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Joint Compilation of Scala and Java Sources

5
Jan
2009

One of the features that the Groovy people like to flaunt is the joint compilation of .groovy and .java files.  This is a fantastically powerful concept which (among other things) allows for circular dependencies between Java, Groovy and back again.  Thus, you can have a Groovy class which extends a Java class which in turn extends another Groovy class.

All this is old news, but what you may not know is the fact that Scala is capable of the same thing.  The Scala/Java joint compilation mode is new in Scala 2.7.2, but despite the fact that this release has been out for more than two months, there is still a remarkable lack of tutorials and documentation regarding its usage.  Hence, this post…

Concepts

For starters, you need to know a little bit about how joint compilation works, both in Groovy and in Scala.  Our motivating example will be the following stimulating snippet:

// foo.scala
class Foo
 
class Baz extends Bar

…and the Java class:

// Bar.java
public class Bar extends Foo {}

If we try to compile foo.scala before Bar.java, the Scala compiler will issue a type error complaining that class Bar does not exist.  Similarly, if we attempt the to compile Bar.java first, the Java compiler will whine about the lack of a Foo class.  Now, there is actually a way to resolve this particular case (by splitting foo.scala into two separate files), but it’s easy to imagine other examples where the circular dependency is impossible to linearize.  For the sake of example, let’s just assume that this circular dependency is a problem and cannot be handled piece-meal.

In order for this to work, either the Scala compiler will need to know about class Bar before its compilation, or vice versa.  This implies that one of the compilers will need to be able to analyze sources which target the other.  Since Scala is the language in question, it only makes sense that it be the accommodating one (rather than javac).

What scalac has to do is literally parse and analyze all of the Scala sources it is given in addition to any Java sources which may also be supplied.  It doesn’t need to be a full fledged Java compiler, but it does have to know enough about the Java language to be able to produce an annotated structural AST for any Java source file.  Once this AST is available, circular dependencies may be handled in exactly the same way as circular dependencies internal to Scala sources (because all Scala and all Java classes are available simultaneously to the compiler).

Once the analysis phase of scalac has blessed the Scala AST, all of the Java nodes may be discarded.  At this point, circular dependencies have been resolved and all type errors have been handled.  Thus, there is no need to carry around useless class information.  Once scalac is done, both the Foo and the Baz classes will have produced resultant Foo.class and Baz.class output files.

However, we’re still not quite done yet.  Compilation has successfully completed, but if we try to run the application, we will receive a NoClassDefFoundError due to the fact that the Bar class has not actually been compiled.  Remember, scalac only analyzed it for the sake of the type checker, no actual bytecode was produced.  Bar may even suffer from a compile error of some sort, as long as this error is within the method definitions, scalac isn’t going to catch it.

The final step is to invoke javac against the .java source files (the same ones we passed to scalac) adding scalac’s output directory to javac’s classpath.  Thus, javac will be able to find the Foo class that we just compiled so as to successfully (hopefully) compile the Bar class.  If all goes well, the final result will be three separate files: Foo.class, Bar.class and Baz.class.

Usage

Although the concepts are identical, Scala’s joint compilation works slightly differently from Groovy’s from a usage standpoint.  More specifically: scalac does not automatically invoke javac on the specified .java sources.  This means that you can perform “joint compilation” using scalac, but without invoking javac you will only receive the compiled Scala classes, the Java classes will be ignored (except by the type checker).  This design has some nice benefits, but it does mean that we usually need at least one extra command in our compilation process.

All of the following usage examples assume that you have defined the earlier example in the following hierarchy:

  • src
    • main
      • java
        • Bar.java
      • scala
        • foo.scala
  • target
    • classes

Command Line

# include both .scala AND .java files
scalac -d target/classes src/main/scala/*.scala src/main/java/*.java

javac -d target/classes 
      -classpath $SCALA_HOME/lib/scala-library.jar:target/classes 
       src/main/java/*.java

Ant

<target name="build">
    <scalac srcdir="src/main" destdir="target/classes">
        <include name="scala/**/*.scala"/>
        <include name="scala/**/*.java"/>
    </scalac>
 
    <javac srcdir="src/main/java" destdir="${scala.library}:target/classes" 
           classpath="target/classes"/>
</target>

Maven

One thing you gotta love about Maven: it’s fairly low on configuration for certain common tasks.  Given the above directory structure and the most recent version of the maven-scala-plugin, the following command should be sufficient for joint compilation:

mvn compile

Unfortunately, there have been some problems reported with the default configuration and complex inter-dependencies between Scala and Java (and back again).  I’m not a Maven…maven, so I can’t help too much, but as I understand things, this POM fragment seems to work well:

<plugin>
    <groupId>org.scala-tools</groupId>
    <artifactId>maven-scala-plugin</artifactId>
 
    <executions>
        <execution>
            <id>compile</id>
            <goals>
            <goal>compile</goal>
            </goals>
            <phase>compile</phase>
        </execution>
 
        <execution>
            <id>test-compile</id>
            <goals>
            <goal>testCompile</goal>
            </goals>
            <phase>test-compile</phase>
        </execution>
 
        <execution>
            <phase>process-resources</phase>
            <goals>
            <goal>compile</goal>
            </goals>
        </execution>
    </executions>
</plugin>
 
<plugin>
    <artifactId>maven-compiler-plugin</artifactId>
    <configuration>
        <source>1.5</source>
        <target>1.5</target>
    </configuration>
</plugin>

You can find more information on the mailing-list thread.

