Haskell: Expression Evaluation

Assignment 1

It exists.

Lazy Evaluation

Consider the evaluation of the expression fst (1+2, 3+4). One of these two evaluation strategies must happen:

fst (1+2, 3+4)
== fst (3, 3+4)
== fst (3, 7)
== 3
fst (1+2, 3+4)
== 1+2
== 3

Basically: do we have to evaluate 3+4 or not?

Lazy Evaluation

In the equivalent C, Python, etc, the answer is clear: 3+4 gets evaluated.

In Haskell, we can try giving an infinite list as the second argument and confirm that it does not get evaluated.

Prelude> fst (1+2, 3+4)
Prelude> fst (1+2, [1..])

Lazy Evaluation

Haskell is lazy: it delays evaluation of any calculation as long as possible.

Whenever you use a calculated value, you are really just passing around a reference to the calculation that will create it.

When Haskell actually needs the value (e.g. has to display it on the screen), it starts working through the calculation, doing just enough work to get the result.

Lazy Evaluation

e.g. consider an operation on an infinite list, like take 2 [1..]. The take function is defined something like:

take 0 _ = []
take n (x:xs) = x : take (n-1) xs

The calculation is evaluated like this:

take 2 [1..]
== take 2 (1:[2..])            -- arg 2 matches x:xs?
== 1 : take (2-1) [2..]        -- recursive case
== 1 : take 1 [2..]            -- arg 1 matches 0?
== 1 : take 1 (2:[3..])        -- arg 2 matches x:xs?
== 1 : 2 : take (1-1) [3..]    -- recursive case
== 1 : 2 : take 0 [3..]        -- arg 1 matches 0?
== 1 : 2 : [] == [1, 2]        -- base case

Lazy Evaluation

Lazy evaluation means that we can define huge/​complicated values, but if we only need small parts, Haskell will only calculate that.

*Main> let bigResults = [bigCalculation i | i<-[1..100000]]
*Main> take 3 bigResults
*Main> bigResults !! 5

e.g. you might do this in assignment 1: define infinite result sets, but extract the ones you want.

Lazy Evaluation

A not-yet-evaluated expression is called a thunk. Haskell generates and stores thunks as needed. When a value is needed, its thunk resumes evaluation.

Thunks that don't need to be evaluated get discarded, like [3..] in the previous example.

Lazy Evaluation

This can lead to calculations actually being done at unexpected times.

*Main> let x = bigCalculation 1000 -- returns immediately
*Main> let y = x+1                 -- returns immediately
*Main> y                           -- takes time

Be careful when you think that was really fast: maybe it didn't actually happen.

Controlling Laziness

Lazy evaluation can be handy, but too much lazy evaluation can be a problem. It's wasteful to store many thunks in memory if we definitely need them evaluated.

Consider foldl, which is lazily evaluated like this:

foldl (+) 0 [1,2,3]
== foldl (+) (0+1) [2,3]
== foldl (+) ((0+1)+2) [3]
== foldl (+) (((0+1)+2)+3) []
== ((0+1)+2)+3
== (1+2)+3
== 3+3
== 6

Controlling Laziness

None of the addition gets done until we actually display the value (or use it in some other way). Many thunks are (invisibly) stored until then.

This is often bad for foldl, so there is a non-lazy (strict) version, foldl' in the Data.List module:

Prelude> foldl (+) 0 [1..100000000]
*** Exception: stack overflow
Prelude> import Data.List
Prelude Data.List> foldl' (+) 0 [1..100000000]

Controlling Laziness

The evaluation of foldl' will be more like this:

foldl' (+) 0 [1,2,3]
== foldl' (+) (0+1) [2,3]
== foldl' (+) 1 [2,3]
== foldl' (+) (1+2) [3]
== foldl' (+) 3 [3]
== foldl' (+) (3+3) []
== foldl' (+) 6 []
== 6

Controlling Laziness

There are less thunks stored with foldl', so it's more efficient if you actually need all of the results.

If the fold produced a list and you only need some elements, then foldl' could be less efficient because it does unnecessary calculations.

tl;dr: In Haskell you have the choice of which things get calculated. Now you have to make the choice.

