Posted by Eric Kidd
Mon, 05 Mar 2007 09:32:00 GMT
Monads are a remarkably powerful tool for building specialized programming languages. Some examples include:
But there’s a bunch of things I don’t understand about monads. In each case, my confusion involves some aspect of the underlying math that “bubbles up” to affect the design of specialized languages.
(Warning: Obscure monad geeking ahead.)
Commutative monads
A “commutative monad” is any monad where we can replace the expression:
…with:
…without changing the meaning. Examples of commutative monads include Reader and Rand. This is an important property, because it might allow us to parallelize the commonly-used sequence function across huge numbers of processors:
sequence :: (Monad m) => [m a] -> m [a]
Simon Peyton Jones lists this problem as Open Challenge #2, saying:
Commutative monads are very common. (Environment,
unique supply, random number generation.) For these, monads over-sequentialise.
Wanted: theory and notation for some cool compromise.
Commutative monad morphisms
Monad morphisms are the category theory equivalent of Haskell’s monad transformers. Haskell’s monad transformers can be expressed as monad layerings, which correspond to the monad morphisms of category theory.
Many complicated monads break down into a handful of monad transformers, often in surprising ways.
But composing monad transformers is a mess, because they interact in poorly-understood ways. In general, the following two types have very different semantics:
FooT (BarT m)
BarT (FooT m)
If FooT and BarT commute with each other, however, the two types would be equivalent. This is helpful when building large stacks of monad transformers.
Chung-chieh Shan encountered a related problem when applying monad morphisms to build a theory of natural language semantics:
It remains to be seen whether monads would provide the appropriate
conceptual encapsulation for a semantic theory with broader coverage. In
particular, for both natural and programming language semantics, combining monads—or perhaps monad-like objects—remains an open issue that
promises additional insight.
Monad morphisms and abstract algebra
Dan Piponi has been drawing some fascinating connections between monad morphisms and abstract algebra. See, for example:
This approach seems to throw a lot of light on monad morphisms—but at least in my case, the light only highlights my confusion.
Of the three problems listed here, this is the one most likely to be discussed in a textbook somewhere. And a solution to this problem would likely help significantly with the other two.
So, my question: Does anybody have any books, papers or ideas that might help untangle this mess?
Update: Be sure to see the comment thread on the second Dan Piponi post above and Chung-chieh Shan’s excellent bibliography on monad transformers.
Tags Haskell, Math, Monads
Posted by Eric Kidd
Sat, 03 Mar 2007 09:02:00 GMT
(Refactoring Probability Distributions: part 1, part 2,
part 3, part 4)
The world is full of messy classification problems:
- “Is this order fraudulent?”
- “It this e-mail a spam?”
- “What blog posts would Rachel find interesting?”
- “Which intranet documents is Sam looking for?”
In each case, we want to classify something: Orders are either valid or
fraudulent, messages are either spam or non-spam, blog posts are either
interesting or boring. Unfortunately, most software is terrible at
making these distinctions. For example, why can’t my RSS reader go out and
track down the 10 most interesting blog posts every day?
Some software, however, can make these distinctions.
Google figures out when I want to watch a movie, and shows me specialized
search results. And most e-mail clients can identify spam with over
99% accuracy. But the vast majority of software is dumb, incapable of
dealing with the messy dilemmas posed by the real world.
So where can we learn to improve our software?
Outside of Google’s shroud
of secrecy, the most successful classifiers are spam filters. And most modern
spam filters are inspired by Paul Graham’s essay A Plan for Spam.
So let’s go back to the source, and see what we can learn. As it turns out, we can formulate a lot of the ideas in A Plan
for Spam in a straightforward fashion using a Bayesian
monad.
Functions from distributions to distributions
Let’s begin with spam filtering. By convention, we divide messages into
“spam” and “ham”, where “ham” is the stuff we want to read.
data MsgType = Spam | Ham
deriving (Show, Eq, Enum, Bounded)
Let’s assume that we’ve just received a new e-mail. Without even looking
at it, we know there’s a certain chance that it’s a spam. This gives us
something called a “prior distribution” over MsgType.
> bayes msgTypePrior
[Perhaps Spam 64.2%, Perhaps Ham 35.8%]
But what if we know that the first word of the message is “free”? We can
use that information to calculate a new distribution.
> bayes (hasWord "free" msgTypePrior)
[Perhaps Spam 90.5%, Perhaps Ham 9.5%]
The function hasWord takes a string and a probability
distribution, and uses them to calculate a new probability distribution:
hasWord :: String -> FDist' MsgType ->
FDist' MsgType
hasWord word prior = do
msgType <- prior
wordPresent <-
wordPresentDist msgType word
condition wordPresent
return msgType
This code is based on the Bayesian monad from part 3. As before,
the “<-” operator selects a single item from a probability
distribution, and “condition” asserts that an expression is true. The
actual Bayesian inference happens behind the scenes (handy, that).