Buildr

Joint compilation for mixed Scala / Java projects has been a long-standing request of mine in Buildr’s JIRA.  However, because it’s not a high priority issue, the developers were never able to address it themselves.  Of course, that doesn’t stop the rest of us from pitching in!

I had a little free time yesterday afternoon, so I decided to blow it by hacking out a quick implementation of joint Scala compilation in Buildr, based on its pre-existing support for joint compilation in Groovy projects.  All of my work is available in my Buildr fork on GitHub.  This also includes some other unfinished goodies, so if you want only the joint compilation, clone just the scala-joint-compilation branch.

Once you have Buildr’s full sources, cd into the directory and enter the following command:

rake setup install

You may need to gem install a few packages.  Further, the exact steps required may be slightly different on different platforms.  You can find more details on Buildr’s project page.

With this highly-unstable version of Buildr installed on your unsuspecting system, you should now be able to make the following addition to your buildfile (assuming the directory structure given earlier):

require 'buildr/scala'
 
# rest of the file...

Just like Buildr’s joint compilation for Groovy, you must explicitly require the language, otherwise important things will break.  With this slight modification, you should be able to build your project as per normal:

buildr

This support is so bleeding-edge, I don’t even think that it’s safe to call it “pre-alpha”.  If you run into any problems, feel free to shoot me an email or comment on the issue.

Conclusion

Joint compilation of Java and Scala sources is a profound addition to the Scala feature list, making it significantly easier to use Scala alongside Java in pre-existing or future projects.  With this support, it is finally possible to use Scala as a truly drop-in replacement for Java without modifying the existing infrastructure beyond the CLASSPATH.  Hopefully this article has served to bring slightly more exposure to this feature, as well as provide some much-needed documentation on its use.

Filed in Scala | Comments (4)

What is Hindley-Milner? (and why is it cool?)

29
Dec
2008

Anyone who has taken even a cursory glance at the vast menagerie of programming languages should have at least heard the phrase “Hindley-Milner”.  F#, one of the most promising languages ever to emerge from the forbidding depths of Microsoft Research, makes use of this mysterious algorithm, as do Haskell, OCaml and ML before it.  There is even some research being undertaken to find a way to apply the power of HM to optimize dynamic languages like Ruby, JavaScript and Clojure.

However, despite widespread application of the idea, I have yet to see a decent layman’s-explanation for what the heck everyone is talking about.  How does the magic actually work?  Can you always trust the algorithm to infer the right types?  Further, why is Hindley-Milner is better than (say) Java?  So, while those of you who actually know what HM is are busy recovering from your recent aneurysm, the rest of us are going to try to figure this out.

Ground Zero

Functionally speaking, Hindley-Milner (or “Damas-Milner”) is an algorithm for inferring value types based on use.  It literally formalizes the intuition that a type can be deduced by the functionality it supports.  Consider the following bit of psuedo-Scala (not a flying toy):

def foo(s: String) = s.length
 
// note: no explicit types
def bar(x, y) = foo(x) + y

Just looking at the definition of bar, we can easily see that its type must be (String, Int)=>Int.  As humans, this is an easy thing for us to intuit.  We simply look at the body of the function and see the two uses of the x and y parameters.  x is being passed to foo, which expects a String.  Therefore, x must be of type String for this code to compile.  Furthermore, foo will return a value of type Int.  The + method on class Int expects an Int parameter; thus, y must be of type Int.  Finally, we know that + returns a new value of type Int, so there we have the return type of bar.

This process is almost exactly what Hindley-Milner does: it looks through the body of a function and computes a constraint set based on how each value is used.  This is what we were doing when we observed that foo expects a parameter of type String.  Once it has the constraint set, the algorithm completes the type reconstruction by unifying the constraints.  If the expression is well-typed, the constraints will yield an unambiguous type at the end of the line.  If the expression is not well-typed, then one (or more) constraints will be contradictory or merely unsatisfiable given the available types.

Informal Algorithm

The easiest way to see how this process works is to walk it through ourselves.  The first phase is to derive the constraint set.  We start by assigning each value (x and y) a fresh type (meaning “non-existent”).  If we were to annotate bar with these type variables, it would look something like this:

def bar(x: X, y: Y) = foo(x) + y

The type names are not significant, the important restriction is that they do not collide with any “real” type.  Their purpose is to allow the algorithm to unambiguously reference the yet-unknown type of each value.  Without this, the constraint set cannot be constructed.

Next, we drill down into the body of the function, looking specifically for operations which impose some sort of type constraint.  This is a depth-first traversal of the AST, which means that we look at operations with higher-precedence first.  Technically, it doesn’t matter what order we look; I just find it easier to think about the process in this way.  The first operation we come across is the dispatch to the foo method.  We know that foo is of type String=>Int, and this allows us to add the following constraint to our set:

X  \mapsto  String

The next operation we see is +, involving the y value.  Scala treats all operators as method dispatch, so this expression actually means “foo(x).+(y).  We already know that foo(x) is an expression of type Int (from the type of foo), so we know that + is defined as a method on class Int with type Int=>Int (I’m actually being a bit hand-wavy here with regards to what we do and do not know, but that’s an unfortunate consequence of Scala’s object-oriented nature).  This allows us to add another constraint to our set, resulting in the following:

X  \mapsto  String

Y  \mapsto  Int

The final phase of the type reconstruction is to unify all of these constraints to come up with real types to substitute for the X and Y type variables.  Unification is literally the process of looking at each of the constraints and trying to find a single type which satisfies them all.  Imagine I gave you the following facts:

  • Daniel is tall
  • Chris is tall
  • Daniel is red
  • Chris is blue

Now, consider the following constraints:

Person1 is tall

Person1 is red

Hmm, who do you suppose Person1 might be?  This process of combining a constraint set with some given facts can be mathematically formalized in the guise of unification.  In the case of type reconstruction, just substitute “types” for “facts” and you’re golden.

