Leslie Lamport: Video course on TLA+

Only for Microsoft employees, unfortunately! If you work at Microsoft, send me an email (alias is same as username here) and I'll get you added to the TLA+ Outlook group for future course announcements and such.

IMO not very. It's easy in TLA+ to write things like "randomly select some bijective function from the power set of N to [0, 1]", which can be really handy for speccing but isn't exactly programming-friendly. I personally think of this as a good thing, actually. TLA+ can be simpler and more expressive than it would be if it had to also be translatable.

So you can't guarantee your implementation matches your spec. All the more reason to write unit tests!

That's fantastic. Would be a great contribution to people interested in formal methods. How's your email?

Great question! So, TLA+ does have a machine-checked proof system called TLAPS, but its intended use is quite a bit different than Coq. While Coq is very well-suited for doing deep reasoning about mathematics and proving all sorts of stuff about whatever mathematical construct you care to define, TLAPS is focused much more on "mile wide, inch deep" type proofs required to demonstrate correctness of algorithms. Most algorithms use very simple mathematical properties, but have a bunch of edge cases and cyclomatic complexity which TLAPS is well-equipped to handle.

There might come a day where TLAPS is as powerful as Coq for writing general mathematical proofs (indeed, better ways of writing proofs and mathematics seem to be a passion of Dr. Lamport's, see his paper How to Write a 21st Century Proof[0]), but for now it's far, far easier to prove even simple statements about the Natural numbers in Coq.

[0] http://lamport.azurewebsites.net/pubs/proof.pdf

This is a complicated question. I am currently writing a blog post series on the theory of TLA+ that I will publish in May that, among other things, touches precisely on this point. But I will try to quickly point out some areas of difference, but the most important thing to remember is that both Coq and TLA+ -- as far as specifying and verifying software is concerned (not general math) -- are universal formalisms. Ultimately, anything you can do in TLA+ you can do in Coq and vice-versa. Also, when using mechanical proof to verify software, the proof techniques and the actual work is pretty much the same.

1. Difference in theory -- the theory of computation in Coq and HOL is based on functional programming; especially in Coq (I know very little about HOL), algorithms are functions in the FP sense (lambda expressions). In TLA, algorithms are state machines, and there's a richer notion of relationship between algorithms. This makes some things easier in TLA: Instead of programs being equal extensionally or definitionally, they can be equal or similar in many ways. In fact, in TLA there's a partial order relation on algorithms that reflects the refinement/abstraction relation. Also, it means that different kinds of algorithms -- sequential, concurrent, parallel, realtime etc., are all defined in the same way using the exact same ideas. This also makes it very easy to specify some properties -- such as worst-case complexity -- that are tricky in Coq. Having said that, people working on realtime systems in Coq implemented a simple version of TLA in Coq.

2. Differences in goals and audience -- Coq and HOL are ultimately general mathematical proof assistants. TLA+ is a tool for reasoning about digital systems. Coq and HOL can obviously reason about programs and TLA+ can prove general math theorems, but the focus is different. For example, Coq is used by logicians to explore new logics and mathematical foundations; you don't want to use TLA+ for that. On the other hand, TLA+ is used by "ordinary" engineers in industry working on "ordinary" software, while Coq and HOL are virtually unused in industry, and in the few cases they are, it's almost always in conjunction with academics and academic projects.

3. Differences in tools/abilities -- Coq can generate runnable code and be used for end-to-end verification. TLA+ cannot generate code, and while it has been used by academics for end-to-end verification, that is most certainly not its intended use. It is intended to be used in modeling large, complex systems at an abstraction level that's significantly higher than the code to find errors in design. As far as general math proofs are concerned, Coq is more capable than TLAPS, the TLA+ proof system. On the other hand, TLA+ has a model-checker that makes its proof checker almost irrelevant for large, complex specifications. As the users of TLA+ usually don't require a 100% watertight end-to-end specification, they're content to get the less than 100% guarantee of the model checker in exchange for what is at least a one order of magnitude reduction in effort.

In short, I would say that tools like Coq have a more academic focus, and are in general more interesting to academics and less to engineers, while TLA+ has a more industrial focus, and is of more interest to engineers and less to academics.

But I will repeat and say that while there are big difference in theory, goals and usage, ultimately when working on verifying software, the challenge is in thinking about your system mathematically and understanding how and why it works (or not). This work is similar both in effort and technique no matter what formalism you choose to use.