ATS-Postiats VS lean4

This is one of my two favorite books in The Little ...er series. The other is The Rational Schemer. These are two of the most advanced books in the series.

The Little Typer provides an introduction to dependent types. These can by used to guarantee things like "applying 'concat' to a list of length X and list of length Y returns a list of X+Y". It is also possible, to some extent, to use dependent types to replace proof tools like Coq. Two interesting languages using dependent types are:

- Idris. This is basically "strict Haskell plus dependent types": https://www.idris-lang.org/)

- ATS. This is a complex systems-level language with dependent types: http://www.ats-lang.org/

The Rational Schemer shows how to build a Prolog-like logic language as a Scheme library. This is a very good introduction to logic programming and the implementation of backtracking and unification is fascinating.

This is an excellent series overall, but these two books are especially good for people who are interested in unusual programming language designs. I don't expect dependent types or logic programming to become widely-used in the next couple generations of mainstream languages, but they're still fascinating.

Not being ATS

> any safety checks put into the competing language will have a runtime cost, which often is unacceptable

This is completely wrong. The best counterexample is probably ATS http://www.ats-lang.org which is compatible with C, yet also features dependent types (allowing us to prove arbitrary statements about our programs, and check them at compile time) and linear type (allowing us to precisely track resource usage; similar to Rust)

A good example is http://ats-lang.sourceforge.net/DOCUMENT/ATS2CAIRO/HTML/c36.... which uses the Cairo graphics library, and ends with the following:

> It may seem that using cairo functions in ATS is nearly identical to using them in C (modulo syntatical difference). However, what happens at the level of typechecking in ATS is far more sophisticated than in C. In particular, linear types are assigned to cairo objects (such as contexts, surfaces, patterns, font faces, etc.) in ATS to allow them to be tracked statically, that is, at compile-time, preventing potential mismanagement of such objects. For instance, if the following line:

      val () = cairo_surface_destroy (sf) // a type error if omitted

For a low level language in which you actually need to prove that your code doesn't cause UB, see http://www.ats-lang.org/

Here's the homepage fo the language: http://www.ats-lang.org/. The trick to finding results about with google is to search "ATS programming language".

In SPARK's case, you have to state your invariants in even greater precision than in Rust, and naturally it has worse inference. That's okay, the same happens in a certain language with Atrocious Type Syntax.

Yes, things like mentioned in the post can be expressed and checked statically, as demonstrated by languages like Idris and ATS. ATS might be even more relevant as it's an imperative language too, it can get rather low-level (like talking about properties of C runtime functions) while proving required properties statically, and it includes a solver for certain amount of arithmetics so that you don't need to prove obvious mathematical identities to the compiler. http://www.ats-lang.org/

Recently replaced by Lean, though.

https://github.com/cedar-policy/cedar-spec

https://lean-lang.org

Nice read! Rust has pushed, and will continue to push, the limits of practical, bare metal, memory safe languages. And it's interesting to think about what's next, maybe eventually there will be some form of practical theorem proving "for the masses". Lean 4 looks great and has potential, but it's still mostly a language for mathematicians. There has been some research on AI constructed proofs, which could be the best of both worlds because then the type checker can verify that the AI generated code/proof is indeed correct. Tools like Kani are also a step forward in program correctness.

Yeah, I believe they said intend for it to be used as a general purpose programming language. I used it to complete Advent of Code last year.

There are some really interesting features for general purpose programming in there. For example: you can code updates to arrays in a functional style (change a value, get a new array back), but if the refcount is 1, it updates in place. This works for inductive types and structures, too. So I was able to efficiently use C-style arrays (O(1) update/lookup) while writing functional code. (paper: https://arxiv.org/abs/1908.05647 )

Another interesting feature is that the "do" blocks include mutable variables and for loops (with continue / break / return), that gets compiled down to monad operations. (paper: https://dl.acm.org/doi/10.1145/3547640 )

And I'm impressed that you can add to the syntax of the language, in the same way that the language is implemented, and then use that syntax in the next line of code. (paper: https://lmcs.episciences.org/9362/pdf ). There is an example in the source repository that adds and then uses a JSX-like syntax. (https://github.com/leanprover/lean4/blob/master/tests/playgr... )

My latest inspiration to use code to a) replace my video editor, b) learn the basics of EDM production and c) understand a few topics in higher maths. This might sound very strange given there are specialised tools for these jobs. There's iMovie / Adobe Premier for video, there's GarageBand and FL studio for music and old good pen and pencil for math proofs. But these tools have three big limitations. First they have a lot of idiosyncratic learning, you have to spend quite some time getting used to these tools and my experience is that this time is quite upsetting. In contrast, you only have to learn to code one, maybe spend a few hours getting used to the syntax of another language. I'm not sure if that's true for most people but it was true for me using the tools mentioned above and wanted a place to discuss and see other people ideas and experiments. The second issue is that all these custom-made tools, are not composing easily. I can't search for all math proofs that used a single theorem. I can't create a plugin for iMovie and apply it to all my videos. I can't pick easily pick a rhythm from the internet and build upon for fun. There's also the issue of costs and version control, all tools I'm using today are open source and my work is stored in my repositories. This way I can create branches and test my ideas and I'm also confident that I can work in these projects in years.

Dafny and Whiley are two examples with explicit verification support. Idris and other dependently typed languages should all be rich enough to express the required predicate but might not necessarily be able to accept a reasonable implementation as proof. Isabelle, Lean, Coq, and other theorem provers definitely can express the capability but aren't going to churn out much in the way of executable programs; they're more useful to guide an implementation in a more practical functional language but then the proof is separated from the implementation, and you could also use tools like TLA+.

https://dafny.org/

https://whiley.org/

https://www.idris-lang.org/

https://isabelle.in.tum.de/

https://leanprover.github.io/

https://coq.inria.fr/

http://lamport.azurewebsites.net/tla/tla.html