Controlling Laziness

The $! operator can be used to force strict evaluation of an argument. This is most useful in recursive calls where you know you'll need the value, but would get a lot of thunks otherwise.

myPowerTailRec, myPowerTailStrict :: Int -> Int -> Int -> Int
myPowerTailRec a _ 0 = a
myPowerTailRec a x y = myPowerTailRec (x*a) x (y-1)

myPowerTailStrict a _ 0 = a
myPowerTailStrict a x y = (myPowerTailStrict $! (a*x)) x (y-1)

These are identical but the $! forces a*x to be calculated strictly.

Controlling Laziness

Compare: same calculations but thunk built or not.

myPowerTailRec 1 2 2
== myPowerTailRec (2*1) 2 (2-1)       -- recursive case
== myPowerTailRec (2*1) 2 1           -- arg 3 matches 0?
== myPowerTailRec (2*(2*1)) 2 (1-1)   -- recursive case
== myPowerTailRec (2*(2*1)) 2 0       -- arg 3 matches 0?
== 2*(2*1)                            -- base case
== 2*2                                -- evaluate thunk
== 4                                  -- evaluate thunk
myPowerTailStrict 1 2 2
== myPowerTailStrict $! (2*1) 2 (2-1) -- recursive case
== myPowerTailStrict 2 2 (2-1)        -- strict eval of arg 1
== myPowerTailStrict 2 2 1            -- arg 3 matches 0?
== myPowerTailStrict $! (2*2) 2 (1-1) -- recursive case
== myPowerTailStrict 4 2 (1-1)        -- strict eval of arg 1
== myPowerTailStrict 4 2 0            -- arg 3 matches 0?
== 4                                  -- base case

Controlling Laziness

The seq function can also control lazy evaluation. Calling…

seq a b

… returns b, but forces strict evaluation of a. It can be used like this to force a let/​where value to be strictly evaluated:

myPowerSeq a _ 0 = a
myPowerSeq a x y = seq newacc (myPowerSeq newacc x (y-1))
    where newacc = a*x

Controlling Laziness

Like any other optimization, $! and seq should only be used where needed. i.e. you tried something without them; it was slow because of laziness; being less lazy helps.

GHC's optimizer can often insert strict evaluation automatically where it's helpful.

Controlling Laziness

We can try it with many versions of the dumb power calculator. Greg's script to see what's happening:

ulimit -v 4194304  # limit everything to 4GB memory
rm *.hi *.o strict; ghc -O0 strict.hs
./strict 5000000   # all succeed
./strict 50000000  # -Tail, -Foldl out of memory
rm *.hi *.o strict; ghc -O2 strict.hs
./strict 50000000  # -Strict, -Seq, -Tail, -Foldl, -Foldl' ~same
./strict 500000000 # myPower, -Foldr out of memory

Function Application

Aside: The ($!) is the strictly-evaluated sibling of ($) which is function application, but lazy. Just like every other function call we've ever done.

Why? The ($) has a different precedence and association than passing an argument to a function. That can make big function compositions more readable.

Function Application

e.g. Suppose we're using map and divisors and (*) together like this (for some reason):

funnyDivisors n   = map pred (divisors (n*2))

That's a lot of parens. This is equivalent:

funnyDivisors' n  = map pred $ divisors $ n*2

The ($) style is often more readable (but right-to-left) when you're combining multiple functions to get something done.

Function Application

Note: ($) combines a function and an argument; (.) combines two functions. All of these are the same function:

funnyDivisors n   = map pred (divisors (n*2))
funnyDivisors' n  = map pred $ divisors $ n*2
funnyDivisors'' n = (map pred) . divisors . (*2) $ n
funnyDivisors'''  = (map pred) . divisors . (*2)

Which is easier to understand (quickly and correctly)?

The ($!) is the strict version of this idea.

Pure Functions

All functions we have seen in Haskell have been pure functions. That is:

  1. Given the same arguments, they always return the same results.
  2. They have no side effects: do not modify the external state of the program.

#2 was easy: there are no variables to modify.

Pure Functions

#1 was a slight lie: function results depend on their arguments and free variables. A free variable is any value used that isn't a function argument.

e.g. foldl is free here; xs is bound.

mySum xs = foldl (+) 0 xs

e.g. in Assignment 1, pwReduce has pwLength free.

