Are there any provable real-world languages? (scala?)

Yes, there are languages designed for writing provably correct software. Some are even used in industry. Spark Ada is probably the most prominent example. I've talked to a few people at Praxis Critical Systems Limited who used it for code running on Boings (for engine monitoring) and it seems quite nice. (Here is a great summary / description of the language.) This language and accompanying proof system uses the second technique described below. It doesn't even support dynamic memory allocation!

My impression and experience is that there are two techniques for writing correct software:

• Technique 1: Write the software in a language you're comfortable with (C, C++ or Java for instance). Take a formal specification of such language, and prove your program correct.

  If your ambition is to be 100% correct (which is most often a requirement in automobile / aerospace industry) you'd be spending little time programming, and more time proving.

• Technique 2: Write the software in a slightly more awkward language (some subset of Ada or tweaked version of OCaml for instance) and write the correctness proof along the way. Here programming and proving goes hand in hand. Programming in Coq even equates them completely! (See the Curry-Howard correspondence)

  In these scenarios you'll always end up with a correct program. (A bug will be guaranteed to be rooted in the specification.) You'll be likely to spend more time on programming but on the other hand you're proving it correct along the way.

Note that both approaches hinges on the fact you have a formal specification at hand (how else would you tell what is correct / incorrect behavior), and a formally defined semantics of the language (how else would you be able to tell what the actual behavior of your program is).

Here are a few more examples of formal approaches. If it's "real-world" or not, depends on who you ask :-)

• The Coq Theorem Prover: Programming with Coq (technique 2)
• ESC/Java (technique 1, neither sound nor complete though)
• Spec# (technique 1, neither sound nor complete though)
• Why / Krakatoa (technique 2, based on Java)

I know of only one "provably correct" web-application language: UR. A Ur-program that "goes through the compiler" is guaranteed not to:

• Suffer from any kinds of code-injection attacks
• Return invalid HTML
• Contain dead intra-application links
• Have mismatches between HTML forms and the fields expected by their handlers
• Include client-side code that makes incorrect assumptions about the "AJAX"-style services that the remote web server provides
• Attempt invalid SQL queries
• Use improper marshaling or unmarshaling in communication with SQL databases or between browsers and web servers

In order to answer this question, you really need to define what you mean by "provable". As Ricky pointed out, any language with a type system is a language with a built-in proof system which runs every time you compile your program. These proof systems are almost always sadly impotent — answering questions like String vs Int and avoiding questions like "is the list sorted?" — but they are proof systems none-the-less.

Unfortunately, the more sophisticated your proof goals, the harder it is to work with a system which can check your proofs. This escalates pretty quickly into undecidability when you consider the sheer size of the class of problems which are undecidable on Turing Machines. Sure, you can theoretically do basic things like proving the correctness of your sorting algorithm, but anything more than that is going to be treading on very thin ice.

Even to prove something simple like the correctness of a sorting algorithm requires a comparatively sophisticated proof system. (note: since we have already established that type systems are a proof system built into the language, I'm going to talk about things in terms of type theory, rather than waving my hands still more vigorously) I'm fairly certain that a full correctness proof on list sorting would require some form of dependent types, otherwise you have no way of referencing relative values at the type level. This immediately starts breaking into realms of type theory which have been shown to be undecidable. So, while you may be able to prove correctness on your list sorting algorithm, the only way to do it would be to use a system which will also allow you to express proofs which it cannot verify. Personally, I find this very disconcerting.

There is also the ease-of-use aspect which I alluded to earlier. The more sophisticated your type system, the less pleasant it is to use. That's not a hard and fast rule, but I think it holds true for the most part. And as with any formal system, you will often find that expressing the proof is more work (and more error prone) than creating the implementation in the first place.

With all that out of the way, it's worth noting that Scala's type system (like Haskell's) is Turing Complete, which means that you can theoretically use it to prove any decidable property about your code, provided that you have written your code in such a way that it is amenable to such proofs. Haskell is almost certainly a better language for this than Java (since type-level programming in Haskell is similar to Prolog, while type-level programming in Scala is more similar to SML). I don't advise that you use Scala's or Haskell's type systems in this way (formal proofs of algorithmic correctness), but the option is theoretically available.

Altogether, I think the reason you haven't seen formal systems in the "real world" stems from the fact that formal proof systems have yielded to the merciless tyranny of pragmatism. As I mentioned, there's so much effort involved in crafting your correctness proofs that it's almost never worthwhile. The industry decided a long time ago that it was easier to create ad hoc tests than it was to go through any sort of analytical formal reasoning process.

You're asking a question a lot of us have asked over the years. I don't know that I have a good answer, but here are some pieces:

• There are well-understood languages with the possibility of being proven in use today; Lisp via ACL2 is one, and of course Scheme has a well-understood formal definition as well.

• a number of systems have tried to use pure functional languages, or nearly pure ones, like Haskell. There's a fair bit of formal methods work in Haskell.

• Going back 20+ years, there was a short-lived thing for using hand-proof of a formal language which was then rigorously translated into a compiled language. Some examples were IBM's Software Cleanroom, ACL, Gypsy, and Rose out of Computational Logic, John McHugh's and my work on trustworthy compilation of C, and my own work on hand-proof for C systems programming. These all got some attention, but none of them made it much into practice.

An interesting question to ask, I think, is what would sufficient conditions be to get formal approaches into practice? I'd love to hear some suggestions.

Typed languages prove the absence of certain categories of fault. The more advanced the type system, the more faults they can prove the absence of.

For proof that a whole program works, yes, you're stepping outside the boundaries of ordinary languages, where mathematics and programming meet each other, shake hands and then proceed to talk using Greek symbols about how programmers can't handle Greek symbols. That's about the Σ of it anyway.

real world uses of such provable languages would be incredibly difficult to write and understand afterwoods. the business world needs working software, fast.

"provable" languages just dont scale for writing/reading a large project's code base and only seem to work in small and isolated use cases.