The Future of TLA+ [pdf]

A more interesting example would be

f == CHOOSE f \in [Int \X Int -> Int]: \A > \in DOMAIN f: f[x, y] = f[y, x]

Which is expressing that `f` is some commutative function, but we don't care which. Could be multiplication, could be addition, could be average, could be euclidian distance from origin, could just be the 0 function.

P.S.

There's another interesting distinction to be made between Idris, a programming language that can also express a lot of maths, and TLA+, a mathemtatical language that isn't a programming language.

Languages like Idris make a sharp distinction between the type level and the object or computation level. The type level can express something analogous to the propositions you can express with TLA+, but these propositions must be filled or "realised" with content at the object level, and that content is required, even at the syntax level, to be a program, i.e. something that is executable.

In TLA+, in contrast, there's only a "type level". You can define refinement relationships between different specifications -- i.e. one is a more detailed description of the other -- but there is no requirement that any level needs to be executable. It may just so happen that some specification are detailed enough to extract a computation from (same goes for mathematics in general), but that's not a requirement.

BTW, it is interesting to note (as I do here in more detail: https://pron.github.io/posts/tlaplus_part3#algorithms-and-pr...) that quite often even simple algorithms -- I give QuickSort as an example -- that are described in a way that's detailed enough for a human to implement and perhaps even a computer could extract a reasonable computation from, are still not a program. The QuickSort algorithm says nothing about how to pick a pivot because any choice would still be an implementation of QS, even though a program must pick a pivot somehow and some choices may be better than others, nor does it describe in what order to sort the two segments -- they could be sorted in any order or even in parallel. For there to be a program, the computer must ultimately be told these details. Still, the algorithm is specified without them, as they don't matter to the algorithm itself, and it can be formally specified in TLA+ without them. If you choose, you may specify a particular and less abstract implementation of QS in TLA+ and check that it is, indeed, an implementation of the more abstract algorithm.

This could be seen as analogous to the object level in a programming language with dependent types, but the difference is that in TLA+ it's not required nor is there a clear distinction between a level of detail that is executable and one that isn't.

I believe this is really the tragedy of formal verification tools. Everybody wants a tool as robust as a compiler. At the same time, nobody wants to invest into development of such tools. Microsoft Research 20 years ago was probably an exception to that. The other companies wish to immediately hide these tools and the benchmarks behind the IP and closed source. As a result, we have early stage MVPs that are developed by 1-3 people.

> You can write hello world in TLA+:

That's not a good example, but before I get to that, you are again using logical implications in the wrong direction. You can describe physical systems, including programs in mathematics. What makes mathematics not the same as physics or programming is that you can also describe things that aren't physical or computable. It's like you're trying to prove that New York is the United States by pointing out that if you're in NY you're in the US.

Let me write that another way: programming ⊊ mathematics.

So you can write all programs in mathematics; you cannot do all mathematics in a programming language. Therefore, to know whether TLA+ is programming or mathematics the question is not whether you can describe programs in TLA+ but whether you can describe the maths that isn't programs, and you can.

Now, the reason that your example is not good is that the PrintT operator is defined in TLA+ [1] like so:

    PrintT(x) ≜ TRUE

It has no meaning in TLA+ other than the value TRUE. It's just another way of writing TRUE. The computer program TLC, however, prints some output when it encounters the PrintT operator in a specification, but that behaviour is very much not the meaning that operator has in TLA+. TLC can equally print "hello" whenever it encounters the number 3 in a mathematical formula. There are other TLC tricks that can be used to manipulate it to do clever stuff [2], but that's all specific to TLC and not part of the TLA+ language.

You could, however, have pointed out that you can specify a Hello, World program in TLA+, only that is unsurprising because you'd specify it in mathematics, and mathematics can also specify a Hello, World program. Here's how I would specify such a program in TLA+ (ignoring liveness for the sake of simplifying things):

    VARIABLE console
    Spec ≜ console = "" ∧ □[console' = "Hello, World!"]_console

It means that a variable named console (which we'll take to represent the content of the console) starts out blank, and at some future point its content becomes "Hello, World!" and it is not changed further.

You could also specify it another way, say, represent the console as a function of time (I'll make it discrete for simplicity):

    console[t ∈ Nat] ≜ IF t = 0 THEN "" ELSE "Hello, World!"

[1]: https://github.com/tlaplus/tlaplus/blob/0dbe98d51d6f05c35630...

[2]: I've even given a talk about such tricks: https://youtu.be/TP3SY0EUV2A

> No, it is defined like so

No, you are looking at the TLC source code, not TLA+ definitions. If you read the TLC documentation, it makes it very clear that it is not TLA+, but a software tool for model checking a certain subset of TLA+ and has some programming-like features that are not in TLA+. Other TLA+ tools also focus on other TLA+ subsets and have their own features, and they're not necessarily consistent with one another.

> This is the same strategy that Haskell uses for its primops

It really isn't, because TLC isn't the TLA+ compiler/interpreter. In TLA+ PrintT is just synonym for TRUE. We can talk about TLC if you like, but you shouldn't confuse the two.

> I guess you didn't understand my point that it would be easy to extend TLA+/TLC to more primops

TLA+ and TLC are two different things. TLA+ is a formal language for writing mathematical specifications. TLC is one of the several tools for analysing TLA+ specifications written in a subset of TLA+.

> I don't care what "the TLA+ language" is

That would sort-of made sense if TLC was the canonical tool for analysing TLA+ but it isn't (certainly not in the sense that ghc is the canonical Haskell compiler). In SANY, TLAPS, and Apalache (another TLA+ model checker), PrintT is just TRUE. In our work on the TLA+ toolbox, we're always careful to separate what's TLA+ and what are the features of the software tools, and in the materials we produce about TLA+ and the TLA+ tools these differences are explained.

In other words, you -- as someone who doesn't know TLA+ -- may not care about what TLA+ is and what TLC is, but those who use TLA+ very much care (though maybe not in the first week), because that distinction is required to put the language to good use.

If you choose to learn TLA+, I'm confident you'll come to agree. Everything I've said here will become clear once learn TLA+, TLC, and TLAPS (or even just TLA+). Those who actually use TLA+ very much do care what "the TLA+ language" is and what are some TLC features, because they often want to use multiple tools to do different things with their specifications, and PrintT only prints stuff when you're using TLC, not, say, when you're using TLAPS or Apalache. Once you gain experience with TLA+ you understand which subset of it can be checked by TLC and which isn't, and you write certain specifications with the intent of checking them in TLC and others for other uses, as you know that they fall outside TLC's subset of TLA+.

I think this short talk I gave and I linked to before (https://www.youtube.com/watch?v=TP3SY0EUV2A) gives a sense of the different ways you use TLA+ and how you combine them. It shows how you use the language as you do ordinary maths -- by manipulating formulas on paper -- and then use the results in conjunction with TLC to obtain some particular kind of automation. In fact, it is about how to use TLA+ to rewrite some TLA+ specifications so that they could be checked in some interesting way by TLC, i.e. the whole premise is that TLA+ users are confronted by the differences between TLC and TLA+, and it shows how to use certain deductions in practice to put TLC into some interesting uses for kinds of TLA+ specifications that TLA+ users would normally think fall outside the bounds of TLC. Anyway, anyone who actually uses TLA+ is -- and needs to be -- very much aware of the differences between TLA+ and TLC, recognises their different roles.