March 21, 2017

Lately I have been playing around a bit with porting some code samples from Fun with Type Functions to Scala. I will be referring to the pdf typefun.pdf linked there as ‘the pdf’.

I will explain how I did section 4.1 ‘Typed sprintf’.

Setting

Let’s explain what we want to achieve (paraphrasing from the pdf).

We want to make a typed printing function inspired by the not type-safe C function sprintf .

What this function does is print out a formatted string, but it can take a variable amount of parameters to be printed as well.

So for example sprintf("Name %s") requires an extra String argument, so we would want it to be a String => String . But we would want sprintf("Name=%s, Age=%d") to be a String => Int => String , since it takes an extra Int parameter. I assume you can extrapolate some other examples yourself.

The pdf then goes on to explain how it can be encoded in Haskell using type families. I will show how I ported this code to Scala.

Representation

We will represent what we want to print (in the pdf called an F ) as follows:

We can request to print a literal, we call this Lit . It must also take a String argument, this is the literal we want to print

. It must also take a argument, this is the literal we want to print We can request to print a value, we call this Val . This can have any type so we make it have a generic type. We also encode a way to print this value as an A => String function.

. This can have any type so we make it have a generic type. We also encode a way to print this value as an function. We need to be able to represent arbitrary combinations of the previous two, we call this Cmp . With Cmp we can combine two F s to create a bigger F .

We end up with this definition:

sealed trait F case class Lit ( str : String ) extends F case class Val [ A ]( printer : A => String ) extends F case class Cmp [ F1 <: F , F2 <: F ]( f1 : F1 , f2 : F2 ) extends F

In this way we could encode the name/age example printing as follows:

val valInt = Val [ Int ]( _ . toString ) val valStr = Val [ String ]( identity ) val nameAge = Cmp ( Cmp ( Lit ( "name=" ), valStr ), Cmp ( Lit ( ", age=" ), valInt ))

A little cumbersome, but it does the job.

Now our type family TPrinter will be represented like this:

sealed trait TPrinter [ A <: F , X ] { type Out def print ( f : A , k : String => X ) : Out }

We can see this as: given a type A and X we will have a type Out corresponding to it. A type-level function in a way.

Our TPrinter also has a function print which takes an F (our representation to print) and a k : String => X . It will then return something of type Out , which in the name/age example should be String => Int => String .

The function k will be String => String (or identity ) at the top level, but will become more important when we get to the implementation of the Cmp case for printing.

We’re also gonna be using the Aux pattern to be able to refer to the Out type in implicit parameters:

object TPrinter { type Aux [ A <: F , X , Out0 ] = TPrinter [ A , X ] { type Out = Out0 } }

Implementing the Type Family

Remember when I referred to the the type family as a type-level function? This might come across clearer when you see what behaviour we expect:

// some example behaviour: // given a Lit and a String, we want to receive type String implicitly [ TPrinter.Aux [ Lit , String , String ]] // but also: given a Lit and an Int => String, we want to receive type Int => String implicitly [ TPrinter.Aux [ Lit , Int => String , Int => String ]] // given a Val[Int] and a String, we want to receive type Int => String implicitly [ TPrinter.Aux [ Val [ Int ] , String , Int => String ]] // but also: given a Val[Int] and an Int => String, we want to receive type Int => Int => String implicitly [ TPrinter.Aux [ Val [ Int ] , Int => String , Int => Int => String ]] // given a Cmp[Lit, Val[Int]] and a String we want to receive type Int => String implicitly [ TPrinter.Aux [ Cmp [ Lit , Val [ Int ]] , String , Int => String ]] // given a Cmp[Cmp[Lit, Val[String]], Cmp[Lit, Val[Int]]] // and a String we want to receive type String => Int => String // notice this first type has the same type as `nameAge` implicitly [ TPrinter.Aux [ Cmp [ Cmp [ Lit , Val [ String ]] , Cmp [ Lit , Val [ Int ]]] , String , String => Int => String ]]

More generally we want the following to hold:

When we have A = Lit then Out = X . A literal doesn’t change the type of our print function.

Implementing this is quite straightforward. The print function will execute the k on the given literal string.

implicit def tPrinterLit [ X ] : TPrinter.Aux [ Lit , X , X ] = new TPrinter [ Lit , X ] { type Out = X override def print ( f : Lit , k : ( String ) => X ) : X = k ( f . str ) }

When we have A = Val[V] then Out = V => X . A value adds another typed parameter at the start of our function.

This is also fairly straightforward. We return a function which takes the typed parameter and when called will use the given printer to print this parameter as a String .

implicit def tPrinterVal [ X , A ] : TPrinter.Aux [ Val [ A ] , X , A => X ] = new TPrinter [ Val [ A ] , X ] { type Out = A => X override def print ( f : Val [ A ], k : ( String ) => X ) : A => X = ( x : A ) => k ( f . printer ( x )) }

When we have A = Cmp[F1, F2] then Out = something which combines the output of the printer function for F1 and F2 .

Ouch this one seems a bit harder to express, let’s make the implementation and leave some generics open and see if the type checker can help us.