Pure Functions

Since free variables cannot be modified (i.e. are actually constants), we can still claim these are pure functions.

i.e. if pwLength is defined a certain way, it will be the same forever (until we re-compile).

Pure Functions

Insisting on pure functions makes anything with a state difficult. All of these depend on external state:

  • file I/O: file contents on disk;
  • user I/O: results of typing/​clicking;
  • random numbers: random seed set by hard-to-predict external inputs.

Functions doing these things must all be non-pure: results inherently depend on more than just the arguments.

Pure Functions

Doing non-pure things in Haskell requires monads. [More later.]

Purity is what allows lazy evaluation. Haskell can choose to calculate a value or not since there are no side effects to worry about.

No side effects means there's no reason to evaluate code unless we really need the return value.

Concurrent Programming

Concurrent program: several things can be happening independently.

Concurrent programming is hard in most programming languages: threads, semaphores, mutexes, deadlock, ….

But the hard part is usually managing the shared state: shared variables must be locked and unlocked when used.

Concurrent Programming

With pure functions in Haskell, there's no state to share. It should be easy to do things concurrently.

Sadly, because of lazy evaluation, it won't be completely automatic. Haskell always calculates the one value it needs next: it won't do multiple things because it's so aggressively lazy.

Concurrent Programming

There are functions in Control.​Parallel and Control.​Parallel.​Strategies to help you specify what can run concurrently.

We need to express: how lazy/​strict we will be, and what should be done concurrently?

Concurrent Programming

Recall seq a b: strictly evaluate a and return b (but maybe calculating b first). For concurrent execution, we have:

  • pseq a b: like seq but evaluates a before returning b.
  • par a b: evaluate a and b concurrently, and return b.

Typical usage: use these to express do A and B in parallel and then combine them.

Concurrent Programming

Suppose we have two values that take a long (but similar) time to compute:

calcA = a + b
    where a = calculation 1
          b = calculation 2

We can compute them in concurrently before adding:

calcB = (a `par` b) `pseq` (a + b)
    where a = calculation 1
          b = calculation 2

It should use two processor cores and take about half as long.

Concurrent Programming

GHC has to be asked to allow parallel execution, something like:

ghc -O2 -with-rtsopts="-N8" -threaded concurrent1.hs

Concurrent Programming

A higher-level option: parMap from Control.​Parallel.​Strategies. It's like map but will evaluate in parallel.

… but you have to specify how to evaluate each element. An execution strategy specifies how non-lazy to be.

Concurrent Programming

For example, this will be single-threaded:

calcC = map slowCalc [0..100]

This is equivalent but can use several cores:

calcD = parMap rseq slowCalc [0..100]

Concurrent Programming

It's still not trivial to make fast concurrent code.

We have to defeat Haskell's lazy evaluation.

Breaking the problem into chunks that are too small causes overhead coordinating threads. e.g. these take about the same amount of time to complete:

calcE = map fastCalc [0..1000000]
calcF = parMap rseq fastCalc [0..1000000]

Concurrent Programming

Even if things can be done in parallel easily, there are still some decisions to be made. In particular: what size chunks of work should be done in parallel?

Too small: too much overhead starting/​stopping/​communicating. Too big: not enough parallelism.

See the Concurrent Fibonacci example for an exploration.


Since Haskell is non-imperative and lazy, it's usually not clear in what order (or if) code will actually be evaluated.

Sometimes that's nice: can work with infinite data structures; can be expressive in interesting ways.

Sometimes it's painful: controlling thunks; writing concurrent code.

Sometimes it's a disaster: reading parts of a file (non-pure functions; out-of-order evaluation could give incorrect results).


Monads give you a way to think of your code in sequence (but usual lazy evaluation rules can apply, depending on the monad).

A monad is basically a wrapper to a type.


An example: the IO monad wraps a type to indicate that its value has something to do with input or output.

The getLine function reads a line of input from the user and has a type that indicates it gives an string with an input/​output origin:

getLine :: IO String

An IO String is a string, but wrapped so we know that the function might return a different value next time it's called.


Another example: the Maybe monad wraps a type but allows representation of failure.

In exercise 4, the findElt function returns a Maybe Int: either the position in a list or failure.


