Procedure Typing for Scala

Procedure Typing for Scala


Alexander Kuklev∗ , Alexander Temerev‡

* Institute of Theoretical Physics, University of Göttingen

‡ Founder and CEO at Miriamlaurel Sàrl, Geneva

April 10, 2012


Functions and procedures

In programming we have:
– pure functions;
– functions with side eﬀects (AKA procedures).


Functions and procedures

In programming we have:
– pure functions;
– functions with side eﬀects (AKA procedures).

Scala does not diﬀerentiate between them:
– both have types A => B .


But it should!


But it should!
Static side eﬀect tracking enables
– implicit parallelisability;


But it should!
Static side eﬀect tracking enables
– implicit parallelisability;
– compile-time detection of a whole new class of problems:
(resource acquisition and releasing problems, race conditions,
deadlocks, etc.).


Short list of applicable methodologies:
Kleisli Arrows of Outrageous Fortune (2011, C. McBride)
Capabilities for Uniqueness and Borrowing (2010, P. Haller, M. Odersky)
Static Detection of Race Conditions [..] (2010, M. Christakis, K. Sagonas)
Static Deadlock Detection [..] (2009, F. de Boer, I. Grabe,M. Steﬀen)
Complete Behavioural Testing of Object-Oriented Systems using
CCS-Augmented X-Machines (2002, M. Stannett, A. J. H. Simons)
An integration testing method that is proved to ﬁnd all faults
(1997, F. Ipate, M. Holcombe)

Specifying procedure categories

We propose a new syntax
where a function deﬁnition may include a category it belongs to:

A =>[Pure] B – pure functions;
A =>[Proc] B – procedures.

There’s a lot more than Pure and Proc

There is a whole lattice of categories between Pure and Proc :

Logged: procedures with no side effects besides logging;
Throws[E]: no side effects besides throwing exceptions of type E ;
Reads(file): no side effects besides reading the file ;
etc.

Extensible approach

An effect system should be extensible.
⇒ We must provide a way to define procedure categories.

Procedure categories are binary types like Function[_,_] or
Logged[_,_] 1

1
Definition of parameterized categories, e.g. Throws[E] or Reads(resource),
is also possible with the help of type lambdas and/or type providers.

Extensible approach

An effect system should be extensible.
⇒ We must provide a way to define procedure categories.

Procedure categories are binary types like Function[_,_] or
Logged[_,_] 1 equipped with some additional structure using an
associated type class.

1
Definition of parameterized categories, e.g. Throws[E] or Reads(resource),
is also possible with the help of type lambdas and/or type providers.

Extensible approach

Syntax details

– A =>[R] B R[A,B]
– A => B Function[A,B] , i.e. type named “Function” from the
local context, not necessarily the Function from Predef2 .

2
(A, B) should also mean Pair[A,B] from the local context, as they
must be consistent with functions: (A, B) => C ∼ A => B => C .
=

Extensible approach

Proposed syntax for deﬁnitions

def process(d: Data):
=>[Throws[InterruptedException]] Int = { ...

// Procedure types can be dependent
def copy(src: File, dest: File):
=>[Reads(src), Writes(dest)] { ...

// Pre- and postconditions can be treated as effects too:
def open(file: File):
=>[Pre{file@Closed}, Post{file@Open}] { ...

Last two examples rely on recently added dependent method types.
(N.B. Such stunts are hard to implement using type-and-eﬀect systems.)

Deﬁning procedure categories

How to deﬁne a procedure category?


First of all, it should be a category in the usual mathematical sense,
i.e. we have to provide procedure composition and its neutral.
trait Category[Function[_,_]] {
def id[T]: T => T
def compose[A, B, C](f: B => C, g: A => B): A => C
}


To give an example, let’s model logged functions on pure functions:
type Logged[A, B] = (A =>[Pure] (B, String))

object Logged extends Category[Logged] {
def id[T] = {x: T => (x, "")}
def compose[A, B, C](f: B => C, g: A => B) = {x: A =>
val (result1, logOutput1) = g(x)
val (result2, logOutput2) = f(result1)
(result2, logOutput1 + logOutput2)
}
}

Besides their results, logged functions produce log output of type
String. Composition of logged functions concatenates their logs.


