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Automated reasoning in F#, Scala, Haskell, C++, and Julia

• adamzubi700 11y ago
  http://googler700.blogspot.com/2015/04/project-list.html http://googler700.blogspot.com/2015/04/project-list.html
• cbsmith 11y ago
  The boost::variant example seems quite painful. I'd use boost::hana instead.
• bargl 11y ago
  Was the text at the bottom hidden for anybody other then me? I had to highlight it to see it. It seemed like a Easter egg, but I'm not sure. It was below "Sum types and pattern matching are awesome."
  • bglusman 11y ago
    Was also hidden for me, agree, small funny easter egg
  • jaytaylor 11y ago
    Nice find! How did you catch that?
    • bargl 11y ago
      I wish I had a cool answer but it was just an accident.
  • harperlee 11y ago
    Perhaps it is an easy way to detect copies of the article...
  • phdpqc 11y ago
    Yes, it is an Easter egg. I thought it would take more time for people to notice. How well :P
• cwyers 11y ago
  I loved the post, but I will say it's incredibly frustrating to see an interesting book referenced, to go looking for the book and to find that a USED copy on Amazon goes for $130.
  • profquail 11y ago
    It's also available on Safari Books if you have a subscription (e.g., through your workplace). I agree, it's a pricy book though.
    • cwyers 11y ago
      We don't, but it's not like it's something I couldn't ask for if there was enough value there, so that's good to know. (Although I don't think Safari works with any e-ink readers, certainly not my Kindle.)
  • deleted 11y ago
    [deleted]
  • brudgers 11y ago
    It's the sort of book that always makes me curious: Is it as awesome as its title suggests?
    • logicallee 11y ago
      simple logic dictates that if it weren't awesome it wouldn't sell used for $130. Curiosity alone or a few references in some article or wherever won't drive a used price that high!
      • cossatot 11y ago
        A few professors requiring it to be used in a class will certainly drive it up that high, combined with the limited printing done by academic publishers. It's pretty common and pretty unfortunate...
• profquail 11y ago
  If anyone's interested in more examples, I have a complete F# port of the example code from The Handbook of Practical Logic and Automated Reasoning I did a while back with a couple of other F# devs: https://github.com/jack-pappas/fsharp-logic-examples https://github.com/jack-pappas/fsharp-logic-examples
• bryanjos 11y ago
  I enjoyed that post so much, I took a couple of minutes and made an Elixir version https://github.com/bryanjos/arithmetic https://github.com/bryanjos/arithmetic. The book is pretty expensive, but I want to get a copy.
• one-more-minute 11y ago
  It's also worth checking out Julia's Match.jl package for more Haskell/OCaml-like pattern matching: https://matchjl.readthedocs.org/en/latest/ https://matchjl.readthedocs.org/en/latest/
• yummyfajitas 11y ago
  One neat feature which you can do in Scala (and probably C++), but which is probably more difficult in other languages, is impose constraints at the type level which force the underlying algebraic expression to already be (partially) simplified. Suppose we represented Add instead as: case class Add(exprs: NonEmptyList[Expr with NotAdd]) extends Expr case class Const(value: Int) extends Expr with NotAdd case class ... extends Expr with NotAdd Now nested Add's are a compile time error, and your expressions are automatically simpler. I'm not sure how easy this is to do in a language without subtypes. I imagine it could be implemented with typeclasses in Haskell somehow (probably using existential types), but I doubt it would be as easy. (Any Haskell gurus care to comment?) To see a more detailed illustration of how it would work, consider this free boolean algebra implementation: https://github.com/wingify/Oldmonk/blob/master/src/main/scala/com/vwo/oldmonk/free/FreeBool.scala https://github.com/wingify/Oldmonk/blob/master/src/main/scal...
