Higher-Order Java Parallelism, Part 2: Parallel List Transformations

With a Callable monad and Parallel Strategies in hand, we’re ready to construct some general-purpose parallel functions of a higher order.

Something we’ll want to do quite often in parallel programming is run a function over a whole list of values, in parallel. So often, in fact, that it would be very tedious and repetitive to write a loop every time we want to do this, sparking off threads for each function call, and collecting the results. Instead, what we’re going to develop next is a kind of parallel list functor. That is, a higher-order function that will turn any function into a function that operates on every element of a list simultaneously.

First, let’s get some terminology out of the way, since I’ve been accused of not defining my terms.

A higher-order function is simply a function that takes a function as an argument.

A functor is any type T for which there exists a higher-order function, call it fmap, that transforms a function of type F<A,B> into a function F<T<A>,T<B>>. This fmap function must also obey the laws of identity and composition such that the following expressions return true for all x, p, and q:

  fmap(identity()).f(x) == x
  fmap(compose(p, q)).f(x) == fmap(p).f(fmap(q).f(x))

Here, identity is a first-class function of type F<A,A> that just returns its argument, and compose is function composition. Composition is defined to mean that compose(p, q).f(x) is the same as p.f(q.f(x)).

For example, Functional Java’s List class is a functor, because it comes equipped with a method List<B> map(F<A,B> f) which obeys the laws above. The implementation of List’s map method is trivial: It iterates over the list, calling the given function for each element, and returns the list of the results.

There’s a lot of information about functors and monads on the Internet, and I’ve written about them before, with examples in Java. Tony Morris also wrote a great post where he explains functors as “function application in an environment”. That’s a good way of looking at things.

I also want to make it clear, if you’re just now tuning in, that I’m using classes from Functional Java in the code examples, in addition to classes from the J2SE concurrency library. Basically only Callable and Future are from the J2SE library. Strategy, List, and the Callables utility class are from Functional Java. The code used here is from the latest trunk head revision of that library and may differ from the current official release.

Now, let’s get to the money.

Mapping Over a List in Parallel

We’re going to make a very similar kind of thing to a list functor, except it’s going to map a function over a list in parallel. Our parallel list transformation will take the form of a static method called parMap. It will take any existing function and turn it into a parallel function on lists. That is, if we already have written a function that takes an Integer and returns a String, we can pass that function to parMap, and it will return us a function that takes a list of Integers and returns a list of Strings. This means that nothing has to be rewritten or refactored to support the mapping of a function over a list in parallel. The parMap function will convert our existing code to parallel code, at runtime.

Here’s parMap:

  public <B> F<List<B>, Callable<List<A>>> parMap(final F<B, A> f) {
    return new F<List<B>, Callable<List<A>>>() {
      public Callable<List<A>> f(final List<B> as) {
        return sequence(as.map(compose(par(), callable(f))));
      }
    };
  }

That’s a lot of power for so little code. Let’s read through what it does. It takes a function f and returns a new function that takes a list called as. This new function transforms f with Callables.callable to wrap its return type in a callable (or: callable(f) applies the Kleisli arrow for Callables to f), and composes that with a function called par(). It then maps the resulting composition over the list as, which will result in a list of Callables, which we turn into a Callable of a List with the sequence function. The only new thing here is par(), which is a very simple instance method on Strategy:

  public F<Callable<A>, Callable<A>> par() {
    return compose(Strategy.<A>obtain(), f());
  }

So par() is just the Strategy’s function composed with the obtain() method that we created in Part 1. This turns the Future back into a Callable so that we can manipulate it in a lazy manner. This little guy is the meat of parMap. That is, parMap is basically just mapping par() over a list.

The result of parMap(myFun).f(myList), then, is a Callable of a list that, when called, will give us the results of calling myFun on every element of myList in parallel. What’s more, it will work for any kind of parallelisation Strategy, and it will return immediately (remember, Callable is lazy), ready for us to make further calculations on the results even while those results are being calculated.

I think that’s pretty cool.