Linear functional composition is not enough.
We want to construct arbitrary circuits.

(This is the key step in enabling implicit parallelisability.)


To make arbitrary circuits, we need just one additional operation
besides composition:
def affix[A, B, C, D](f: A => B, g: C => D): (A, C) => (B, D)


In case of pure functions, affix is trivial:
– the execution of f and g is independent.

In case of procedures affix is not-so-trivial:
– have to pass the eﬀects of f to the execution context of g ;
– execution order can be signiﬁcant.


Thus, procedures belong to a stronger structure than just a
category, namely a structure embracing the aﬃx operation.

Such a structure is called circuitry.


A circuitry is a closed monoidal category with respect to the affix
operation, where affix splits as follows:
trait Circuitry[F[_,_]] extends PairCategory[F] {
def passr[A, B, C](f: A => B): (A, C) => (B, C)
def passl[B, C, D](g: C => D): (B, C) => (B, D)
override def affix[A, B, C, D](f: A => B, g: C => D) = {
compose(passl(g), passr(f))
}
}

+ =


For the mathematicians among us:
trait PairCategory[F[_,_]] extends Category[F] {
type Pair[A, B]
def assoc[X, Y, Z]: ((X, Y), Z) => (X, (Y, Z))
def unassoc[X, Y, Z]: (X, (Y, Z)) => ((X, Y), Z)

type Unit
def cancelr[X]: (X, Unit) => X
def cancell[X]: (Unit, X) => X
def uncancelr[X]: X => (X, Unit)
def uncancell[X]: X => (Unit, X)

def curry[A, B, C](f: (A, B) => C): A => B => C
def uncurry[A, B, C](f: A => B => C): (A, B) => C
def affix[A, B, C, D](f: A => B, g: C => D): (A, B) => (C, D)
}


type Pair[A, B]

type Unit

}

Don’t panic!


type Pair[A, B]

type Unit

}

Don’t panic!
In most cases the default Pair and Unit work perfectly well.
⇒ No need to understand any of this, just use with Cartesian .


Elements of circuitries are called generalised arrows.

Besides procedures, circuitries provide a common formalism for:
– reversible quantum computations;
– electrical and logical circuits;
– linear and aﬃne logic;
– actor model and other process calculi.

Circuitries provide the most general formalism for computations, see
“Multi-Level Languages are Generalized Arrows”, A. Megacz.


We are talking mostly about procedure typing, so we are going to
consider some special cases:

Arrow circuitries3 : circuitries generalising =>[Pure] .

Executable categories: categories generalising to =>[Proc] .

Procedure categories: executable cartesian4 procedure circuitries.

3
AKA plain old “arrows” in Haskell and scalaz.
4
i.e. having cartesian product types.


trait ArrowCircuitry[F[_,_]] extends Circuitry[F] {
def reify[A, B](f: A =>[Pure] B): A => B
... // With reify we get id and passl for free
}

trait Executable extends Category[_] {
def eval[A, B](f: A => B): A =>[Proc] B
// eval defines the execution strategy
}

trait ProcCategory[F[_,_]] extends ArrowCircuitry[F] with
Executable with Cartesian {
... // Some additional goodies
}


It’s time to give a full deﬁnition of =>[Logged] :

object LoggedCircuitryImpl extends ProcCategory[Logged] {
def reify[A, B](f: A =>[Pure] B) = {x: A => (f(x), "")}
}
def passr[A, B, C](f: A => B): = {x : (A, C) =>
val (result, log) = f(x._1)
((result, x._2), log)
}
def eval[A, B](p: A => B) = {x: A =>
val (result, log) = p(x)
println(log); result
}
}


It’s time to give a full deﬁnition of =>[Logged] :

object LoggedCircuitryImpl extends ProcCategory[Logged] {
def reify[A, B](f: A =>[Pure] B) = {x: A => (f(x), "")}
}
def passr[A, B, C](f: A => B): = {x : (A, C) =>
val (result, log) = f(x._1)
((result, x._2), log)
}
def eval[A, B](p: A => B) = {x: A =>
val (result, log) = p(x)
println(log); result
}
}

Wasn’t that easy?


Additionally we need a companion object for Logged[_,_] type.

