Let's create a very small fragment of a programming language:
data Expr = Num Int -- atom
| Str String -- atom
| Op BinOp Expr Expr -- compound
deriving (Show)
data BinOp = Add | Concat
deriving (Show)
And an interpreter for it:
interp x@(Num _) = x
interp x@(Str _) = x
interp (Op Add a b) = Num (i a + i b)
where i x = case interp x of Num a -> a
interp (Op Concat (Str a) (Str b)) = Str (a ++ b)
Our very quick round of prototyping gave us a tiny interpreter that actually seems to work:
>> interp (Op Add (Num 2) (Num 3))
Num 5
Please help me to spot some problems with my interpreter!
We can construct ill-formed expressions ("add a Num to a Str").
Our interpreter crashes on these expressions, because we (quite reasonably) didn't take their possible existence into account.
Here's a slightly modified version of our language:
data Expr a = Num Int
| Str String
| Op BinOp (Expr a) (Expr a)
deriving (Show)
-- This is unchanged.
data BinOp = Add | Concat
deriving (Show)
We've introduced a type parameter here...
...But we never actually use it to represent a value of whatever type a is.
Let's see where that takes us.
Here is our modified interpreter.
interp x@(Num _) = x
interp x@(Str _) = x
interp (Op Add a b) = Num (i a + i b)
where i x = case interp x of Num a -> a
interp (Op Concat a b) = Str (i a ++ i b)
where i x = case interp x of Str y -> y
Our only change is to apply interp recursively if we're asked to perform a Concat.
We could have done this in our original interpreter, so that can't be the real fix. But what is?
What's the type of the rewritten interp?
The interpreter function now has this type:
interp :: Expr a -> Expr a
But we know from the definitions of Expr and BinOp that we never use a value of type a. Then what purpose does this type parameter serve?
Recall the type of Expr:
data Expr a = ...
| Op BinOp (Expr a) (Expr a)
Let's think of that a parameter as expressing our intent that:
The operands of an Op expression should have the same types.
The resulting Expr value should also have this type.
data Expr a = ...
| Op BinOp (Expr a) (Expr a)
In fact, the type system will enforce these constraints for us.
The first step in making all of this machinery work is to define some functions with the right types.
These two functions will construct atoms (values that can't be reduced any further) in our language:
num :: Int -> Expr Int
num = Num
str :: String -> Expr String
str = Str
These two functions construct compound expressions:
add :: Expr Int -> Expr Int -> Expr Int
add = Op Add
cat :: Expr String -> Expr String -> Expr String
cat = Op Concat
Notice that each one enforces the restriction that its parameters must be compatible.
One we have our functions defined, the last step is to lock our world down.
Here's what the beginning of my module looks like:
module Interp
(
Expr, -- type constructor
interp, -- interpreter
num, str, -- atom constructors
add, cat, -- expression constructors
) where
Notice that we've exercised careful control over what we're exporting.
We export the Expr type constructor, but none of its value constructors.
Users of our module don't need BinOp, so we don't export that at all.
Consequences of exporting only the type constructor for Expr:
Clients cannot use the value constructors to create new values.
The only way for a client to construct expressions is using our handwritten "smart constructor" functions with their carefully chosen types.
Clients cannot pattern-match on an Expr value. Our internals are opaque; we could change our implementation without clients being able to tell.
These are in fact the completely standard techniques for creating abstract data types in Haskell. So where does the type parameter come in?
Due to our judicious use of both abstraction and that type parameter:
This additional safety comes "for free":
We don't need runtime checks for ill-formed expressions, because they cannot occur.
Our added type parameter never represents data at runtime, so it has zero cost when the program runs.
When we refer to a type parameter on the left of a type definition, without ever using values of that type on the right, we call it a phantom type.
We're essentially encoding compile-time data using types, and the compiler computes with this data before our program is ever run.
We've already seen the very handy MVar type, which represents a "blocking mutable box": we can put a value in or take one out, but we'll block if we put when full or take when empty.