Nested Parallelism

Sometimes the problem at hand is not quite so flat that it can be solved with just a map over a list. You might have a need for nested traversal of lists, where for every element in a list you traverse another list. We can model that behaviour with a higher-order function very similar to parMap, except that it takes a function that generates a list. This higher-order function is parFlatMap:

  public static <A, B> Callable<List<B>> parFlatMap(final Strategy<List<B>> s,
                                                    final F<A, List<B>> f,
                                                    final List<A> as) {
    return fmap(List.<B>join_()).f(s.parMap(f).f(as));
  }

In this definition, fmap is from the Callable monad, as described in Part 1. Notice that parFlatMap provides an example use of parMap, using it to turn the given function into a parallel function. The call to parMap with that function will actually yield a Callable<List<List<B>>> (can you see why?), so we use a final join lifted into the Callable environment with fmap, to turn the List of Lists into just a List.

The parFlatMap function is analogous to a parallel list monad, and it works very much like a parallel version of nested loops. For example, given a list of Points pts, a distance function dist that takes two Points and returns the distance between them, and a Strategy<Double> s, we can use parFlatMap to calculate the distance between every Point and every other, in parallel:

  Callable<List<Double>> distances = parFlatMap(s, new F<Point, List<Double>>() {
    public List<Double> f(final Point p) {
      return pts.map(dist.f(p));
    }
  }, pts);

The inner “loop” is represented by List.map, so we’re traversing the list serially, multiplying every element in parallel with each element in sequence. I.e. the outer loop is parallel and the inner loop is serial. It’s quite possible to replace map with parMap, and I’ll leave that as an exercise for the reader (hint: you will need both fmap and join). Note that the call to parFlatMap will return immediately, and the computation will be carried out in the background.

Position-Wise Parallelism and Concurrent Folding

Another function I want to talk about is the parZipWith function. By contrast to parFlatMap, it is like a parallel version of a single loop over two lists. Values from each list are taken at matching positions and the pairs are fed through the argument function all at the same time:

  public static <A, B, C> F2<List<A>, List<B>, Callable<List<C>>> parZipWith(final Strategy<C> s,
                                                                             final F2<A, B, C> f) {
    return new F2<List<A>, List<B>, Callable<List<C>>>() {
      public Callable<List<C>> f(final List<A> as, final List<B> bs) {
        return sequence(as.zipWith(bs, compose(Callables.<B, C>arrow(), curry(f))).map(s.par()));
      }
    };
  }

You will note that parZipWith simply uses zipWith from Functional Java’s List class. There’s some manouvering required to turn the argument function into the right form. The curry method turns the F2<A,B,C> into the equivalent F<A,F<B,C>> (sometimes called a Curried function). The arrow function is the first-class Kleisli arrow for Callables. That just means that composing it with f yields a version of f which returns a Callable.

Let’s take an example of using parZipWith to do useful work. If we had a list of Points representing a path, and we wanted to get the total length of the path, we could zip the list and its tail (every element except the first) with the dist function and take the sum of the results:

  Double length = parZipWith(s, dist).f(pts, pts.tail()).call().foldLeft(curry(sum), 0);

For brevity’s sake, I’m pretending that there’s already a function called sum, which is just an F2<Double, Double, Double> that returns the sum of its two arguments. If foldLeft is something new to you, then have a look at one implementation of it. It’s a more general abstraction of a for-loop than map() is. Folding a list {1,2,3} with the sum function (and 0) can be thought of as replacing all the commas with plusses to yield [0+1+2+3]. We could have just as well iterated over the list, adding each value to a total, instead of using a foldLeft function. But we didn’t, for a reason that will become apparent presently.

There’s something about the above example that’s dissatisfying. We’re getting the distances in parallel, but then call() sticks out like a sore thumb. The computation will block until the list of distances is ready, then it will fold the list with the sum function. What’s worse, call() might throw an Exception. There’s a way we can avoid all that by traversing the list of distances while they’re being calculated. What we need to do is somehow lift the foldLeft function into the Callable environment. The call to foldLeft above turns a List<Double> into a Double. What we want is to avoid the call to call(), and turn our Callable<List<Double>> into a Callable<Double> directly.

This is really easy to do, and it’s a good thing we’re using a folding function rather than a for-loop. Here’s what we do:

  Callable<List<Double>> edges = parZipWith(s, dist).f(pts, pts.tail());
  Callable<Double> length = par(fmap(List.<B, B>foldLeft().f(curry(sum)).f(0)).f(edges));

There. No exceptions, and no waiting for results. This code will return immediately and perform the whole computation in the background. The part that calculates the edge lengths is the same as before, and we’re keeping that in a variable edges for clarity. The other line may require some explanation. We’re getting a first-class foldLeft function from the List class, partially applying it to yield a function that folds a List<Double> into a Double. Then we’re promoting it with fmap into the Callable monad. The promoted function then gets applied to edges, and finally the whole fold is evaluated concurrently by passing it to par.

