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November 18, 2012 / cdsmith

CodeWorld and the Future

I’ve been thinking a lot the last few weeks about the future of CodeWorld.  I’m writing this to share my vision of where things are going, and why.  I’d appreciate feedback from anyone with thoughts on the future, different ideas, and so on.

And Overview of the Past

For anyone new to the ideas, I should review a little about what CodeWorld is.  CodeWorld is the name for the curriculum I built and taught last school year.  It has the goal of teaching abstract and mathematical reasoning skills through computer programming.  Throughout the course of the year, students use a web site to write descriptions of pictures, then animations, and finally video games, and run them in the web browser to immediately see the results.

I taught a pilot program last school year in a neighborhood school, and it was a huge success.  I have high hopes of getting something going again in the future.

What Happened…

Many people have noticed that CodeWorld stopped in its tracks some time during the summer.  What ended up happening was a fairly big life change for me: I’ve changed jobs (now working for Google on YouTube!), moved (to San Francisco from Colorado), and have been focused on personal changes for a while.  Unfortunately, that meant I had to cancel the plans to teach CodeWorld at three schools in Colorado this school year.  I hope, though, this is only a temporary setback!  I still have high hopes for the future.

Part 1: The Technology

In terms of technology, CodeWorld is built on several things:

  1. The Haskell programming language.  This is a language that’s very well suited for CodeWorld because it expresses things in a way that’s very declarative and consistent with the mental models students should bring into algebra and other abstract mathematics classes.
  2. The Gloss library.  This library, originally developed for introductory college programming classes, is a great fit for many of the same reasons.  It’s based on Haskell, and again does everything declaratively.  It’s also very concise, not needing a lot of wordy boilerplate to get something quickly on the screen.
  3. The SafeHaskell feature of GHC.  This allows the server to safely compile and run code written by students without worrying that it might delete critical files and such.

In some ways, the set of technology chosen for CodeWorld is perfect.  Specifically, it let me get a web site up and running in about a week of work, right before the start of the class last school year.  That saved a lot of hassle, and made the class a lot more successful.  In other ways, though, each of these choices is lacking in a few ways.

  1. Haskell is a great choice of language, but it’s very difficult to target Haskell to run in web browsers.  For this reason, CodeWorld actually runs students’ programs way over in a data center, far from their computers.  This worked and was easy to do (great!), but it also means programs run very slowly, and the server gets bogged down if it’s used by too many people at once (not so good).
  2. Gloss is the closest thing we have to a perfect choice.  Even that, though, has its limitations.  Because Gloss has a number of users who want to do more advanced things (like use it to show off parallel processing APIs in Haskell), there are some technical pieces in the back end that are difficult to fit in with a web browser as well.  Also, mainly due to backward compatibility, there are inconsistencies in the API (circle to draw the outline of a circle versus rectangleWire to do the same with a rectangle, for example), inconsistencies with what students will see in math classes (angles are clockwise, for example, in the official implementation), and lost opportunities to make small tweaks that will more clearly emphasize declarative thinking.  We definitely want something like Gloss, but there are a few details that it would help to tweak in some incompatible ways.
  3. SafeHaskell turns out to be problematic for a variety of reasons.  Aside from the performance problems from running on the server in the first place, it’s really only a small part of a solution to the problems of running untrusted code.  For example, it has no protections against using excessive resources and slowing down the server until it’s unusable.

So I’ve been evaluating a new technical path.  This new approach uses Chris Done’s language called Fay, which is a subset of Haskell specifically designed to run in web browsers.  It has all of those advantages of Haskell that I mentioned above, but works better in a web browser.  I’ve been porting the Gloss library over to Fay, but taking liberties to modify some of the inconsistent behaviors and lost opportunities as I go.  SafeHaskell then becomes entirely unnecessary, as the code is no longer running on the server.  It’s running in the student’s web browser instead, where JavaScript is already thoroughly sandboxed by companies that put lots of resources into the problem.

This technology migration isn’t done yet, and there’s still a lot of little syntax stuff that’s frustratingly unsupported by Fay… but I’m working on it, helping out the Fay project a little but, too, and it’ll get there!

Part 2: The Presentation

In addition to changing technologies, I’ve been putting a lot of thought into what I want the future of the system to look like.  I am looking at how to make the platform more appealing when I am not personally living right by the school and able to pop over and help with any students who are having problems.

One side of this is definitely a redesign of the web site.  The site needs examples, it needs to look more finished, and it needs documentation!  I’m working on a mock-up of the future web site, and while my web design skills are definitely lacking, I think I’m happy with the concept so far.  You can take a look, if you want, and let me know what you think.

Another side actually was started over the summer, and that’s putting together a comprehensive teacher’s guide, including worksheets, exercises, and organization of the content.  This needs to be more than just a web site; it needs to be an organized approach to presenting the ideas and guiding students through the learning process.  I’m strongly considering making a somewhat involved video series to present the concepts, as well.

Part 3: Community

The third and final part of this is putting the pieces in place for a sustainable community of people working together.  I don’t have any specific plans here, yet… but I’ll just say that I see positive things ahead.  I’m now working with tens of thousands of brilliant people, many of whom care very much about education, in a company that encourages its employees to experiment with technology in ways that aren’t focused on just one product.  I’m also within a short drive of the Khan Academy, and plenty of other groups that care a lot about math education.

Some things I’d like to try here:

  1. Getting together students who build things with CodeWorld to meet and share their creations.
  2. Helping students make the transition from developing things for “a school project” to sharing them with friends and others, and possibly even offering their creations in various app stores or markets.  (Student creations in CodeWorld actually tend to be fairly similar to the casual game market for smartphones and such…)
  3. Creating more resources for teachers who want to incorporate this into their schools.  I’d love to hold seminars and such for teachers!  Please ask me to do this, once the pieces are in place!

So that’s where CodeWorld is, and where it’s hopefully going.  I’m still very excited about the future of this effort, and looking forward to making some more progress here soon.

April 18, 2012 / cdsmith

Why Do Monads Matter?

(A Side Note: I’ve been formulating the final thoughts on this post for about a week now. In an entirely unrelated coincidence, a good friend of mine and fellow Haskell programmer, Doug Beardsley, ended up writing two posts about monads over the weekend as well. Weird! But don’t fret; this isn’t really the same thing at all. I’m not writing to teach Haskell programmers how to use monads. I’m writing about a kind of intuition about why these concepts turn out to matter in the first place. You won’t find much here by way of how to program in Haskell.)

Category Theory for Software Development?

Match made in heaven? Or abstraction distraction?

If you’re a software developer, have you heard about monads and wondered what they were? Have you tried to learn Haskell, and struggled with them? Wondered why people worry so much about them? Have you watched the videos from Microsoft’s “Channel 9″ and heard a bunch of researchy Microsoft folk talk about them, but had trouble relating them to your day-to-day programming experience?

Or if you’re interested in mathematics, have you heard murmurs in the past about how category theory interests computer science people? Looked for some clear statement of why we care, and what problems we might be interested in? Wondered if it’s really true at all? Perhaps you are like a friend of mine (and a first-rate algebraist, too, so it’s entirely reasonable to have these questions) who asked me about this a year or so ago, remembered hearing a lot of excitement in the early 90s about category theory and computer science, but never heard whether it had really panned out or was a dead end?

These are the kinds of questions I begin with. My goal is to demonstrate for you, with details and examples:

  • Where category-based intuition and ideas, and monads in particular, come from in computer programming.
  • Why the future of programming does lie in these ideas, and their omission in today’s mainstream languages has cost us dearly.
  • What the state of the art looks like in applying category-based ideas to problems in computer programming.

If you’re coming into this without a knowledge of category theory, never fear; this may be one of the gentlest introductions to the idea of categories and monads that you will find. But you’ll want to slow down and take a moment to understand the definition of a category and related ideas like function composition; these are absolutely crucial. Then you want to completely skip or just skim through the section called “What’s This Got To Do With Monads?” where I tell you how what we’re talking about here relates to the traditional math meaning of monads. Don’t worry, you don’t need to know that at all.

On the other hand, if you’re a mathematician, you may want to skim the bits where I review basic category theory, and just dig in where I am talking about the computer programming perspective. Just be forewarned, my introduction to monads will be via Kleisli categories, so take a minute when we get to that part and make sure you’re familiar with how the relationship works out.

Ready? Here goes!

Computer Programming and Functions: A Tenuous Relationship

Quick quiz: Do computer programmers use functions?

Ask any computer programmer you know, and you will hear: YES! Functions are some of the most basic tools computer programmers use. Then you’ll get odd looks, for asking such a silly question. Of course computer programmers use functions. That’s like asking if carpenters use nails! Right?

