[kazinator]

> It boggles my mind why more people are not involved in this direction.

Because only a vanishingly rare amount of the incorrectness in compiled C programs comes from a bug in the compiler.

Also, some code coaxes a desired behavior out of the object code while bending the rules of the language.

The compiler being proven over correct code (i.e. free of any undefined behaviors) is useless in that situation, unless the prover actually works with an extended language definition (specific to that compiler) which defines those behaviors.

[110011]

> Because only a vanishingly rare amount of the incorrectness in compiled C programs comes from a bug in the compiler.

True.

What I meant to say, but didn't, is, why aren't more people working on ensuring that this provability bubbles up from the compiler to the program itself. I come across a lot of work in specialized areas on this topic but haven't seen anyone have a go integrating formal proofs and a widely used programming language. At the moment this looks like a pipe dream because of the obvious objection (and I hope this disappears "soon") that currently it is too cumbersome to have to prove your programs, let alone write them in the first place.

[seanmcdirmid]

The biggest problem in formal program verification is the lack of a spec to verify against for most programs. There are projects like ur web that define correctness more broadly to get around that.

[pron]

That's certainly a problem, but it's not the biggest problem. Writing a formal spec is no harder than writing tests (and arguably easier). It doesn't have to be perfect -- the more correctness conditions the better. Of course, a complete spec -- one that says, anything that satisfies this is a correct implementation of the system -- is best, but it's really not required to start getting real benefits.

I'd say that the biggest problem is the intractability of verification. Xavier Leroy has said that his style of verification -- end-to-end (i.e., global correctness conditions verified all the way down to source- and machine-code) and employing formal proofs -- is only suitable to the smallest and simplest programs (he claims it's suitable to programs hundreds to thousands of lines long, and even he had to take a few shortcuts in CompCert, e.g., he eschewed some termination proofs). Model-checking fares only a little better, and static analysis -- which scales to arbitrarily-sized programs -- can only verify specific, and mostly local correctness properties. Other approaches can tractably verify global correctness conditions, but only against high-level descriptions of the algorithm/system, and don't reach all the way down to code.

AFAIK, the most robust, and still scalable, formal verification employed in real-world programs of considerable size is that employed by Altran UK. They verify global properties against a high-level spec (verified mostly with model-checking, and possibly some proofs), plus local properties at the code level (verified mostly automatically with SMT solvers with a small number of manual proofs), but leaving a gap in between. This could very well become easier and scale well, and could become affordable (as it automates relatively well) yet still extremely useful, as end-to-end verification is hardly ever required. They're using Z for the high-level spec and Ada SPARK for the code-level spec, but other tools could be used (e.g. TLA+/Alloy for the high level, and JML/ACSL/Clojure Spec/Spec# for the code level).

I've personally successfully used refinement mappings in TLA+, which allow you to move between different levels of the system description, with the different specs as well as the refinement mapping itself checked with a model checker. This is indeed harder than just checking a single description, but still in the realm of affordability. Taking refinements all the way down to the source level is still far from affordable, though.

[neel_k]

> [...] even he had to take a few shortcuts in CompCert, e.g., he eschewed some termination proofs.

This turned out to be a premature optimization, actually! The CakeML compiler did full functional correctness proofs for their register allocator and report that it wasn't significantly harder than Compcert's translation validation.

> Writing a formal spec is no harder than writing tests (and arguably easier).

This, I'm much less sure of. IME, the really hard thing about correctness proofs is that specifications and proofs often require genuinely new ideas that didn't occur in the code.

For example, the spec of something like a sorting routine is that the output returns a list in increasing order, and that the output is a permutation of the input. The concept of a permutation doesn't occur explicitly in the implementation of the sort, and so the need to make this concept explicit can stop learners dead in their tracks.

I taught Agda (a dependently-typed programming language) to undergraduates, and found that Agda-the-programming language was actually easier to teach than Haskell (the IDE is better and there's less magic). But Agda-the-proof-assistant was hard to teach, mostly AFAICT because I don't know how to teach having mathematical ideas well.