In our case, the unification of our set of type constraints is fairly trivial.  We have exactly one constraint per value (x and y), and both of these constraints map to concrete types.  All we have to do is substitute “String” for “X” and “Int” for “Y” and we’re done.

To really see the power of unification, we need to look at a slightly more complex example.  Consider the following function:

def baz(a, b) = a(b) :: b

This snippet defines a function, baz, which takes a function and some other parameter, invoking this function passing the second parameter and then “cons-ing” the result onto the second parameter itself.  We can easily derive a constraint set for this function.  As before, we start by coming up with type variables for each value.  Note that in this case, we not only annotate the parameters but also the return type.  I sort of skipped over this part in the earlier example since it only sufficed to make things more verbose.  Technically, this type is always inferred in this way.

def baz(a: X, b: Y): Z = a(b) :: b

The first constraint we should derive is that a must be a function which takes a value of type Y and returns some fresh type Y’ (pronounced like “why prime“).  Further, we know that :: is a function on class List[A] which takes a new element A and produces a new List[A].  Thus, we know that Y and Z must both be List[Y'].  Formalized in a constraint set, the result is as follows:


X  \mapsto  (Y=>Y’ )

Y  \mapsto  List[Y’ ]

Z  \mapsto  List[Y’ ]

Now the unification is not so trivial.  Critically, the X variable depends upon Y, which means that our unification will require at least one step:


X  \mapsto  ( List[Y’ ]=>Y’ )

Y  \mapsto  List[Y’ ]

Z  \mapsto  List[Y’ ]

This is the same constraint set as before, except that we have substituted the known mapping for Y into the mapping for X.  This substitution allows us to eliminate X, Y and Z from our inferred types, resulting in the following typing for the baz function:

def baz(a: List[Y']=>Y', b: List[Y']): List[Y'] = a(b) :: b

Of course, this still isn’t valid.  Even assuming that Y' were valid Scala syntax, the type checker would complain that no such type can be found.  This situation actually arises surprisingly often when working with Hindley-Milner type reconstruction.  Somehow, at the end of all the constraint inference and unification, we have a type variable “left over” for which there are no known constraints.

The solution is to treat this unconstrained variable as a type parameter.  After all, if the parameter has no constraints, then we can just as easily substitute any type, including a generic.  Thus, the final revision of the baz function adds an unconstrained type parameter “A” and substitutes it for all instances of Y’ in the inferred types:

def baz[A](a: List[A]=>A, b: List[A]): List[A] = a(b) :: b

Conclusion

…and that’s all there is to it!  Hindley-Milner is really no more complicated than all of that.  One can easily imagine how such an algorithm could be used to perform far more complicated reconstructions than the trivial examples that we have shown.

Hopefully this article has given you a little more insight into how Hindley-Milner type reconstruction works under the surface.  This variety of type inference can be of immense benefit, reducing the amount of syntax required for type safety down to the barest minimum.  Our “bar” example actually started with (coincidentally) Ruby syntax and showed that it still had all the information we needed to verify type-safety.  Just a bit of information you might want to keep around for the next time someone suggests that all statically typed languages are overly-verbose.

Filed in Scala | Comments (21)

The Joy of Concatenative Languages Part 3: Kindly Types

22
Dec
2008

In parts one and two of this series, we dipped our toes into the fascinating world that is stack-based languages.  By this point, you should be fairly familiar with how to construct simple algorithms using Cat (the language we have been working with) as well as the core terminology of the paradigm.  In fact, with just the information given so far, you could probably go on to be productive with a real-world concatenative language like Factor.  However, the interest does not just stop there…

One of the interesting challenges in programming language design is the construction of a type system.  So as to clear up any possible misconception before it arises, this is how Pierce defines such a thing:

A type system is a tractable syntactic method for proving the absence of certain program behaviors by classifying phrases according to the kinds of values they compute.

For Java, which has a comparatively weak type system, this usually means preventing you from accidentally using a String as if it were an int.  In other words, Java’s type system generally proves the absence of things like NoSuchMethodError and similar.  C#, which has a slightly more-powerful type system, can also prove the absence of most NullPointerException(s) when code is written in a correct and idiomatic fashion.  Scala goes even further with pattern matching…need I go on?  The point is that type systems do different things in different languages, so the definition needs to be flexible enough to reflect that.

In this article, we’re going to look at how we can define a type system for a functional (meaning that we have quotations) concatenative language.  In a comment on the first part of this series, it was suggested that the task of typing stack-based languages is a fairly trivial one.  This is true, but only to a certain point.  As we will see, there are dragons lurking in the conceptual shadows, waiting for us to disturb their sleep.