The truth, though, is a bit more complicated. To a mathematician, a function is just an association of input values to output values… and that is all! Any two functions that associate the same input values to the same output values are the same. Yes, you can represent functions by formulas (sometimes, anyway), but you can also represent them with just tables of inputs and outputs, or if they are functions between real numbers, as graphs. If you ask computer programmers for examples of functions, though, you will start hearing about some pretty bizarre things. I call these the “I must have skipped that day of calculus” functions. These are things that computer programmers are quite happy referring to as functions, but that to a mathematician are not really functions at all!

  • “Functions” that return randomly chosen numbers… and if evaluated several times, will give a different answer each time.
  • “Functions” that return one answer on Sundays, but a different answer on Mondays, yet another on Tuesdays, and so on.
  • “Functions” that cause words to appear on some nearby computer screen every time you calculate their values.

What’s going on here? Most computer programmers go about their lives happily calling these things functions, but really they are something different. But wait a second! They do have quite a lot in common with functions. Namely, they have: (a) parameters, representing their domain; and (b) return values, representing their range. (Many computer programmers are happy to talk about functions that have no parameters, or no return values… but there’s no need to be overly picky here. We can just regard their domains and ranges as one-element sets, so that no actual information is conveyed, but we can keep up appearances.)

Even more importantly, these “functions” share one more thing with the functions of mathematicians: they are constantly being composed by taking the result from one function and passing it along as a parameter to another function. When I say composed, I mean it almost exactly in the basic mathematics sense of function composition: (f · g)(x) = f(g(x)). In fact, the whole reason our “functions” exist at all is to be composed with each other! Once upon a time, in the early days of computers, we liked to keep track of information by just sticking it in known places in the computer’s memory; but all this shared knowledge about where to find information made it hard to write parts separately and fit them together, so we mostly switched to this idea of functions and composition instead.

Here’s the executive summary so far:

  1. When computer programmers talk about functions, they do not mean exactly what mathematicians do.
  2. What they do mean is the idea of having inputs (domains), outputs (ranges), and most importantly composition.

Along Came The Category…

So in the previous section, we ended up with our hands full of things that sort of look like functions. They have domains and ranges, and they can be composed. But at the same time, they are not functions in the mathematics sense. Baffling? No, not really. Mathematicians deal with stuff like that a lot. They have a name for systems of function-esque things of exactly that form. That name is… cue the drumroll, please… CATEGORIES!

In math-speak, categories are:

  1. collections of “objects” (you should think of sets),
  2. and “arrows” (you should think of functions between sets),
  3. where each arrow has a domain and a range,
  4. each object has an “identity” arrow (think of the identity function, where f(x) = x)
  5. and arrows can be composed when the domains and ranges match up right.

Before we agree to call something a category, we also throw in a few rules, such as if you compose any function with an identity, it doesn’t actually change, and composing functions obeys the associative property. These should be unsurprising, so if they seem strange to you, please take a moment, grab a pencil, and try working it out using the definition of function composition earlier: (f · g)(x) = f(g(x)), and simplifying.

The nice thing about categories is this: it’s not just some pointless abstraction that a bunch of mathematicians made up. Categories are defined that way because people have looked at literally hundreds of things that all look sort of like functions with domains and ranges and compositions. Things from algebra, like groups and rings and vector spaces; things from analysis, like metric spaces and topological spaces; things from combinatorics, like elements of partially ordered sets and paths in graphs; things from formal logic and foundations, like proofs and propositions. Almost without fail, they can be described using all the ideas we just looked at! In short, categories are the right intuition for talking about composing things with domains and ranges, which is exactly the situation we’re in.

The Four Horsemen of the Catapocalypse

Now you can see why categories come into the picture: they are the right intuition for things that maybe aren’t functions, but can be composed like functions. But just because a category exists doesn’t mean it’s worth talking about. What makes this worth talking about is that the category-related ideas aren’t just there, but actually express common concerns for computer programmers.

It’s now time to get a little more specific, and introduce the four examples that will guide us the rest of the way through this exploration. Each example highlights one way that the “functions” used by computer programmers might be different from the functions that mathematicians talk about. These examples represent actual kinds of problems that computer programmers have run into and solved, and we’ll look more at the practical side of them later. For now, we’ll just be happy getting familiar with the general ideas.

The First Horseman: Failure

The first problem is failure. Computer programmers do lots of things that might fail. Reading from files (they might not exist, or on a computer with more than one user, they might not be set to allow you to read them), talking over the internet (the network might be broken or too slow), even just doing plain old calculations with a large amount of data (you might run out of memory). Because of this, dealing with failure is a constant concern.

In general, in modern computer programming tools, it’s always understood that a function might fail. You may get an answer, but you also may get a reason that the task could not be completed. When that happens, programmers are responsible for dealing with it responsibly: letting someone know, cleaning up the leftover mess in computer memory from a half-complete task, and just otherwise putting the pieces back together. A major factor in programming techniques or tools is how easy they make it for programmers to cope with the constant possibility of failure.

The Second Horseman: Dependence

The second problem is dependence on outside information. While functions of mathematics are nice and self-contained, computer programmers often don’t have that luxury. Computer programs are messes of configuration. Even simple mobile phones have pages and pages of settings. What language does the user speak? How often should you save off a copy of their work? Should you encrypt communication over the network? Rare is the application today that doesn’t have a “Settings” or “Preferences” menu item. In many other contexts, too, computer programs depend on information that is a sort of “common knowledge” throughout the application, or some part of the application.

Ways of dealing with this have progressed through the ages. When everything was stored in well-known memory locations anyway, it was easy enough to just look there for information you need; but that led to problems when different parts of a program needed different information and sections of programs could step on each other’s toes. The massively influential technique known as object-oriented programming can be seen as partly an attempt to solve exactly this problem by grouping functions into a context with information that they depend on. The simplest and most flexible answer would be to just pass the information around to all the functions where it is needed… but when that’s a lot of places, passing around all those parameters can be very, very inconvenient.

The Third Horseman: Uncertainty

The third problem is uncertainty, also known as non-determinism. A normal function associates an input to an output. A non-deterministic function associates an input to some number of possible outputs. Non-determinism is less well-known than the first two problems, but possibly only because it hasn’t yet seen a convincing solution in a general purpose language! Consider:

  • Theoretical computer science talks about non-determinism all the time, because it’s the right approach for discussing a lot of computational problems, ranging from parsing to search to verification.  That language just hasn’t made its way into the programming practice.
  • Non-determinism comes up when querying, searching, or considering many possible answers. These are precisely the places that programmers end up relying on a variety of domain specific languages, ranging from SQL to Prolog, and more recently language-integrated technologies like LINQ.
  • Even with specialized languages for heavy-duty querying and search tasks, we still end up writing a lot of our own nested looping and control structures for the purpose of looking through possibilities when it’s not worth crossing that language barrier.  This kind of thing is responsible for some of the more complex code structures you find these days.

While the first two problems of failure and dependence are at least partly solved by current mainstream programming languages, non-determinism is as yet solved mostly by special-purpose sub-languages, with LINQ as the notable exception.

The Fourth Horseman: Destruction

Finally, the fourth problem is destruction.  Evaluating a math-type function is observable only in that you now know the answer.  But in computer programming, functions can have permanent effects on the world: displaying information, waiting on responses from other computers or people, printing documents, even quite literally exploding things, if they are running on military systems!  Because of this, things that aren’t specified in mathematics, like the order in which evaluation happens, matter quite a lot here.

The destructive nature (by which we just mean having effects that can’t be undone) of computer programming functions has plenty of consequences. It makes programming more error-prone. It makes it harder to divide up a task and work on different parts simultaneously, such as you might want to do with a modern multi-core computer, because doing the parts in the wrong order might be incorrect. But at the same time, these destructive effects are in a sense the whole point of computer programming; a program that has no observable effects would not be worth running! So in practically all mainstream programming languages, our functions do have to cope with the problem of destruction.

Back To The Function

Now we’ve seen the faces of some problems we find in the computer programming world. We build software that might fail, has to deal with a ton of extra context, models non-deterministic choice, and sometimes has observable effects on the world that constrain when we can perform the computation.

It may now seem that we’ve left the nice and neat world of mathematical functions far behind. We have not!  On closer inspection, we’ll see that if we can just squint hard enough, each of these quasi-functions can actually be seen as true, honest-to-goodness functions after all.  There is a cost, though. To turn them into real functions, we need to change the range of those functions to something else.  Let’s see how it works for each of our function types in turn:

Functioning With Failure

Our first example of pseudo-functions were those that might fail. It’s not hard to see that a function that could fail is really just a function whose results include two things:

  • successes, which are the intended possible results; and
  • failures, which are descriptions of why the attempt failed.

So for any set A, we’ll define a new set called Err(A) to be just A together with possible reasons we might have failed. Now a possibly failing function from a set A to a set B is really just an ordinary function from A to Err(B).

Functioning With Dependence

Our second type of pseudo-functions were those that depended on information that they got from the world around them: perhaps preferences or application settings. We play a similar trick here, but for a set A, we will define the set Pref(A) to be the set of functions from application settings to the set A. Now watch closely: a function in context from A to B is just an ordinary function from A to Pref(B). In other words, you give it a value from the set A, and it gives you back another function that maps from application settings to the set B.