A monad must be able to do a few operations on its values:

  • Wrap and unwrap its values (e.g. convert between 2 and Just 2)
  • Chain together operations from one monad value to another.


The chaining of operations is done by the >>= operator (think of it as and then):

(>>=) :: (Monad m) => m a -> (a -> m b) -> m b
  1. It takes the result of the previous step (of type m a);
  2. unwraps it automatically (to type a);
  3. a function takes the unwrapped value and produces the result of this step (a function a -> m b);
  4. The m b is the result of this step.


There's nothing inherently imperative about what (>>=) does: it's all still functional operations.

But the programmer can think in steps and express imperative thoughts. A specific monad implementation can implement the operations in whatever way is correct. (e.g. either strictly evaluate to force execution order, or not.)


Writing with the (>>=) is painful, so there's a shortcut with the do syntax. This code reads two lines of input, and returns the second:

secondLine :: IO String
secondLine = do
    line1 <- getLine
    line2 <- getLine
    return line2

That is exactly equivalent to:

secondLine' :: IO String
secondLine' =
  getLine >>= (\line1 -> getLine >>= (\line2 -> return line2))


secondLine :: IO String
secondLine = do
    line1 <- getLine
    line2 <- getLine
    return line2

Here, the result of the first getLine becomes an argument to the rest of the code named line1; same with line2.

The return function wraps a non-monad value in the appropriate monad. Here, it transforms line2 :: String to an IO String.


I think return is named wrong: it doesn't stop a function's execution and indicate the result (like in every other language where you've seen the word). It wraps a value up in a monad.

If return was instead named wrapInMonad, people would understand it.


Doing y <- f x takes the monad-wrapped result of f x, unwraps it and calls it y as an argument to the following code.

And return wraps a non-monad value in a monad. So doing y <- f x followed by return y is pointless. This is equivalent:

secondLine'' :: IO String
secondLine'' = do
    line1 <- getLine


Your job when writing monadic code:

  • Each statement is a function that takes the unwrapped values and produces a monad value.
  • The last statement is the result; return can be used to wrap a non-monad value if necessary.
  • If you have a step that produces a non-monad value, you can use let as you would anywhere else.


e.g. read a line of input and print the result of map succ.

succInput :: IO ()
succInput = do
    text <- getLine
    let succtext = map succ text
    putStrLn succtext

getLine returns IO String; text is that unwrapped to a String; succtext is built by a (non-Monad) Haskell function call and is a String; putStrLn takes a String, does IO, and returns nothing (i.e. returns IO ()).

getLine returns a monad so use <-, but map returns a non-monad so <- doesn't make sense.


Or we could use the wrapping of return and unwrapping of <- to make the code more uniform:

succInput' :: IO ()
succInput' = do
    text <- getLine
    succtext <- return $ map succ text
    putStrLn succtext

… which may be silly, but the result is the same.



  • You can think imperatively, but Haskell can still be very functional.
  • A monad knows how to chain the operations together.
  • Values can be wrapped/​unwrapped in monads: the monad takes care of it.

Monad: Maybe

As said earlier: Maybe is a monad, but the assignments don't use its monad functionality.

The rule for (>>=) with Maybe: if a step returns a Just value, the calculation continues to the next step; if it returns Nothing, each subsequent step is skipped and returns Nothing.

Monad: Maybe

An example using the findElt from the exercises: find the position of three things in the list, and either return all of them or Nothing:

findThree v1 v2 v3 xs = do
    pos1 <- findElt v1 xs
    pos2 <- findElt v2 xs
    pos3 <- findElt v3 xs
    return (pos1, pos2, pos3)
*Main> findThree 1 2 3 [1,6,2,5,3,4]
Just (0,2,4)
*Main> findThree 1 2 3 [1,6,2,5,4]

Monad: Maybe

findThree v1 v2 v3 xs = do
    pos1 <- findElt v1 xs
    pos2 <- findElt v2 xs
    pos3 <- findElt v3 xs
    return (pos1, pos2, pos3)

If the findElt succeeds, the value is unwrapped (e.g. Just 2 becomes 2) and passed along. If any fails, the whole function returns Nothing.

The return is necessary because pos1pos3 are non-monad values, and we have to wrap up our final result as a Maybe.

