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
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?)
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].
An inliner might break NotAdd, but why would it need to preserve it? Presumably an inliner is meant to be a homomorphism, not an isomorphism - i.e., it only goes one way.
Also, FreeBool[A] (the example given) actually handles substitution quite well. The `flatMap` operation will properly flatten things out:
If you're not careful about how you encode a property, the type system will _require_ you to preserve that property throughout each step of the computation, regardless of what sort of morphism that computation is. (It's an endomorphism.) Now that may be simple and straightforward when dealing with structural recursion over a boolean language, but if you want to write an inliner for Haskell, for example, it's probably a better idea to drop invariants from types (but not from the computation, of course), run the algorithm, and reintroduce invariants to the types afterwards.
Again, that example's way overshooting the complexity needed to prove the point. Even a seemingly simple invariant like the red-black balancing property is one that is impractical to preserve in this fashion without a proof assistant. And even then its practicality is still dubious.
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
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.
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.)
If you live close to a college or university, check to see if they have it. Almost all schools will let you into their libraries and let you browse (and photocopy) materials while you're there. Checking them out may be another issues, but you should ask about that while you're there as well.
If you're frequently passing by a university library, see if they have a shelf where they sell old books. Mine does, and I've scored some sweet stuff there, e.g. Dunford and Schwartz' Linear Operators Part 1 & 2 together for $8, which looks to be $150 + shipping for used copies from Amazon ($400 new). As you can probably tell, there's an arb opportunity there as well.
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!
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.
More like it is old enough that what number of copies were printed are owned. I personally perceive value, and I am willing to buy in at certain prices for the pay-off in knowledge and convenience. For instance, driving to a college, spending umpteen hours copying pages for a sub-standard replica of a book that I can have delivered for $130? I will buy it and do something else with the time saved.
I was looking at buying an old Lisp book, 'Common LISP Modules: Artificial Intelligence in the Era of Neural Networks and Chaos Theory'. I already own a copy, but not where I am living for now. It was $3 on Amazon. There was a thread on HN where I mentioned the book, and the author himself wanted a copy, so he ordered it.
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].
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?