As confusing as that might sound, a function whose range is another function is really just a function of two parameters, except that it takes its parameters one at a time! Take a minute to convince yourself of this. The conversion between these two equivalent ideas is sometimes called “currying”. So by changing the range of our function, we actually effectively added a new parameter, and it now receives the application settings as a parameter. Remember that except for being inconvenient (we’ll deal with that later), that’s exactly what we wished for.

Functioning With Uncertainty

This is perhaps the most obvious example of all. Our third type were those that represent non-determinism: instead of one specific answer, they have many possible answers. This is easy enough: for each set A, define P(A) to be the power set of A, whose members are themselves sets of values of A. Then a non-deterministic function from A to B is just an ordinary function from A to P(B).

Functioning With Destruction

Our final trick is to deal with functions that have destructive effects. Here we’ll need to be a bit more elaborate in constructing a new range: for each set A, we define IO(A) (standing for input/output, which captures the notion of effects that interact with the rest of the world). An element of the set IO(A) is a list of instructions for obtaining a member of A. It is not a member of A, merely a way to obtain one, and that procedure might have any number of observable effects.

Now we play the same trick and change the range: a destructive function from A to B is just an ordinary plain old mathematical function from A to IO(B). In other words, if you give me an A, then as a plain old function I can’t actually do the steps to get a B, but I can certainly tell you what they are.

But what about composition? It’s great to be back in the world of plain functions, but remember what got us here in the first place? We liked functions because we liked composition; but it seems we’ve now lost composition! If I have a possibly failing function from A to B, and another from B to C, well now I’ve turned them into functions from A to Err(B) and then B to Err(C). Those function domains and ranges don’t match up, and I can’t compose them!

Oh no…

Hold Your Horses, Heinrich Kleisli to the Rescue!

Well, all is not lost. I just haven’t yet told you how to compose these “special” functions.

Because some math dude found these things before us, we call our “special” functions by a name: Kleisli arrows.  There are two things going on here at once, so keep your eyes open: first, Kleisli arrows are just plain old ordinary functions, but with weird-looking ranges. Since they are just functions, you can compose them as functions, and that’s just fine.  But at the same time, they are “special”, and we can compose them as Kleisli arrows, too.

Remember what we decided earlier? The right way to think about composition is by talking about a category. Sets are a category, and that’s fine if you want plain function composition. But now we want a new kind of category, too. It’s called the Kleisli category. If you don’t remember what all the parts of a category are, take a second to review them. To define a category, I need objects, arrows, identities, and composition.

  • To keep things simple, the objects in this new category will be the same: they are just sets of things.
  • The arrows in this category are, unsurprisingly, the Kleisli arrows.
  • I haven’t told you yet what the identities and composition look like, so let’s do that next.

First, we look at failure. We’re given a failure Kleisli arrow from A to B, and one from B to C. We want to compose them into a Kleisli arrow from A to C. In other words, we have an ordinary function from A to Err(B), and a function from B to Err(C), and we want one from A to Err(C).  Take a minute to think about what to do.

The central idea of error handling is that if the first function gives an error, then we should stop and report the error. Only if the first function succeeds should we continue on to the second function, and give the result from that (regardless of whether it’s an error or a success).

To summarize:

  1. If g(x) is an error, then (f · g)(x) = g(x).
  2. If g(x) is a success, then (f · g)(x) = f(g(x)).

To complete the definition of a category, we also need to decide about the identity Kleisli arrows. These are the ones that don’t do anything, so that if you compose them with any other Kleisli arrow, it doesn’t change the other one. Identities are functions from A to Err(A), and it turns out these are just the functions f(x) = x, just like for sets. Notice that means they never return an error; only a successful result.

I’ll run more briefly through the remaining three examples, but I encourage readers who aren’t yet clear on how this will work to write them out in more detail and use this opportunity to become more comfortable with defining a second category of Kleisli arrows.

Next we have Klesli arrows for dependence, which are functions from A to Pref(B).  Recall that adding the Pref to the range is equivalent to adding a new parameter for the application preferences. The key idea here is that if I have two functions that both need to know the application preferences, I should give the same preferences to both. Then composing two of these Kleisli arrows just builds a new function that gets the extra preferences parameter, and passes the same one along to the two component parts. And identities? A Kleisli identity will get that extra preferences parameter, but will ignore it and just return its input anyway.

The Kleisli arrows for uncertainty, or non-determinism, are functions from A to P(B), the power set of B. The key idea for non-determinism is that at each stage, we want to try all possible values that might exist at this point, and collect the results from all of them. So the composition calculates the second function for each possible result of the first, then the resulting possibilities are merged together with a set union. The identities, of course, aren’t really non-deterministic at all, and just return one-element sets containing their input.

Finally, Kleisli arrows for destructive effects are functions from A to IO(B). The key idea here is to combine instructions by following them in a step-by-step manner: first do one, then the next. So the composition writes instructions to perform the first action, look up the second action from the result, and then perform the second action, in that order. A Kleisli identity here is just an instruction to do nothing at all and announce the input as the result. So for each of the four motivating examples, we created a new category, the Kleisli category.

These new categories have their own function-like things, and related ideas of composition and identities, that express the unique nature of each specific problem. By using the appropriate notion of composition in the right Kleisli category, you can solve any of these long-standing computer programming problems in a nice composable way.

And that’s why you should care about monads.

Monads?!? Oh yes, I should mention that we’ve just learned about monads. We simply forgot to use the word.

What’s This Got To Do With Monads?

This section is for those of you who want to know how the stuff we said earlier are related to monads as they are understood in mathematics.  If you open Wikipedia, or most category theory textbooks, and look up monads, they won’t look very much like what we just did. You’ll see something about an endofunctor, and two natural transformation, and properties about commuting triangles and squares.

We haven’t talked about functors at all, much less natural transformations… so how could we have possibly learned about monads? It turns out there’s more than one way to describe monads.  The one we’ve just gone through is an entirely valid one. The shifts we made to the ranges of our functions earlier – Err, Pref, P, and IO – are actually examples of monads. To make sure they are monads in the conventional math way, we’d have to work pretty hard: first, prove that they are functors. Then build two natural transformations called η and µ, and prove that they are natural. Finally, prove the three monad laws.

But wait, there’s an easier way!  Heinrich Kleisli, whom we’ve already met from the categories earlier, pointed out that if you can build a category like the ones we did in the last section, whose arrows are just functions with a modified range, then your category is guaranteed to also give you a monad. That’s quite convenient, because as computer programmers, we tend to care a lot more about our Kleisli arrows than we do about a mathematician’s idea of monads.  Remember, those Kleisli arrows are exactly the modified notion of functions that we were already using, long before we ever heard a word about category theory!  And Kleisli tells us that as long as composition works the way we expect with our Kleisli arrows (namely, that it’s associative and the identities act like identities), then all that other stuff we’re supposed to prove to show we have a monad just happens for us automatically.

Still, it’s an interesting side question to look at the relationship between the two. I won’t give all the details, but I’ll give the structure, and then leave the interested reader with some familiarity with category theory to fill in the proofs of the relevant properties. We’ll use Err as our monad, just to pick a specific example, but nothing here is specific to Err.

  1. We start with Err, which is already a map from objects to objects. But the traditional definition of a monad also requires that it be a functor. That is, given a function f from A to B, I need a way to construct a function Err(f) from Err(A) to Err(B). I do it as follows: in the underlying category (not the Kleisli category, just the category of sets), I find an identity function from Err(A) to Err(A). Then I find a Kleisli identity from B to Err(B). I compose that Kleisli identity in the underlying category with f, and get a function from A to Err(B). I can now do a Kleisli composition of the identity from Err(A) to Err(A) and the function from A to Err(B), and get a function from Err(A) to Err(B). That’s the one I’ll call Err(f).
  2. Next, I need a natural transformation η, from the identity functor to Err. This is easy: the components of η are the Kleisli identities.
  3. Finally, I need a natural transformation µ from Err² to Err. To get the component of µ at A, I take the identity functions in the underlying category from Err (Err A) to Err (Err A), and then from Err A to Err A, and I combine them with Kleisli composition to get a function from Err (Err A) to Err A. This is the component of µ.

The construction in the opposite direction is easier. Given a monad Errwith ? and µ, the Kliesli category is constructed as follows.

  1. The identities are just the components of η.
  2. Given a function f from A to Err(B) and a function g from B to Err(C), I compose the two as µ · Err(g) · f.

Again, the details and the proofs of the appropriate monad and category laws are left to the reader. I hope this brief aside has been useful. I now return to using the word “monad” but talking about monads via Kleisli categories.

Joining The Monadic Revolution

Once again, let’s pause to sum up.