Simple Expressions

Let’s start out with typing something simple.  Consider the following program:

 

For those of you reading the RSS, what you see between the previous paragraph and this is exactly what I intended to write: nothing at all.  In a concatenative language, the empty program is usually considered to be valid.  After all, it takes a stack as input and returns the exact same stack.  We could replicate the semantics of this program by writing “dup pop“, but why bother?

The empty program has the following type:

(->)

Or, more properly:

('A -> 'A)

To the left of the -< we have what I like to call the “input constraints”: what types must be on the stack coming into the program (or phrase).  To the right of the arrow are the “output constraints”: what types will be on the stack when we’re done.  For reasons which will become clear later on, 'A in this case represents the whole input stack (regardless of what it contains).  Since we never change anything on the stack (the program is, after all, empty), the output stack has whatever type the input stack was given.  Another way of writing this type would be as follows: 

* -> *

This literally symbolizes our intuition that the empty program has no input or output constraints.  However, this is somewhat less correct notationally since it implies that the input and output stacks are unrelated.  In fact, I would go so far as to say that this notation is wrong.  The only reason it is produced here is to serve as a memory aid.  For the remainder of the article, we will be using Cat’s notation for types.

Let’s look at something a little less trivial.  Consider the following program one word at a time:

1 2 +

Remember that an integer literal (or any literal for that matter) is just a function which pushes a specific constant onto the stack.  Let’s assign types based on what we expect the input/output constraints of these functions to be.  Note: I will be using the colon (:) notation to denote a type.  This isn’t conventional coming from C-land, but it is the gold standard of formal type theory:

1 : ('A -> 'A Int)
2 : ('A -> 'A Int)
+ : ('A Int Int -> 'A Int)

This is all very intuitive.  Integer literals work on any stack and just produce that stack with a new Int pushed onto the top.  Both 1 and 2 have the same type, which is a good sign that we’re on the right track.

The + word is a little more interesting.  Its runtime semantics are as follows: pop two integers off the stack, add them together and then push the result back on.  This word will not be able to execute without both integer values on the top of the stack.  Thus, it only makes sense that its input constraints be some stack with two values of type Int at the top.  Likewise, when we’re done, those two integers will be gone and a new Int will be pushed onto the remainder of the stack which was given to us.  Remember that 'A represents any stack, even if it is completely empty.

Coming back to our program, we can see that it is well typed by simply string together the types we have generated.  Starting from the top (using * to symbolize the empty stack):

Word Input Stack Output Stack
1 * * Int
2 * Int * Int Int
+ * Int Int * Int

Do you see how the input stack of each word matches the output stack of the previous?  In this case, this sort of one-to-one matching indicates that the program is well-typed, producing a final stack with a single Int on it.  If we actually run this program, we would see that the evaluation matches the assigned types.

First-Order Functions

This is fine for a simple addition program, but what if we throw functions into the mix?  Consider the same program we just analyzed wrapped up within a function:

define addSome {
  1 2 +
}
 
addSome

Here we define a function which has as a body the program we have already analyzed.  Down at the bottom of our new program, we actually call this function.  Here is the question: what type does the addSome word have?

To answer this question, look back at the table above and consider the Input Stack for the first word in concert with the Output Stack for the last.  Putting these two types together yields the following type for the aggregated whole:

1 2 + : ('A -> 'A Int)

These words (or “phrase”) takes any stack as input, and then through some manipulation produces a single Int on top of that stack as a result.  The stack may grow and shrink within the function, but at the end of the day, only the Int remains.  As we would expect, this matches the runtime semantics perfectly.

Given the fact that the phrase “1 2 +” has the type ('A -> 'A Int), it is reasonable to assign that same type to the function which contains it.  Thus, we can type-check the addSome program in a simple, one-row table:

Word Input Stack Output Stack
addSome * * Int

At the start of execution, the input stack to any program is *, or the empty stack.  However, this is fine with our type checker, since the program has 'A — or any stack — for its input parameters.

This is all so nice and intuitive, so let’s consider the case where we have a function which actually takes some parameters.  Specifically, let’s consider the following definition:

define addTwice {
  + +
}

At runtime, this function will take three values off the stack and then add them all together.  It is the Cat equivalent of the following in Scala:

def addTwice(a: Int, b: Int, c: Int) = a + b + c

The question is: how do we assign this (the Cat function) a type?  As we have done before, let’s look at the types of the individual words:

+ : ('A Int Int -> 'A Int)
+ : ('A Int Int -> 'A Int)

Not much help there.  Let’s try making a table:

Word Input Stack Output Stack
+ * Int Int * Int
+ * Int Int * Int

It’s tempting to look at this and just assign addTwice the type of ('A Int Int -> 'A Int).  However, this would be a mistake.  Notice the problem with our table above: the Input Stack type of the second word does not match the Output Stack of the first.  In other words, this program does not immediately type-check.

The problem is the second word is accessing more of the stack than the first.  We’re effectively “deferring” a parameter access until later in the function, rather than grabbing everything right away and threading the processing through from start to finish.  This is a perfectly reasonable pattern, but it plays havoc with our naive type system.

