# Lazy Evaluation

## Evaluation Strategies

How are programs evaluated? There are a number of strategies for evaluating a program. For example, the expression (4+8) * (15 + 16) can be evaluated in different ways:

(4+8) * (15 + 16)
=> 12 * (15+16)
=> 12 * 31
=> 372

-- or

(4+8) * (15 + 16)
=> (4 + 8) * 31
=> 12 * 31
=> 372


The final value when reducing an expression (it cannot be reduced further) is the normal form, 372 in this case. No matter how the expression is reduced, the normal form is the same. Haskell's type system prevents us from writing anything that cannot reduce to normal form.

A sub-expression (anything partially reduced that can still be reduced further) is called a redex, short for reducible expression. Evaluation strategies only matter when there are multiple redexes, otherwise there is only one route we can take to evaluate an expression.

## Strict Evaluation

A programming language is strict if the arguments of the function are evaluated before the function is called.

Evaluating fac 500 using a strict method:

fac :: Int -> Int
fac n = fac' n 1

fac' :: Int -> Int -> Int
fac n m = case n of
0 -> m
_ -> fac' (n-1) (n*m)

fac 500      -- a redex, function application
=> fac' 500 1   -- another redex
=> fac' (500-1) (500*1)     -- 3 redexes, two multiplications and function application
=> fac' 499 (500*1)     -- two redexes now as 500-1=499 is now in normal form
=> fac' 499 500         -- now only one redex
=> fac' (499-1) (499*500) -- back to 3 redexes
... -- this goes on for a while


Call-by-value means that all function arguments are reduced to their normal forms (values), and then passed as such to the function. The call-by-value strategy is an example of strict evaluation. This is the evaluation strategy used by most programming languages: Java, JS, PHP, C/C++, OCaml, F#, Python, Scala, Swift. Note that some of these are also functional languages.

Haskell, on the other hand, is far superior. It is non-strict: aka lazy.

## Call-by-name

A non-strict evaluation strategy by which expressions given to functions as arguments are not reduced before the function call is made.
Expressions are only reduced when their value is needed. Same example as before:

fac 2
=> fac' 2 1  -- still a redex here
=> case 2 of
0 -> 1
_ -> fac' (2-1) (2*1)   -- the function call is expanded to its expression
=> fac' (2-1) (2*1) -- left with 3 redexes now
=> case 2-1 of
0 -> 2*1
_ -> fac' ((2-1)-1) ((2-1) * (2*1)) -- a lot of redexes, but we don't need to know the value of any except the one in the case expression. this one is evaluated but not the others
=> case 1 of
0 -> 2*1
_ -> fac' ((2-1)-1) ((2-1) * (2*1)) -- something actually got evaluated, as we needed it's value. we still have a lot of redexes though


Note how that the same argument ((2-1)) is there 3 times, but it is only evaluated when it is needed. This means that it is evaluated possibly more than once, as it may be needed more than once at different points. With call-by-value (strict), an expression is only reduced once but will only ever be reduced once, but with call-by-name (lazy), expressions may end up being evaluated more than once.

## Sharing

Sharing avoids duplicate evaluation. Arguments to functions are turned into local definitions, so that when an expression is evaluated, any expressions that are identical are also evaluated. The same example again, using both call-by-name and sharing:

fac' :: Int -> Int -> Int
fac' n m = case n of
0 -> m
_ -> let x = n-1
y = n*m
in fac' x y

-- the compiler has replaced the expression arguments with let-bound definitions

fac 2
=> fac' 2 1
=> case 2 of
0 -> 1
_ -> let x0 = 2-1
y0 = 2*1
in fac' x0 y0 --expressions bound to variables

=> let x0 = 2-1
y0 = 2*1 -- two redexes
in fac' x0 y0
=> let x0 = 2-1
y0 = 2*1
in case x0 of
0 -> y0
_ -> let x1 = x0-1
y1 = x0 * y0
in fac' x1 y1 -- even more redexes and bindings
-- x0 can be replaced by 1, which evaluates the expresion in all places where x0 is used


Can think of let or where bindings as storing expressions in memory in such a way that we can refer to them from elsewhere using their names.