Hopefully you can see what an immensely powerful tool for parallel programming functional style and higher-order functions are, even in Java. We were able to develop a couple of one-liners above that are terser, clearer, more maintainable, more re-usable (and more type-safe) than anything we could write in imperative object-oriented style using producer/consumer, factories, inversion of control, etc. There’s no secret sauce, no special syntax, no functional fairy dust. It’s just plain old Java with a handful of very useful interfaces.

In the third installment of this series, we will develop a tiny light-weight library for threadless Actors that can juggle millions of simultaneous computations in only a few threads. Stay close.

Note: The code herein is adapted from a Haskell library by Phil Trinder, Hans-Wolfgang Loidl, Kevin Hammond et al.

Higher-Order Java Parallelism, Part 1: Parallel Strategies and the Callable Monad

Now that even budget desktop and laptop computers are shipping with multi-core processors, it’s more important than ever to design programs so that they can take advantage of parallel processing. If you’re already writing software in Erlang or Parallel Haskell, then lucky you. But if you’re writing in Java (and are unable or unwilling to take up Scala), you will need to get organized.

Since Java 5, the JDK comes with a concurrency library that makes concurrent programming a lot easier than it used to be, and the library has been improved further in Java 6. This provides some basic abstractions that take care of the nitty-gritty of thread management for us. In this short series of articles, I’m going to show how we can build on top of that library to achieve concurrent programming of a higher order. We will employ some design patterns and simple abstract building blocks from functional programming. This style of writing concurrent programs will afford us the ability to:

• Compose ordinary functions into concurrent programs.
• Decouple parallel behavior from the algorithm itself.
• Turn existing code into parallel code, without refactoring, at runtime.
• Work with hypothetical results of concurrent computations, before they finish.

We will not use any locking or explicit synchronization. Basic familiarity with the java.util.concurrent package is assumed but not required. Familiarity with generics is highly recommended, and I recommend reading parts 1 and 2 of my Lazy Error Handling series as it explains some preliminaries. Most of the source code herein is already part of the Functional Java library, and I’ll use a fair bit of interfaces and classes from it, which I’ll introduce as we go along.

When we’re through, I will have demonstrated that a functional style of programming promotes code-reuse and modularity of parallel programs, above and beyond what can be achieved with the canonical object-orientated style.

The Callable Monad

The Callable interface is new in Java 5. It’s similar to the old Runnable interface that we all know and love, except that its call() method returns a result, and it may throw an Exception. If you followed my Lazy Error Handling articles, you will immediately notice that Callable is nearly identical to the Thrower interface described in that series, and you’ll know that it is indeed a potential monad. Treating Callable as a monad will allow us to work with it at a higher level of abstraction, letting us compose Callables, chain them, and project computations into them.

As you may already know, we need three ingredients (and only these three) to have a monad:

1. Type construction. This comes free with the Callable interface, since it takes a type parameter (i.e. it’s generic).
2. A unit function. This allows us to turn any value into a Callable that returns that value again.
3. A binding function. This allows us to chain together Callables with functions, without actually calling them.

Here is the unit function for Callable (we will refer to it as unit in the text to avoid ambiguity):

  public static <A> Callable<A> callable(final A a) {
    return new Callable<A>() {
      public A call() {
        return a;
      }
    };
  }

And here is its binding function:

  public static <A, B> Callable<B> bind(final Callable<A> a, final F<A, Callable<B>> f) {
    return new Callable<B>() {
      public B call() throws Exception {
        return f.f(a.call()).call();
      }
    };
  }

Note: The F interface is from Functional Java. It represents a first-class function (a function that can be treated like any other value). F<A, B> has one method that takes an argument of type A and returns a value of type B.