That’s where circuitry-speciﬁc primitives should be deﬁned.
object Logged {
val log: Logged[Unit, Unit] = {s: String => ((),s)}
}


Other circuitry-speciﬁc primitives include:
– throw and catch for =>[Throws[E]]

– shift and reset for =>[Cont]

– match/case and if/else for =>[WithChoice]

– while and recursion for =>[WithLoops]

– etc.

Often they have to be implemented with Scala macros (available in
a next major Scala release near you).

Language purification by procedure typing

Note that impure code is localised to the eval method.
Thus, thorough usage of procedure typing localizes
impurities to well-controlled places in libraries.

Except for these, Scala becomes a clean multilevel language,
with effective type systems inside blocks being type-and-effect
systems internal to corresponding circuitries.


Curry-Howard-Lambek correspondence
relates type theories, logics and categories:
For cartesian closed categories:
Internal logic = constructive proposition logic
Internal language = simply-typed λ-calculus
For locally cartesian closed categories:
Internal logic = constructive predicate logic
Internal language = dependently-typed λ-calculus
...


Curry-Howard-Lambek correspondence
relates type theories, logics and categories:
For cartesian closed categories:
Internal logic = constructive proposition logic
Internal language = simply-typed λ-calculus
For locally cartesian closed categories:
Internal logic = constructive predicate logic
Internal language = dependently-typed λ-calculus
...
Informally, the work of A. Megacz provides an extension of it:
For arrow circuitries:
Internal logics = contextual logics
Internal languages = type-and-eﬀect extended λ-calculi


Scala purification/modularization programme

– Design a lattice of procedure categories between Pure and
Proc . In particular, reimplement flow control primitives as
macro5 methods in companion objects of respective categories.

5
One reason for employing macros is to guarantee that scaffoldings will be
completely removed in compile time with no overhead on the bytecode level.



– Implement rules for lightweight eﬀect polymorphism using a
system of implicits à la Rytz-Odersky-Haller (2012)

5



– Implement rules for lightweight eﬀect polymorphism using a
system of implicits à la Rytz-Odersky-Haller (2012)
– Retroﬁt Akka with circuitries internalizing an appropriate actor
calculus + ownership/borrowing system (Haller, 2010).

5

Comparing with other notions of computation

How do circuitries compare to
other notions of computation?

Type-and-eﬀect systems

Type-and-eﬀect systems are the most well studied approach to
procedure typing:


procedure typing: effects are specifiers as annotations for
“functions”; type system is extended with rules for “effects”.


procedure typing: effects are specifiers as annotations for
“functions”; type system is extended with rules for “effects”.

Circuitry formalism is not an alternative, but an enclosure for
them.


Direct implementation of type-and-eﬀect systems
– is rigid (hardly extensible) and
– requires changes to the typechecker.


Direct implementation of type-and-eﬀect systems
– is rigid (hardly extensible) and
– requires changes to the typechecker.

Embedding eﬀects into the type system by means of
the circuitry formalism resolves the issues above.

Monads

Arrows generalise monads

In Haskell, monads are used as basis for imperative programming,
but they are often not general enough (see Hughes, 2000).

Monads are similar to cartesian arrow circuitries. The only
diﬀerence is that they are not equipped with non-linear composition.

Monads

Monads
– do not compose well,
– prescribe rigid execution order,
– are not general enough for concurrent computations.

Monads

Monads
– do not compose well,
– prescribe rigid execution order,
– are not general enough for concurrent computations.

Circuitries were invented to cure this.

Applicatives

Applicatives are a special case of arrows...

If procedures of type =>[A] never depend on effects of other
procedures of the same type, A is called essentially commutative.

Example
=>[Reads(config), Writes(log), Throws[NonBlockingException]]

Essentially commutative arrows arise from applicative functors.
They are flexible and easy to handle: you don’t have to propagate
effects, just accumulate them behind the scenes.

Applicatives

...but not a closed special case!

Composing applicatives may produce non-commutative circuitries
like =>[Reads(file), Writes(file)] .
Procedures of this type are no longer eﬀect-independent: eﬀect of
writes have to be passed to subsequent reads.

Besides these, there are also inherently non-commutative arrows
such as those arising from monads6 , comonads and Hoare triples7 .