Monad: Random Numbers

A function that generates random numbers can't be pure: it must give a different result each time it's called. There is typically some internal state that keeps track of a random seed.

We don't have anywhere for state in Haskell.

Monad: Random Numbers

We can fake it: introduce a random generator state object and pass it around as an argument to everything that generates random numbers.

Haskell does this with StdGen instances defined in the System.Random module.

import System.Random

Monad: Random Numbers

But if we have a random number state, we need to update it. We can only pass the state as an argument and receive the new state in the return value.

threeRand :: [Int]
threeRand = 
    let gen0 = mkStdGen 1234 -- gen0 :: StdGen
        (rand0, gen1) = randomR (1, 100) gen0
        (rand1, gen2) = randomR (1, 100) gen1
        (rand2, _)    = randomR (1, 100) gen2
    in [rand0, rand1, rand2]

Bad things: always uses random seed 1234; lots of managing the StdGen instances.

Monad: Random Numbers

If we want to have an unpredictable seed for the random number generator, we need to look for one: the current time, or network traffic, or processor's true-random value generator. All of those are external state that we have to read.

Read from the outside world: an IO operation. There is a newStdGen function that returns a well-seeded random number generator (StdGen).

newStdGen :: IO StdGen

Monad: Random Numbers

We need to work with the IO monad to get the IO StdGen instance, but can do everything else with the unwrapped StdGen.

threeRand' :: IO [Int]
threeRand' = do
    gen0 <- newStdGen
        (rand0, gen1) = randomR (1, 100) gen0
        (rand1, gen2) = randomR (1, 100) gen1
        (rand2, _)    = randomR (1, 100) gen2
    return [rand0, rand1, rand2]

Monad: Random Numbers

There is a convenience function randomRs that generates a infinite list of random values in a range, and we can have much nicer code:

threeRand'' :: IO [Int]
threeRand'' = do
    gen0 <- newStdGen
    return $ take 3 $ randomRs (1, 100) gen0

Monad: Random Numbers

Another example: generate a bunch of random numbers, and print an ASCII-art histogram to see the distribution. We can generate the random numbers as before:

import System.Random

-- generate n random integers
randInts :: Int -> Int -> Int -> IO [Int]
randInts n minval maxval = do
    gen <- newStdGen
    return $ take n $ randomRs (minval, maxval) gen

Monad: Random Numbers

We can generate the histogram in a purely-functional way. It's easier if we do, so we will.

-- convert a list of values into a histogram
histogram :: (Enum a, Eq a, Ord a) => [a] -> [String]
histogram vals = bars
    counts = [length $ filter (==i) vals
             | i <- [(minimum vals)..(maximum vals)]]
    bars = [take n $ repeat 'X' | n <- counts]

Monad: Random Numbers

Then we can combine them, doing as little monad work as possible.

-- print histogram of randomly-generated values
printHisto :: IO ()
printHisto = do
    vals <- randInts 1000 1 20
    let bars = histogram vals
    mapM_ putStrLn bars

Aside: mapM_ applies the monad action to each list element.

Full code for this example.

Haskell Tips

A few things for the end of our Haskell exploration…

  • Declare types (::), especially with recursive functions. Will get you much better error messages meaning type isn't what you declared instead of types don't match.
  • Don't put function definitions in let/where right away. You can't test those separately.

Haskell Tips

  • Don't think too much about the order calculations happen: let the imperative programming mindset go. Give a clear description of the result you need and let Haskell figure it out.
  • Don't repeat yourself. If you feel the urge to copy-and-paste some expression, put it in a let/where. Better code, and should only get calculated once.

Haskell Tips

  • Think small.
    • Don't write long expressions: they are too difficult to understand/debug.
    • Write small functions that can be easily checked and debugged.
    • Then combine them.

Functional Programming Context

We have looked at one functional language. Haskell is a good example: purely functional, well-designed.

There are others. Generally common to them: pure functions, recursion, first-class functions. Not universal: lazy evaluation, complete lack of state.

Functional Programming Context

Many languages that aren't truly functional have borrowed some ideas: use of pure functions, list comprehensions, more use of recursion, pattern matching, partial function application, type inference, lazy evaluation.