Given the above two methods, Callable is now a monad. All monads are also functors, so we can define a Callable functor. What I mean by “functor” is that we can define a method to turn any existing function into a function on Callables, or, in other words, to apply any function to a value wrapped in a Callable, without actually calling it (i.e. lazily). This method is called fmap, and we can define it in terms of bind:

  public static <A, B> F<Callable<A>, Callable<B>> fmap(final F<A, B> f) {
    return new F<Callable<A>, Callable<B>>() {
      public Callable<B> f(final Callable<A> a) {
        return bind(a, new F<A, Callable<B>>() {
          public Callable<B> f(final A ab) {
            return new Callable<B>() {
              public B call() {
                return f.f(ab);
              }
            };
          }
        });
      }
    };
  }

Useful operations on Callables, that the methods above allow us to implement, include sequence—which is a method of turning a list of Callables into a single Callable that returns a list—and join, which peels one layer of Callables from a Callable<Callable<A>> so that it becomes just Callable<A>. You will find the source code for those as part of Functional Java.

Parallel Strategies

When working with the java.util.concurrent library, you normally don’t work with Threads directly. You might instead implement Callable<A> and submit instances of your implementation to an ExecutorService, which yields values of type Future<A> which represents a running computation that yields an A. Future has a method called get() that returns the result as soon as it’s ready. This is a great improvement over managing threads and Runnables ourselves, but it still ties our hands somewhat to the ExecutorService, and encourages tight coupling of our code to the parallelisation library.

Inspired by Haskell’s “Parallel Strategies”, let’s instead work with parallelisation in the abstract. For what is ExecutorService, really? It’s a method of turning Callables into Futures. That means it’s a kind of function, so we can abstract from it, using a function type: F<Callable<A>, Future<A>>. Such a function can then be backed by an ExecutorService, or by something else entirely, such as a load-balancing facility that serializes Callables to be executed on remote servers.

We will use a new class, Strategy<A>, that allows us to effectively separate parallelism from the algorithm itself:

  public final class Strategy<A> {

    private F<Callable<A>, Future<A>> f;

    private Strategy(final F<Callable<A>, Future<A>> f) {
      this.f = f;
    }

    public F<Callable<A>, Future<A>> f() {
      return f;
    }

    public static <A> Strategy<A> strategy(final F<Callable<A>, Future<A>> f) {
      return new Strategy<A>(f);
    }

  }

We’ll add a couple of static functions to create simple strategies:

  public static <A> Strategy<A> simpleThreadStrategy() {
    return strategy(new F<Callable<A>, Future<A>>() {
      public Future<A> f(final Callable<A> p) {
        final FutureTask<A> t = new FutureTask<A>(p);
        new Thread(t).start();
        return t;
      }
    });
  }

  public static <A> Strategy<A> executorStrategy(final ExecutorService s) {
    return strategy(new F<Callable<A>, Future<A>>() {
      public Future<A> f(final Callable<A> p) {
        return s.submit(p);
      }
    });
  }

One of the neat things that working with Strategies as functions allows us to do is use the Callable monad to compose them with existing functions. Any function can be lifted into the Callable monad using fmap, and then composed with a Strategy to yield a concurrent function. Moreover, we can use Strategies to convert existing functions to concurrent functions. The following method on Strategy will take any function and return the equivalent function that executes concurrently. Calling such a function will give you a Future value from which you can get the computed result whenever it’s ready.

  public <B> F<B, Future<A>> lift(final F<B, A> f) {
    final Strategy<A> self = this;
    return new F<B, Future<A>>() {
      public Future<A> f(final B b) {
        return self.f().f(new Callable<A>() {
          public A call() {
            return f.f(b);
          }
        });
      }
    };
  }

Alert readers will note that lift represents the Kleisli arrow for the Future monad. As for most monads, this arrow is a very useful kind of thing (see Lazy Error Handling, Part 2). In the Future monad, the Kleisli arrow provides parallel function application. If you already have a function f, then calling lift(f).f(x) will apply that function to x while simultaneously continuing with the next statement in the current program.

Lazy Futures

For functions involving Futures, we generally always want to have the Future type in the codomain (on the right-hand side, in the return type). If you think about it, you’ll see that functions that take Futures in their arguments won’t be able to do much without blocking on Future.get(). However, that doesn’t mean we can’t compose Future-valued functions together. It just means that we can only compose two of them together in such a way that we either wait for the first Future to obtain a value before firing off the second one, or we have to spark a new thread that does nothing but wait on the first Future. We’re much better off composing Callables and then turning those into Futures as needed. In fact, we might want to wrap values of type Future<A> inside of a Callable<A> again so that we can manipulate their return values while they are running:

  public static <A> Callable<A> obtain(final Future<A> x) {
    return new Callable<A>() {
      public A call() throws Exception {
        return x.get();
      }
    };
  }

And this takes us full circle back into the Callable monad, where we can compose computations, bind them to functions and map functions over them, all lazily. Which means: without actually asking what their values are until we absolutely need to know.