[pron]

> Agda-the-proof-assistant was hard to teach, mostly AFAICT because I don't know how to teach having mathematical ideas well.

One of the (several) problems I find with dependently-typed languages is that they don't easily allow you to separate specification from verification [1]. When using a language like TLA+, it's very easy to separate the two. I actually think it's easier to teach mathematical thinking with TLA+ than without any formal language at all, as you can get feedback as to what has gone wrong, i.e, you can "play" with the math, and use the model checker before writing a proof to get a counterexample. I agree that teaching how to think mathematically is still required and still hard, but there's much less accidental complexity in the way.

Another advantage is that complex concepts are only encountered later [2] (e.g., you can cleanly separate safety from the much more complex liveness). The only downside (for teaching) is that the proofs are declarative, so you don't get to teach actual syntactic manipulation of terms (which is, arguably, not important unless you want to teach low-level proofs). Of course, the computational theory is TLA rather than lambda calculus, so it's irrelevant if you want to teach that formal system in particular. On the other hand, advanced and very powerful concepts such as refinement are very elegant and relatively simple.

[1]: Another is that proof is pretty much the only form of verification (of specification given as types), which is both extremely costly and very rarely required, and yet another is that complicated matters are encountered very early on.

[2] I've been playing with Lean for some weeks, and I still don't understand what object `id` (the identity "function", polymorphic over all types in all universes) is; all I know is that it's not an actual function. I'm not sure whether it's the same or not in Agda.

[neel_k]

HN seems to have eaten my first reply, so here's an abbreviated version.

It's easy to separate specification from verification with dependent types. You can (and I did) show students how to take ordinary simply-typed functional programs, give a specification as a different type, and then do a verification by producing a term of the appropriate specification type. Then, you can use this to motivate intrinsically-typed strategies, and (a) show the students that intrinsic/extrinsic is a gradient rather than a dichotomy, and (b) show them how to exploit this gradient systematically.

As for liveness, in a purely functional language, basically the only interesting liveness property is termination, and this didn't pose a challenge for the students. Indeed, it was the opposite -- I got a lot of questions about why Haskell lacked a termination checker. (They had in their minds that termination checking was an impossible dream, and so seeing a system with a termination checker that was both usable and useful really fired their imaginations.)

Specs were still hard. A proof-based approach is actually desirable for teaching good spec-writing skills, because proofs are more sensitive to the details of the spec. Eg, property-based testcase generation is happy with the equation that the reverse of the reverse of a list is the same list, but it's too weak an induction hypothesis for a proof. So it strongly motivates you to get to the characterizing equation for reversal (that reversing the concatenation of two lists is the concatenation of their reverses in the opposite order). (And it's still good for testcase generation.)

Unfortunately, I don't know enough small, self-contained examples where you have to strengthen the induction hypothesis to get the proof to go through. Ideally I'd want about 60 or 70 good examples of this (to set problems for a course) and, alas, I'd be hard-pressed to come up with a tenth that many satisfying examples.

[xgk]

> separate safety from the much more complex liveness

True, and with program logics, you can choose partial correctness vs total vs generalised correctness.

Itt's even worse in practise. With many contemporary dependently typed languages you'll have to worry about termination even during programming, even before thinking about verification.

The Curry-Howard prover advocates seem to regard this as a solved problem ("just distinguish programs from proofs by way of a typing system, problem solved") but has this solution been implemented in any mature systems as of 2017?

[catnaroek]

> use the model checker before writing a proof to get a counterexample.

Lovely, they should have taught me algebraic geometry that way. /s

Trying to write proofs without knowing a priori the truth value of the proposition to be proved builds character, and it's an important part of developing mathematical maturity. It forces you to develop intuition for what could or could not be true.

[xgk]

> to separate specification from verification

Exactly.

Program logics are superior in this regard: the encourage rather than inhibit separation of concerns. (Not to mention: program logics are more developed and cover a large class of computing paradigms, while dependent types have not grown much beyond the pure functions ghetto.)

I put forward this point to dependent types luminaries all the time. Its difficult to get more than polite silence in response.

