This is an article I wrote several years ago. It is part of the “Functional Programming in Haskell” online course. It discusses one of the aspects of functional programming that I like in particular, the fact that the entire language can be build starting from the lambda calculus.
In a functional language, there are only functions
Although it might seem that a language like Haskell has a lot of different objects and constructs, they can all be reduced to functions. We will demonstrate how variables, tuples, lists, conditionals, Booleans and numbers can all be constructed from lambda functions. The article assumes some familiarity with Haskell, but here is a quick introduction.
Haskell: a quick introduction
Haskell is whitespace-sensitive like Python, but has a markedly different syntax. Because everything is a function, there is no keyword to mark a function; because there is only lexical scope, there is no need for any special scope identifiers. Function arguments are separated by spaces; anonymous functions are called lambda functions and have a special syntax to identify them:
-- named function
square x = x*x
-- lambda function bound to a named variable
anon_square = \x -> x*x
Several of the examples use the let ... in ... construct, which behaves as a lexically scoped block:
let_square x a b c = let
x2 = a*x*x
x1 = b*x
x0 = c
in
x2+x1+x0
x0, x1 and x2 are in scope only in the expression after the in keyword.
Haskell is statically typed, and the type of a function or variable is written in a separate annotation, for example:
isEmpty :: [a] -> Bool
isEmpty lst = length lst == 0
The isEmpty function has a type signature, identified by ::, that reads “isEmpty is a function from a list of anything to a Boolean”. Types must be written with an initial capital. The a is a type variable which can take on any type, as explained in my post on algebraic data types. Most of the time, you don’t need to write type declarations as Haskell works them out automatically. In the rest of the article, I’m not focusing on the types so I have omitted them.
Variables and let
Haskell uses let expressions to define variables used in a final expression, for example:
let
n = 10
f x = x+1
in
f n
We can rewrite this to use only one variable per let:
let
n = 10
in
let
f x = x+1
in
f n
Now we rewrite any named functions (f) as lambda functions:
let
n = 10
in
let
f = \x -> x+1
in
f n
Then we rewrite the let expressions themselves as lambdas, first the inner let:
let
n = 10
in
(\f -> f n) (\x -> x+1)
We do this by turning the variable in the let part of the expression (f) into a parameter of a lambda function (\f -> ...). The body of the function is the expression after the in (f n). Then we apply this lambda function to the expression bound to the variable (\x -> x+1).
Then we rewrite outer let in the same way:
( \n -> ((\f -> f n) ( \x -> x+1 )) ) 10
This expression consists only of lambda expressions, which shows that variables and let-expressions are just syntactic sugar for lambda expressions.
Tuples
Haskell has tuples, also called record types or product types, ordered collections of expressions of potentially different types:
tp = (1,"2",[3])
The tuple notation is syntactic sugar for a function application:
tp = mkTup 1 "2" [3]
The tuple construction function can again be defined purely using lambdas:
mkTup x y z = \t -> t x y z
What we do here is to use the elements of the tuple as the arguments of a lambda function. So what mkTup returns is also a lambda function, in other words mkTup is a higher-order function. Now we rewrite the mkTup named function as lambda function as well:
mkTup = \x y z -> \t -> t x y z
So our tuples are now also encoded purely as lambda functions.
The same goes for the tuple accessor functions:
fst tp = tp (\x y z -> x)
snd tp = tp (\x y z -> y)
Let’s see what happens here: the argument tp of fst is a function: \t -> t x' y' z'. We now apply this function to another function, \x y z -> x:
(\t -> t x' y' z') (\x y z -> x)
Applying the function gives:
(\x y z -> x) x' y' z'
And so the result will of course be x', which is indeed the first element of the tuple.
Lists
Lists can be defined in terms of the empty lists [] and the cons operation (:).
ls = [1,2,3]
Rewriting this using : and []:
ls = 1 : 2 : 3 : []
Or using cons explicitly:
ls = cons 1 (cons 2 (cons 3 []))
Defining cons
We can define cons using only lambda functions as
cons = \x xs ->
\c -> c x xs
We’ve used the same approach as for the tuples: cons returns a lambda function. So we can write a list as:
ls = cons 1 (...)
= \c -> c 1 (...)
We can also define head and tail using only lambdas, similar to what we did for fst and snd above:
head ls = ls (\x y -> x)
tail ls = ls (\x y -> y)
The empty list
We can define the empty list as follows:
[] = \f -> true
This is a lambda function which always returns true, regardless of its argument. The definitions for true and false are given below under Booleans. With this definition we can check if a list is empty or not:
isEmpty lst = lst (\x xs -> false)
Let’s see how this works. A non-empty list is always defined as:
lst = x:xs
which with our definition of (:) (i.e. cons) is:
lst = (\x xs -> \c -> c x xs) x xs
= \c -> c x xs
And therefore:
isEmpty lst
= isEmpty (\c -> c x xs)
= (\c -> c x xs) (\x xs -> false)
= false
isEmpty []
= isEmpty (\f -> true)
= (\f->true) (\x xs -> false)
= true
And so we have a pure-lambda definition of lists, including construction, access and testing for empty.
