Modularization of Algorithms on
Complex Data Structures

An Encoding of
Typesafe
Extensible
Functional
Objects

Jonathan Immanuel Brachthäuser

Duality

Allowing modularly defined
folds (tree traversals) by choosing a 
first-class extensible representation of
algebras.

Duality

Allowing modularly defined
folds (tree traversals) by choosing a 
first-class extensible representation of
algebras.

Allowing modularly defined
unfolds (object definitions) by choosing a 
first-class extensible representation of
coalgebras.

vs.

Extensibility at Runtime

In code

trait CoffeeMachine {
  def makeCoffee: Coffee
}
trait Grinder {
  def grind(beans: Beans): Powder
}
trait Steamer {
  def steam: Steam
  def foam(m: Milk): MilkFoam
}
trait WaterLineConnected { self: CoffeeMachine => }
trait EspressoMachine { self: CoffeeMachine with Compressor => }

Possible in Scala

new CoffeeMachine with Grinder with Steamer {}

Not Possible In scAla

val cm = new CoffeeMachine
cm.makeCoffee
cm extends(Grinder) extends(Steamer)

Aspects of modularity

Static Modularity

Temporal Modularity

Examples

Runtime Component Adaption

Incremental Construction

Monkey Patching

Debugging / Tracing

Our Solution

An Encoding of Functional Objects that is Typesafe and Extensible

Our Solution

An Encoding of Functional Objects that is Typesafe and Extensible

Why an encoding?

An encoding of FUNCTIONAL OBJECTS THAT IS TYPESAFE AND EXTENSIBLE

  • Helps understanding a language feature by using already known semantics
  • Can be used as a library, does not require a special compiler
    • Allows reuse of existing infrastructure
    • Flexibility to modify and experiment with the semantics

Why an encoding?

An encoding of FUNCTIONAL OBJECTS THAT IS TYPESAFE AND EXTENSIBLE

  • Helps understanding a language feature by using already known semantics
  • Can be used as a library, does not require a special compiler
    • Allows reuse of existing infrastructure
    • Flexibility to modify and experiment with the semantics

Encoding objects
In Scala?

An encoding of FUNCTIONAL OBJECTS THAT IS TYPESAFE AND EXTENSIBLE

What are Objects?

An encoding of FUNCTIONAL OBJECTS THAT IS TYPESAFE AND EXTENSIBLE

Interface 

trait Counter {
  def get: Int
  def inc: Unit
}

Implementation 

trait CounterImpl extends Counter {
  var count: Int // state
  def get: Int = count
  def inc: Unit = { count += 1 }
}

Instantiation 

def newCounter(init: Int): Counter = 
  new CounterImpl {  var count = init }

Infinite tree of observations

An encoding of FUNCTIONAL OBJECTS THAT IS TYPESAFE AND EXTENSIBLE

val c = newCounter(0) // c0
c.get //=> 0
c.inc // c1
c.get //=> 1
c.inc // c2
// ...

ScalA

Encoding

An encoding of FUNCTIONAL OBJECTS THAT IS TYPESAFE AND EXTENSIBLE

trait Counter {
  def get: Int
  def inc: Unit
}
trait CounterF[State] {
  def get: Int
  def inc: State
}

Interface

Endofunctor

ScalA

Encoding

An encoding of FUNCTIONAL OBJECTS THAT IS TYPESAFE AND EXTENSIBLE

trait Counter {
  def get: Int
  def inc: Unit
}
trait CounterF[State] {
  def get: Int
  def inc: State
}
trait CounterImpl extends Counter {
  var count: Int
  def get: Int = count
  def inc: Unit = { count += 1 }
}
val CounterImpl = count => new CounterF[Int] {
  def get: Int = count
  def inc: Int = count + 1
}

Interface

Endofunctor

Implementation

Coalgebra

ScalA

Encoding

An encoding of FUNCTIONAL OBJECTS THAT IS TYPESAFE AND EXTENSIBLE

trait Counter {
  def get: Int
  def inc: Unit
}
trait CounterF[State] {
  def get: Int
  def inc: State
}
trait CounterImpl extends Counter {
  var count: Int
  def get: Int = count
  def inc: Unit = { count += 1 }
}
val CounterImpl = count => new CounterF[Int] {
  def get: Int = count
  def inc: Int = count + 1
}
def newCounter(init: Int): Counter = 
  new CounterImpl {  var count = init }
def newCounter(init: Int): Fix[CounterF] = 
  unfold(CounterImpl, init)

Interface

Endofunctor

Implementation

Coalgebra

Instantiation

Unfolding

Coalgebraic Encoding
In a Nutshell

An encoding of FUNCTIONAL OBJECTS THAT IS TYPESAFE AND EXTENSIBLE

val obj: Fix[F] = unfold(impl: S => F[S], init: S)

fixed point

coalgebra

inital state

interface functor

Modularization

Split object implementation into modular components.
Use late bound self-references to articulate dependencies.