  • Computer programmers like to work by composing some things together, which we call functions.
  • They aren’t functions in the obvious way… but they do make up a category.
  • Actually, they are functions after all, but only if you squint and change the ranges into something weirder.
  • The category that they form is called a Kleisli category, and it’s basically another way of looking at monads.
  • These monads / Kleisli categories nicely describe the techniques we use to solve practical problems.

It’s not just about those four examples, either.  Those are typical of many, many more ideas about programming models that can be described in the same framework. I think it’s fair to sum up and say, at this point, that someone interested in studying and analyzing programming languages and models should be familiar with some ideas from category theory, and with monads in particular.

But still, what about the humble computer programmer, who is not designing a new language, is not writing research papers analyzing programming languages, but just wants to solve ordinary everyday problems?  That’s a fair question.  As long as monads remain just a mathematical formalism for understanding what computer programmers mean by functions, the practicing computer programmer has a good claim to not needing to understand them.

It’s becoming clear, though, that monads are on their way into practical programming concerns, too.  In the past, these Kleisli arrows, the modified notions of “function” used by computer programmers, were built into our programming languages.  Functions in C used a Kleisli arrow, and C++ functions used a different one.  The language specification would tell us what is and what is not possible using a function in this language, and if we wanted something different, too bad.  Maybe once a decade, we’d make the swap to a brand new programming language, and bask in the warm rays of some new language features for a while.

The Past: Error Handling

Consider the Err monad, which gave us functions that might fail and report their failure in structured ways.  Modulo some details and extensions, this is basically structured exception handling. Looking to history, programmers worked without exception handling in their programming languages for many years. Of course, languages like C are all Turing complete, and can solve any possible computational problem, proper error handling included. But we don’t apply categories to think about possible computations; categories are for thinking about composition. Without exception handling in the notion of a “function” that’s provided by languages like C, programmers were left to do that composition by hand.

As a result, any C function that could fail had to indicate that failure using a return value.  In many cases, conventional wisdom built up saying things like “return values are for indicating success or failure, not for giving back answers”. Coding conventions called for most if not all function calls to be followed with if statements checking for failure, and the resulting code was borderline unreadable.  This was the heyday of flowcharts and pseudo-code, because no one expected to be able to understand real code at a glance!  In reality, though, programmers only checked for errors when they thought they was possible, and a lot of errors went undetected. Programs were often unreliable, and likely untold billions of dollars spent on extra development work and troubleshooting.

What was the reason for this?  It’s quite simple: the C programming language and others of its time provided an insufficient kind of Kleisli arrow!  If their Kleisli arrow had included the functionality from the Err monad we defined above, this could have been avoided.  But the notion of what a function means in C is fixed, so the answer was to deal with it, and eventually migrate to a different programming language, rewriting a lot of software, and likely costing another untold billions of dollars.

The Present: Global Variables and Context

What about the Pref monad, and others like it? As discussed earlier, this is about defining computations in a larger context of available information and state of the world.

In the past, we had global variables, the slightly more modern equivalent of just storing information at a known place in computer memory. Quick and dirty, but even 30 years ago, programmers knew they were the wrong answer, and wouldn’t be manageable for larger programs. Object oriented programming tried to alleviate the problem a little, by having functions run in a specific “object” that serves as their context, and that was implicitly passed around at least within the implementation of the object itself. To get this, everyone effectively had to change programming languages to get a better Kleisli arrow again.  But even so, object-oriented languages don’t give a perfect answer to this problem.

The Near Future (/ Present): Purity, Effects, Parallelism, Non-Determinism, Continuations, and More!

This point is about the future, but I’ll start out by pointing out that everything here is already possible, but just requires an appropriate choice of programming language!

One current challenge for the computer programming community is finding effective ways to handle parallelism. Ironically, while past examples have focused on the problem of putting too little power into a language’s Kleisli arrow, the problem this time is too much!  Plain (also known as “pure”) functions present lots of opportunities for parallelism. When code is executed in parallel, it may run faster, or if the parallelism is poorly designed it may even run slower, but in any case it will certainly still give the same answer. But when the Kleisli arrow incorporates destructive updates, that is no longer the case. Now parallelism is risky, and might give unexpected or incorrect results due to so-called race conditions.

We can’t just remove destructive updates from a language’s Kleisli arrow, though.  A program that has no observable effects at all isn’t useful. What is useful is the ability to separate the portions of code that perform destructive update from those that just compute pure functions.  So for the first time, we need a language with more than one kinds of Kleisli arrow, in the same language!

There is already at least one language that offers precisely this. Programmers in the Haskell language can build their own monads, and work in the Kleisli category of a monad of their choosing.  The programming language offers a nice syntax for making this approach readable and easy to use.  If something might fail, you can throw it in Err.  If it needs access to the application settings, throw it in Pref.  If it needs to do input or output, throw it in IO.  Haskell web application frameworks and similar projects start by defining an appropriate monad with the appropriate features for that kind of application.

Another current trend in the computer programming community is toward building more domain-specific programming models. The language Erlang became popular specifically for providing a new programming model with advantages for parallelism.  Microsoft’s .NET framework incorporates LINQ, which offers a programming model that’s better for bulk processing and querying of collections of data.  Rails popularized domain-specific languages for web applications.  Other languages offer continuations as a way to more easily build specify computations in a more flexible way.  All of these are examples of working in new and different Kleisli arrows that capture exactly the model appropriate for a given task.

It comes down to this: If we believe that there is one single notion of “function” that is most appropriate for all of computer programming, then as practical programmers we can find a language that defines functions that way, and then forget about the more general idea of monads or Kleisli arrows as a relic of theoreticians.  But it’s not looking that way.  The programming community is moving quickly toward different notions of what a function means for different contexts, for different tasks, even for different individual applications.  So it’s useful to have the language, the tools, and the intuition for comparing different procedural abstractions.  That’s what monads give us.

Abstraction Over Monads

Using a language with a choice of monads offers some other advantages here, too.  It gives us back our abstraction.  In Haskell, for example, it’s possible to write code that is applicable in multiple different monads. A surprising amount of the programming done with one monad in mind actually has meaning in very different monads! For example, consider the following Haskell type:

sequence :: Monad m => [m a] -> m [a]

What this means is that for any monad, which we’ll call M, sequence converts from a list of values of M(A) into M(List(A)), the monad applied to lists themselves.  Let’s take a minute to consider what this means for each of our four examples. For Err, it takes a list of results that might be failures, and if any of them are failures, it fails; but if not, then it gives back a list of all the results.  It’s basically a convenient way to check a whole list of computations for a failure. For Pref, it takes a single set of application preferences, and distributes that to everything in the list, giving back a list of the results. For the power-set monad, P, it would take a list of sets, and give back a set of all the ways to choose one item from each set.  And for IO, it takes a list of instruction cards, and gives back the single card with instructions for doing all of them in turn.  Amazingly, this one function, which had only one implementation, managed to make sense and do something useful for all four of our examples of monads!

Along with a choice of monads comes the ability to abstract over that choice, and write meaningful code that works in any monad that you do end up choosing.

Between all of these forces, I predict that within the next ten years, software developers will be expected to discuss monads in the same way that most developers currently have a working vocabulary of design patterns or agile methodologies.

Beyond Monads: More Categorical Programming

While most of this has been about monads, I don’t want to leave anyone with the impression that monads are the only influence of categories in computer programming.  All of the following ideas have found their way into programming practice, mostly (so far) within the Haskell programming language community because of its flexibility and a deep academic culture and history.

  • Monad transformers are a powerful technique for combining the effects of more than one monad to build rich and powerful programming models.
  • Functors and applicative functors (a.k.a. strong lax monoidal functors for mathematicians) are weaker than monads, but more widely applicable.
  • Other kinds of categories that are not Kleisli categories can often be defined and composed to solve specific problems. Freyd categories are also useful.

I’ll stop there, but only as an encouragement to look more into the various abstractions from category theory that programmers have found useful. A good starting point is the (Haskell-specific) Typeclassopedia by Brent Yorgey. That’s just a door into the many possibilities of applying category-based ideas and intuitions in computer programming.

But I hope I was able to convey how these ideas aren’t just made up, but are actually the natural extension of what computer programmers have been doing for decades.

February 14, 2012 / cdsmith

Juggling in Haskell and Gloss

Just sharing what we’ve been talking about in my classes at LSV (Little School on Vermijo) the past few days.  It all started when a math teacher I follow on Google+ posted a link to this video on the mathematics of juggling, by Cornell professor Allen Knutson.

