Derivatives of algebraic data structures: An excellent tutorial

Posted by Eric Kidd Fri, 20 May 2011 20:01:00 GMT

Last month, the folks at Lab49 explained how to compute the derivative of a data structure. This is a great example of how to write about mathematical subjects for a casual audience: They draw analogies to well-known programming languages, they follow a single, well-chosen thread of explanation, and there’s a clever payoff at the end.

The Lab49 blog post is, of course, based on two classic papers by Conor McBride, and Huet’s original paper The Zipper.

If you’re interested in real-world applications of this technique, there’s a great explanation in the final chapter of Learn You a Haskell for Great Good. If you’re interested in some deeper mathematical connections, see the discussion at Lambda the Ultimate.

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What do these fixed points have in common?

Posted by Eric Kidd Thu, 12 May 2011 12:09:00 GMT

A question asked while standing in the shower: What do all of the following have in common?

  1. Banach and Brouwer fixed points. If you’re in Manhattan, and you crumple up a map of Manhattan and place it on the ground, at least one point on your map will be exactly over the corresponding point on the ground. (This is true even if your map is larger than life.)
  2. The fixed points computed by the Y combinator, which is used to construct anonymous recursive functions in the lambda calculus.
  3. The Nash equilibrium, which is the stable equilibrium of a multi-player game (and one of the key ideas of economics). See also this lovely—if metaphorical—rant by Scott Aaronson.
  4. The eigenvectors of a matrix, which will still point in the same direction after multiplication by the matrix.

At what level of abstraction are all these important ideas really just the same idea? If we strip everything down to generalized abstract nonsense, is there a nice simple formulation that covers all of the above?

(I can’t play with this shiny toy today; I have to work.)

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Ubiquitous Hoogle

Posted by Eric Kidd Mon, 01 Sep 2008 10:33:00 GMT

Ubiquity is an experimental Firefox plugin. It’s a “graphical command line” similar to QuickSilver on the Macintosh.

You can easily add your own commands to Ubiquity. The following article shows how to create a Hoogle search command that looks up Haskell functions by name or by type signature.

Searching for putStr

You can press Return or click on one the links in the preview.

Read more...

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Probability monads at Hac 07 II

Posted by Eric Kidd Tue, 02 Oct 2007 07:50:00 GMT

From October 5-7, I’ll be at the Haskell Hackathon in Freiburg.

I’ll be working on probability monads, attempting to turn my various blog articles into a real Haskell library.

Some resources:

If you were a peer reviewer, or gave me feedback on the paper, my humble thanks–and apologies. I haven’t had a chance to revise the paper yet, and so your feedback is not yet included.

See you at the Hackathon!

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Freiburg in October: Scheme, Dylan, and probability monads

Posted by Eric Kidd Tue, 18 Sep 2007 12:36:00 GMT

Good morning! I’ll be in Freiburg for several events this October, including CUFP and the Haskell Hackathon.

Commercial Users of Functional Programming (CUFP)

On October 4th, I’ll be speaking at CUFP ’07, describing the use of Scheme in a real-world multimedia engine. Some likely topics:

  1. How we switched to Scheme (and why refactoring is your friend).
  2. How our artists learned to program in Scheme (it’s all about the tools).
  3. The tension between functional programming and objects: Can we have both?

Dylan Beer Night

Once upon a time, I dreamt of generic functions and built RPMs for Gwydion Dylan.

Some current and former Dylan hackers are hoping to meet in Freiburg, most likely on October 4th. If you’re at ICFP or one of the workshops, we’d love to hear from you.

Haskell Hackathon

I’ll be at the Haskell Hackathon from Friday to Sunday.

Perhaps it’s time to whip Control.Monad.Probability into shape?

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Bowling in Haskell: A response to Ron Jeffries

Posted by Eric Kidd Sat, 28 Apr 2007 11:53:00 GMT

Bowling is a tricky game to score. It’s just complicated enough to act as a good programming exercise. And Ron Jeffries has performed this exercise many times, in C#, Smalltalk, and other languages. He’s been searching for a tidy and elegant solution, one which makes the rules of bowling as clear as possible.