Even though MVars are the fastest blocking concurrent structure in the industry (they made the the Kessel Run in less than twelve parsecs!), we don't always want blocking semantics.
For cases where we want non-blocking updates, there's the IORef type, which gives us mutable references.
import Data.IORef
newIORef :: a -> IO (IORef a)
readIORef :: IORef a -> IO a
writeIORef :: IORef a -> a -> IO ()
modifyIORef :: IORef a -> (a -> a) -> IO ()
Application writers are often faced with a question like this:
There are of course many ways to address this sort of problem.
Let's consider one where we use a reference to a piece of config data.
Any code that's executing in the IO monad can, if it knows the name of the config reference, retrieve the current config:
curCfg <- readIORef cfgRef
The trouble is, ill-behaved code could clearly also modify the current configuration, and leave us with a debugging nightmare.
Let's create a new type of mutable reference.
We use a phantom type t to statically track whether a piece of code is allowed to modify the reference or not.
import Data.IORef
newtype Ref t a = Ref (IORef a)
Remember, our use of newtype here means that the Ref type only exists at compile time: it imposes no runtime cost.
Since we are using a phantom type, we don't even need values of our access control types:
data ReadOnly
data ReadWrite
We're already in a good spot! Not only are we creating compiler-enforced access control, but it will have zero runtime cost.
To create a new reference, we just have to ensure that it has the right type.
newRef :: a -> IO (Ref ReadWrite a)
newRef a = Ref `fmap` newIORef a
Since we want to be able to read both read-only and read-write references, we don't need to mention the access mode when writing a type signature for readRef.
readRef :: Ref t a -> IO a
readRef (Ref ref) = readIORef ref
Of course, code can only write to a reference if the compiler can statically prove (via the type system) that it has write access.
writeRef :: Ref ReadWrite a -> a -> IO ()
writeRef (Ref ref) v = writeIORef ref v
This function allows us to convert any kind of reference into a read-only reference:
readOnly :: Ref t a -> Ref ReadOnly a
readOnly (Ref ref) = Ref ref
In order to prevent clients from promoting a reference from read-only to read-write, we do not provide a function that goes in the opposite direction.
We also use the familiar technique of constructor hiding at the top of our source file:
module Ref
(
Ref, -- export type ctor, but not value ctor
newRef, readOnly,
readRef, writeRef
) where
What does this type signature mean?
something :: a -> a
What are all of the possible behaviours of a code with this type?
What about this signature?
another :: [a]
What does this type signature mean?
something :: a -> a
We know that for all possible types a, this function accepts a value of that type, and returns a value of that type.
We clearly cannot enumerate all possible types, so we equally clearly cannot create all (or indeed any) values of these types.
Therefore, if we exclude crashes and infinite loops, the only possible behaviour for this function is to return its input.
In fact, Haskell provides a keyword, forall, to make this quantification over type parameters more explicit:
something :: forall a. a -> a
The same "universal quantification" syntax works with typeclass constraints:
something :: forall a. (Show a) -> String
Here, our quantifier is "for all types a, where the only thing we know about a is what the Show typeclass tells us we can do".
These forall keywords are implied if they're not explicitly written.
Love 'em or hate 'em, everybody has to deal with databases.
Here are some typical functions that a low-level database library will provide, for clients that have to modify data concurrently:
begin :: Connection -> IO Transaction
commit :: Transaction -> IO ()
rollback :: Transaction -> IO ()
We can create a new transaction with begin, finish an existing one with commit, or cancel one with rollback.
Typically, once a transaction has been committed or rolled back, accessing it afterwards will result in an exception.
Clearly, these constructs make it easy to inadvertantly write bad code.
oops conn = do
txn <- begin conn
throwIO (AssertionFailed "forgot to roll back!")
-- also forgot to commit!
We can avoid rollback and commit forgetfulness with a suitable combinator:
withTxn :: Connection -> IO a -> IO a
withTxn conn act = do
txn <- begin conn
r <- act `onException` rollback txn
commit txn
return r
All right! The code running in act never sees a Transaction value, so it can't leak a committed or rolled back transaction.