6
e.g. Tx = transaction monad, Cont = continuation passing monad.
7
pre- and postconditioned arrows.

Traditional imperative approach

Can we do everything available in imperative
languages with arrows and circuitries?



Any imperative code can be reduced to compose and affix.



The reduction process is known as variable elimination, it can be
understood as translation to a concatenative language like Forth.



(The concatenative languages’ juxtaposition is an overloaded operator reducing
to either compose or affix depending on how operands’ types match.)



(The concatenative languages’ juxtaposition is an overloaded operator reducing
to either compose or affix depending on how operands’ types match.)

But! Writing code this way can be quite cumbersome.

Do-notation

Deﬁning procedure categories is easy
enough. How about using them?
We develop a quasi-imperative notation8 and implement it using
macros.
Our notation shares syntax with usual Scala imperative code...
...but has diﬀerent semantics: it compiles to a circuit of
appropriate type instead of being executed immediately.

8
Akin to Haskell’s do-notation, but much easier to use.

Do-notation

Deﬁning procedure categories is easy
enough. How about using them?
We develop a quasi-imperative notation8 and implement it using
macros.
Our notation shares syntax with usual Scala imperative code...
...but has diﬀerent semantics: it compiles to a circuit of
appropriate type instead of being executed immediately.

Circuit notation for Scala is the topic of the part II...

8
Akin to Haskell’s do-notation, but much easier to use.

Do-notation

Do-notation example
...but here’s a small example to keep your interest

Even pure functions have a side effect: they consume time.
=>[Future] is an example of a retrofitting procedure category9 .

=>[Future] {
val a = alpha(x)
val b = beta(x)
after (a | b) {
Log.info("First one is completed")
}
after (a & b) {
Log.info("Both completed")
}
gamma(a, b)
}

9
its reify is a macro, so any procedures can be retrofitted to be =>[Future].

Do-notation

Literature:
– The marriage of eﬀects and monads, P. Wadler, P. Thiemann
– Generalising monads to arrows, J. Hughes
– The Arrow Calculus, S. Lindley, P. Wadler, and J. Yallop
– Categorical semantics for arrows, B. Jacobs et al.
– What is a Categorical Model of Arrows?, R. Atkey
– Parameterized Notions of Computation, R. Atkey
– Multi-Level Languages are Generalized Arrows, A. Megacz

Syntax for Circuitires

Part II: Syntax for Circuitires

A cup of coﬀee?

Implicit Unboxing

How do you use an arrow (say f: Logged[Int, String] ) in
present Scala code?

println(f(5)) seems to be the obvious way, but that’s
impossible, application is not deﬁned for f.

To facilitate such natural notation, we need implicit unboxing.

Implicit Unboxing

Preliminaries
A wrapping is a type F[_] equipped with eval[T](v: F[T]): T
and reify[T](expr: => T): F[T] (reify often being a macro) so
that
eval(reify(x)) ≡ x and
reify(eval(x)) ≡ x for all x of the correct type.

Implicit Unboxing

Preliminaries
A wrapping is a type F[_] equipped with eval[T](v: F[T]): T
and reify[T](expr: => T): F[T] (reify often being a macro) so
that
eval(reify(x)) ≡ x and
reify(eval(x)) ≡ x for all x of the correct type.

A prototypical example where reify is a macro is Expr[T]. Example
with no macros involved is Future[T] (with await as eval).

Implicit Unboxing

Preliminaries
Implicit unboxing is this: whenever a value of the wrapping type
F[T] is found where a value of type T is accepted, its eval is called
implicitly.

In homoiconic languages (including Scala), all expressions can be
considered initially having the type Expr[T] and being unboxed into
T by an implicit unboxing rule Expr[T] => T .

Implicit Unboxing

Syntax proposal
Let’s introduce an instruction implicit[F] enabling implicit
unboxing for F in its scope.

Implicit contexts can be implemented using macros:
– macro augments the relevant scope by F.reify as an implicit
conversion from F[T] to T;
– F.eval is applied to every occurrence of a symbol having or
returning type F[T] which is deﬁned outside of its scope.

Implicit Unboxing

Code that uses futures and promises can be made much more
readable by implicit unboxing.