  • tikhonj 11y ago
    This feels like something you could do with GADTs in Haskell and OCaml. Here's your example rendered with GADTs: {-# LANGUAGE DataKinds, GADTs, KindSignatures #-} data IsAdd = IsAdd | NotAdd data Expr :: IsAdd -> * where Add :: [Expr 'NotAdd] -> Expr 'IsAdd Const :: Integer -> Expr 'NotAdd ... -> Expr 'NotAdd You could also use True and False instead of Add and IsAdd, or even type-level string literals as arbitrary symbols: data Expr :: Symbol -> * where Add :: [Expr "NotAdd"] -> Expr "IsAdd" Const :: Integer -> Expr "NotAdd"
  • seliopou 11y ago
    Something like this could be implemented in Haskell or OCaml using GADTs[0]. You can add a phantom type parameter to the expression type and use that indicate whether or not the expression is well-formed according to some predicate--in your example, "non-nested added-ness." In fact this technique is often used as an introductory example to the use of GADTs, except the well-formed-ness property that's typically presented is that the expression is well-typed. It's possible to express multiple properties using the same phantom type parameter is both languages. However, as you sort of alluded to, this encoding is much simpler if your language supports subtyping. Since OCaml does support subtyping, you can use polymorphic variants[1] for this purpose. For example, if you had the NotAdd property as well as the NotMult property, you could write down the type for an operation that depends on just the NotAdd operation like this: val notAddOp : ([> `NotAdd] as 'a) expr -> 'a t ... and write down an operation that depends on both like this: val bothOp : ([> `NotAdd | `NotMult] as 'a) expr -> 'a u In Haskell, things would get more complicated. The type of the first operation would like more like this: notAddOp : Contains NotAdd a -> Expr a -> T a ... where the first argument is acting like a proof witness that the type parameter 'a in some sense "contains" tye type NotAdd. Similarly again for the second, except with two witnesses: bothOp : Contains NotAdd a -> Contains NotMult a -> Expr a -> T a I haven't thought it completely through, but I'm almost certain that a combination of type classes and GCH extensions would allow you to turn those proof witness arguments into type class contexts. Anyways, doing this sort of encoding of properties in types is all well and good until you start considering more realistic examples. Even in the one you presented, it's going to cause you problems if, say, you want to write an inliner for your language. Substituting an expression for a variable within a "NotAdd" addition may very well break the "NotAdd" property. This means that your inliner has to be aware of that property, so that it can preserve it while doing its job. In other words, the option of writing code that will break an invariant, and then immediately recover the invariant, is no longer on the table when you take this approach to verifying the correctness of your code. That may seem bad, until you try to verifiably balanced red-black tree without learning Coq and reading this[2]. Life's full of trade-offs. [0]: https://en.wikipedia.org/wiki/Generalized_algebraic_data_type https://en.wikipedia.org/wiki/Generalized_algebraic_data_typ... [1]: https://realworldocaml.org/v1/en/html/variants.html#polymorphic-variants https://realworldocaml.org/v1/en/html/variants.html#polymorp... [2]: http://www.cs.princeton.edu/~appel/papers/redblack.pdf http://www.cs.princeton.edu/~appel/papers/redblack.pdf
• steego 11y ago
  I know it's outside the scope of this post, but I'd like to add that F# adds a really nice feature called Active Patterns which are incredibly handy when your expression tree and logic become more complicated. In the author's example, he matches on a few rules. A zero rule, an identity rule, an evaluation rule, and the default like so: let simplify1 e = match e with | Add (Const 0, x) | Add (x, Const 0) | Mul (x, Const 1) | Mul (Const 1, x) -> x | Mul (x, Const 0) | Mul (Const 0, x) -> Const 0 | Add (Const a, Const b) -> Const (a + b) | Mul (Const a, Const b) -> Const (a * b) | _ -> e Active Patterns let you define these patterns as reusable blocks like so: let (|IdentityExpr|_|) = function | Add (Const 0, x) | Add (x, Const 0) | Mul (x, Const 1) | Mul (Const 1, x) -> Some(x) | x -> None It has the effect of making your pattern matching code more readable with a syntax that somewhat resembles a Prolog horn clause. As a bonus, you can mix and match these patterns when doing advanced reasoning. let simplify1 e = match e with | IdentityExpr(x) -> x | ZeroExpr() -> 0 | ConstantExpr(expr) -> Const(eval expr) | _ -> e
  • stuhood 11y ago
    In scala, the body of a match expression is a PartialFunction, and PartialFunctions compose similarly.
  • junkyarddog 11y ago
    Have to agree that Active Patterns are an incredibly useful F# feature.
• paulkoer 11y ago
  If someone is wondering about the relationship between sum types (discriminated unions / case classes) and the visitor pattern, I've written about this here [1]. [1] http://paulkoerbitz.de/posts/Sum-Types-Visitors-and-the-Expression-Problem.html http://paulkoerbitz.de/posts/Sum-Types-Visitors-and-the-Expr...
• barsonme 11y ago
  I got bored and wrote it in Go[1]. It's about as ugly as you'd expect. [1] - https://gist.github.com/EricLagerg/beb586714cd9f509c812 https://gist.github.com/EricLagerg/beb586714cd9f509c812
• white-flame 11y ago
  This is constant propagation, probably one of the smallest slivers of "automated reasoning" that could arguably still be called as such. Personally, I don't think it falls under that umbrella. Could this get a title change?
• kaychaks 11y ago
  My attempt to port it in Erlang https://gist.github.com/kaychaks/61b9b860646a6a47375d https://gist.github.com/kaychaks/61b9b860646a6a47375d
• mullr 11y ago
  Notably absent is anything in the lisp family, which excel at symbolic manipulation. Here's a Clojure version, both vanilla and with core.match. https://gist.github.com/mullr/d655db18ebb7aa77a62e https://gist.github.com/mullr/d655db18ebb7aa77a62e