The line between functional and non-functional languages can be blurry.

Functional + Imperative

Much of what we have learned about functional programming can be applied to non-functional languages.

Many of the idioms from Haskell can be translated to other languages. Many constructs in modern programming languages (and libraries) may make more sense with some Haskell experience.

Functional + Imperative

A filter and map in Ruby:

[1,2,3].select{|x| x!=2}.collect{|x| x*10} == [10, 30]

A list comprehension in Python:

[x*10 for x in [1,2,3] if x!=2] == [10, 30]

Also, Python's functools module and itertools module.

Functional + Imperative

LINQ in C# blends these ideas with SQL ideas:

int[] numbers = { 1, 2, 3 };
var results = from n in numbers
              where n != 2
              select n * 10;

These more functional coding styles are often easier to read/​write (and sometimes faster) than the equivalent for loop.

Functional + Imperative

Java Streams (since Java 8) allow very functional-looking interaction with collections. For example, given a List we can map, filter, and reduce (==fold):

List<Integer> larger = values.stream().map(i -> i+1)
List<Integer> larger_even = values.stream().map(i -> i+1)
    .filter(i -> i%2 == 0).collect(Collectors.toList());
Integer total = values.stream().reduce(0, (a, b) -> a+b);

Streams demand that the operations are non-interfering and stateless: pure. (And associative for .reduce: it could be folding from the left or right.)

Functional + Imperative

And streams are lazily evaluated. Here, the third line takes the time: until then, the stream is build but not evaluated.

Stream<Integer> stream1 = values.stream();
Stream<Integer> stream2 = stream1.map(StreamsExample::slowFunction);
List<Integer> res3 = stream2.collect(Collectors.toList());

The Streams implementation can then optimize for better memory locality: do all of the operations in one pass through the collection.

Functional + Imperative

Or stream operations can be done in parallel with almost no change:

List<Integer> res1 = values.stream()
List<Integer> res2 = values.parallelStream()

This was ≈6 times faster on my 6 core processor.

Functional + Imperative

Or similarly, the algorithm library for C++, or the C++20 ranges library, or the C++23 plan for ranges.

Or Rust's Iterator trait: lazy operations on anything iterable.

Basically, if you're writing a loop in an imperative language, stop and ask yourself if you're being old-fashioned. If the loop is iterating over a collection (list, array, table, etc), you probably should re-write as the language's equivalent of a list comprehension, map, reduce, etc.

Functional + Imperative

Maybe more important: you have gotten used to writing code with pure functions. In imperative languages, you can create non-pure functions.

  • Can have side-effects: print or do other I/O, modify argument objects, etc.
  • Can depend on more than arguments: global variables, file contents, etc.

Functional + Imperative

But non-pure functions are often harder to test and debug.

You have to control/​analyze not only function arguments, but broader state. State is much harder to control: you have to examine any code that modifies it.

Calling non-pure functions concurrently isn't easy. (e.g. Python's multiprocessing.Pool.map)

Solution: write pure functions as much as possible.

Functional + Imperative

Writing pure functions in an imperative language:

  • Pass in all info you need in arguments.
  • Don't modify external state (e.g. don't change arguments or external variables).
  • Return whatever results you have.
  • Write small, pure, easy-to-test functions.

… as much as possible.

Functional + Imperative

There are going to be cases there that's not realistic advice. If you're writing non-pure functions:

  • Be clear about what parts of your code are non-pure (in comments or docs); minimize those parts.
  • Be as local as possible: accessing or modifying class properties is still better than working with global variables.

Functional + Imperative

The overall message: functional code is often better code, no matter what language you're using.

Some compilers can take advantage of it too. e.g. GCC's pure and const attributes let a compiler optimize the way a function is called.

Functional + Imperative

There are places in imperative languages where you must write (almost?) pure functions.

  • task queues (Celery, Resque) where you ask that some code be evaluated later in a job queue: code has to be essentially pure since it is run in a separate process from where it's called.
  • distributed computation (e.g. MapReduce, Spark, Flink, Dask) where your computation can be spread across many servers with no/​little shared state.

Functional + Imperative

Imperative languages are generally strictly evaluated (not lazy), but there are places where lazy evaluation has been introduced because it made sense. e.g.