The solution is to merge the input constraints across both words.  The first word (+) requires two Int(s) to be on the top of the stack.  When it is done, those Int(s) are gone and a single Int has taken their place.  The second word (again +) also requires two Int(s) on the stack.  We only have one that we know of (the output Int from the first word), so we must unify the constraints and merge things back “up the chain” as it were.  In other words, our first word (+) will require not just two Int(s) on the stack but three: two for itself and one for the second word (+).  Our corrected table will look something like the following:

Word Input Stack Output Stack
+ * Int Int Int * Int Int
+ * Int Int * Int

With this new table, all of the Input and Output stacks match, which means that the type is valid and can prove runtime evaluation.  Thus, based on this whole song and dance, we can assign the following type:

addTwice : ('A Int Int Int -> 'A Int)

As expected, this function takes not two, but three Int(s) on the stack and returns the remainder of that stack with a new Int on top.

Polymorphic Words

One mildly-annoying issue that we have just skated over is the problem of polymorphism.  Consider the following two programs:

42 pop

And this…

"fourty-two" pop

The question is: what type do we assign to pop?  We can easily make the following two assertions:

42           : ('A -> 'A Int)
"fourty-two" : ('A -> 'A String)

If we attempt to use this information to type-check the first program (assuming that it is sound), we will arrive at the following type for pop:

pop : ('A Int -> 'A)

That’s intuitive, right?  All that we’re doing here is taking the first value off of the stack (an Int, in the case of the first program) and throwing it away, returning the remainder of the stack.  However, if we use this type, we will run into some serious troubles type-checking the second program:

Word Input Stack Output Stack
"fourty-two" * * String
pop * Int *

Since pop has type ('A Int -> 'A) (as we asserted above), it is inapplicable to a stack with String on top.  Note that we can’t just push these constraints “up the chain”, since it is a case of direct type mismatch, rather than a stack of insufficient depth.  In short: we’re stuck.

The only way to solve this problem is to introduce the concept of parametric types.  Literally, we need to define a type which can be instantiated against a given stack, regardless of what type happens to match the parameters in question.  Java calls this concept “generics”.  Rather than giving pop the overly-restrictive type of ('A Int -> 'A), we will instead allow the value on top of the stack to be of any type (not just Int):

pop : ('A 'a -> 'A)

Note the fact that 'A and 'a are very separate type variables in this snippet.  'A represents the “rest of the stack”, while 'a represents a specific type which just happens to be on top of the input stack.  Using this new, more flexible type, we can produce tables for both of our earlier programs:

Word Input Stack Output Stack
42 * * Int
pop * Int *

 

Word Input Stack Output Stack
"fourty-two" * * String
pop * String *

Everything matches and the world is once again very happy.  Note that we can also apply this parametric type concept to the slightly more interesting example of dup:

dup : ('A 'a -> 'A 'a 'a)

In other words, dup says that whatever type is on top of the stack when it starts, that type will be on top of the stack twice when it is finished.  Just like pop, this type can be instantiated against any stack with at least one type, regardless of whether that type is Int, String, or anything else for that matter.

Higher-Order Functions

We’ve seen how to type-check simple phrases, as well as first-order functions with deferred stack access and the occasional polymorphic word.  However, there is one particularly troublesome aspect of concatenative type systems which we have completely ignored: functions which take quotations off the stack.  In other words: what type do we assign to apply?  Consider the following function:

define trouble {
  apply
}

At runtime, trouble will pop a quotation and then evaluate it against the remainder of the stack.  Intuitively, we need to have some way of representing the type of a quotation, but that’s not even the most serious problem.  Somehow, we need to constrain the quotation to itself accept exactly the stack which remains after it is popped.  We also need to find some way of capturing its output type in order to compute the final output type of trouble.

More concretely, we can make a first attempt at assigning a type for trouble.  The underscores (_) illustrate an area where our type system is incapable of helping us:

trouble : (_ (_ -> _) -> _)

It’s very tempting to just throw an 'A in there and be done with it, but the truth is that for this type expression, there is no “unused stack”.  We don’t really know how much (or how little) of the stack will be used by the quotation; it could pop five elements, twenty or none at all.  It literally needs access to the remainder of the input stack in its entirety, otherwise the expression is useless.  Enter stack polymorphism…

Just as we needed a way to represent any single type in order to type-check pop and dup, we now need a way to represent any stack type in order to type-check apply.  Fortunately, the answer is already nestled within our pre-established notation.  Consider the type of +:

+ : ('A Int Int -> 'A Int)

We have been taking this to mean “any stack with two Int(s) on top resulting in that same stack with only one Int“.  This is true, but we’re being a little hand-wavy about the meaning of “any stack” and how it relates to 'A.  When we really get down to it, what’s happening here is 'A is being instantiated against a particular input stack, whatever that stack happens to be.  When we were type-checking + +, the first word instantiated 'A not to mean the empty stack (*), but rather a stack with at least one Int on it.  This was required to successfully type the second +.

We can very easily extend this notational convenience to represent generalized stack parameters.  Rather than being instantiated to specific types, stack parameters are instantiated to some stack in its entirety.  Just as with type parameters, wherever we see that instantiated stack parameter within a type expression, it will be replaced with whatever stack type it was assigned.  Thus, we can assign trouble the following type:

trouble : ('A ('A -> 'B) -> 'B)

In other words, trouble takes some stack A which has a quotation on top.  This quotation accepts stack A itself and returns some new stack B.  Note that we don’t really know anything about B.  It could be related to A, but it might not be.  The final result of the whole expression is this new stack B.