      * * *

It's partly a social problem: The community working on dependently typed languages and tools typically have most of their programming experience with pure functional languages. Exaggerating only a bit, here is the typical trajectory of a formal methods researcher:

- As an undergraduate writes a lambda-calculus interpreter in Lisp/Scheme/Racket/Haskell

- As a PhD Student writes their own pure language with wacky typing system, and demonstrate its usefulness by embedding a lambda-calculus interpreter in it.

- As a postdoc verifies a lambda-calculus interpreter in Coq/Agda.

- As an assistant professor writes their own Curry-Howard-based interactive prover in some dependently typed language and demonstrate its usefulness by verifying a lambda-calculus interpreter in it.

- After tenure, gets their students to do the above.

2018's POPL has a type-theory in type-theory paper, but none on how programming language theory can improve SAT solving ...

It's an echo chamber. I recently reviewed a grant proposal by some of the more famous dependent types researchers. They proposed using HoTT to verify computational effects. The grant proposal claimed (1) that there is currently no known way to verify effectful languages and (2) that embedding generalisations of effect monads in HoTT is the way to overcome this problem. It seems like these people have not ever heard of program logics?

   * * *

A second reason is more serious: the real problem of formal verification is not this-language/that-logic. In day-to-day verification, it doesn't really matter that much whether you are using this/that approach. You need to set up the right definitions and invariants. That's the hard part. Whether you express them in some program logic or dependently typed system doesn't matter much, you can usually transliterate between them.

The real problem is automation.

Better autmation means dramatically better SAT solvers (modern SMT solvers are heavily reliant of SAT solvers). It's just unclear where 10x, 100x, 1000x speedups in SAT solvers should come from. We don't even know how to parallelise SAT solvers well. In the absence of dramatic progress in SAT solvers, what can we do other than tinkering with dependently typed languages and their relatives?

[tom_mellior]

> This turned out to be a premature optimization, actually! The CakeML compiler did full functional correctness proofs for their register allocator and report that it wasn't significantly harder than Compcert's translation validation.

Not sure if you're saying that using a translation validation approach for register allocation was a premature optimization. But if that's what you're saying, you're wrong: The paper describing the register allocation validator explains that this replaced a register allocator in Coq that was proved directly. The validator approach is much smaller and simpler than this direct proof was. So it was an actual optimization after the fact.

It's possible that CakeML does it better, of course.

[neel_k]

That's interesting! I heard this from some CakeML people, who told me they decided to go with a full proof because it wasn't much more work for them. Maybe Compcert has a fancier register allocator architecture that benefits more from translation validation?

[catnaroek]

Clearly, the only way forward is to make programming attractive to mathematicians, since making mathematics attractive to programmers has largely failed.

[pron]

I'm not at all sure mathematicians would make better programmers. One of the mistakes, IMO, some proof assistants make in their design philosophy is that since, from the perspective of some formal logics, writing programs and proofs are "the same thing", their is a corresponding equivalence of the two activities. This is somewhat analogous to concluding that since both programming and writing involve entering words into a text editor, the two are the same. But the fact that, from some specific perspective, both mathematicians and programmers write proofs does not mean that the activities are similar. Both physicists and mathematicians solve equations, but no one would say that physics is math. Mathematicians manipulate objects that are very different from those that programmers manipulate. Mathematicians manipulate objects that are very regular, and reasoning about them is largely tractable, while programmers manipulate very irregular and very intractable objects. Ostensibly, the difference is only in measure, but a difference in any robust complexity measure is usually a difference in quality. Even the actual proofs are very different. Mathematical proofs are usually short but mathematically deep, while program proofs are mathematically shallow and uninteresting, but very, very long and full of detail. Not to mention that the motivations are completely different. The goal of a programmer is to write a program that meets the requirements, which may include some distribution of acceptable bugs that depends on their severity (i.e. a major, but non-catastrophic bug occurring once a month and some minor bugs occurring daily are acceptable), and increasing the cost of programming by an order of magnitude in order to make all bugs provably absent is the wrong thing to do. In math, the requirements are completely different.

