're all more similar to one another than differen, but it may be too terse if you don't have any prior background.

> As I said, at best I might be interested in HOL Isabelle. And even that is doubtful, so far Agda and Coq appear more useful for what I'm interested in.

Sure. In general, Lean/Coq/HOL are more geared toward research -- inventing new logics, formalizing "high" mathematics -- and TLA+ is more geared toward industry use and verification of software and hardware, and because that's what I need, that's the one I use most. But it really doesn't matter which you learn -- they're all much more similar to one another than different. Their most important differences are in user-friendliness and their tooling. I would recommend Lean over Coq and Agda, though; I think it subsumes both, and its documentation and tutorials are better, so it's easier to learn. I enjoyed TLA+ and Lean more than Coq and HOL, and I haven't tried Agda. But this could be a matter of personal preference. Again, they're all more similar than different.

[permeakra]

>It doesn't, though.

read this: https://serokell.io/blog/dimensions-and-haskell-introduction , then look into the referenced libraries.

Have fun arguing with objective reality.

As for resource safety, Haskell is yet to follow Rust and introduce linear types, but many cases are covered by bracket pattern https://wiki.haskell.org/Bracket_pattern which can be enforced on type level using some tricks with ST monad. BTW, it is very similar to RAI discipline suggested by many C++ guidelines, but here it can be enforced.

Again, have fun arguing with objective reality.

>This is true for all formal methods, but here, too, most of them beat deductive proofs.

Sure, feel free to feel superior about that. The problem is, again, type system can be embedded into programming language and allow for quite a bit of guarantees even without full blown dependent types. Java has static typing for a reason.

Sure, to enforce some properties using type system we need an uncomfortably deep delve into type theory or use awkwardly polymorphic code. However, a lot of properties can be enforced with relatively low cost, and Haskell is a long-running research project on what properties are practical to enforce this way.

So, why I need to use proof assistants for properties that can be enforced via practical type system? I really don't see a reason.

[pron]

> read this: https://serokell.io/blog/dimensions-and-haskell-introduction , then look into the referenced libraries.

Dude, I actually work with dependent type proof assistants (Lean), with a HOL proof assistants (Why3), and FOL/ZFC proof assistant (TLA+'s TLAPS), as well as with a model checker for TLA+ (TLC); I've also worked with a model checker for Java in the past. I have read several books on the subject, and have written two (both available online: https://pron.github.io/tlaplus, and https://pron.github.io/computation-logic-algebra). I've also trained engineers in formal methods.

> Have fun arguing with objective reality.

You've read a few blog posts on one aspect of one formal method, and seem to have not even heard of the majority of software correctness research. You certainly don't seem to have practiced formal methods in industry even using one method, let alone more than one. You don't even seem to know what first-order logic is, what higher-order logic is, and I'm sure that if I asked you the difference between HOL and Martin-Löf type theory, or between classical and constructive mathematics you won't even know what I'm talking about; ditto if I asked you what it means for a type theory to have a set-theoretical interpretation (as Lean's, and maybe Agda's -- I don't know -- type theories do). You've read a few blog posts about a small part of one of the most complex, refractory topics in all of computer science and software engineering. You don't know what the objective reality is.

That you can prove some stuff with dependent types has been true for at least 30 years now. That other approaches, of which you seem to know nothing, scale much better in practice to the point that industry rarely uses deductive proof for anything but small, niche problems and uses model checkers all the time to verify some of the software your life depends on is just a fact. If you don't know what the alternatives are, how can you argue with what I'm saying?

> Sure, feel free to feel superior about that.

I don't feel superior. It seems that, unlike you, I actually use deductive proofs for verification, as well as model checkers, because, unlike you, I actually practice formal verification. But, knowing the tools, I know what works and when. I'm not researching one approach or another; I use which one currently works best, and the reality is pretty clear.

> The problem is, again, type system can be embedded into programming language

Contract systems are also embedded into programming languages. You seem to know one thing (know of, seems more accurate) and argue it's the best, when you have no idea what other things are out there.

> and allow for quite a bit of guarantees even without full blown dependent types. Java has static typing for a reason.

Sure, that because static types are awesome! There are some things -- simple, compositional properties -- for which nothing else works better. Type systems beat the alternatives for these simple properties and, at least currently, lose to the alternatives when deeper properties are involved. Correctness is complicated, and different properties require different tools. That is why, when deep properties are involved, I prefer separating the specification method from the verifiation method. Contract systems are embedded into programming languages and work alongside simple types. They then allow you to choose the best verification tool for the job: you can use deductive proofs, like depdendent types do, and other verification methods when those work better (which is the majority of cases). Dependent types tie you down to the verification method that is the most restrictive and least scalable.

> So, why I need to use proof assistants for properties that can be enforced via practical type system? I really don't see a reason.

You don't. The two are equivalent. I.e., they are equally the least scalable verification method we know. But contract systems, unlike depdent types, allow you to choose: you can use deductive proofs or other, cheaper, more scalable approaches. You need them because deductive proofs -- the only verification approach that dependent types give you -- just don't scale. But I don't need to convince you. I encourage you to learn Idris/HOL/Lean/TLA+/Coq -- whichever one you like (better yet, learn more than one) -- and try writing deductive proofs for large programs. Try different kinds of programs, too: sequential, interactive, concurrent and distributed. Knock yourself out. Perhaps when you speak from experience you'll have a better grasp of what objective reality is. In the process, you will learn a lot, so I truly wish you the best of luck!

But take one piece of advice: don't become dogmatic about something until you've actually looked around. If you argue that X is better than Y and Z because X is all you know, you just sound like a cultist. Learn first, argue later.

[dwohnitmok]

Wow. Did not anticipate such a long reply thread. Man it looks like I really should write up a comparison of my (amateur) experiences with TLA+ (TLC + TLAPS), Dafny, Idris, and Liquid Haskell.

Also permeakra, regardless of whether you agree with the points that pron is making here, if you ever find yourself interested in TLA+, I would highly recommend his four part TLA+ series (potentially as an intermediate step after e.g. Lamport's video course). It's absolutely fabulous.

[permeakra]

> if you ever find yourself interested in TLA+

Unlikely. Though I might be interested in HOL Isabelle eventually.