If we have multiple pieces of evidence, we can apply them one at a time.
Each piece of evidence will update the probability distribution produced by
the previous step:
hasWords [] prior = prior
hasWords (w:ws) prior = do
hasWord w (hasWords ws prior)
The final distribution will combine everything we know:
> bayes (hasWords ["free","bayes"] msgTypePrior)
[Perhaps Spam 34.7%, Perhaps Ham 65.3%]
This technique is known as the naive Bayes classifier. Looked at from the right angle, it’s surprisingly simple.
(Of course, the naive Bayes classifier assumes that all of our evidence is independent. In theory, this is a pretty big assumption. In practice, it works better than you might think.)
But this still leaves us with a lot of questions: How do we keep track of
our different classifiers? How do we decide which ones to apply? And do
we need to fudge the numbers to get reasonable results?
In the following sections, I’ll walk through various aspects of Paul
Graham’s A Plan for Spam, and show how to generalize it. If you
want to follow along, you can download the code using Darcs:
darcs get http://www.randomhacks.net/darcs/probability
Tags Haskell, Math, Monads, Probability, ProbabilityMonads, Spam
Posted by Eric Kidd
Thu, 22 Feb 2007 18:11:00 GMT
Part 3 of Refactoring Probability Distributions.
(Part 1: PerhapsT,
Part 2: Sampling functions)
A very senior Microsoft developer who moved to Google told
me that Google works and thinks at a higher level of abstraction than
Microsoft. “Google uses Bayesian filtering the way Microsoft uses the if
statement,” he said. -Joel Spolsky
I really love this quote, because it’s insanely provocative
to any language designer. What would a programming language look
like if Bayes’ rule were as simple as an if statement?
Let’s start with a toy problem, and refactor it until Bayes’ rule is baked
right into our programming language.
Imagine, for a moment, that we’re in charge of administering drug tests for
a small business. We’ll represent each employee’s test results (and drug use) as follows:
data Test = Pos | Neg
deriving (Show, Eq)
data HeroinStatus = User | Clean
deriving (Show, Eq)
Assuming that 0.1% of our employees have used heroin recently, and that our test is 99%
accurate, we can model the testing process as follows:
drugTest1 :: Dist d => d (HeroinStatus, Test)
drugTest1 = do
heroinStatus <- percentUser 0.1
testResult <-
if heroinStatus == User
then percentPos 99
else percentPos 1
return (heroinStatus, testResult)
percentUser p = percent p User Clean
percentPos p = percent p Pos Neg
percent p x1 x2 =
weighted [(x1, p), (x2, 100p)]
This code is based our FDist monad, which is in turn based on
PFP. Don’t worry if it seems slightly mysterious; you can think of the
“<-” operator as choosing an element from a probability
distribution.
Running our drug test shows every possible combination of the two
variables:
> exact drugTest1
[Perhaps (User,Pos) 0.1%,
Perhaps (User,Neg) 0.0%,
Perhaps (Clean,Pos) 1.0%,
Perhaps (Clean,Neg) 98.9%]
If you look carefully, we have a problem. Most of the employees who test
positive are actually clean! Let’s tweak our code a bit, and try to zoom
in on the positive test results.
Read more...
Tags Haskell, Math, Monads, Probability, ProbabilityMonads, Recommended
Posted by Eric Kidd
Wed, 21 Feb 2007 23:53:00 GMT
In Part 1, we cloned PFP, a library for computing with probability distributions. PFP represents a distribution as a list of possible values, each with an associated probability.
But in the real world, things aren’t always so easy. What if we wanted to pick a random number between 0 and 1? Our previous implementation would break, because there’s an infinite number of values between 0 and 1—they don’t exactly fit in a list.
As it turns out, Sungwoo Park and colleagues found an elegant solution to this problem. They represented probability distributions as sampling functions, resulting in something called the λ◯ calculus. (I have no idea how to pronounce this!)
With a little bit of hacking, we can use their sampling functions as a drop-in replacement for PFP.
A common interface
Since we will soon have two ways to represent probability distributions, we need to define a common interface.
type Weight = Float
class (Functor d, Monad d) => Dist d where
weighted :: [(a, Weight)] -> d a
uniform :: Dist d => [a] -> d a
uniform = weighted . map (\x -> (x, 1))
The function uniform will create an equally-weighted distribution from a list of values. Using this API, we can represent a two-child family as follows:
data Child = Girl | Boy
deriving (Show, Eq, Ord)
child :: Dist d => d Child
child = uniform [Girl, Boy]
family :: Dist d => d [Child]
family = do
child1 <- child
child2 <- child
return [child1, child2]
Now, we need to implement this API two different ways: Once with lists, and a second time with sampling functions.