[catnaroek]

Ur/Web doesn't redefine “correctness”, since, by definition, your program specification is the definition of what it means for your program to be “correct”. Ur/Web merely guarantees the absence of certain behaviors in programs written in it.

[nickbauman]

Formal proof of behavior is hard and probably specious.

There cannot be a formally logical explication of code for even "common sense problems", like, say, how to safely drive a car. It doesn't exist because it's probably impossible but it's at least not feasible. Instead we use sampling of real world data to train models to solve this problem. This is the story of Data over Logic.

[pron]

Most software is very different from one that drives cars, but even in that case, while a complete specification (i.e., one that fully describes what driving a car safely is) is very hard, you can still quite easily specify many very useful correctness conditions. For example, you could state that the car must never go on the sidewalk or move to oncoming traffic unless it's trying to avoid some accident; similarly for running a red light, going over a crosswalk while there are pedestrians on it, or going over 100MPH. Verifying each of those would make your program safer, and that's the goal of formal methods -- not to make the program perfect, but better.

A complete formal specification of simpler programs or program components -- say banking systems or specific controllers in the car -- are not only possible and feasible but actually used in practice. A good number (I don't know if it's the majority or not) of avionics programs are formally specified and verified (although normally using model checkers rather than proofs). Having said that, different kinds of software can benefit from formal methods to different degrees. I'd say that if you can write tests for it, then you can write at least partial specs for it, too, but whether or not doing so is worth it varies by circumstance (in the cases I've used formal methods recently, it was not for more correctness but actually because it was cheaper than tests).

Also, we must separate formal specification from formal proof. Formal specification means writing what the program must or must not do in some precise formal language. Formal proof is one of many forms of verifying that specification. It provides the most confidence, but it is also by far the most expensive. Most programs can benefit greatly from formal specifications but can settle for weaker, but far cheaper, forms of verifying them.

[Jweb_Guru]

> Formal proof of behavior is hard and probably specious.

Utter nonsense. Formal proofs are specious because you can't do a formal proof of problems that are so ill-defined, we don't even have algorithms that solve them in enough cases to be road-safe yet? If you restrict your focus to software that actually exists, almost all of it is amenable to formal modeling.

[nickbauman]

You're making a philosophical argument about problem definition being the reason why we can't use formal proofs for many common sense problems. I wish you the best of luck.

My claim is that proving that software is correct using formal logical proofs is tautologically specious unless you extend the software to the problem it's trying to solve. Once those problems are of sufficient complexity (or interest, I might add), logic fails.

[Jweb_Guru]

A best-in-class, hard-real-time, preemptible operating system has been formally verified (along with many other components). CompCert exists. Iris progresses apace. Every day, newer and more complex formal systems are proved correct. I don't really buy the "logic fails at sufficient complexity" argument. By all accounts, it's succeeded where people have put in the effort.

[angry_octet]

Well, it does define a subset of C... Anything in that subset will run as specified in a correct physical machine. So I don't get your point? Are you saying that a verified compiler is useless if you write buggy software? Because that isn't really a criticism.

Have you read anything about CompCert C or Coq?

[110011]

What OP is saying is if you invoke undefined behavior in your program (like the Heartbleed bug for example) it doesn't matter that in the end the compiler is correct. And this does account for the vast majority of bugs (programmer error compared to compiler error).

[catnaroek]

> if you invoke undefined behavior in your program (like the Heartbleed bug for example) it doesn't matter that in the end the compiler is correct.

This is not true. It means that you can safely omit the compiler writer from the list of people to burn at the stake.

[kazinator]

> It means that you can safely omit the compiler writer from the list of people to burn at the stake.

Not if the problem is due to a compiler change which breaks something that has worked for decades and in other compilers too.

Say we have an open source distro full of programs, and a compiler change breaks some program that hasn't been touched in years so that it misbehaves in situations where old builds of that program do not.

This means the compiler writers aren't testing with a full distro, and possibly that they don't care.

"We can do anything we want within the limits set out by the ISO language spec, and that spec alone; let the downstream consumers sort out whatever happens."