So we have accomplished what we set out to do in this first part of the series. We’ve written a Callable monad that lets us compose existing code into concurrent programs. We’ve implemented parallel strategies to run those programs in parallel, without regard to the actual parallelisation implementation. We have a Kleisli arrow that lets us run ordinary functions concurrently with each other, and finally we have a way of taking concurrently running computations back into the Callable monad. Not bad for a day’s work.

In the next part of this series, we will employ what we’ve built here to develop some higher-order parallel functions, including a parallel list functor. I hope you join me.

Lazy Error Handling in Java, Part 3: Throwing Away Throws

In parts 1 and 2, I covered how we could create first-class functions in Java that declare, in their types, the exceptions they might throw. I talked about exploiting a functor to lift any existing function so that it carries exception information with it, and how to compose functions of that kind by using a monad. In this third and last installment, we’re going to fully abandon checked exceptions and embrace the functional style of handling errors.

First, let’s examine what exceptions really are. During the evaluation of some expression (some function), unexpected conditions or data are encountered, with which it is undefined how to proceed. That is to say, for some inputs, the result of the computation is undefined. For example, when a parseInt function encounters a character that isn’t a digit (or a minus sign), what do we do? Well, we don’t know exactly, so we need some way to indicate failure. It would be invalid to return an integer. One could return null, but that isn’t exactly reliable, nor correct, and well, it would be lying since the type signature promises an integer.

So far, I’ve shown how we can use a Thrower monad to indicate that a computation can error. But the Thrower monad doesn’t do anything except wrap Java’s existing exception handling with a datastructure. And what do you think Java’s existing error handling is? It’s a monad. No, really, it is. In fact, Java is one big monad with bells on.

Note that throws changes the type of a method so that it may throw exceptions. Sound familiar? That’s what Thrower does (see part 1). Java’s throws keyword is the Kleisli arrow (see part 2) for Java’s exceptions functor. Binding is done with a combination of try and a Java keyword called “;” which I’m sure you use all the time without even batting an eye. You can then think of catch as a function that takes a monadic value and “extracts” from it, performing case analysis on the result.

Why did we need Thrower then? The difference is that throws needs to be hard-coded at method definition time, while our Kleisli arrow, which we called Partial, transforms function values “at runtime”.

Returning Errors

We now find that we can take this idea further. We don’t actually need the throws monad at all, and hence, we don’t need Thrower either. Don’t get me wrong. I like the Trower monad. I use the Thrower monad in production. But let’s explore what we can do to approach a more functional style of error handling, just for grins.

Instead of throwing Exceptions, we can just collapse error types into the return type. We’ll use a monad called Either, which looks very similar to Thrower except that errors are returned, not thrown. The Functional Java library luckily has Either already defined:

  public abstract class Either<A, B>

Either is the disjoint union of two types. A value of type Either<A, B> simply wraps a value of either type A or type B. The Either class comes with two static data constructors:

  public static <A, B> Either<A, B> left(final A a)

  public static <A, B> Either<A, B> right(final B b)

We’ll use the standard convention that left values represent failure, and that right values represent success.

Let’s see how we might use this data type for error handling. It’s really quite straightforward. The following declares the return type of parseInt to be either an Exception or an Integer (the left value can actually be any type, but… baby steps).

  public Either<Exception, Integer> parseInt(String s);

Oh, but wait. We’ve lost the laziness. Functional Java’s Either, by itself, is not lazy like Thrower is. If we do want laziness, we have to wrap the computation in a closure. For example, we can use fj.P1, which is just a generic interface with one method _1() which takes no arguments.

  public P1<Either<Exception, Integer>> parseInt(String s);

That’s better. Raising errors is done by returning a left value:

  return new P1<Either<Exception, Integer>>() {
    public Either<Exception, Integer> _1() {
      return Either.left(new Exception("You broke it!"));
    }
  };  