Recursion on lists
Now that we can test for the empty list we can define recursions on lists such as foldl, map etc.:
foldl f acc lst =
if isEmpty lst
then acc
else foldl f (f (head lst) acc) (tail lst)
and
map f lst =
let
map' f lst lst' =
if isEmpty lst
then (reverse lst')
else map' f (tail lst) (head lst: lst')
in
map' f lst []
with
reverse lst = (foldl (\acc elt -> (elt:acc)) [] lst
The definitions of foldl and map use an if-then-else expression which is defined below under Conditionals.
#### List concatenation
With foldl and reverse it is easy to express list concatenation:
(++) lst1 lst2 =
foldl (\acc elt -> (elt:acc)) lst2 (reverse lst1)
#### The length of a list
To compute the length of a list we need integers, they are defined below. We increment the lent counter for every element of the list consumed by the fold.
length lst = foldl calc_length 0 lst
where
calc_length _ len = inc len
Conditionals
We have used conditionals in the above expressions:
if cond then if_true_exp else if_false_exp
Here cond is an expression returning either true or false, these are defined below.
We can write the if-then-else clause as a pure function:
ifthenelse cond if_true_exp if_false_exp
Booleans
To evaluate the condition we need to define Booleans as lambda functions:
true = \x y -> x
false = \x y -> y
The Boolean is a function selecting the expression corresponding to true or false. With this definition, the if-then-else becomes simply:
ifthenelse cond if_true_exp if_false_exp =
cond if_true_exp if_false_exp
Basic Boolean operations: and, or and not
Using ifthenelse we can define and, or and not:
and x y = ifthenelse x (ifthenelse y true) false
or x y = ifthenelse x true (ifthenelse y true false)
not x = ifthenelse x false true
Boolean equality: xnor
We note that to test equality of Booleans we can use xnor, and we can of course define xor in terms of and, or and not:
xor x y = (x or y) and (not x or not y)
and
xnor x y = not (xor x y)
Signed Integers
The common way to define integers in the lambda calculus is as Church numerals. Here we take a different approach, but it is of course equivalent. We define an integer as a list of Booleans, using thermometer code, and with the following definitions:
We define unsigned 0 as a 1-element list containing false. To get signed integers we simply define the first bit of the list as the sign bit. We define unsigned and signed versions of 0:
u0 = false:[]
0 = +0 = true:u0
-0 = false:u0
For convenience we define also:
isPos n = head n
isNeg n = not (head n)
isZero n = not (head (tail n))
sign n = head n
Integer equality
The definition of 0 makes the integer equality (==) easier:
(==) n1 n2 = let
s1 = head n1
s2 = head n2
b1 = head (tail n1)
b2 = head (tail n2)
if (xnor s1 s2) then
if (and (not b1) (not b2))
then true
else
if (and (not b1) (not b2))
then (==) (s1:(tail n1)) (s2:(tail n2))
else false
else false
Negation
We can also easily define negation:
neg n = (not (head n)):(tail n)
Increment and Decrement
For convenience we define also define increment and decrement operations:
inc n = if isPos n
then true:true:(tail n)
else if isZero n
then 1
else false:(tail (tail n))
dec n = if isZero n
then -1
else if isNeg n
then false:true:(tail n) n
else true:(tail (tail n))
Addition and Subtraction
General addition is quite easy:
add n1 n2 = foldl add_if_true n1 n2
where
add_if_true elt n1 = if elt then true:n1 else n1
In the same way, subtraction is also straightforward:
sub n1 n2 = foldl sub_if_true n1 n2
where
sub_if_true elt n1 = of elt then (tail n1) else n1
Multiplication
An easy way to define multiplication is by defining the replicate and sum operations:
replicate n m =
let
repl n m lst = if n==0
then lst
else repl (dec n) m (m:lst)
in
repl n m []
sum lst = foldl add 0 lst
Then multiplication simply becomes
mult n m = sum (replicate n m)
In a similar way we can define integer division and modulo.
Floats, Characters and Strings
We note that floating-point numbers and characters use an integer representation, and strings are simply lists of characters. So we don’t need to do any additional work to represent them, and the operations on them are analogous to the ones defined above.
Conclusion
In this way, we have defined a language with variables, (higher-order) functions, conditionals and recursion. We can manipulate lists and tuples of integers, floats, chars and strings. And yet it consists of nothing more than lambda functions!