This concept is remarkably powerful.  With it in combination with the other types we have already examined, we can type check the entirety of Cat and be assured of the absence of type-mismatch and stack-underflow errors.  Considering the fact that Cat is almost exactly as powerful as Joy, that’s a pretty impressive feat.

From a theoretical standpoint, things get even more interesting when we consider the type of the following function:

define y {
  [dup papply] swap compose dup apply
}

This has the following type:

y : ('A ('A ('A -> 'B) -> 'B) -> 'B)

As you may have guessed by the name, this is the Y-combinator1, one of the most well-known mechanisms for producing recursion in a nameless system.  Note that this definition looks a little different from the pure-untyped lambda calculus (call-by-name semantics):

λf . (λx . f (x x)) (λx . f (x x))

What I’m trying to point out here is the fact that Cat is able to leverage its type system to assign a type to the Y-combinator.  This is something which is literally impossible in System F, a typed form of lambda-calculus.  In fact, the only way to type-check this function in a lambda-calculus-derivative system would be to add recursive types.  Cat is able to get by with a very much non-recursive type definition, something which I find fascinating in the extreme.

Update: The above paragraph is somewhat misleading.  It turns out that Cat actually does use a recursive type under the surface to derive the non-recursive type for y.  Specifically:

dup papply : ('A ('B ('B self -> 'C) -> 'C) -> 'A ('B -> 'C))

On a further theoretical note, the device in Cat’s type system which allows this power is in fact the stack type variable (e.g. 'A).  These stack types are conceptually quite similar to the type parameters we used in typing pop (e.g. 'a), but still in a very separate domain.  In fact, stack types have a different kind than regular types.  This is not to say that Cat employs higher-kinds such as Scala’s (e.g. * => *), but it does have two very different type kinds: stacks and values.

And yet, it is not kinds in and of themselves which allows for the typing of the Y-combinator.  Fω is essentially System F with higher-kinds, and yet it is still incapable of handling this tiny little expression.  Most interesting indeed…

Conclusion

As you can see, type systems and concatenative languages do fit together nicely, but it takes a lot more effort than one would initially expect.  While typing simple expressions is easy enough, the waters are muddied as soon as higher-order functions and even deferred stack access enters the mix.  This is an extremely fertile area for research, where a lot of interesting ideas are being developed.  For example, John Nowak’s 5th attempts to apply a type system to the stack-based paradigm, but in a very different way than Cat.

I hope you enjoyed this mini-series of articles on concatenative languages.  While they are a bit of a backwater in the programming language menagerie, I think that studying them can be a very instructive experience.  Furthermore, there remain some problems that are very nicely expressed in languages like Cat while being extremely unwieldy in more conventional languages like Scala.  Despite the obscurity of concatenative languages, it never hurts to have an extra language on hand, ready for those times when it really is the best tool for the job.

1 Technically, this is a little different from the Y-combinator used in conventional lambda-calculus (it executes the quotation rather than returning a fixed-point).  However, conceptually it is the same idea.

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The Joy of Concatenative Languages Part 2: Innately Functional

15
Dec
2008

In part one of this series, I introduced the concept of a stack-based language and in particular the syntax and rough ideas behind Cat.  However, to anyone coming into concatenative land for the first time, my examples likely seemed both odd and unconvincing.  After all, why would you ever use point-free programming when everyone else seems to be sold on the idea of name binding?  More importantly, where do these languages fit in with our established menagerie of language paradigms?

The answer to the first question really depends on the situation.  I personally think that the best motivation for concatenative languages is their syntax.  If you want to create an internal DSL, there will be no language better suited to it than one which is concatenative, Cat, Factor or otherwise.  This is because stack-oriented languages can get away with almost no syntax whatsoever.  They say that Lisp is a syntax-free language, but this holds even more strongly for languages like Cat.  Well, that and you don’t have to deal with all the parentheses…

The second question is (I think) the more interesting one: how do we classify these languages and what sort of methodologies should we apply?  At first glance, Cat (and other languages like it) seem to be quite imperative in nature.  After all, you have a single mutable stack that any function can modify.  However, if you turn your head sideways and blink twice, you begin to realize that concatenative languages are really much closer to the functional side of the oyster.

Consider the following Cat program:

define plus { + }
define minus { - }
 
7 2 3
plus minus

Trivial, but to the point.  This program first adds the numbers 2 and 3, then subtracts the result from 7.  Thus, the final result is a value of 2 on the stack.  The only twist is that we have defined functions plus and minus to do the dirty work for us.  This wasn’t strictly necessary, but I wanted to emphasize that + and - really are functions.  We could express the exact same program in Scala:

def plus(a: Int, b: Int) = a + b
def minus(a: Int, b: Int) = a - b
 
minus(7, plus(2, 3))

Do you see how the consecutive invocations of plus and minus in Cat became composed invocations in Scala?  This is where the term “concatenative language” derives from: the whole program is just a series of function compositions.  Wikipedia’s article on Cat has a very nice, mathematical description:

Two adjacent terms in Cat imply the composition of functions that generate stacks, so the Cat program f g is equivalent to the mathematical expressions and , where x is the stack input to the expression.

Strictly speaking, a concatenative language could be implemented without a stack, but such an implementation would likely be a bit harder to use than the average stack-based language.