An encoding of FUNCTIONAL OBJECTS THAT IS TYPESAFE AND EXTENSIBLE

Modularization

Split object implementation into modular components.
Use late bound self-references to articulate dependencies.

An encoding of FUNCTIONAL OBJECTS THAT IS TYPESAFE AND EXTENSIBLE

trait OpenCoAlg[Self[_], Prov[_], S] {
  def apply[T](priv: Lens[T, S]): CoAlg[Self, T] => CoAlg[Prov, T]
}

provided interface

required interface

late bound self-reference

type CoAlg[F[_], S] = S => F[S]

Modularization

Split object implementation into modular components.
Use late bound self-references to articulate dependencies.

An encoding of FUNCTIONAL OBJECTS THAT IS TYPESAFE AND EXTENSIBLE

trait OpenCoAlg[Self[_], Prov[_], S] {
  def apply[T](priv: Lens[T, S]): CoAlg[Self, T] => CoAlg[Prov, T]
}

provided interface

required interface

late bound self-reference

type CoAlg[F[_], S] = S => F[S]
type CompleteCoAlg[F[_], S] = OpenCoAlg[F, F, S]

An encoding of FUNCTIONAL OBJECTS THAT IS TYPESAFE AND EXTENSIBLE

trait SkipF[State] {
  def skip: State
}

Example For Static Extensibility

An encoding of FUNCTIONAL OBJECTS THAT IS TYPESAFE AND EXTENSIBLE

trait SkipF[State] {
  def skip: State
}
val SkipImpl = new OpenCoAlg[CounterF, SkipF, Unit] {
  def apply[S](priv: Lens[S, Unit]) = self => state => new SkipF[S] {
    def skip = self.apply(self.apply(state).inc).inc
  }
}

Example For Static Extensibility

dependency on Counter

using it in the implementation

An encoding of FUNCTIONAL OBJECTS THAT IS TYPESAFE AND EXTENSIBLE

val CounterImpl = ...
val SkipImpl = ...

val both = compose(
  CounterImpl, 
  SkipImpl)

Instantiating the Composite

What about Extensibility?

An encoding of FUNCTIONAL OBJECTS THAT IS TYPESAFE AND EXTENSIBLE

An encoding of FUNCTIONAL OBJECTS THAT IS TYPESAFE AND EXTENSIBLE 

val c0 = unfold(CounterImpl, 0)
val c1 = c0.out.inc

An encoding of FUNCTIONAL OBJECTS THAT IS TYPESAFE AND EXTENSIBLE 

val c0 = unfold(CounterImpl, 0)
val c1 = c0.out.inc
val c2 = c1.extend(SkipImpl, ())
val c3 = c2.out.skip

The type changes: Fix[CounterF] to Fix[CounterF with SkipF]

An encoding of FUNCTIONAL OBJECTS THAT IS TYPESAFE AND EXTENSIBLE

unfold(co1, s1)  extend(co2, s2)

is implemented by

unfold(compose(co1, co2), (s1, s2))

The extensibility is enabled by...

... defining static composition of coalgebra components:

    1.   by pointwise  application with the tupled state,

    2.   composing the resulting interface implementations.

 

... implementing extend in terms of compose, thus

NoW Possible In scAla

val cm1 = unfold( CoffeeMachine )
val cm2 = cm1.makeCoffee
val cm3 = cm2 extends(Grinder) extends(Steamer)

Extensions to the ENcoding

-    Allow references to the extended base coalgebra,
     imitating super-calls.

-    Allow selective open-recursion by passing the current
     as well as the late bound self-reference.

Conclusions

We have shown

    -  how to encode extensible objects in Scala.

 

It seems

   -  that object algebras can be dualized meaningfully.

 

In the future

    -  we would like to investigate the duality formally,

    -  improve the usability by optimizing performance and
       syntactic overhead.

Thank you!

Modularizing Algorithms on Complex Data Structures

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