[catnaroek]

> Not if the problem is due to a compiler change which breaks something that has worked for decades and in other compilers too.

The compiler writer is within their rights to perform any changes that don't contravene the language specification, whether it breaks other people's code or not.

> "We can do anything we want within the limits set out by the ISO language spec, and that spec alone; (...)"

Yes.

> "(...) let the downstream consumers sort out whatever happens."

It's the language user's responsibility to make sure they write meaningful programs.

[dhekir]

To complement: Leroy recently worked on a research project, Verasco, to formally verify a static analyzer for C (in the likes of Astree, Frama-C, Polyspace, etc.) that could plug that hole: if the analyzer reports "no undefined behaviors", then you are sure your program compiled with CompCert will result in a bug-free program. Turns out that it is very hard to scale such analyses, but with more research they may end up working for real-world programs.

[kazinator]

> then you are sure your program compiled with CompCert will result in a bug-free program.

That is simply not true. A UB-free program which calculates e is incorrect, if the task was to calculate pi. Anything downstream from that program which uses its output as if it were pi is going to break.

Correctness is only with respect to a specification.

Specifications can have issues. Aside from being vague or contradictory, they can neglect to include important requirements, such as being completely silent on the treatment of bad inputs. A program with an exploitable security flaw outside of its intended operation can in fact be correct, because those inputs lie outside of its requirements: i.e. just like a compiler, the program has its own undefined behaviors. Also, the requirements for a program can explicitly call for undefined behavior, like "divide a number by zero to demonstrate/test the platform-specific handling of the situation".

Calling a function outside of ISO C that is not defined in the C program itself is also undefined behavior, so these proven programs cannot use any platform libraries without relaxing what "undefined behavior" means.

[angry_octet]

That is a super pedantic nit picking response. You well knew what was meant. Everyone here knows the meaning of V&V.

[kazinator]

Problem is, some undefined behaviors are actually correct. Some are documented extensions (a concept straight out of ISO C), and some are just nonportable situations that empirically work.

These could be handled in the prover; you just need an extended language definition. Or maybe not. Depending on what is being relied upon, it could be difficult.

In any case, a prover based on equating "correct" with "strictly conforming ISO C" is going to be of severely limited use. It might be good for some module-level assurance, like that some linked list or hash table module is perfect or whatever.

[Blaisorblade0]

But you are aware real-world compilers are exploiting more and more UB and keep breaking existing software? Maybe you are, but the comment does not seem to reflect that.

Either way: The good thing is, after compiler writers start exploiting a kind of UB they add options like `-fno-strict-aliasing` and pinky swear that option preserves the UB. I agree a useful checker should support real-world relevant C dialects.

[kazinator]

> * Are you saying that a verified compiler is useless if you write buggy software?*

Not exactly, but if we look at it that way, how is it wrong?

I write buggy software; what do you write?

With the approach I take to writing software, the methods by which I flush out my bugs find compiler issues also in the same stroke.

If I didn't write buggy software, I could ditch testing, right?

Except, oops, not without a proven toolchain.

[angry_octet]

Well, if I was writing something which had really serious failure consequences (space, aviation, crypto, bitcoin wallets, nuclear systems, etc) I think I would go to great lengths to minimise the unnecessary complexity, remove entire bug classes, etc.

Now, a compiler is not a simple thing, and one which does optimisation steps is doing something complicated to code one may already by reasoning about in complicated ways.

I've encountered a number of compiler bugs, but usually in the sense of it crashing or doing something violently wrong. I've never encountered one in terms of it doing something very subtly wrong, but I consider it likely that that could be going on, but that I have no way to test for it. The only way would be to have dual implementations producing matching output, and it would still be dependent on the test vectors, and even then it could be a common mode error. So, for me, it would be very useful to be able to remove the compiler from the list of possibly buggy components. In fact you could say that this is the first C compiler ever, the others compilers have been for some close cousin of C.

So, I don't really thing your argument holds. Testing is notoriously hard to get right, and testing which simultaneously accounts for complex compiler interactions seems implausible. Especially compared to 'use a verified compiler and stop worrying if the compiler got it wrong'.