Key Constructs and Functions in Haskell
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In a Functional Language, There are Only Functions
Although it might seems that a language like Haskell has a lot of different objects and constructs, we can express all of them in terms of functions. Watch the video to get a little insight, then read on to find out more …
Variables and let
let
n = 10
f x = x+1
in
f n
 One variable per let =>
let
n = 10
in
let
f x = x+1
in
f n
 Rewrite f as lambda =>
let
n = 10
in
let
f = x > x+1
in
f n
 Rewrite inner let as lambda =>
let
n = 10
in
(f > f n) (x > x+1)
 Rewrite outer let as lambda =>
( n > ((f > f n) ( x > x+1 )) ) 10
So variables and let expressions are just syntactic sugar for lambda expressions.
Tuples
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
The same goes for the tuple accessor functions:
fst tp = tp (x y z > x)
snd tp = tp (x y z > y)
Lists
Lists can be defined in terms of the empty lists []
and the cons
operation (:)
.
ls = [1,2,3]
Rewrite using : and [] =>
ls = 1 : 2 : 3 : []
Or using cons =>
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
Thus
ls = cons 1 (...)
= c > c 1 (...)
We can also define head
and tail
using only lambdas:
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
The definitions for true
and false
are given below under Booleans.
Then we can check if a list is empty or not:
isEmpty lst = lst (x xs > false)
A nonemptylist is always defined as:
lst = x:xs
which with our defintion of (:)
is
lst = (x xs > c > c x xs) x xs
= c > c x xs
Thus,
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
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 ifthenelse expression which is defined below under Conditionals.
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.
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 ifthenelse clause as a pure function:
ifthenelse cond if_true_exp if_false_exp
Booleans
To evaluate the condition we need to define booleans:
true = x y > x
false = x y > y
With this definition, the ifthenelse 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)
xnor x y = not (xor x y)
Signed Integers
We define an integer as a list of booleans, in thermometer encoding, and with the following definitions:
We define usigned 0 as a 1element 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 b1 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 integer division and modulo can be defined.
Floats, Characters and Strings
We note that floats and chars use an integer representation, and strings are simply lists of chars. So we now have a language that can manipulate lists and tuples of integers, floats, chars and strings.
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Functional Programming in Haskell: Supercharge Your Coding
Functional Programming in Haskell: Supercharge Your Coding
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