The combination of call-by-name and sharing is known as lazy evaluation, which is the strategy haskell uses. Nothing is evaluated until it is needed, and work is only ever done once. (Strict evaluation is done sometimes if the compiler decides to, so it is technically non-strict instead of lazy.)

An example, using haskell's lazy evaluation strategy:

length (take 2 (map even [1,2,3,4]))
=> length (take 2 (even 1 : map even [2,3,4])) -- check argument is non-empty list
=> length (even 1 : take (2-1) (map even [2,3,4])) -- even 1 cons'd to take 1 of map
=> 1 + length (take (2-1) (map even [2,3,4])) --know length is at least 1, take out
=> 1 + length(take 1 (map even [2,3,4]))
=> 1 + length (take 1 (even 2 : map even [3,4])) --another map call
=> 1 + (1 + length (take (1-1) (map even [3,4])) -- length again
=> 1 + (1 + length []) --take 0 so empty list
=> 1 + 1 + 0 -- return 0
=> 2 -- done


Note how half the map wasn't evaluated, because haskell knew we only cared about the first 2 elements. However this trace doesn't show any of the internal bindings haskell makes for sharing expressions. The compiler does this by transforming the expression:

length (take 2 (map even [1,2,3,4]))
-- becomes
let
xs = take 2 (map even [1,2,3,4])
in length xs
-- becomes
let
ys = map even [1,2,3,4]
xs = take 2 ys
in length xs
-- becomes
let
ys = map even (1:(2:(3:(4:[]))))
xs = take 2 ys
in length xs
-- finally
let
zs4 = 4:[]
zs3 = 3:zs4
zs2 = 2:zs3
zs  = 1:zs2
ys  = map even zs
xs  = take 2 ys
in length xs


In this representation, everything is let bound it it's own definition, and nothing is applied except to some literal or to another let bound variable. The representation in memory looks something like this:

These things in memory are called closures. A closure is an object in memory that contains:

• A pointer to some code that implements the function it represents (not shown)
• A pointer to all the free variables that are in scope for that definition
• A free variable is any variable in scope that is not a parameter

The closures form a graph, where the closures all point to each other.

Another example, using map:

map :: (a -> b) -> [a] -> [b]
map _ [] = []
map f (x:xs) = f x : map f xs

-- removing all syntactic sugar, done by compiler

map = \f -> \arg ->
case arg of
[]      -> []
(x: xs) -> let
y  = f x
ys = map f xs
in (y:ys)


Using this definition of map to evaluate the expression from before (length (take 2 (map even [1,2,3,4]))):

let
zs4 = 4:[]
zs3 = 3:zs4
zs2 = 2:zs3
zs  = 1:zs2
xs  = map even zs
ys  = take 2 xs
in length ys
-- new closures allocated by map, using 2nd case of map function
let
zs4 = 4:[]
zs3 = 3:zs4
zs2 = 2:zs3
zs  = 1:zs2
y0 = even 1
ys0 = map even zs2 -- new closures
xs  = y0 : ys -- updated to be a cons cell
ys  = take 2 xs
in length ys


The graph of closures representing this:

Things can be evaluated strictly in haskell, if you want. This is prefereable in some cases for performance reasons. The \$! operator forces strict function application. The version of the function below forces the recursive call to be evaluated first. fac' :: Int -> Int -> Int fac' 0 m = m fac' n m = (fac' \$! (n-1)) (n*m)


## Infinite Data Structures

Laziness means data structures can be infinite in haskell. This is also facilitated by the lack of call stack, as there is no "max recursion depth" like in strict languages.

from :: Int -> [Int]
from n = n : from (n+1)


This function builds an infinite list of a sequence of Ints, starting with the Int passed. An example usage, showing how lazy evaluation works with it:

take 3 (from 4)
=> take 3 (4 : from 5)
=> 4 : take 2 (from 5)
=> 4 : take 2 (5 : from 6)
=> 4 : 5 : take 1 (from 6)
=> 4 : 5 : take 1 (6 : from 7)
=> 4 : 5 : 6 : take 0 (from 7)
=> 4 : 5 : 6 : []
=> [4,5,6]


The infinite evaluation is short-circuited, as the compiler knows it only needs the first 3 elements.