Coming back to my original premise: concatenative languages are functional in nature.  Absolutely everything in Cat is a function.  Operators, words, even numeric literals like “3” are actually functions at the conceptual level.  Additionally, Cat, Joy and Factor all offer a mechanism for treating functions as first-class values:

2 3
[ + ]
apply

The square-bracket ([]) syntax is representative of a quotation.  Literally this mean “create a function of the enclosed words and place it as a value on the stack”.  We can pop this function off the stack and invoke it by using the apply word.  Incidentally, you may have noticed that this syntax is remarkably close to that which is used in if conditionals:

5 0 <
[ "strange math" ]
[ "all is well" ]
if

This syntax works because if isn’t conceptually a language primitive: it’s just another function which happens to take a boolean and two quotations off the stack.  For the sake of efficiency, Cat does indeed implement if as a primitive, but this was a deliberate optimization rather than an implementation forced by the language design.  Untyped Cat (see Part 3) is equivalent in power to the pure-untyped lambda calculus, and as our friend Alonzo Church showed us, if-style conditionals are easily accomplished:

TRUE = λa . λb . a
FALSE = λa . λb . b

IF = λp . λt . λe . p t e

Yeah, maybe we’re drifting a bit off-point here…

Higher-Order Programming

So if Cat is just another functional programming language, then we should be able to implement all of those higher-order design patterns that we’ve come to know and love in languages like Scala and ML.  To see how, let’s look at implementing some simple list manipulation functions in Cat.  The easiest would be to start with append, which pops two lists off of the stack and pushes a new list which is the end-to-end concatenation of the two originals:

define append {
  empty
  [ pop ]
  [
    uncons
    [append] dip
    cons
  ]
  if
}

This function first starts by checking to see if the top list is empty.  If so, then just pop it off the stack and leave the other right where it is.  Appending an empty list should always yield the original list.  However, if the head list is not empty, then we need to work a bit.  First, we decompose it into its tail and head, which are pushed onto the stack in order by the uncons function.  Next, we need to recursively append the tail with our second list on the stack.  However, the head of the list from uncons is in the way on top of the stack.  We could use stack manipulation to move things around and get our lists up to the head of the stack, but dip provides us with a handy, higher-order shortcut.  We temporarily remove the top of the stack, invoke the quotation “[append]” against the remainder and then push the old top back on top of the result.

The dip operation is surprisingly powerful, making it possible to completely live without either variables or multiple stacks.  Any non-trivial Cat program will need to make use of this handy function at some level.

Once we have the old head and the new appended-list on the stack, all we need to do is put them back together using cons.  This function leaves a new list on the stack in place of the old list and head element.  This Cat program is almost precisely analogous to the following ML:

fun append ls nil = ls
  | append ls (hd :: tail) = hd :: (append ls tail)

Personally, I find the ML a lot easier to read, but that’s just me.  Obviously it’s a lot shorter, but as it turns out, our Cat implementation, while intuitive, was sub-optimal.  Cat already implements append in the guise of the cat function, and it is far more concise than what I showed:

define cat {
  swap [cons] rfold
}

It’s almost frightening how short this is: only three words.  It’s not as if rfold is doing anything mysterious either; it’s just a simple right-fold function that takes a list, an initial value and a quotation, producing a result by traversing the entire list.  We can use something similar back in ML-land, achieving an implementation which is arguably equivalent in subjective elegance:

val append = foldr (op::)

Moving on, we can also implement a length function in Cat, this time using fold to tighten things up:

define length {
  0 [ pop 1 + ] fold
}

You’ll notice that we have to mess around a bit in the quotation in order to avoid the first “parameter”, the current element of the list (which we do not need).  Expressing this in ML yields a very similar degree of cruft:

val length = foldl (fn (n, _) => n + 1) 0

Conclusion

The important take-away from this tangled morass of an article is the fact that Cat is a highly functional language, capable of easily keeping up with some of the stalwart champions of the paradigm.  More significantly, this is a trait which is shared by all concatenative languages.  Rather than throwing away all of the old wisdom learned in language design, stack-based languages build on it by providing an alternative view into the world of functions.

In the next (and final) article of the series, we will take a brief look at the challenges of applying a type system to a concatenative language and the fascinating techniques used by Cat to achieve just that.

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The Joy of Concatenative Languages Part 1

8
Dec
2008

Concatenative languages like Forth have been around for a long time.  Hewlett-Packard famously employed a stack-based language called “RPL” on their HP-28 and HP-48 calculators, bringing the concept of Reverse Polish Notation to the mainstream…or as close to the mainstream as a really geeky toy can get.  Surprisingly though, these languages have not seen serious adoption beyond the experimental and embedded device realms.  And by “adoption”, I mean real programmers writing real code, not this whole interpreted bytecode nonsense.

This is a shame, because stack-based languages have a remarkable number of things to teach us.  Their superficial distinction from conventional programming languages very quickly gives way to a deep connection, particularly with functional languages.  However, if we dig even deeper, we find that this similarity has its limits.  There are some truly profound nuggets of truth waiting to be uncovered within these murky depths.  Shall we?

Trivial aside: I’m going to use the terms “concatenative” and “stack-based” interchangeably through the article.  While these are most definitely related concepts, they are not exactly synonyms.  Bear that in mind if you read anything more in-depth on the subject.

