Can Wombat do Lazy?

I went to a nice talk by Paul Bone: Haskell Sucks. He pointed out various issues with lazy evaluation. He wants a language to be strict by default, with laziness as an option. So I wondered if Wombat could do laziness just as it currently is.

If we have a type X then we want a type Lazy X. Clearly Lazy X is implementedBy a Unit=>X which gets evaluated if and when we need the value. The archetypal lazy construct [in some unspecified language] is:

lazy allNats = cons(1,  map (1+) allNats)

but we can't do that in Wombat. Instead we need something like

`allNats =  lazy { cons(1, map (1+) allNats) }

And here we see a problem. Any operation which contains a Lazy operand has to not do its normal thing, but instead produce a new Lazy object. Well at least every operation in Wombat is a function call (more or less), and if we assume that then we just need a rule like this:

forall _f:_X=>_Y (  `_f (_lx:Lazy _X) = lazy {_f (unlazy _lx)}  )

For every value of type _X=>_Y it creates a new value of type (Lazy _X)=>(Lazy _Y).

Unfortunately that isn't going to do it. Because when we want to unpack a lazy thing to get some part of it out, we want the rest to be left lazy. You need to keep calling unlazy on the bits you are interested in till you get what you want. There seem to be problematic issues with that. Why do we expect that there will be only one way to unpack a lazy value into some combination of lazy values? In our example it seeems obvious that we want to unpack our Lazy (List Int) into a pair of a Lazy Int and a Lazy (List Int), but do we know that is the only way to unpack it? I conclude that a laziness option has to be built in to the language, and I'm inclined to wonder whether it is actually possible to do that in a way that has perfectly clear semantics.

1+2

1+2

We have:

• A type Int (that has a compile time equivalent).
• A self-declaring name system so that we have WCTL producing something internal that is roughly:
• ``1 = magic “1”;
• ``2 = magic “2”;
• [note that the double ` means that the name can not be redeclared.]
• The + operator converts to something internal that is roughly:
• Semigroup.reqdProp.binop.default(1,2)
  [but it’s not a textual translation: no operators involved.]
• We have a Tuple which is a List(Type)=>Type.
• And, of course, Type is a Type (in a higher Universe, but we ignore that).
• So we need Proc, a Tuple[Type Type]=>Type
• And List, a Type=>Type
• And we need Atom, a Type [but we don’t need the self-declaring names starting with a “.”.]
• Int conformsTo Semigroup [in the default way in this case].
• .binop.default has type Tuple[X X]=>X for some X conformsTo Semigroup
• So we need to find a type X that is the unique type which 1 and 2 belong to, which conformsTo Semigroup, and which is below any other such (in the isA hierarchy). Yes: it’s an initial object in an obvious (ahem) category. And the answer is … Int.
• So we somehow get Int.behaviour.Semigroup.binop.default(1,2) [I’m a bit vague about how this happens].
• Which converts to Int.property.add(1,2)
• As a .property of Int, add has to be implemented for each implementation
• Both 1 and 2 are created using the Int64 implementation
• So we call Int64.property.add(1,2), and check for overflow and finding none, convert the Int64 result to Int (conceptually) and return that.
• That being all, we call Printable.reqdProp.toString and print that string (since Int conformsTo Printable).

So writing just enough of the compiler for 1+2 will not be trivial.

Holes: ideas and puzzles

[note that holes have been brought in to version 0.0.4 of the doc, but in a simpler way than this.]

An example from the overview:

firstCase someExpresion of [
{ `x, ok = $; … }
{ `x, error(timeout,`extraInfo) = $; … }
{ `x, `otherError = $; … }]

The idea is that when the error is a timeout we extract the extraInfo and do something with that. The error identifier is acting as a procedure here, and maybe it is one, just building up an appropriate structure:

`error = { (`errorCode,`extraInfo) = $; (.error, errorCode, extraInfo) }

Here we return a 3-tuple with an atom first component used to identify it as an error structure. This might not be a very likely methodology, but it is conveniently illustrative.