We're not out of the woods yet!
High-performance web apps typically use a dynamically managed pool of database connections.
getConn :: Pool -> IO Connection
returnConn :: Pool -> Connection -> IO ()
It's a major bug if a database connection is not returned to the pool at the end of a handler.
So we write a combinator to handle this for us:
withConn :: Pool -> (Connection -> IO a) -> IO a
withConn pool act =
bracket (getConn pool) (returnConn pool) act
Nice and elegant. But correct? Read on!
In a typical database API, once we enter a transaction, we don't need to refer to the handle we got until we either commit or roll back the transaction.
So it was fine for us to write a transaction wrapper like this:
withTxn :: Connection -> IO a -> IO a
On other other hand, if we're talking to a database, we definitely need a connection handle.
query :: Connection -> String -> IO [String]
So we have to pass that handle into our combinator:
withConn :: Pool -> (Connection -> IO a) -> IO a
Unfortunately, since withConn gives us a connection handle, we can defeat the intention of the combinator (sometimes accidentally).
What is the type of this function?
evil pool = withConn pool return
Here, we are using the newtype keyword to associate a phantom type with the IO monad.
newtype DB c a = DB {
fromDB :: IO a
}
We're going to run some code in the IO monad, and pass around a little extra bit of type information at compile time.
Let's create a phantom-typed wrapper for our earlier Connection type:
newtype SafeConn c = Safe Connection
Where are these phantom types taking us?
The easiest place to start to understand with a little use of our new code, in the form of a function we'll export to clients.
This is just a wrapper around the query function we saw earlier, making sure that our newtype machinery is in the right places to keep the type checker happy.
safeQuery :: SafeConn c -> String -> DB c [String]
safeQuery (Safe conn) str = DB (query conn str)
Notice that our phantom type c is mentioned in both our uses of SafeConn c and DB c: we're treating it as a token that we have to pass around.
Our library will not be exporting the value constructors for SafeConn or DB to clients. Once again, this newtype machinery is internal to us!
Here, we'll use our earlier exception-safe withConn combinator. Recall its type:
withConn :: Pool -> (Connection -> IO a) -> IO a
To make it useful in our new setting, we have to wrap the Connection, and unwrap the DB c that is our act to get an action in the IO monad.
withSafeConn pool act =
withConn pool $ \conn ->
fromDB (act (Safe conn))
It's not at all obvious what this is doing for us until we see the type of withSafeConn.
Here's a burly type for you:
{-# LANGUAGE Rank2Types #-}
withConnection :: Pool
-> (forall c. SafeConn c -> DB c a)
-> IO a
We've introduced a universal quantifier (that forall) into our type signature. And we've added a LANGUAGE pragma! Whoa. Duuude.
Relax! Let's not worry about those details just yet. What does our signature seem to want to tell us?
We accept a Pool.
And an "I have a connection, so I can talk to the database now" action that accepts a SafeConn c, returning a value a embedded in the type DB c.
Not so scary after all. Well, except for the details we're ignoring.
Let's start with the obviously bothersome part of the type signature.
(forall c. SafeConn c -> DB c a)
This is the same universal quantification we've seen before, meaning:
Our "I can haz connection" action must work over all types c.
The scope of c extends only to the rightmost parenthesis here.
Putting it back into context:
withConnection :: Pool
-> (forall c. SafeConn c -> DB c a)
-> IO a
The type variable c can't escape from its scope, so a cannot be related to c.
withConnection :: Pool
-> (forall c. SafeConn c -> DB c a)
-> IO a
Because SafeConn c shares the same phantom type as DB c, and the quantified c type cannot escape to the outer IO, there is no way for a SafeConn c value to escape, either!
In other words, we have ensured that a user of withConnection cannot either accidentally allow or force a connection to escape from the place where we've deemed them legal to use.
