Rewriting Process Algebra, Part 3: FreeACP Implementation

This is the third part of my progress report on a rewriting-based implementation of SubScript, FreeACP. This part covers the architecture of FreeACP I came up with so far while implementing the rewriting engine for SubScript.

If you have not read the previous parts of this report, you are advised to do so before reading this one:


Process algebra expressions are modeled as ordinary Trees. A Tree has a higher-kinded type argument to it, S[_]. S is the boxed type of a suspended computation's result, for example Future[_].

There are the following notorious subclasses of Tree:

  • Operators: Sequence and Choice.
  • Terminal cases: Result and its subclasses: Success and Failure, representing ε and δ respectively.
  • Suspend - carries a suspended computation.

A suspended computation is carried by Suspend nodes as S[Tree[S]] and should be interpreted as follows: There is some computation running under S and the result of this computation is another process algebra expression Tree[S].

For example, an ordinary AA's evaluation can be represented as S[Result[S]] and should be read as follows: There is some action running under S, and if it is successful (no exception happens), the result is ε, otherwise δ.


Sets of rewriting and suspension axioms are defined as partial functions and are straightforward to read.

Some points to note about them:

  • Rewriting axioms take a Tree[S] and return a Tree[S] as a result - they just rewrite a given tree.
  • Suspension axioms take a Tree[S] and return a List[S[Tree[S]]]. List reflects that there may be a need to make a choice between several trees. S[Tree[S]] means that the trees which the current one should be rewritten to are not readily available and are computed in S.


These axioms are applied in a loop until a terminal case is reached, as described in the theory in Part 2.

Suspension type as a free object

If one has a Tree[S], they do not have much choice but to execute it under S. This may not always be desirable: For instance, one may have a Tree[Eval], but want to execute it in parallel via Future.

Alternatively, S may stay constant, but the way of execution under S may not. A good example of this is the setText(textField, string) AA from our GUI example: We agreed that it sets the text of textField to string under a particular GUI framework we are working under. But what if we want our program to work under several GUI frameworks? Each of them will have its own implementation of textField and the way to set its text will differ between the frameworks.

For this reason, the function that runs the trees, runM, can take a natural transformation, S ~> G, using which one can specify how to map a suspended computation S to a suspended computation G.

Further increasing flexibility, we may even have a default implementation of S as a free object, in style of the free monad.

LanguageT as a free S

The pattern is as follows: All the expressions for FreeACP are written with suspension type S equal to LanguageT by default. These expressions are of type Tree[LanguageT].

LanguageT is a free object - it does not do anything by itself, but remembers the operations you tried to perform on it. It does this by reifying all the operations done on it into case classes. For example, == MapLanguage(t, f).

Next, the user can select whichever S they want to execute their program under, define a natural transformation LanguageT ~> S (roughly, LanguageT[Tree[LanguageT]] => S[Tree[LanguageT]]) and use it in the runM method to execute the LanguageT instances. This natural transformation is called the compiler, it compiles your program written under LanguageT to a concrete suspension type S.

In our example of the setText AA, one may define the following class to represent its action:

case class SetText[TF](textField: TF, string: String) extends LanguageT[Result[LanguageT]]

It contains all the data necessary to set the text of the text field, but does not say anything about how to do it. Then, one can define a different natural transformation LanguageT ~> S for each GUI framework they are working under, each specifying the way this particular framework performs this action. This way, a GUI controller can be written once and executed on many GUI frameworks.

For example, such a natural transformation under Swing may look like:

new (LanguageT ~> Future) {
  override def apply[A](t: LanguageT[A]): Future[A] =
    t match {
      case SetText[TextField](textField, string) =>
        Future {
          Swing.onEDTWait { textField.text = string }

The point to notice here is that all the things specific to the GUI framework are encapsulated in this compiler, and hence one program can be executed under many GUI frameworks, provided one has proper compilers for these frameworks.

Compilers for LanguageT

Default compiler

There is a number of default subclasses of LanguageT that reify operations that are used in the suspension axioms (recall that the suspension axioms rely on map: (A => B) => (S[A] => S[B]) and suspend: A => S[A]) and hence are necessary for the LanguageT to be used as a suspension type.

Hence, there is also a default compiler for such subclasses.

Internally, the compiler is a partial function, specifying how to translate various subclasses of LanguageT to F. For example, case MapLanguage(t1, f) => is the line handling MapLanguage, a reification of the map method called on LanguageT. All it does is mapping t1 by f using a where variable F is of type Functor[F]. In essence, the compiler describes how to delegate the map to the functor of F, whatever this F is.

In its definition, the default compiler declares some implicit arguments: implicit F: Functor[F], S: Suspended[F]. This means that internally it delegates some operations to a Functor and a Suspend type classes and hence depends on them. So if they are not in scope, there's nothing to delegate to and hence one can't instantiate the compiler for F. This also works another way around: whatever your F, if you have a Suspended and a Functor for it, you will be able to compile your program under that F.

One can get the default compiler for F if and only if one has Functor[F] and Suspended[F].

Compiler framework

It is possible to compose compilers for different subclasses of LanguageT. One can define their own subclass of LanguageT and add a compiler for that subclass, this way extending the expressive power they have.

By default, one can only use an atomic action to create a suspended computation, but one can define more primitives to write process algebra expressions with, as in the setText() example above.

The simplest example is a say primitive that just prints something to the console. The pattern for say is as follows: first, we define a subclass of LanguageT - wherever we use it, we want it to mean "print the payload to the console". Next, we define a compiler capable of executing that subclass under some F. This compiler has a very similar structure to the default one, except that it can only handle one case of the language: SayContainer.

Once defined, all the compilers can be composed to form a single compiler to be used in runM. This is done via the compiler function. Since all the "small" compilers return an Option[F[Tree[LanguageT]]], the "large" compiler iterates through the list of the "small" ones until a Some(_) is returned.

Also, the "small" compilers, or PartialCompilers are of type Compiler[F] => LanguageT ~> OptionK[F, ?] (Compiler[F] is LanguageT ~> F, and OptionK[F[_], A] is Option[F[A]]). This means that we are able to use the "large" compiler from the "small" ones, so that we can invoke it recursively.


FreeACP is still a work in progress. The theory and architecture are far from perfect and hopefully will endure substantial changes in future. However, the expressive power it grants is promising and worth exploring.