So we all watched the video, and talked about it… and then about midnight last night, I realized this would make a perfect introduction to Gloss simulations, which we’re learning about in our Haskell programming class!  Fast forward about 20 minutes of hacking, and we have this short program:

import Graphics.Gloss
thePattern = [5,2,5,1,2]
initial g = (-1, 0.0, [], cycle thePattern)
step dt (hand, time, balls, pattern) = (newhand, newtime, newballs, newpattern)
    where (throw, newtime) = properFraction (time + dt)
          newhand    = if throw == 1 then -hand else hand
          thrown     = if throw == 1 then [ (hand, newtime, head pattern) ]
                                     else []
          newpattern = if throw == 1 then tail pattern else pattern
          newballs   = [ (bhand, btime + dt, height)
                       | (bhand, btime, height) <- balls,
                         btime + dt < fromIntegral height ]
                       ++ thrown
draw (hand, time, balls, pattern) = pictures [
    juggler,
    pictures [ ball b | b <- balls ]
    ]
ball (bhand, btime, height) = translate (50*x) (50*y) (circleSolid 10)
    where t = 1 - 2 * (btime / fromIntegral height)
          x = if even height then bhand else bhand * t
          y = if height < 3 then 0 else fromIntegral (height - 1) * (1 - t^2)
juggler = pictures [
    line [(-50, 0), (0, 25), (50, 0)],
    line [(-30, -100), (0, -50), (30, -100)],
    line [( 0, 25), (0, -50)],
    translate 0 50 (circle 25)
    ]

Feel free to check it out.

This isn’t the best code in the world, because I deliberately write it to avoid as many ideas as I can that we haven’t talked about in my Haskell class.  Among those, for example, are any kind of user-defined data types, or any structured data other than tuples and lists!  I’ve also used list comprehensions when maps and filters might be clearer, for the same reason: we’ve done list comprehensions in class!  Those are coming soon, but for the time being, it’s not too hard to see what’s going on here.  A couple notes about the world type:

  • The state of world comprises four pieces of information: which hand will throw next, how much time has passed since the last throw, what balls are in the air, and what pattern is coming up next.  Since we don’t have user-defined types and it makes the math easy, I’ve described hands as -1 for left, and +1 for right.
  • The balls in the air form a list, but each element of that list is another tuple, containing the hand the ball was thrown from, how long it has been in the air, and its height.

So like all Gloss simulations, we need to specify three things:

  1. The initial state.  This is the left hand, no time since the last throw, no balls in the air, and the pattern sitting ahead of us.  The cycle function turns a finite list into an infinite repeating one.
  2. The step function.  This is where most of the logic sits… there are a lot of bits, but it does what you’d expect!  Every second, a ball is thrown, and the hand switches sides.  We update the balls to add to their time in the air, and keep only the ones that haven’t landed.  When a ball is thrown, we add it.  And each time we drop a number from the pattern.
  3. The draw function.  This draws the juggler and all of the balls.  The balls are drawn with some faux-physicsy thing that, frankly, is just the result of fiddling and isn’t an accurate physics simulation.  I can get away with that when teaching middle schoolers!  Hey, at least they follow parabolas; just not all with the same acceleration.  In class I just skimmed over the math for where the ball appears, since the class right now is about the state type, and drawing is something they’ve been doing all year anyway.

Definitely a lot of fun.  I encourage you to copy and paste the code above into http://dac4.designacourse.com:8000/sim and fiddle with the patterns and see what they look like!

October 10, 2011 / cdsmith

Build Your Own Simple Random Numbers

Liam O’Connor got me thinking about the best way to explain the idea of a pseudo-random number generator to new programmers.  This post is my answer.  If you already understand them, there won’t be anything terribly new here.  That said, I enjoy clean examples even for easy ideas, so if you do too, then read on!

Note: The title may have caused some confusion.  I’m not suggesting you use the trivial algorithms provided here for any purpose.  Indeed, they are intentionally over-simplified to make them more understandable.  You should read this as an explanation of the idea of how generating random numbers works, and then use the random number generators offered by your operating system or your programming language, which are far better than what’s provided here.

The Problem

Suppose you’re writing a puzzle game, and you need to choose a correct answer.  Or suppose you are writing a role-playing game, and need to decide if the knight’s attack hits the dragon or deflects off of its scales.  Or you’re writing a tetris game, and you need to decide what shape is going to come next.  In all three of these situations, what you really want is a random number.  Random numbers aren’t the result of any formula or calculation; they are completely up to chance.

Well, here’s the sad truth of the matter: computers can’t do that.  Yes, that’s right.  Picking random numbers is one of those tasks that confound even the most powerful of computers.  Why?  Because computers are calculating machines, and we just said that random numbers aren’t the result of any calculation!

Of course, you’ve probably played games on a computer before that seem to pick numbers at random, so you may not believe me.  What you’re seeing, though, aren’t really random numbers at all, but rather pseudo-random numbers.  Pseudo-random numbers are actually the result of a mathematical formula, but one designed to be so complicated that it would be hard to recognize any pattern in its results!

Writing a Pseudo Random Number Generator

A lot of smart people actually spend a lot of time on good ways to pick pseudo-random numbers.  They try a bunch of different complicated formulas, and try to make sure that patterns don’t pop up.  But we can build a simple one pretty easily to pick pseudo-random numbers from 1 to 10.  Here it is, in the programming language Haskell:

random i = 7 * i `mod` 11

Since it’s a function, it needs to have an input.  It then multiplies that input by 7, and then finds the remainder when dividing by 11.  We’ll give it the previous number it picked as input, and it will give us back the next one.  Suppose we start at 1.  Then we get the following:

random  1 ->  7
random  7 ->  5
random  5 ->  2
random  2 ->  3
random  3 -> 10
random 10 ->  4
random  4 ->  6
random  6 ->  9
random  9 ->  8
random  8 ->  1

Let’s look at the range of answers.  Since the answer is always a remainder when dividing by 11, it’ll be somewhere between 0 and 10.  But it should be pretty easy to convince ourselves that if the number we give as input is between 1 and 10, then 0 isn’t a possibile answer: if it were, then we’d have found two numbers, both less than 11, that multiply together to give us a multiple of 11.  That’s impossible because…. 11 is prime.  So we’re guaranteed that this process picks numbers between 1 and 10.  It seems to pick them in a non-obvious order with no really obvious patterns, so that’s good.  We appear to have at least a good start on generating random numbers.

Notice a couple things:

  • We had to pick somewhere to start.  In this case, we started out by giving an input of 1.  That’s called the seed.  If you use the same seed, you’ll always get the exact same numbers back!  Why?  Because it’s really just a complicated math problem, so if you do the same calculation with the same numbers, you’ll get the same result.
  • To get the next number, we have to remember something (in our case, the last answer) from the previous time.  That’s called the state.  The state is important, because it’s what makes the process give you different answers each time!  If you didn’t remember something from the last time around, then you’d again be doing the same math problem with the same numbers, so you’d get the same answer.

Doing Better By Separating State

Unfortunately, our random number generator has a weakness: you can always predict what’s coming next, based on what came before.  If you write tetris using the random number generator from earlier, your player will soon discover that after a line, they always get an L shape, and so on.  What you really want is for your game to occasionally send them a line followed by a T, or even pick two lines in a row from time to time!

How do we do this?  Well, the next answer that’s coming depends on the state, so our mistake before was to use the previous answer as the state.  The solution is to use a state that’s bigger than the answer.   We’ll still be looking for random numbers from 1 to 10, but let’s modify the previous random number generator to remember a bigger state.  Now, since state and answer are different things, our random function will have two results: a new state, and an answer for this number.

random i = (j, ans)
    where j   = 7 * i  `mod` 101
          ans = (j - 1) `mod` 10 + 1  -- just the ones place, but 0 means 10

That says take the input, multiply by 7, and find the remainder mod 101.  Since 101 is still prime, this will always give answers from 1 to 100.  But what we really wanted was a number from 1 to 10, just like the one we had before.  That’s fine: we’ll just take the ones place (which is between 0 and 9) and treat 0 as 10.  The tens place doesn’t really change the answer at all, but we keep it around to pass back in the next time as state.

Let’s see how this works:

random  1 -> ( 7,  7)
random  7 -> (49,  9)
random 49 -> (40, 10)
random 40 -> (78,  8)
random 78 -> (41,  1)
random 41 -> (85,  5)
random 85 -> (90, 10)
random 90 -> (24,  4)
random 24 -> (67,  7)
random 67 -> (65,  5)
random 65 -> (51,  1)

Excellent!  Now instead of going in a fixed rotation, some numbers are picked several times, and some haven’t been picked yet at all (but they will be, if we keep going), and you can no longer guess what’s coming next just based on the last number you saw.  In this random number generator, the seed was still 1, and the state was a number from 1 to 100.

People who are really interested in good random numbers sometimes talk about the period of a pseudo-random number generator.  The period is how many numbers it picks before it starts over again and gives you back the same sequence.  Our first try had a period of 10, which is rather poor.  Our second try did much better: the period was 100.  That’s still pretty far off, though, from the random number generators in most computers, the period of which can be in the millions or billions.