`extraInfo is a hole waiting to be filled. What can we make of it being a part of the parameter we pass to error? And holes naturally appear in other places (such as pigworker’s amazing derivatives of data structures). And we’ll see others below. We want to handle them in a uniform way.

The easy thing, if we can make it work, is to see holes as just another type. Let’s write -T for a hole of type T. If X convertsTo Y, then we see that -Y convertsTo -X. If Int convertsTo Rational, then a hole for a Rational can take an Int. We want this to also work for the more important isA relation, and let’s assume that before getting to how it might work.

Extrapolating wildly based on our notation, it is natural to assume that Empty = -Empty, and the other hole types are below that, with -Any at the bottom.

An interesting thing happens when you consider the type Assignable(X) of mutables that can hold a value of type X. Naturally Assignable(X) convertsTo X, so that we can conveniently use the value. But now we see that we can also have Assignable(X) convertsTo -X, which will allow us to use our Assignable as a hole and assign new values using more general hole-oriented code.

The isA rule is that properties must always decrease as we go up. To fit this in we will say that holes types have negative properties. Some types (such as Assignables, or the tuples we looked at above) have mixed positive and negative character. The isA rule now says that, as we go up the isA hierarchy the number of positive properties must go down, while the number of negative properties must go up.

I’m not sure if this is all going to work, but it is fun.

[update: This suggests that an explicit closure should be an Any=>-Any, and maybe other things.]

[upupdate: A cute thing, if this all works is the ancient assignment statement becomes:
someHole := someExpression
and this works in all hole contexts, and its use for Assignables becomes a special case.]

procedures: - combining and adjoint pairs

After the previous post, I feel a need to put things on a slightly firmer footing. This is a start.

Combinable

We define a Behaviour Combinable with a property combine that is a procedure of type SemiSet X=>X. Informally (ignoring side effects):

• Every type that is Distinguishable is Combinable. The SemiSet is a Set if X is Distinguishable. If the set has one element then return it, else fail.
• Procedure type P=>Q are combinable if Q is. The result of combining them is a procedure which runs all procedures and returns the combination of the successful results.
• A Union is combinable if the alternatives are. If the parameters are in one branch only then it is the combination in that branch. If the values are all in the intersection of 2 or more branches then that intersection must support combining (since properties are lost but not gained as we go up the isA hierarchy).
• A tuple with combinable component types is combinable: just combine each component separately.
• Etc.

All types considered in Wombat documentation to date are Combinable, and types which aren’t Combinable will have poor semantics. For example we could imagine a well behaved form of approximate numbers which are defined by a range of Rational numbers. We could say that two are Combinable if their ranges overlap, in which case the combine operation creates a range covering both ranges. But standard floating point numbers are not going to be Combinable in any sensible way.

AdjointPair(X,Y)

These used to be called inverse pairs. That was misleading because it suggested symmetry between the two directions. The new name is also slightly misleading, suggesting that it can be seen as an example of adjoint functors from Category Theory, after which it is named because it seems similar.

If ap is an AdjointPair(X,Y) then it has 2 procedures:

1. ap.to:(X=>Y) never fails for an x:X, and is effectively 1-1 (injection / monomorphism / section) from X to Y, since one can (almost) get back to x with the other half of the pair;
2. ap.from:(Y=>X) applied to ap.to(x) may not get back to x, but it will get back to something that is compatible with x, so that they could be combined. In the common case where X is Distinguishable it does get back to x. Conversely ap.from will fail (Fail.next) if its parameter does not come from an X, except that if X is a procedure type it might instead return a procedure that fails when called.

