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).

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


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.).


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 usingCCS-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)

Procedure Typing for ScalaSpecifying 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.

Procedure Typing for ScalaSpecifying procedure categoriesThere’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.

Procedure Typing for ScalaSpecifying procedure categoriesExtensible approach

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

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

equipped with some additional structure using anassociated type class.

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


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

Procedure categories are binary types like Function[_,_] orLogged[_,_] 1 equipped with some additional structure using anassociated type class.

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


Syntax details

– A =>[R] B R[A,B]

– A => B Function[A,B] , i.e. type named “Function” from thelocal 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 .


Proposed syntax for definitions

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

// Procedure types can be dependentdef 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.)

Procedure Typing for ScalaDefining 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 => Tdef 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 typeString. 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 operationbesides 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 acategory, 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 Unitdef cancelr[X]: (X, Unit) => Xdef cancell[X]: (Unit, X) => Xdef uncancelr[X]: X => (X, Unit)def uncancell[X]: X => (Unit, X)

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

}

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 toconsider some special cases:

Arrow circuitries3: circuitries generalising =>[Pure] .

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

Procedure categories: executable cartesian4 procedure circuitries.

3AKA plain old “arrows” in Haskell and scalaz.4i.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] withExecutable with Cartesian {

... // Some additional goodies}


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

type Logged[A, B] = (A =>[Pure] (B, String))

object LoggedCircuitryImpl extends ProcCategory[Logged] {def reify[A, B](f: A =>[Pure] B) = {x: A => (f(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)

}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 ina next major Scala release near you).

Procedure Typing for ScalaLanguage purification by procedure typing

Note that impure code is localised to the eval method.

Thus, thorough usage of procedure typing localizesimpurities to well-controlled places in libraries.

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


Curry-Howard-Lambek correspondencerelates type theories, logics and categories:For cartesian closed categories:

Internal logic = constructive proposition logicInternal language = simply-typed λ-calculus

For locally cartesian closed categories:Internal logic = constructive predicate logicInternal language = dependently-typed λ-calculus...

Informally, the work of A. Megacz provides an extension of it:For arrow circuitries:

Internal logics = contextual logicsInternal languages = type-and-effect extended λ-calculi


Scala purification/modularization programme

– Design a lattice of procedure categories between Pure andProc . In particular, reimplement flow control primitives asmacro5 methods in companion objects of respective categories.

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

– Retrofit Akka with circuitries internalizing an appropriate actorcalculus + ownership/borrowing system (Haller, 2010).

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

Procedure Typing for ScalaComparing with other notions of computation

How do circuitries compare toother notions of computation?

Procedure Typing for ScalaComparing with other notions of computationType-and-effect systems

Type-and-effect systems are the most well studied approach toprocedure 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 forthem.


Type-and-effect systems are the most well studied approach toprocedure 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 forthem.


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 ofthe circuitry formalism resolves the issues above.

Procedure Typing for ScalaComparing with other notions of computationMonads

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 onlydifference is that they are not equipped with non-linear composition.

Procedure Typing for ScalaComparing with other notions of computationMonads

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

Circuitries were invented to cure this.

Procedure Typing for ScalaComparing with other notions of computationApplicatives

Applicatives are a special case of arrows...

If procedures of type =>[A] never depend on effects of otherprocedures 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 propagateeffects, just accumulate them behind the scenes.

Procedure Typing for ScalaComparing with other notions of computationApplicatives

...but not a closed special case!

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

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

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

Procedure Typing for ScalaComparing with other notions of computationTraditional imperative approach

Can we do everything available in imperativelanguages with arrows and circuitries?

Any imperative code can be reduced to compose and affix.

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

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

But! Writing code this way can be quite cumbersome.

Procedure Typing for ScalaDo-notation

Defining procedure categories is easyenough. How about using them?

We develop a quasi-imperative notation8 and implement it usingmacros.

Our notation shares syntax with usual Scala imperative code......but has different semantics: it compiles to a circuit ofappropriate type instead of being executed immediately.

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

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


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)

}

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


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

Procedure Typing for ScalaSyntax for Circuitires

Part II: Syntax for Circuitires

A cup of coffee?

Procedure Typing for ScalaSyntax for CircuitiresImplicit Unboxing

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

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

To facilitate such natural notation, we need implicit unboxing.


Preliminaries

A wrapping is a type F[_] equipped with eval[T](v: F[T]): Tand reify[T](expr: => T): F[T] (reify often being a macro) sothat

eval(reify(x)) ≡ x andreify(eval(x)) ≡ x for all x of the correct type.

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


Preliminaries

Implicit unboxing is this: whenever a value of the wrapping typeF[T] is found where a value of type T is accepted, its eval is calledimplicitly.

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


Syntax proposal

Let’s introduce an instruction implicit[F] enabling implicitunboxing 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.


Code that uses futures and promises can be made much morereadable 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())

}


Now this can be recast without any unintuitive empty parentheses:

flow {z << x + yif (v > u) println("z = " + z)

}


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(", ")

}

Procedure Typing for ScalaSyntax for CircuitiresPurifying Scala

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

As it evaluates =>[Logged], it should be =>[Logged] itself. Thisallows 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.


Purity Declaration

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

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

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


Inside of @pure implicit unboxing for arrows becomesimplicit circuit notation, which is operationallyindistinguishable, but semantically different.

Procedure Typing for ScalaSyntax for CircuitiresCircuit 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).


Example:

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

}

Result:

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


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


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

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


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

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

=>[Future] {val n = fafter(n) g

}


Without after , =>[Future] and other similar circuitries respectonly 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.


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.)


Blocks as ObjectsFor the sake of composability, blocks should be treated asanonymous 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 , butits anonymous descendant with a public member partialResult .


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.)


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

Thank you!

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