We will need the TPrinter instances for F1 and F2 . These will be printerF1 : TPrinter.Aux[F1, XF1, OF1] and printerF2 : TPrinter.Aux[F2, XF2, OF2] .

Eventually we will need to print the two sides and return an X again. This is k(s1 + s2) and has signature endStr : String => String => X .

If we have the first part as a String (let’s say s1 ) we can create the printer function for the second part by doing printerF2.print(f.f2, s2 => endStr(s1, s2)) . This returns us the printer function determined by the second part OF2 , so print2 : String => OF2 .

Then we can create the definition for the overall printer function printerF1.print(f.f1, s1 => print2(s1) . This has type OF1 , the output type of the first TPrinter.

Let’s write out the first attempt. We can see the type checker isn’t quite happy yet.

// WARNING: doesn't compile, for fixed version see below implicit def tPrinterCmp [ X , F1 <: F , F2 <: F , XF1 , XF2 , OF1 , OF2 , OUT ] ( implicit printerF1 : TPrinter.Aux [ F1 , XF1 , OF1 ] , printerF2 : TPrinter.Aux [ F2 , XF2 , OF2 ] ) : TPrinter.Aux [ Cmp [ F1 , F2 ] , X , OUT ] = new TPrinter [ Cmp [ F1 , F2 ] , X ] { type Out = OUT override def print ( f : Cmp [ F1 , F2 ], k : ( String ) => X ) : OUT = { def endStr ( s1 : String , s2 : String ) : X = k ( s1 + s2 ) def print2 ( s1 : String ) = printerF2 . print ( f . f2 , // we have : String => XF2 , we need : String => X s2 => endStr ( s1 , s2 )) def print1 = printerF1 . print ( f . f1 , // we have : String => OF2 , we need : String => XF1 s1 => print2 ( s1 )) // we have : OF1 , we need : OUT print1 } }

Following the guidance of the type checker gets us at the correct result, but let’s reason about it a little bit more.

OF2 will give us the type of the second part of the print function. This will be (A =>)* String , where ()* denotes zero or more occurrences.

We want the first printer part to add types at the start of the OF2 type, as in (B =>)* (A =>)* String , this is OF1 .

If XF1 = OF2 then we have the behaviour we want:

in case of Lit , (B =>)* is empty and we get back (A =>)* String for OF1 .

, is empty and we get back for . in case of 1 Val[V] we get back V => (A =>)* String for OF1

we get back for etc…

Then we can also see that OUT must be equal to OF1 , since OF1 is the output type we want for the printer function for the Cmp case.

Since there is no recursive lookup for the tprinterF2 case our XF2 must be X . Ok, this explanation isn’t that intuitive but the type checker insists on having it like this, so let’s comply.

Doing all the substitutions gives us:

implicit def tPrinterCmp [ X , F1 <: F , F2 <: F , OF1 , OF2 ] ( implicit printerF1 : TPrinter.Aux [ F1 , OF2 , OF1 ] , printerF2 : TPrinter.Aux [ F2 , X , OF2 ] ) : TPrinter.Aux [ Cmp [ F1 , F2 ] , X , OF1 ] = new TPrinter [ Cmp [ F1 , F2 ] , X ] { type Out = OF1 override def print ( f : Cmp [ F1 , F2 ], k : ( String ) => X ) : OF1 = { def endStr ( s1 : String , s2 : String ) : X = k ( s1 + s2 ) def print2 ( s1 : String ) = printerF2 . print ( f . f2 , s2 => endStr ( s1 , s2 )) def print1 = printerF1 . print ( f . f1 , s1 => print2 ( s1 )) print1 } }

And lo and behold! We have implemented our very own typed printing function.

Using the function

How can we use our function? Well that’s easy, like this:

println ( implicitly [ TPrinter.Aux [ Cmp [ Cmp [ Lit , Val [ String ]] , Cmp [ Lit , Val [ Int ]]] , String , String => Int => String ] ]. print ( nameAge , identity )( "ruben" )( 23 ))

Not convinced on the usability? Okay let’s create an implicit class for some nicer ops

implicit class printerOps [ A <: F ]( f : A ) { def print [ X , O ]( k : String => X )( implicit tp : TPrinter.Aux [ A , X , O ]) : O = tp . print ( f , k ) def sprintf [ O ]( implicit tp : TPrinter.Aux [ A , String , O ]) : O = tp . print ( f , identity ) }

Now we can do this (I have to use .apply since otherwise it thinks I’m passing the implicit TPrinter instance):

println ( nameAge . sprintf [ String => Int => String ]. apply ( "ruben" )( 23 ))

Okay that’s a bit cooler, but if we leave out the type annotation then it doesn’t compile anymore, which is a bit of a shame.

But! The guys at dotty have done their job well and using the dotty compiler we can write the example like this:

println ( nameAge . sprintf . apply ( "ruben" )( 23 ))

Ok that’s looking more like it. Now if only we could also parse "name=%s, age=%d" as it’s corresponding F type (at compile-time), then we would have something pretty close to the original C function, but type-safe. This is probably possible by implementing type-level String parsing, but I’ll leave that for another time…

Check out the full code for this post on this scalafiddle.