The most obvious place where AdjointPair is used is in an isA declaration. Now if X isA Y via apxy, and P isA Q via appq, then there is an induced relation:

Y=>P isA X=>Q via AdjointPair(
   { `yp = $=:Y=>P; { appq.to (yp (apxy.to $)) }} : (Y=>P)=>(X=>Q)},
   { `xq = $=:X=>Q; { appq.from (xq (apxy.from $)) }} : (X=>Q)=>(Y=>P)})

The 2nd (.from) procedure doesn’t itself fail, but instead it returns a procedure that might fail if given a Y that isn’t an X. The key point is that if you start with a Y=>P, convert to an X=>Q, then convert back, then you get back to a procedure that has the same semantics, just some extra unnecessary checks thrown in. One might well say that this transformation is a natural one. Maybe if we define an appropriate category then these really are an adjoint pair in the Category Theory sense. Anybody who can see that and explain it will get a permanent acknowledgement in Wombat’s main documentation. For extra merit: What then is the associated monad?

Adventures in combining procedures

Gaussian Integers are just complex numbers with integer real and imaginary parts. Why don't they just call them Complex Integers? That's what I'll do here.

Suppose we have types Int, Rational and ComplexInteger, such that Int isA Rational and Int isA ComplexInteger.

Imagine a procedure that takes a Union(Rational,ComplexInteger). Any well behaved such procedure is going to give the same result for an Int parameter, whether it arrives as a Rational (with denominator 1) or as a ComplexInteger (with imaginary part 0). Indeed by transitivity, Int isA Union(Rational,ComplexInteger), so it can just come as itself. For definiteness let's suppose the procedure returns the magnitude as an AlgebraicNumber. The traditional hacky approach is to use the equivalent of Wombat's firstCase:

  `magnitude =
    { firstCase $ of [
        { $=`p:ComplexInteger; AlgebraicNumber.sqrt(p.real**2 + p.imag**2) }
        { $=`p:Rational; AlgebraicNumber.abs(p) }
    ]}

Int parameter's will always take the first exit (even if Rational!). The more respectable way to code this is to use the case operator:

  `magnitude =
    { case $ of [
        { `p = $=:ComplexInteger; AlgebraicNumber.sqrt(p.real**2 + p.imag**2) }
        { `p = $=:Rational; AlgebraicNumber.abs(p) }
    ]}

Note that the procedures are still given as a list here, but are actually a set in this instance (with no order), or rather a SemiSet, but we can use list operator because List X convertsTo SemiSet X.

In this 2nd implementation using case, when magnitude is given an Int parameter it will (in theory) run both procedures to conclusion and check that they return the same result. However a runtime check might not be necessary if the compiler can work out that both return the same answer, and in that  case it can choose the more efficient.

The case operator is precisely the mechanism for combining procedures. So we can imagine a class of generic procedures such that, if the procedure is defined for types X and Y, then it is also automatically defined for Union(X,Y). Wombat supports such generic procedures via properties of types. In this case it is natural to make .magnitude a method by making it a case in the type's .asProc static property. This is convenient, allowing one to write expr.magnitude. Other static properties of types can be used for generic procedures that don't fit the .asProc pattern. Except that Wombat doesn't do duck-typing. So the automatic combination of .asProc and other properties only applies where both types get the property from the same Behaviour, but we'll gloss over that for the moment.

Let's see if we can prove in an informal way that this works correctly. Suppose we have, as well we might, a type ComplexRational such that there are declared relations: Rational isA ComplexRational and ComplexInteger isA ComplexRational. The rules say that Union(Rational,ComplexInteger) isA ComplexRational. Suppose (simplifying) that the .magnitude operation for the Union is formed by combining the operations for Rational and ComplexInteger. We know that the .magnitude operation for Rational is compatible with that for ComplexRational, and ditto for ComplexInteger. Now consider the operation of .magnitude on a Union value. If the Rational branch fails then the ComplexInteger branch is the one that wins, and it is compatible with the result for ComplexRational. Conversely if the ComplexInteger case fails the Rational branch result will be compatible. If both succeed then the two results are compatible with each other, and each is separately compatible with the .magnitude of ComplexRational. The combination then has to be compatible as well. Hopefully I can make a less hand-wavy proof one day.

I was going to continue with a discussion of folds. For associative operations, foldl and foldr can be combined (and maybe we can add other potentially parallel variants), and one can imagine adding a little magic pixie-dust that will allow the compiler to pick the best for various available implementations of List. However that will have to wait for another day.