Sometimes you’re able to call functions that might fail, recovering with a default value:

  P1<Either<Exception, Integer>> a = parseInt(s);
  int addFive = (a._1().isLeft() ? 0 : a._1().right().value()) + 5;

Other times, you will be catching exceptions thrown by other people’s code and converting them to Either:

  public static P1<Either<Exception, Double>> divide(double x, double y) {
    return new P1<Either<Exception, Double>>() {
      public Either<Exception, Double> _1()
        try {
          return Either.right(x / y);
        } catch (Exception e) {
          return Either.left(e);
        }
      }
    };
  }

An error is sent to the caller simply by returning it. In the example below, the parse errors are explicitly caught and re-raised, while any errors raised by the divide method are implicitly returned.

  public static P1<Either<Exception, Double>> parseAndDivide(String sx, String sy) {
    final P1<Either<Exception, Integer>> x, y;
    x = parseInt(sx); y = parseInt(sy);
    return new P1<Either<Exception, Double>>() {
      public Either<Exception, Double> _1() {
        if (x.isLeft() || y.isLeft()) {
          return Either.left(x.isLeft() ? x.left().value() : y.left().value());
        }
        return divide(x.right().value(), y.right().value());
      }
    };
  }

Either is a monad, so we can lift existing functions into it, compose Eithers, sequence them, just like we did with Throwers. Here I’m taking an existing sine function and reusing it for values of type Either<Exception, Double>:

  F<Double, Double> sin = new F<Double, Double>() {
    public Double f(final Double a) {
      return Math.sin(a);
    };}

  P1<Either<Exception, Double>> z =
    divide(x, y).map(Either.rightMap_().f(sin));

If you need to work with more than one kind of error at a time, you can. Simply nest them on the right, like so: Either<E1, Either<E2, A>>. This is something we could not do with Thrower.

Nothing to Declare

Now, we’ve been returning Exceptions, but what does an Exception give you, really? You get some information about the failure, like an Exception type name, an error message, a stack trace, and possibly a nested Exception. If you know how to recover from an error, are you going to need the stack trace and the error message? No. You don’t actually have to return Exceptions to indicate errors if you don’t want to. You could just as well return String, Integer, fj.Unit, or your favourite data structure. Or you could return… Nothing.

There’s a simpler monad than Either which is often used in functional programming. It’s called Option (sometimes known as Maybe). An example is Functional Java’s fj.data.Option class, which I’ll demonstrate briefly here. Option<A> is a lot like a List<A> that can either be empty, or contain a single value of type A. For example:

  public static Option<Double> divide(double x, double y) {
    if (y == 0) {
      return Option.none();
    }
    return Option.some(x / y);
  }

This is similar to returning null on failure, except that callers won’t be surprised if the method fails to return a value. It’s declared in the method signature, after all. Option.none() has the correct type and will respond to methods declared on the Option class. If you want a lazy Option, simply wrap it in a P1 like we did with Either.

Not surprisingly, Option comes equipped with monadic methods like fmap (or simply “map”) to lift an F<A,B> to an F<Option<A>, Option<B>>, “bind” to chain Option-valued functions together, and “join” to collapse an Option<Option<A>> into an Option<A>. Option’s unit function is the static method Option.some. The instance method some() returns the value in the Option, if any. This last one is the Option equivalent to Thrower.extract().

So, now that we’re using Option and Either to indicate recoverable errors, what about unrecoverable errors? Those are the kinds of errors that we’re not going to catch, nor expect any reasonable application to catch. They should result in immediate and visible failure. For example, we might decide that it’s not reasonable to recover from a critical function if it returns nothing:

  Option<MyAppConfig> config = getAppConfig();
  if (config.isNone()) {
    throw new Error("No configuration!");
  } else {
    // Do stuff with config here.
  }

At this point, we have completely obviated Java’s checked exceptions mechanism. Recoverable errors are simply returned, and unrecoverable errors are thrown all the way out. We can now construct computations, and even if they might fail we can pass them around, compose them together, extract values from them, etc. All without a throws keyword in sight. The circle is complete.

That’s the end of Lazy Error Handling in Java. We’ve gone everywhere from throwing exceptions lazily, to higher-order chaining and manipulation of exception-throwing computations, to doing away with checked exceptions altogether. I hope you have found this short series educational and as fun to read as it was to write.

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