Standard Haskell types and functions have just one scope for universal quantification.
foo :: forall a b. a -> b -> a
When an extra level of scoping for universal quantification is introduced, this is called a rank-2 type.
fnord :: forall b. (forall a. a -> a) -> b
(Normal types are thus called rank-1 types.)
Although widely used, rank-2 types are not yet a part of the Haskell standard, hence our use of a pragma earlier:
{-# LANGUAGE Rank2Types #-}
What expressions can we write that have this type?
[forall a. a]
What about this one?
[forall a. (Enum a) => a]
Or this?
[forall a. (Num a) => a]
Do we have time to talk about how to write a Monad instance for DB?
We've now seen several cases where phantom types and rank-2 types let us use the compiler to automatically prevent ourselves from writing bad code.
We can also use them to introduce safe, controlled mutation into our programs.
A typical lament of a functional programmer:
:-(
:-(
Of course, in the worst case, we can emulate a flat, mutable memory with a purely functional map, thus incurring only O(log n) of additional overhead.
Enter the ST monad!
import Control.Monad.ST
This defines for us a function with a glorious rank-2 type:
>> :t runST
runST :: (forall s. ST s a) -> a
Since we've only just been introduced to rank-2 types, we know exactly what this implies:
What happens in the ST monad stays in the ST monad.
Nevertheles, we can obtain a pure result when we run an action in this monad. That's an exciting prospect!
The STRef type gives us the same mutable references as IORef, but in the ST monad.
import Control.Monad.ST
import Data.STRef
whee :: ST s Int
whee z = do
r <- newSTRef z
modifySTRef r (+1)
readSTRef r
Let's try this in ghci:
>> runST (whee 1)
2
Thanks to chaining of the universally quantified s, there is no way for an STRef to escape from the ST monad, save by the approved route of reading its current value with readSTRef.
newSTRef :: a -> ST s (STRef s a)
readSTRef :: STRef s a -> ST s a
For working with large collections of uniform data, the usual representation in most languages is an array.
The longtime standard for working with arrays in Haskell is the Array type, from the array package, but I don't like it: it has an API that is simultaneously bizarre, too general, and puny.
I much prefer its modern cousin, the vector package:
vector provides a vastly richer API than array.
A Vector is one-dimensional and indexed by Ints counting from zero, so it's easy to reason about.
An Array is indexed by an instance of the Ix class, can have arbitrary bounds, and makes my brain hurt.
The vector package provides two "flavours" of vector type:
Vector types are immutable.
MVector types can be modified in either the ST or IO monad, and cannot be read by purely functional code.
Within these flavours, there are two "families" of vector type:
Unboxed vectors are tightly packed in contiguous memory. They are very fast, but it is only possible to create unboxed vectors of certain types, and an unboxed vector can't store thunks.
Normal vectors are boxed, just like ordinary Haskell values. Any value can be stored in a plain old vector, at the cost of an additional level of indirection.
We can thus have an immutable unboxed vector, a mutable boxed vector, and so on.
The classic Haskell implementation of a "quicksort":
import Data.List (partition)
qsort (p:xs) = qsort lt ++ [p] ++ qsort ge
where (lt,ge) = partition (<p) xs
qsort _ = []
This isn't really a quicksort, because it doesn't operate in-place.
We can apply our newfound knowledge to this problem:
import qualified Data.Vector.Unboxed.Mutable as V
import Control.Monad.ST (ST)
quicksort :: V.MVector s Int -> ST s ()
quicksort vec = go 0 (V.length vec)
where
{- ... -}
recur left right
| left >= right = return ()
| otherwise = do
idx <- partition left right
(left + (right-left) `div` 2)
recur left (idx-1)
recur (idx+1) right
(Remember, vec is in scope here.)
partition left right pivotIdx = do
pivot <- V.read vec pivotIdx
V.swap vec pivotIdx right
let loop i k
| i == right = V.swap vec k right >>
return k
| otherwise = do
v <- V.read vec i
if v < pivot
then V.swap vec i k >>
loop (i+1) (k+1)
else loop (i+1) k
loop left left
We can even use this in-place sort to efficiently perform an in-place sort of an immutable array!