In the past, though, Jeffries has been a bit skeptical of Haskell implementations of bowling:

The recursive [Haskell] solution, however, is questionable on more fundamental grounds. A game of bowling consists of ten frames, not less or more, and the “ten-ness” of the game is not represented in the recursive solutions at all. Even if we let that slide, the recursive solutions make it a bit hard to understand what’s going on.

Let’s see if we can do better. No knowledge of bowling is required–if we do this right, our program should be at least as clear as an English-language version of the rules.

Along the way, we’ll encounter lazy lists, an interesting recursion combinator, and Hoogle, the Haskell search engine.

The rules of bowling

In bowling, we roll balls down a lane, trying to knock down pins. If we know how many pins we knock down with each ball, we can compute the final score. So our program looks something like this:

-- Pins knocked down by each ball.
type Balls = [Int]

-- Number of points scored.
type Score = Int

scoreGame :: Balls -> Score
scoreGame balls = ???

But how do we implement scoreGame?

Scoring a frame

A bowling game is divided into 10 frames. Ordinary frames consist of 1 or 2 balls. The 10th frame may have an additional 1 or 2 bonus balls, which we discuss below.

To score an individual frame, we need to do two things: (1) calculate the score for our frame, and (2) figure out where the next frame starts. Our scoring function will return both pieces of information:

-- Score one frame and return the rest.
scoreFrame :: Balls -> (Score, Balls)

If we knock down all 10 pins with the first ball in a frame (x1), we call it a strike, and move on to the next frame immediately. But we also get a bonus—we’re allowed to count balls y1 and y2 from the next frame towards this frame’s score:

scoreFrame (x1:    y1:y2:ys) | x1 == 10 =
  (x1+y1+y2, y1:y2:ys)  -- Strike

If we knock down all the pins using two balls (x1 and x2), we call it a spare. And we get to count one ball from the next frame as our bonus:

scoreFrame (x1:x2: y1:ys) | x1+x2 == 10 =
  (x1+x2+y1, y1:ys)     -- Spare

If we don’t manage to knock all 10 pins with two balls, we call it an open frame. And we don’t get any bonus:

scoreFrame (x1:x2: ys) =
  (x1+x2,    ys)        -- Open frame

What happens if we have a strike or a spare in the 10th frame? We get to roll our bonus balls anyway. Conventionally, these extra balls are recorded as part of the 10th frame (making it 3 balls long), but they’re really just phantom balls hanging off the end of the game.

Next, we need to turn scoreFrame into a recursive function.

Read more...

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Robot localization using a particle system monad

Posted by Eric Kidd Thu, 19 Apr 2007 20:43:00 GMT

Refactoring Probability Distributions: part 1, part 2, part 3, part 4, part 5

Welcome to the 5th (and final) installment of Refactoring Probability Distributions! Today, let’s begin with an example from Bayesian Filters for Location Estimation (PDF), an excellent paper by Fox and colleagues.

In their example, we have a robot in a hallway with 3 doors. Unfortunately, we don’t know where in the hallway the robot is located:

The vertical black lines are “particles.” Each particle represents a possible location of our robot, chosen at random along the hallway. At first, our particles are spread along the entire hallway (the top row of black lines). Each particle begins life with a weight of 100%, represented by the height of the black line.

Now imagine that our robot has a “door sensor,” which currently tells us that we’re in front of a door. This allows us to rule out any particle which is located between doors.

So we multiply the weight of each particle by 100% (if it’s in front of a door) or 0% (if it’s between doors), which gives us the lower row of particles. If our sensor was less accurate, we might use 90% and 10%, respectively.

What would this example look like in Haskell? We could build a giant list of particles (with weights), but that would require us to do a lot of bookkeeping by hand. Instead, we use a monad to hide all the details, allowing us to work with a single particle at a time.

localizeRobot :: WPS Int
localizeRobot = do
  -- Pick a random starting location.
  -- This will be our first particle.
  pos1 <- uniform [0..299]
  -- We know we're at a door.  Particles
  -- in front of a door get a weight of
  -- 100%, others get 0%.
  if doorAtPosition pos1
    then weight 1
    else weight 0
  -- ...

What happens if our robot drives forward?

Read more...

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How to make Data.Set a monad

Posted by Eric Kidd Thu, 15 Mar 2007 22:29:00 GMT

…and how to fake Lisp macros with Template Haskell

(I wrote this article in response to a comment by sigfpe. You may find it pretty dry reading, unless you want to build domain-specific languages in Haskell. Proceed at your own risk.)