Real World Pseudo-Random Number Generators

Our two toy pseudo-random number generators were fun, but you wouldn’t use them in real programs.  That’s because operating systems and programming languages already have plenty of ways to generate pseudo-random numbers.  And those were created by people who probably have more time to think about random numbers than you do!  But some of the same ideas come up there.  For example, consider this (specialized) type signature for the random function in the Haskell programming language:

random :: StdGen -> (Int, StdGen)

Look familiar?  StdGen is the state, and choosing a random Int gives you back the Int, and a new StdGen that you can use to get more pseudo-random numbers!  Many programming languages, including Haskell, also have “global” random number generators that remember their state automatically (in Haskell, that is called randomIO), but under the covers, it all comes down to functions like the ones we’ve written here… except a lot more complex.

Where To Get a Seed

We’ve still left one question unanswered: where does the seed come from?  So far, we’ve always been using 1 for the seed, but that means that each time the program runs, it will get the same numbers back.  So we end up with a similar situation to what we saw before, where players will realize that a game starts with the same sequence of random events each time.

To solve this problem, the seed should come from somewhere that won’t be the same each time.  Here are two different ways to seed a random number generator.

  1. Mostly, pseudo-random number generators are seeded from a clock.  Imagine if you looked at the second hand on a clock, used it to get a number from 1 to 60, and used that for your seed.  Then the game would only act the same if it started at the same number of seconds.  Even better, you could take the number of seconds since some fixed time in the past, so you’d get an even bigger difference in seeds.  (Entirely by coincidence, computers often use the number of seconds since January 1, 1970.)
  2. You might try to get a good seed from details of the way the user uses your program.  For example, you can look at the exact place the user first clicks the mouse, or exactly how much time passes between pressing keys.  They will most likely not be exact, and click a few pixels off or type ever so slightly slower, even if they are trying to do exactly the same thing.  So, again, you get a program that acts differently each time.  This is called using entropy.

Most of the time, using the computer’s built-in clock is okay.  But suppose you’re making up a code word.  It would be very bad if someone could guess your code word just by knowing when you picked it!  (They would also need to know how your computer or programming language picks random numbers, but that’s not normally kept secret; they can probably find that out pretty easily.)  Computer security and privacy often depends on picking unpredictable random numbers — ones that people snooping on you won’t be able to guess.  In that case, it’s important that you use some kind of entropy, and not just the clock.  In fact, when security is at stake, you can use entropy to modify the state as well, to make sure things don’t get too predictable.  Most operating systems have special ways of getting “secure” random numbers that handle this for you.

Another example of entropy:

If you play the game Dragon Warrior for the Nintendo, but use an emulator instead of a real Nintendo, then you can save a snapshot of your game before you fight a monster, memorize what the monsters are going to do, and figure out exactly the right way to respond.  When you load the game from the snapshot and try again, as long as you do the same things, the monster will respond in exactly the same way!  That’s because the snapshot saves the state of the random number generator, so when you go back and load from the snapshot, the computer picks the same random numbers.  So if a fight against a monster is going well but you make a disastrous move at the end, you can load your snapshot and repeat the exact same fight up to that point.

The same trick doesn’t work in Dragon Warrior 2 (or later ones), though!  Why not?  Because the company that makes the game started using entropy in their sequel.  So now little things like exactly how long you wait between pressing buttons will change the game.  Since you can’t possibly time everything exactly the same down to hundredths or thousandths of a second, the task is hopeless, and you have to just take your chances and trust to luck.

So as you can see, random numbers can become a very tricky topic.  But ultimately it’s all just a complicated formula, a seed, and a state.

October 9, 2011 / cdsmith

Just for fun: Hangman with gloss-web

So this will be brief.  Ivan Miljenovic pointed me to a discussion about writing games with kids, and the question came up about how clearly one could write a hangman game.  To I took that a challenge, and I present to you: Hangman in gloss-web!

Edit: I’ve now added a random number generator to the signature of the initial worlds, so it even chooses words at random.

The code:

import Graphics.Gloss
import Graphics.Gloss.Interface.Game

import System.Random

import Data.Char

possibleWords = [
    "cat", "hat", "zebra", "delicious", "water",
    "elephant", "granite", "sunset", "dragon"
    ]

data World = World {
    word    :: String,
    strikes :: Int,
    guesses :: [Char]
    }

initial g = World (possibleWords !! i) 0 []
  where (i, _) = randomR (0, length possibleWords - 1) g

lost world = strikes world == 6
won  world = all (`elem` guesses world) (word world)
over world = won world || lost world

event (EventKey (Char c) Down _ _) world
  | not (isLetter c)                     = world
  | over world                           = world
  | toLower c `elem` guesses world       = world
  | toLower c `elem` word world          = world { guesses = c : guesses world }
  | otherwise                            = world { strikes = strikes world + 1,
                                                   guesses = c : guesses world }
event _                            world = world

step time world = world

draw world = pictures [ frame, hangee world, answer world ]

frame = pictures [
    line [ (100, -50), (-100, -50), (-100, 200), (0, 200), (0, 150) ],
    line [ (-75, -50), (-100, -25) ]
    ]

hangee world = pictures (take (strikes world) parts)
  where parts = [ hangeeHead, hangeeBody, leftArm, rightArm, leftLeg, rightLeg ]

hangeeHead = translate 0 125 (circle 25)
hangeeBody = line [ (0, 100), (  0, 30) ]
leftArm    = line [ (0,  90), (-30, 60) ]
rightArm   = line [ (0,  90), ( 30, 60) ]
leftLeg    = line [ (0,  30), (-30,  0) ]
rightLeg   = line [ (0,  30), ( 30,  0) ]

answer world = translate (-50) (-200)
             $ scale 0.3 0.3
             $ color (wordColor world)
             $ text (letters world)

wordColor world
  | won  world = dark green
  | lost world = red
  | otherwise  = black

letters world = concat [
    if c `elem` guesses world then [c, ' '] else "_ "
      | c <- word world
    ]
And, here's what it looks like:

To run the code, visit http://dac4.designacourse.com:8000, choose Game, and copy and paste the code above into the editor area.  Or visit this link to just play the game.  Have fun!

October 6, 2011 / cdsmith

Haskell For Kids: Week 8

Welcome to the Week 8 summary of my Haskell class for kids.  If you’re just joining us, feel free to browse through weeks 1, 2, 3, 4, 5, 6, and 7.

A Brief Review

Just to set the stage: we spent quite a few weeks on pictures, and just last week we started on animations.  We learned:

  • Whereas pictures were just variables, animations are functions.
  • They have a single parameter, which is how many seconds since the program started running.
  • The result of the function should be a picture.

To define a function, we give the parameter a name on the left side of the equal sign, right after the function name.  We can then use that name after the equal sign, just like if it were a number.  Here’s an example:

animation t = rotate (60*t) (rectangleSolid 40 200)

The function is called animation.  The parameter is called t.  And after the equal sign, we can use t just like any other number.  For example, we can multiply it by 60, and then rotate the picture by that many degrees.

The result of functions comes from substitution.  So to find out what this animation will look like after 5 seconds, the computer just looks at everything after the equal sign, and replaces any t that it sees with a 5.  So it takes the following steps:

  1. animation 5 – this is what the computer starts out trying to figure out.
  2. rotate (60*5) (rectangleSolid 40 200) – It replaces any t by 5 after the equal sign.
  3. rotate 300 (rectangleSolid 40 200) – This is just math: the computer calculated 60 times 5, and got 300.

So after 5 seconds, the computer will draw the rectangle rotated by 300 degrees.  If you try this out with a few numbers, you’ll notice that the rectangle turns by more and more degrees over time, making it spin.

So What’s the Problem?

We played around with animations that change lots of things: by t in different places, you can make things spin (using rotate), move (using translate), or get bigger or smaller (using scale, or by using t as the parameter to Gloss functions like circle).  But you might have noticed something disappointing about doing anything except rotation: after a while, the numbers get too big to see!  Things move off the edge of the screen, or get so big you can’t see them any more.

Because of this, our programs so far haven’t really done anything except rotate.  To do much more without everything ending up too big or too far off the screen, we’re going to have to get comfortable with a different kind of function…

Functions That Give Back Numbers

Animations are functions that get a number, and turn it into a picture.  But that’s not the only kind of function!  We’ve actually seen quite a few kinds of functions, if you count the ones that Gloss defines for us:

  • circle is a function that turns a number into a picture (just like animations, but the number means something different)
  • light is a function that turns a color into a different color.
  • rotate is a function that turns a number and a picture into different picture (one that’s rotated).
  • polygon is a function that turns a list of points into a picture.

In general, you can think of a function as a machine that turns things into other things:

You feed the parameters in, and you get the result out the other side.  Different functions need different types and different numbers of parameters, and they give you different types of results, too.  So when you want to use a function, the first things you should ask are: (1) How many parameters does it need? (2) What types of things does it want for its parameters?  And, (3) What type of thing is it going to give me back when it’s done?

In the end, you want to create pictures, of course!  So at first you might think that you’re only interested in functions that give you pictures back.  But that would be short-sighted!  Remember the functions light and dark?  They are functions that expect one parameter – a color – and give you back another color, not a picture.  But they’ve still been very useful for building your pictures, haven’t they?