Our building blocks:
thaw :: Vector a -> ST s (MVector s a)
create :: (forall s. ST s (MVector s a)) -> Vector a
thaw creates a new mutable vector, and copies the contents of the immutable vector into it.
create runs an ST action that returns a mutable vector, and "freezes" its result to be immutable, and hence usable in pure code.
import qualified Data.Vector.Unboxed as U
vsort :: U.Vector Int -> U.Vector Int
vsort v = U.create $ do
vec <- U.thaw v
quicksort vec
return vec
The big advantage of the ST monad is that it gives us the ability to efficiently run computations that require mutability, while both the inputs to and results of our computations remain pure.
In order to achieve this, we sacrifice some power:
We can't run arbitrary IO actions. No database accesses, no filesystem, etc.
Other potential sources of nondeterminism (e.g. threads) are thus also off limits.
Originally, this lecture was supposed to be all about the joys of lazy evaluation, but we hijacked much of our time to serve other purposes.
I'm going to talk a little bit about it anyway.
In a minute.
How can we use random numbers to approximate the value of π?
How can we use random numbers to approximate the value of pi?
Take two random numbers, x and y, on the interval [0, 1]
Add their squares: r = x2 + y2
We have a π / 4 probability of r ≤ 1
What can we do with this knowledge?
Haskell supplies a random package that we can use in a purely functional setting.
class Random a where
random :: RandomGen g => g -> (a, g)
class RandomGen g where
next :: g -> (Int, g)
split :: g -> (g, g)
The RandomGen class is a building block: it specifies an interface for a generator that can generate uniformly distributed pseudo-random Ints.
There is one default instance of this class:
data StdGen {- opaque -}
instance RandomGen StdGen
The Random class specifies how to generate a pseudo-random value of some type, given the random numbers generated by a Gen instance.
Quite a few common types have Random instances.
For Int, the instance will generate any representable value.
For Double, the instance will generate a value in the range [0, 1].
Since we want to use a PRNG in pure code, we obviously can't modify the state of a PRNG when we generate a new value.
This is why next and random return a new state for the PRNG every time we generate a new pseudo-random value.
Here's how we can generate a guess at x2 + y2:
guess :: (RandomGen g) => (Double,g) -> (Double,g)
guess (_,g) = (z, g'')
where z = x^2 + y^2
(x, g') = random g
(y, g'') = random g'
Note that we have to hand back the final state of the PRNG along with our result!
If we handed back g or g' instead, our numbers would either be all identical or disastrously correlated (every x would just be a repeat of the previous y).
We can use the getStdGen function to get a handy global PRNG state:
getStdGen :: IO StdGen
This does not modify the state, though. If we use getStdGen twice in succession, we'll get the same result each time.
To be safe, we should update the global PRNG state with the final PRNG state returned by our pure code:
setStdGen :: StdGen -> IO ()
Calling getStdGen and setStdGen from ghci is a pain, so let's write a combinator to help us.
Remember that split method from earlier?
class RandomGen g where
split :: g -> (g, g)
This "forks" the PRNG, creating two children with different states.
The hope is that the states will be different enough that pseudo-random values generated from each will not be obviously correlated.
withGen :: (StdGen -> a) -> IO a
withGen f = do
g <- getStdGen
let (g',g'') = split g
setStdGen g'
return (f g'')
Now we can use our guess function reasonably easily.
>> let f = fst `fmap` withGen (guess . ((,) 0))
>> f
1.2397265526054513
>> f
0.9506331164887969
Here's a useful function from the Prelude:
iterate :: (a -> a) -> a -> [a]
iterate f x = x : iterate f (f x)
Obviously that list is infinite.
Let's use iterate and guess, and as much other Prelude machinery as we can think of, to write a function that can approximate π.
By the way, in case you don't recognize this technique, it's a famous example of the family of Monte Carlo methods.
What aspects of laziness were important in developing our solution?
Providence Salumu