Read more...
Tags Haskell, Math, Monads, Probability, ProbabilityMonads
Posted by Eric Kidd
Wed, 21 Feb 2007 08:06:00 GMT
(Warning: This article is a bit more technical than most of my stuff. It assumes prior knowledge of monads and monad transformers.)
Martin Erwig and Steve Kollmansberger wrote PFP, a really sweet Haskell library for computing with probabilities. To borrow their example:
die :: Dist Int
die = uniform [1..6]
If we roll a die, we get the expected distribution of results:
> die
1 16.7%
2 16.7%
3 16.7%
4 16.7%
5 16.7%
6 16.7%
If you haven’t seen PFP before, I strongly encourage you to check it out. You can use it to solve all sorts of probability puzzles.
Anyway, I discovered an interesting way to implement PFP using monad transformers. Here’s what it looks like:
type Dist = PerhapsT ([])
uniform = weighted . map (\x -> (x, 1))
In other words, Dist can be written by adding some semantics to the standard list monad.
Perhaps: A less specific version of Maybe
First, let’s define a simple probability type:
newtype Prob = P Float
deriving (Eq, Ord, Num)
instance Show Prob where
show (P p) = show intPart ++ "." ++ show fracPart ++ "%"
where digits = round (1000 * p)
intPart = digits `div` 10
fracPart = digits `mod` 10
Thanks to the deriving (Num) declaration, we can treat Prob like any other numeric type.
We can now define Perhaps, which represents a value with an associated probability:
data Perhaps a = Perhaps a Prob
deriving (Show)
Now, this is just a generalization of Haskell’s built-in Maybe type, which treats a value as either present (probability 1) or absent (probability 0). All we’ve added is a range of possibilities in between: Perhaps x 0.5 represents a 50% chance of having a value.
Note that there’s one small trick here: When the probability of a value is 0, we may not actually know it! But because Haskell is a lazy language, we can write:
We’ll need a convenient way to test for this case, to make sure we don’t try to use any undefined values:
neverHappens (Perhaps _ 0) = True
neverHappens _ = False
So Perhaps is just like Maybe. As it turns out, they’re both monads, and they both have an associated monad transformer.
Read more...
Tags Haskell, Math, Monads, Probability, ProbabilityMonads
Posted by Eric Kidd
Sat, 10 Feb 2007 16:14:00 GMT
Syntaxfree is hacking on Martin Erwig’s probability monad. This is one of the coolest monads out there—it allows you to trivially solve all kinds of probability problems.
Mikael Johansson has a good example.
I hope to write a bit more about probability monads soon. There’s already a long post sitting on my hard drive, and some more ideas that I’m still trying to puzzle out.
In the meantime, I’d like to recommend The Haskell Road to Logic, Maths and Programming. There’s an excellent review available.
Tags Haskell, Math, Monads, Probability
Posted by Eric Kidd
Sat, 10 Feb 2007 09:55:00 GMT
Or, how to optimize MapReduce, and when folds are faster than loops
Purely functional programming might actually be worth the pain, if you care about large-scale optimization.
Lately, I’ve been studying how to speed up parallel algorithms. Many
parallel algorithms, such as Google’s MapReduce, have two parts:
- First, you transform the data by mapping one or more functions over each value.
- Next, you repeatedly merge the transformed data, “reducing” it down to a
final result.
Unfortunately, there’s a couple of nasty performance problems lurking here. We really want to combine all those steps into a single pass, so that we can eliminate temporary working data. But we don’t always want to do this optimization by hand—it would be better if the compiler could do it for us.
As it turns out, Haskell is an amazing testbed for this kind of
optimization. Let’s build a simple model, show where it breaks, and then
crank the performance way up.
Trees, and the performance problems they cause
We’ll use single-threaded trees for our testbed. They’re simple enough to demonstrate the basic idea, and they can be generalized to parallel systems. (If you want know how, check out the papers at the end of this article.)