An example: dataﬂows in Akka 2.0. Presently they look like this:
flow {
z << (x() + y())
if (v() > u) println("z = " + z())
}

Implicit Unboxing

Now this can be recast without any unintuitive empty parentheses:
flow {
z << x + y
if (v > u) println("z = " + z)
}

Implicit Unboxing

Back to our Logged example:
implicit[Logged]
def example(f: Int =>[Logged] String, n: Int): List[String] {
f(n).split(", ")
}

Which translates to:
def example(f: Int =>[Logged] String, n: Int): List[String] {
LoggedCircuitryImpl.eval(f)(n).split(", ")
}

Purifying Scala

Now, which procedure category should example() belong to?

As it evaluates =>[Logged], it should be =>[Logged] itself. This
allows its reinterpretation without any usage of eval:
def example(f: Int =>[Logged] String, n: Int): List[String] = {
import LoggedCircuitryImpl._
reify{n} andThen f andThen reify{_.split(", ")}
}

This is now a pure code generating a new circuit of the type
=>[Logged] based on the existing one (f) and some pure functions.

Purifying Scala

Purity Declaration

Let’s introduce @pure annotation which explicitly forbids calling
any functions with side eﬀects and assignments of foreign variables.
This renders the code pure.

Procedure with side eﬀects have to be composed by circuit
composition operations which are pure. The execution of
procedures, which is impure, always lies outside of the scope.

All code examples below are to be read as @pure .

Purifying Scala

Inside of @pure implicit unboxing for arrows becomes
implicit circuit notation, which is operationally
indistinguishable, but semantically diﬀerent.

Circuit notation

Circuit notation, general idea:
– write circuitry type like =>[X] in front of a braced code block;
– the code block will be reinterpreted as a circuitry of the given
type (via macros).

Circuit notation

Example:
=>[Logged] {
f(x) + g(x)
}

Result:
(reify{x} andThen reify(dup) andThen (f affix g) andThen reify{_ + _})

Circuit notation

In presence of implicit[X] every free braced block {...}
which uses external symbols of the type =>[X] should be
treated as =>[X] {...} , an implicit form of circuit syntax.

Circuit notation

The desugaring rules producing operationally indistinguishable
circuits from imperative-style code blocks are quite complicated,
but certainly doable.

To make the other direction possible, we need an additional
operator: after .

Circuit notation

Consider two arrows f: Unit => Unit and g: Unit => Unit .
They can be composed in two ways: f affix g (out-of-order) and
f andThen g (in-order).

affix in circuit notation will obviously look like f; g , though for
andThen we need some new syntax:

=>[Future] {
val n = f
after(n) g
}

Circuit notation

Without after , =>[Future] and other similar circuitries respect
only dataflow ordering, but ignore the order of independent effects
(e.g. writing into a log).

By combining usual imperative notation and after ,
any possible circuit configurations can be achieved.

Circuit notation

Now the example stated above is fully understandable:
=>[Future] {
val a = alpha(x)
val b = beta(x)
after (a | b) {
Log.info("First one is completed")
}
after (a & b) {
Log.info("Both completed")
}
gamma(a, b)
}

( after trivially supports any combinations of ands and ors.)

Circuit notation

Blocks as Objects
For the sake of composability, blocks should be treated as
anonymous classes extending their arrow type:
=>[Future] {
val result = {
@expose val partialResult = compute1(x)
compute2(partialResult)
}
after (result.partialResult) {
Log.info("Partial result ready")
}
}

The result in the after context is not just =>[Future] Int , but
its anonymous descendant with a public member partialResult .

Circuit notation

Of course, it should also work for named blocks:
def lengthyComputation(x: Double): Double = {
var _progress = 0.0 // goes from 0.0 to 1.0
@expose def progress = _progress // public getter
... // _progress is updated when necessary
}

val f = future someLengthyCalculation(x)
while (!f.isDone) {
Log.info("Progress: " + f.progress)
wait(500 ms)
}

(This is a perfect example of what can easily be done with macros.)

Circuit notation

The exact desugaring rules are quite complex (but perfectly real).
We hope these examples gave you some insight how everything
might work.

Thank you!

Procedure Typing for Scala

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