The same thing is true with numbers.  Even though you ultimately want to end up with a picture, it’s very useful to be familiar with some functions that work with numbers, because we use numbers all the time when we’re making pictures.  So we’ll spend a lot of this week talking about functions that work with numbers.

How to Draw a Function on Numbers

When you have a function that expects a number for its only parameter, and gives you back a number as a result, we have a convenient way to draw it.  It might look weird at first, but believe me, it makes things a lot easier!  So here’s how to draw a function.  It only works if the function is from numbers to other numbers.

We start out by drawing two lines: one horizontal, and one vertical, like this:

I’ve called the vertical line f and the horizontal line t.  That’s because normally, the horizontal line will represent the amount of time the program has been running, and we’ve been calling that t.  The f is just short for “function”.

Next, you want to label the tick marks on the lines: for t, just number them 1, 2, 3, 4, and so on.  Zero is where the two lines cross, so the first tick mark after that should be 1.  For the vertical line, how you label it depends on the numbers you’re working with!  Notice it’s got both positive and negative numbers, too.

The next step is to draw some points.  Ask yourself, if you give your function is given the number 0, what number does it give back?  Now draw a point directly above or below the point on the horizontal line that means zero (remember, that’s the point where the lines cross).  How far up or down you draw the line is decided by what the function gives you back.  Now do the same thing for 1, and for 2, and so on…

Finally, once you’ve got a bunch of points, connect them with a smooth line.  That line is how you draw that function.

Example

Let’s try an example.  We’re going to draw the function

f t = t/2 - 2

That means the function is called f, its parameter (the number you give it) is called t, and the number it gives you back is half of t, minus 2.

Let’s draw the lines like we did before, and label the points counting by one in both ways:

The next step is to figure out some values of the function, and draw them as points on the graph:

f t = t/2 - 2
f  0 =  0/2 - 2 = 0.0 - 2 = -2.0
f  1 =  1/2 - 2 = 0.5 - 2 = -1.5
f  2 =  2/2 - 2 = 1.0 - 2 = -1.0
f  3 =  3/2 - 2 = 1.5 - 2 = -0.5
f  4 =  4/2 - 2 = 2.0 - 2 =  0.0
f  5 =  5/2 - 2 = 2.5 - 2 =  0.5
f  6 =  6/2 - 2 = 3.0 - 2 =  1.0
f  7 =  7/2 - 2 = 3.5 - 2 =  1.5
f  8 =  8/2 - 2 = 4.0 - 2 =  2.0
f  9 =  9/2 - 2 = 4.5 - 2 =  2.5
f 10 = 10/2 - 2 = 5.0 - 2 =  3.0
f 11 = 11/2 - 2 = 5.5 - 2 =  3.5
f 12 = 12/2 - 2 = 6.0 - 2 =  4.0
f 13 = 13/2 - 2 = 6.5 - 2 =  4.5

Here’s the graph with the line that connects those points:

How to Read the Drawing

Looking at the graph, you can get a good feeling for what that function does.  Imagine you’re always moving to bigger and bigger numbers of seconds since the program started (because you are!).  Then this function will keep getting bigger and bigger forever.  You can also see that it will keep getting bigger at the same speed.  And you can see that it starts out being -2, and then after 4 seconds it’s 0, and it keeps going up from there.

So what does that mean?

  • If you say rotate (t/2-2), that means start out rotated a little bit clockwise, and then turn counter-clockwise at a constant speed forever.
  • If you say translate (t/2-2) 0, that means start out a little to the left, then move right at a constant speed forever.
  • and so on…

Practice Drawing Functions on Numbers

In class on Tuesday, we did this worksheet together, where we practiced drawing functions that use numbers.

graphs_worksheet

Some Useful Functions on Numbers

You might notice at the end of the worksheet a few functions that we haven’t used yet that are useful:

  • min and max.  These functions actually take two numbers as parameters, and they return one of the two numbers you give them.  So min gives you the smaller of the two numbers, and max gives you the larger of the two numbers.
  • sin.  This function, which is pronounced like “sign”, is useful for making things move back and forth.  It is way more interesting than just that, but we’ll save the more interesting stuff for high school trigonometry classes.  All we want it for is to make things move back and forth.  Keep in mind that sin itself only goes back and forth between -1 and 1, so you probably want to multiple the answer by something bigger!

Want to see what the sin function looks like?  Here you go:

See how graphs of functions help you understand what they do?

Your Task

Because the Little School is off next week for a inter-term break, it will be two weeks until my next summary!  Never fear, though, I’ve got something for you to work on.

Your task is to make an animation of an amusement park ride.  Try to get several kinds of movement in there: rotating in circles, moving back and forth, and anything else you can do!  Those of us in my class are going to make these:

  • Skyler: Roller coaster
  • Grant: Bungee Jump Ride
  • Marcello: Ferris Wheel
  • Sophia: Tilt-a-Whirl
  • Sue: Swinging Pirate Ship
  • Me: Kids Jumping Rope (the example from week 7, but I might tweak it a little)

We’re planning to put them together into a gigantic amusement park picture when we’re done.  If you have friends doing this at the same time, you can, too!  (Do you know how?  It’s just like top-down design: get everyone to send you their programs, but call them something other than animation, then scale them, move them to different places, and combine them using pictures.)

Good luck!  And as always, use the comments to ask for help if you’re stuck.

September 28, 2011 / cdsmith

Haskell for Kids: Week 7

Where We Are

This is the summary of week 7 of my Haskell class for children, aimed at ages 11 through 13 or so.  You can go back and review the previous weeks to catch up.

We’ve now spent six weeks learning some of the Haskell programming language, and making various pictures, from simple stuff to some pretty complicated drawings.  Now it’s time for the next big step: animations!

Using gloss-web for Animations

We’ve done all of our programming using the gloss-web web site at http://dac4.designacourse.com:8000, where you can program and look at the result in a web browser.  So far, everything we’ve done has been in the “Picture” portion of the web site, which you get to by clicking the picture button:

Now we’re ready to move on to the next type of program.  So to get started, click the “Animate” button instead.

This page will look very nearly the same, but don’t be fooled!  The web site now expects from you a different kind of program: one that describes a picture that moves over time.

You’ll remember that the picture mode expected you write a definition for a variable called picture.  Everything else in your program was just there to help you define what picture meant, and when you finished, the program would run by looking up the definition of picture, and drawing it.  Your other definitions were useful only because they let you use those new words to define picture more easily.

The same thing is going on here, but there are two changes:

  1. Instead of picture, you’re defining a word called animation.
  2. Instead of being a description of a picture, animation is a function.  Like all functions, it has a parameter, which is the amount of time that’s passed.

Warning: I’ve said from the beginning not to use Internet Explorer for the programming web site.  It’s probably worked so far if you didn’t listen to me.  But now it will not work any more, so you’ll need to use something else: Firefox, Chrome, Safari, Opera… take your pick, but Internet Explorer won’t work.

Let’s jump in with some examples.

Example 1: Rotating a square

The first example program we’ll write is the one that is given to you by the web site:

import Graphics.Gloss
animation t = rotate (60 * t) (rectangleSolid 100 100)

Here’s how to think about what that means.Think of t as standing for the number of seconds it’s been since you clicked the Run button.

At the very beginning of your animation, it’s been 0 seconds since you pressed Run.  So the program is saying the same thing as: rotate (60*0) (rectangleSolid 100 100).  Remember that * means multiplication.  Of course, 60 times 0 is 0, so this is just rotate 0 (rectangleSolid 100 100).  It draws a rectangle that hasn’t been rotated at all.

But then some time passes.  After half of a second, now t is 0.5.  Now the picture your program is drawing is rotate (60*0.5) (rectangleSolid 100 100).  What is 60 times one half?  It’s 30.  So now the picture is rotate 30 (rectangleSolid 100 100), and it will draw a rectangle rotated by 30 degrees.  This will continue, too.  After a full second, t is 1, and 60 times 1 is 60, so the rectangle will be rotated by 60 degrees.  After 2 seconds, it will be rotated by 60 times 2, or 120 degrees.  As t gets bigger, the rectangle will be rotated more.

Okay, here’s a little quiz: how long will it take for the rectangle to turn all the way around, a full 360 degrees?  That shouldn’t be too hard: you want to find the number that will give 360 when it’s multiplied by 60.  That’s how many seconds it will take.

What about after that?  Can you rotate something more than 360 degrees?  Sure, you can… but you can’t tell that you did it!  Rotating 360 degrees looks just like leaving it alone.  So, for example, when it’s rotated 390 degrees, that’s the same as just rotating 30.  (If you were thinking about this, you might have noticed that you actually can’t even tell if a square has been rotated 90 degrees!  That’s because of the symmetry of the square.  But you can’t tell if any shape has been rotated 360 degrees, no matter if it has symmetry or not.)

Try this example on the web site, and make sure it looks the way you expected.