Haskell’s built-in Monad type has some serious limitations. We can fix those limitations using a number of advanced Haskell techniques, including Template Haskell, Haskell’s closest equivalent to Lisp macros.

We can illustrate the limitations of Monad with an example from math. In set theory, we can define a set by specifying how to compute each element:

{ xy : x ∈ {1,2,4}, y ∈ {1,2,4} }

We can read this as, “the set of all xy, where x is one of {1,2,4}, and y is one of {1,2,4}.” To calculate the answer, we first multiply together all the possible combinations:

1×1=1, 1×2=2, 1×4=4, 2×1=2, 2×2=4, 2×4=8, 4×1=4, 4×2=8, 4×4=16

We then collect up the answers, and—because we’re working with sets–we throw away the duplicates:

{1,2,4,8,16}

Can we do the same thing in Haskell? Well, using Haskell’s list monad, we can write:

listExample = do
  x <- [1,2,4]
  y <- [1,2,4]
  return (x*y)

But when we run this, Haskell gives us lots of duplicate values:

> listExample
[1,2,4,2,4,8,4,8,16]

Our problem: We’re using lists (which can contain duplicate values) to represent sets (which can’t). Can we fix this by switching to Haskell’s Data.Set?

import qualified Data.Set as S

-- This doesn't work.
setExample = do
  x <- S.fromList [1,2,4]
  y <- S.fromList [1,2,4]
  return (x*y)

Unfortunately, this code fails spectacularly. A Haskell monad is required to work for any types a and b:

class Monad m where
  return :: a -> m a
  fail :: String -> m a
  (>>=) :: m a -> (a -> m b) -> m b

But Data.Set only works for some types. Specifically, it requires that values of type a can be ordered:

data (Ord a) => Set a = ...

As it turns out, we can make Data.Set into a monad. But be warned: The solution involves some pretty ugly Haskell abuse.

Read more...

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Monads in 15 minutes: Backtracking and Maybe

Posted by Eric Kidd Mon, 12 Mar 2007 19:39:00 GMT

This morning, a programmer visited #haskell and asked how to implement backtracking. Not surprisingly, most of the answers involved monads. After all, monads are ubiquitous in Haskell: They’re used for IO, for probability, for error reporting, and even for quantum mechanics. If you program in Haskell, you’ll probably want to understand monads. So where’s the best place to start?

A friend of mine claims he didn’t truly understand monads until he understood join. But once he figured that out, everything was suddenly obvious. That’s the way it worked for me, too. But relatively few monad tutorials are based on join, so there’s an open niche in a crowded market.

This monad tutorial uses join. Even better, it attempts to cram everything you need to know about monads into 15 minutes. (Hey, everybody needs a gimmick, right?)

Backtracking: The lazy way to code

We begin with a backtracking constraint solver. The idea: Given possible values for x and y, we want to pick those values which have a product of 8:

solveConstraint = do
  x <- choose [1,2,3]
  y <- choose [4,5,6]
  guard (x*y == 8)
  return (x,y)

Every time choose is called, we save the current program state. And every time guard fails, we backtrack to a saved state and try again. Eventually, we’ll hit the right answer:

> take 1 solveConstraint
[(2,4)]

Let’s build this program step-by-step in Haskell. When we’re done, we’ll have a monad.

Read more...

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8 ways to report errors in Haskell

Posted by Eric Kidd Sat, 10 Mar 2007 19:05:00 GMT

Haskell is a marvellous language, but there are some things I don’t like about it. My least favorite: Haskell has no fewer than 8 different APIs for reporting errors.

To make a bad situation worse, the choice of API varies between popular libraries. To give a particularly unfortunate example, Network.URI.parseURI and Network.HTTP.simpleHTTP report errors in entirely different ways, turning a “download this URL” program into a page of code, nearly half of which is devoted to dealing with various kinds of errors. (The rest is boilerplate that could be refactored into a nice wrapper.)

Let’s begin with a toy function, the simplest possible program that could actually fail:

myDiv x y = x / y

As every algebra student knows, we can’t divide by zero. Using this function as our example, let’s take a look at all the different ways we can implement error-reporting in Haskell.

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