A tree is either empty, or it is
a node with a left child, a value and a right child:
data Tree a = Empty
| Node (Tree a) a (Tree a)
deriving (Show)
Here’s a sample tree containing three values:
tree = (Node left 2 right)
where left = (Node Empty 1 Empty)
right = (Node Empty 3 Empty)
We can use treeMap to apply a function to every value in a
tree, creating a new tree:
treeMap :: (a -> b) -> Tree a -> Tree b
treeMap f Empty = Empty
treeMap f (Node l x r) =
Node (treeMap f l) (f x) (treeMap f r)
Using treeMap, we can build various functions that manipulate
trees:
treeDouble tree = treeMap (*2) tree
treeIncr tree = treeMap (+1) tree
What if we want to add up all the values in a tree? Well, we could write a
simple recursive sum function:
treeSum Empty = 0
treeSum (Node l x r) =
treeSum l + x + treeSum r
But for reasons that will soon become clear, it’s much better to refactor
the recursive part of treeSum into a reusable
treeFold function (“fold” is Haskell’s name for “reduce”):
treeFold f b Empty = b
treeFold f b (Node l x r) =
f (treeFold f b l) x (treeFold f b r)
treeSum t = treeFold (\l x r -> l+x+r) 0 t
Now we can double all the values in a tree, add 1 to each, and sum up the
result:
treeSum (treeIncr (treeDouble tree))
But there’s a very serious problem with this code. Imagine that we’re
working with a million-node tree. The two calls to treeMap
(buried inside treeIncr and treeDouble) will each
create a new million-node tree. Obviously, this will kill our performance,
and it will make our garbage collector cry.
Fortunately, we can do a lot better than this, thanks to some funky GHC
extensions.
Read more...
Tags Haskell, Performance, Recommended
Posted by Eric Kidd
Thu, 08 Feb 2007 20:01:00 GMT
Haskell has been tying my brain in knots. Sure, it keeps teaching me all
sorts of amazing things, but it’s also forcing me to relearn the basics.
Right now, I’m trying to implement simple data structures in Haskell. It’s
challenging, because most typical data structures are based on updating pointers,
and Haskell doesn’t allow that. (Well, I could cheat, and use the IO
monad, but where’s the fun in that?)
Fortunately, Chris Okasaki has done some amazing research into Haskell (and
ML) data structures. He wrote an excellent thesis, which was later
published as Purely Functional Data Structures. It’s a
comprehensive book, well-written, and highly recommended for anyone programming
in a functional language.
Let’s take a look at a data structure from Okasaki’s book: A purely
functional queue. It’s competitive with a traditional queue (on average, at least), but it doesn’t use any pointers.
The queue API
A queue is a list of values to process. We can add values to the back of
the queue, and retrieve values from the front.
In Haskell, we often begin by writing down the types of our functions. If we’re lucky, the
code will fill itself in almost automatically. The first part of our API
is easy:
data Queue a = ???
newQueue :: Queue a
empty :: Queue a -> Bool
The next part is a bit trickier. Since we can’t update our actual
Queue values, our data structure is read-only! We can work
around this by making enq and deq return an
updated version of the Queue:
enq :: Queue a -> a -> Queue a
deq :: Queue a -> (a, Queue a)
We could use this API as follows:
q = newQueue `enq` 1 `enq` 2 `enq` 3
(x,q') = deq q
(y,q'') = deq q'
Now, the tricky bit: Making it run quickly.
Read more...
Tags Haskell
Posted by Eric Kidd
Fri, 02 Feb 2007 22:00:00 GMT
Sometime back in elementary school, I first asked teachers, “What happens when you divide infinity by 2?” Some teachers couldn’t answer, and others told me, “It’s still infinity!”
More recently, a couple of friends were discussing a similar question at lunch: “What happens when you add 1 to infinity?”
Of course I said, “It’s still infinity!”, but I couldn’t explain it much better than my school teachers (at least not without using the word denumerable, which is a good way to ruin a lunch conversation).
And then tonight, while reading a paper about Haskell, I was hit by an evil idea: When in doubt, ask the Haskell interpreter!
Step 1: Counting
First, we need to teach Haskell about the natural numbers. (Why not use Haskell’s built-in integers? Just humor the crazy programmer for a moment, OK?)
A number is either zero, or the successor of another number. We can write that in Haskell as:
data Nat = Zero | Succ Nat
deriving (Show, Eq, Ord)
Math geeks in the audience will recognize this as the Peano arithmetic. The “deriving” keyword tells Haskell to define show and the comparison operators for us.
Using this definition of Nat, we can now define some numbers:
one = Succ Zero
two = Succ one
three = Succ two
four = Succ three
These work the way you’d expect:
*Main> three
Succ (Succ (Succ Zero))
*Main> two < three
True
OK, I threw in that last example just for fun.
Read more...
Tags Hacks, Haskell, Math
Posted by Eric Kidd
Mon, 22 Jan 2007 08:32:00 GMT
Yesterday, I was working on a Haskell program that read in megabytes of data, parsed it, and wrote a subset of the data back to standard output. At first it was pretty fast: 7 seconds for everything.
But then I made the mistake of parsing some floating point numbers, and printing them back out. My performance died: 120 seconds.
You can see similar problems at the Great Language Shootout. Haskell runs at 1/2th the speed of C for many benchmarks, then suddently drops to 1/20th for others.
Here’s what’s going on, and how to fix it.
(Many thanks to Don Stewart and the other folks on #haskell for helping me figure this out!)
Read more...
Tags Haskell, Performance