The Basics

Before we look at some of those “deeper truths of which I speak, it might be helpful to at least understand the fundamentals of stack-based programming.  From Wikipedia:

The concatenative or stack-based programming languages are ones in which the concatenation of two pieces of code expresses the composition of the functions they express. These languages use a stack to store the arguments and return values of operations.

Er, right.  I didn’t find that very helpful either.  Let’s try again…

Stack-based programming languages all share a common element: an operand stack.  Consider the following program:

2

Yes, this is a real program.  You can copy this code and run/compile it unmodified using most stack-based languages.  However, for reasons which will become clear later in this series, I will be using Cat for most of my examples.  Joy and Factor would both work well for the first two parts, but for part three we’re going to need some rather unique features.

Returning to our example: all this will do is take the numeric value of 2 and push it onto the operand stack.  Since there are no further words, the program will exit.  If we want, we can try something a little more interesting:

2 3 +

This program first pushes 2 onto the stack, then 3, and finally it pops the top two values off of the stack, adds them together and pushes the result.  Thus, when this program exits, the stack will only contain 5.

We can mix and match these operations until we’re blue in the face, but it’s still not a terribly interesting language.  What we really need is some sort of flow control.  To do that, we need to understand quotations.  Consider the following Scala program:

val plus = { (x: Int, y: Int) => x + y }
plus(2, 3)

Notice how rather than directly adding 2 and 3, we first create a closure/lambda which encapsulates the operation.  We then invoke this closure, passing 2 and 3 as arguments.  We can emulate these exact semantics in Cat:

2 3
[ + ]
apply

The first line pushes 2 and 3 onto the stack.  The second line uses square brackets to define a quotation, which is Cat’s version of a lambda.  Note that it isn’t really a closure since there are no variables to enclose.  Joy and Factor also share this construct.  Within the quotation we have a single word: +.  The important thing is the quotation itself is what is put on the stack; the + word is not immediately executed.  This is exactly how we declared plus in Scala.

The final line invokes the apply word.  When this executes, it pops one value off the stack (which must be a quotation).  It then executes this quotation, giving it access to the current stack.  Since the quotation on the head of the stack consists of a single word, +, executing it will result in the next two elements being popped off (2 and 3) and the result (5) being pushed on.  Exactly the same result as the earlier example and the exact same semantics as the Scala example, but a lot more concise.

Cat also provides a number of primitive operations which perform their dirty work directly on the stack.  These operations are what make it possible to reasonably perform tasks without variables.  The most important operations are as follows:

  • swap — exchanges the top two elements on the stack.  Thus, 2 3 swap results in a stack of “3 2” in that order.
  • pop — drops the first element of the stack.
  • dup — duplicates the first element and pushes the result onto the stack.  Thus, 2 dup results in a stack of “2 2“.
  • dip — pops a quotation off the stack, temporarily removes the next item, executes the quotation against the remaining stack and then pushes the old head back on.  Thus, 2 3 1 [ + ] dip results in a stack of “5 1“.

There are other primitives, but these are the big four.  It is possible to emulate any control structure (such as if/then) just using the language shown so far.  However, to do so would be pretty ugly and not very useful.  Cat does provide some other operations to make life a little more interesting.  Most significantly: functions and conditionals.  A function is defined in the following way:

define plus { + }

Those coming from a programming background involving variables (that would be just about all of us) would probably look at this function and feel as if something is missing.  The odd part of this is there is no need to declare parameters, all operands are on the stack anyway, so there’s no need to pass anything around explicitly.  This is part of why concatenative languages are so extraordinarily concise.

Conditionals also look quite weird at first glance, but under the surface they are profoundly elegant:

2 3 plus    // invoke the `plus` function
10 <
[ 0 ]
[ 42 ]
if

Naturally enough, this code pushes 0 onto the stack.  The conditional for an if is just a boolean value pushed onto the stack.  On top of that value, if will expect to find two quotations, one for the “then” branch and the other for the “else” branch.  Since 5 is less than 10, the boolean value will be True.  The if function (and it could just as easily be a function) pops the quotations off of the stack as well as the boolean.  Since the value is True, it discards the second quotation and executes the first, producing 0 on the stack.

I’ll leave you with the more complicated example of the factorial function:

define fac {
  dup 0 eq
  [ pop 1 ]
  [ dup 1 - fac * ]
  if
}

Note that this isn’t even the most concise way of writing this, but it does the job.  To see how, let’s look at how this will execute word-by-word (assuming an input of 4):

Stack Word
4

dup
4 4

0
4 4 0

eq
4 False

[ pop 1 ]
4 False [pop 1]

[ dup 1 - fac * ]
4 False [pop 1] [dup 1 - fac *]

if
4

dup
4 4

1
4 4 1

-
4 3

fac (assume magic recursion)
4 6

*

The final result is 24, a value which is left on the stack.  Pretty nifty, eh?

Conclusion

You’ll notice this is a shorter post than I usually spew forth (no pun intended…this time).  The reason being that I want this to be fairly easy to digest.  Concatenative languages (and Cat in particular) are not all that difficult to digest.  They are a slightly different way of thinking about programming, but as we will see in the next part, not so different as it would seem.

Note: Cat is written in C# and is available under the MIT License.  Don’t fear the CLR though, Cat runs just fine under Mono.  If you really want to experiment with no risk to yourself, a Javascript interpreter is available.

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