Scala is scary

This paper about DOT (= future Scala) has stuff that might be relevant to Wombat: http://arxiv.org/abs/1510.05216.

• "extending subtyping to a lattice breaks at least one key metatheoretic property such as narrowing or subtyping transitivity". Apparently they solve the problem by distinguishing run-time values from those that exist at compile time but are never instantiated. This distinction is important in Wombat where infinite things exist at compile time.
• However the name DOT = "Dependent Object Types" makes me wonder. Objects are bags full of variables and variables preclude any sane type hierarchy because they can't behave covariantly or contravariantly with respect to any other type. I.e. if X isA Y then we can't create a rule that Assignable(X) isA Assignable(Y) because suppose we pass an Assignable(X) to a procedure that wants an Assignable(Y): that procedure might assign a Y to it that isn't an X. So I don't know how DOT can have sane semantics. I should add it to my list of "stuff I really should read", but I don't feel strong enough.

Jon Pretty gives another interesting talk: http://rapture.io/talks/slippery-surface/cadiz.html. I should try to understand how his ideas fit in with Wombat. His (and presumably Scala's) idea of intersection types is not the same as Wombat's. 

The reason Wombat should (I expect) work without all the drama of more complex languages is the tough rule on the "isA" hierarchy: As you go up the hierarchy you always get less properties and the properties that are the same are (provably) compatible, and you have more data so that you can (provably) go back down the hierarchy with case statements and get back where you came from.

[Update: I should say something about what "compatible" means. One example is that addition works compatibly as you go from Int to Rational, even though one gives an Int result and the other a Rational result, the results are equal whether you promote the Int or demote the Rational. But it gets more complex when you get to things that don't support equality, particularly procedures. This shows up in the case of .asProc. We can use any type as if it was a procedure. For example "a string".length actually calls String.asProc("a string") with parameter .length. But .asProc is going to do less and less as we go up the hierarchy. So compatibility means that a higher .asProc might fail on a parameter, but if it succeeds it will return the same result (or compatible!!). A .asProc can take a parameter and pass it to the .asProc of all lower types (without having to know what they are) and combine the results. This is the equivalent of Abstract methods in other systems, but it has no problem with the situation where the value belongs to the intersection of multiple lower types.]

Units in Wombat

The way I hope to do units in Wombat is something like this:

There will be a type for length/distance. There will be some predefined values of that type such as metre. If you want 3.0 metres it is 3.0*metre. If you have a distance and you want to write it out as a number, divide by metre: myDist/metre. The same for angles: π*radian=180*degree. It gets a bit ugly for temperatures: zeroC+30.0*cDeg, and ditto date-time.

To be more exact. Assume the numbers we are working with are type Approx. Units will live in a type U. So metre above should really be U.metre, but there is nothing to stop you declaring `metre=U.metre, etc. The Approx number live in U as a degenerate set of units, so when you do myDist/metre you actually end up with type U, not type Approx. However Approx isA U. So, either immediately or at any later time, we can convert our result to Approx without failing.

[Update: This is just because units exist where we have a torsor (http://math.ucr.edu/home/baez/torsors.html): the difference or ratio is a number, but there is no default scale (multiplicative) or no certain starting point (additive), and the two variants seem to often come together. At any rate it is obvious that we want to generalize this and have type U be just a special case of a wider torsor type.]

[Update 2: Wouldn't it be better to have each sort of quantity (distance, mass, time, etc) be a different type? Then the type system could sort everything out, and detect mistakes at compile time, and some things become more convenient (distance1/distance2 is automatically just a number). But there are an infinite number of combinations (mass/(distance**3) is density) so it would be a complex variety of indexed types. This is very typical of the hard choices between static and dynamic typing. Wombat's standard library will try to hit the middle, while allowing others to do very static or very dynamic stuff. In this case a Distance type may not need any additional properties compared to type U restricted to distance. If that is the case then the advantages of having Distance as a separate type can be realised in a simple way using subset types, that is also pretty easy for the programmer to understand, and which let's them go back to U for one-off stuff without having to create new types all the time.]