Example 2: Changing the speed

Let’s make a small modification to the example earlier.  We’ll make the square rotate faster:

import Graphics.Gloss
animation t = rotate (100 * t) (rectangleWire 100 100)

Just like before, t stands for the number of seconds since you clicked Run.  At the very beginning, t is zero, and zero times 100 is still zero.  So it starts in the same place.  But now look what happens: after half a second, it’s rotated by 50 degrees.  After a full second, it’s rotated by 100 degrees.  It will only take about three and a half seconds to make a full turn.

Example 3: Starting at a different angle

Now let’s try to change the angle it starts at, so that it started standing on one point like a diamond.

import Graphics.Gloss
animation t = rotate (100 * t + 45) (rectangleWire 100 100)

Okay, what’s that when t is zero?  It’s rotate 45 (rectangleWire 100 100).  So the square will start out turned on one point.  It will still rotate at the same speed as before, though: 100 degress every second.  Try that and see what it looks like.

Example 4: Moving

Now let’s try moving something instead of rotating it.

import Graphics.Gloss
animation t = translate (50 * t) 0 (circle 25)

Take a moment and try to guess what that will look like.

To figure it out, let’s look at the picture we’ll get at different times.  When the animation first starts, it’s been 0 seconds, and t is 0.  Then this is translate 0 0 (circle 25).  That won’t move the circle at all, so it will still be in the middle of the box.  But now when t is 1 (after the animation has been going for a full second), it will become translate 50 0 (circle 25), so the circle will be a bit off to the right.  In fact, the circle keeps moving right until it runs right off the screen (and even afterward, but then you can’t see it any more).

Try it!

Example 5: Moving faster

Can you guess how to make the circle move faster?  If you guessed that we want to multiply t by a higher number, you’re right!

import Graphics.Gloss
animation t = translate (100 * t) 0 (circle 25)

The circle will still start in the middle. (Why?)  But now after one second, it will have moved twice as far.  It’s moving faster.

We won’t do it as an example, but what if you wanted the circle to start on the left side of the screen?  Hint: it works a lot like example 3!

Example 6: Changing the size

What if you want something to get larger as time goes on?  You can do that, too.  Here’s an example:

import Graphics.Gloss
animation t = circle (20 * t)

Once again, t stands for the number of seconds since you pressed Run.  At the very beginning, your picture will be circle (20 * 0), which is the same thing as circle 0.  A circle with a radius of 0 is too small to see, so you won’t see anything.  But after half a second, it becomes circle (20 * 0.5), which is circle 10, and you will see a circle with a radius of 10.  The circle will keep getting larger: after ten seconds, it will be circle 200.  Eventually, if you’re patient enough, it will grow too big to even show up on the screen.

Try that, and see if it does what you expected.

(You could have also written that using scale: animation t = scale t t (circle 20).  Do you see why those two programs should do the same thing?)

Example 7: Rotating around another point

Sometimes we might want something to rotate, but around the middle of the whole picture (or part of a picture), and not just around its own middle.  The trick is to move it (with translate), and then rotate the picture that you get as a result of that.

import Graphics.Gloss
animation t = rotate (60 * t) (translate 100 0 (circleSolid 25))

See if you can work out what that will look like at different times, and then try it out and check yourself.

Animations and List Comprehensions and Functions

You might have noticed that a lot of the things we’re doing here seem very similar to what we did with list comprehensions.  You’d be right!

Both times, we had variables, and we got different pictures because those variables got different values.  The difference is that with list comprehensions, we put all of those different pictures together at the same time.  Now we’re using different pictures at different times.  But the idea is the same: you use variables to represent numbers that you don’t know yet, and then you can build versions of your picture where those variables mean something different.

We’ve actually used variables like that three times:

  • When we defined functions (like awesome), the parameters got names, and you used those names in the function.
  • In list comprehensions, you used the funny backward arrow: <-, to pick a name and arrange for it to get lots of different values.
  • Now, in animations, you’re again defining a function: a special one, where the parameter is the time.

This is something that will keep happening.  Variables are very important, so you should get used to seeing expressions that have variables in them, and doing something called “substitution”.  That just means answering the question “If t is 5, what does this work out to?”

Top Down Design With Animations

With pictures, we worked on organizing our program by top down design: we started by saying what picture meant, but we sometimes used other words the computer doesn’t know yet.  Then we went back in and filled in what those words meant.  We just kept this up until the program was done.  The nice thing about top-down design was this: you didn’t have to think about your entire program at once.  You could just think about one piece at a time.

We’d like to keep doing top-down design with animations, too.  There’s one thing we have to figure out, though, and that’s what to do with t.  Here’s what we’ll do:

  • When we’re defining a word, if its a piece of the picture that, just looking at that one piece, will have some movement, then we’ll make it a function, taking a parameter called t.
  • When we’re using it later on, we’ll pass t for that parameter.

It’s important to keep these consistent: if you define a function with a parameter, then you also have to give it that parameter when you use the function.  If you define a variable without a parameter, then you aren’t allowed to give it a parameter when you use it.

Let’s look at an example: here’s a program to draw a clock:

import Graphics.Gloss

animation t = clock t

clock t = pictures [
    backPlate,
    minuteHand t,
    secondHand t
    ]

backPlate = color (dark white) (circleSolid 250)

minuteHand t = rotate (-0.1 * t) (line [(0,0), (0,250)])

secondHand t = rotate (-6 * t) (line [(0,0), (0,250)])

We’ll break it down piece by piece:

  1. First, we defined the animation function.  When you’re working in animation mode, you always have to define this, and it always has the parameter t, that means how many seconds since it started.  We say that the animation is a clock.  The clock will have moving parts, so we pass along t.
  2. Since we gave the time, t, to clock earlier as a parameter, our definition of clock has to say it has a parameter, too.  A clock consists of three parts: a back plate, a minute hand, and a second hand.  The back plate does not move, so we don’t pass along t.  But the minute hand and second hand do move, so we’ll be passing the time along to those.
  3. Next we define backPlate.  Remember we didn’t give it a t parameter because it won’t be moving, so this is just a variable.
  4. Now we define minuteHand.  We did give it a parameter, so now we have to say it gets a parameter.  And just like in the earlier examples, we can use that parameter t to make it rotate.  We want the minute hand to rotate 360 degrees every hour, and an hour is 3600 seconds.  So that means we want it to move only a tenth of a degree every second.  Also, we’d like it to move clockwise, so that number has to be negative.  (Remember, positive rotations are counter-clockwise.)
  5. Finally, we define secondHand.  It needs a parameter, too, so we give it the t parameter.  The definition looks the same, but now we want it to rotate 360 degrees every minute (60 seconds).  That works out to 6 degrees every second, and it’s still negative so that the rotation is clockwise.

Hopefully, that doesn’t look too bad!  It’s just the same kind of top-down design we’ve been doing, except with some time thrown in to keep things moving.  You have to ask yourself whether each part has movement in it or not, and you have to consistent about the answer.

This week’s assignment

The assignment we’re working on this week is to make a working model of the solar system.  That is, put the sun in the middle, and the planets around it, and have them rotate around the sun.  A few notes:

  • It doesn’t need to be to scale.  In fact, the planets are so far apart, and so small compared to the distance between them, that if you try to draw everything to scale, you won’t even be able to see it!  So just pick what looks good.
  • If you are ambitious, try to add at least one moon, and give Saturn its rings.The hint here is that moons and rings aren’t too hard, as long as you’re using top-down design.  For example, you should have something called saturn, and instead of making it a circle, make it a pair of circles with one of them stretched a bit.

    Now suppose you’re adding a moon to earth, you probably want a function called something like earthSystem that represents the earth and its moon together.  Since the moon is moving around the earth, it should be a function and have a parameter called t.  When you’re defining earthSystem, you’ll probably want even more variables called earth and moon.  These will probably just be colored circles.

  • Our class here in Colorado Springs actually all decided to make up their own solar system with different planet names.  That’s fine, too.  In fact, the computer doesn’t care what you name things.

Good luck!  And feel free to ask in the comments for help.

Just for fun…

If you’re bored, try this animation.  It’s a preview of what’s coming in the next couple weeks.

import Graphics.Gloss

animation t = pictures [
    jumprope t,
    translate (-210) 0 stickFigure,
    translate ( 210) 0 stickFigure,
    jumper t
    ]

jumprope t = scale 1.75 (1.1 * sin t) rope

rope = line [ (x, 100 * cos (0.015 * x)) | x <- [-100, -75 .. 100 ] ]

stickFigure = pictures [
    line [ (-35, -100), (0, -25) ],
    line [ ( 35, -100), (0, -25) ],
    line [ (  0,  -25), (0,  50) ],
    line [ (-35,    0), (0,  40) ],
    line [ ( 35,    0), (0,  40) ],
    translate 0 75 (circle 25)
    ]

jumper t = translate 0 (max 0 (-50 * sin t)) stickFigure

That’s all for this week.

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