Functional programming using haskell notes iitk

Functional Programming Lecture notes
Piyush P Kurur
October 31, 2011

Lecture 1
Introduction
In this course we will learn functional programming through haskell. The main
motivation of this course is to prepare people for real world programming. To
be a successful haskell programmer, I believe, one needs the uncluttered thought
of a mathematician combined with the pragmatism of an Engineer. One can
learn the basics in a matter of days. However, one requires regular practice to
be an accomplished programmer.
1.1 What will be done in this course ?
Covering the entire language, the mathematical background and the libraries is
impossible in a semester long course. So in the class we will learn the basics of
haskell and some of the mathematical notions. In fact, most of the class will be
devoted to understanding these mathematical concepts. Programming in haskell
is a good exercise to your mathematical muscle and thus this is an important
component of this course. But I hope this course does not end up being a
collection of abstract nonsense. Therefore, we will have regular (i.e. weekly)
ungraded assignments to pull us back to real world.
1.2 Evaluation.
Your grade will depend on your performance in exams (which constitutes 70%
of the marks) and in the semester long project (which takes up the remaining
30% of marks). There will be zero credit for the weekly assignments. However,
they are what will help you build the knowledge.
Projects can be done in a group of 3 or less. Beside the projects have to be
under version control and since it is expected to be in haskell, should be released
3

4 LECTURE 1. INTRODUCTION
as a cabal package. I recommend using darcs as the version control program.
We will NOT discuss darcs or cabal in the course, you will have to learn it on
your own. You may choose on what you want to work on but I will put up a list
of sample projects for you to get an idea of the scope of haskell as an application
development framework.
Besides, I encourage you to use haskell in your other programming needs like
B.Tech project, M.Tech/Ph.D Thesis etc.

Lecture 2
Warming up
We will start with the haskell interpreter ghci. To start the interpreter type
the following on the command line. You might see something like this.
$ ghci
GHCi, version 7.0.3: http://www.haskell.org/ghc/ :? for help
Loading package ghc-prim ... linking ... done.
Loading package integer-gmp ... linking ... done.
Loading package base ... linking ... done.
Loading package ffi-1.0 ... linking ... done.
Prelude>
The Prelude> is the prompt and the interpreter is expecting you to type in
stuﬀ. What you can type is any valid haskell expression. You can for example
use ghci like a calculator.
Prelude> 42
42
Prelude> 2 + 4
6
Prelude> sqrt 2
1.4142135623730951
The line 5 sqrt 2 is a function application, the function sqrt is applied on 2.
In haskell, applying a function f on an argument x is done by placing f on the
left of x. Function application associates to the left, i.e. f x y z means ((f
x) y)z.
5

6 LECTURE 2. WARMING UP
2.1 Types
Although it did not appear to be the case in this limited interaction, Haskell
is a strongly typed language unlike python or scheme. All values in haskell
have a type. However we can almost always skip types. This is because the
compiler/interpreter infers the correct type of the expression for us. You can
find the type of an expression prefixing it with ‘:type’ at the prompt. For
example
Prelude> :type ’a’
’a’ :: Char
In haskell the a type of a value is asserted by using :: (two colons). The
interpreter just told us that ’a’ is an expression of type Char.
Some basic types in Haskell are given in the table below
2.2 Lists and Strings.
A list is one of the most important data structure in Haskell. A list is denoted
by enclosing its components inside a square bracket.
Prelude> :type [’a’,’b’]
[’a’,’b’] :: [Char]
Prelude> :type "abcde"
"abcde" :: [Char]
A string in Haskell is just a list of characters. The syntax of a string is exactly
as in say C with escapes etc. For example "Hello world" is a string.
Unlike in other languages like python or scheme, a list in Haskell can have only
one type of element, i.e. [1, ’a’] is not a valid expression in haskell. A list of
type t is denoted by [t] in haskell. Example the type String and [Char] are
same and denote strings in Haskell.
2.3 Functions.
In haskell functions are first class citizens. They are like any other values.
Functions can take functions as arguments and return functions as values. A
function type is denoted using the arrow notation, i.e. A function that takes an
integer and returns a character is denoted by Integer -> Char.

2.3. FUNCTIONS. 7
Prelude> :type length
length :: [a] -> Int
Notice that the interpreter tells us that length is a function that takes the list of
a and returns an Int (its length). The type a in this context is a type variable,
i.e. as far as length is concerned it does not care what is the type of its list, it
returns the length of it. In Haskell one can write such generic functions. This
feature is called polymorphism. The compiler/interpreter will appropriately
infer the type depending on its arguments
Prelude> length [1,2,3]
3
Prelude> length "Hello"
5
Polymorphism is one of the greatest strengths of Haskell. We will see more of
this in time to come.
Let us now define a simple haskell function.
fac 0 = 1
fac n = n * fac (n - 1)
Save this in a file, say fac.hs and load it in the interpreter
$ ghci fac.hs
[1 of 1] Compiling Main ( fac.hs, interpreted )
Ok, modules loaded: Main.
Main> fac 0
1
Main> fac 4
24
A haskell function is defined by giving a set of equations. A function equation
looks like
f pat_1_1 pat_1_2 ... pat_1_n = expr_1
f pat_2_1 pat_2_2 ... pat_2_n = expr_2
...
f pat_m_1 pat_2_2 ... pat_m_n = expr_m

8 LECTURE 2. WARMING UP
The formal parameters can be patterns. We will define what patterns in detail
latter on but a constant like 0 or a variable like n is a pattern. When a function
is evaluated, its actual arguments are matched with each of the equations in
turn. We cannot explain what matching means in full detail here because we
have not explained patterns yet. However, for constant patterns like 0 or a
variable it is easy. An actual argument can match a constant if and only if
the argument evaluates to that constant. On the other hand a variable pattern
matches any argument. If a function f is defined using n equations, then while
evaluating the function on a set of arguments each equation is tried out in order.
The first equation of f whose formal parameters match the actual argument is
used for evaluation.
To illustrate we consider the function fac that we defined. The expression
fac 0 evaluates to 1 because the actual parameter matches with the formal
parameter 0 in the first equation. On the other hand, in the expression fac 2
the actual argument 2 cannot match the formal argument 0 in the first equation
of fac. Therefore the next equation is tried namely fac n = n * fac (n-1).
Here the formal parameter n matches 2 (recall a variable pattern can match any
argument). Therefore, fac 2 evaluates to 2 * fac (2 - 1) which by recursion
evaluates to 2.
We give two alternate defintions of the factorial function, the first using guards
and the next using if then else.
fac1 n | n == 0 = 1
| otherwise = n * fac1 (n -1)
fac2 n = if n == 0 then 1 else n * fac2 (n -1)
To summarise
1. Haskell functions are polymorphic
2. They can be recursive.
3. One can use pattern matching in their definition.

2.3. FUNCTIONS. 9
Haskell type What they are
Bool Boolean type
Int Fixed precision integer
Char Character
Integer Multi-precision integer

Lecture 3
More on pattern matching.
In the last lecture we, defined the factorial function and illustrated the use of
pattern matching. In this chapter we elaborate on it a bit more.
3.1 Pattern matching on lists.
We have already seen the list type. There is an algebraic way of defining a list.
A list either an empty list or an element attached to the front of an already
constructed list. An empty list in Haskell is expressed as [] where as a list
whose first element is x and the rest is xs is denoted by x:xs. The notation
[1,2,3] is just a syntactic sugar for 1:(2:(3:[])) or just 1:2:3:[] as : is
a right associative operator. We now illustrate pattern matching on list by
giving the example of the map function. The map function takes a function and
a list and applies the function on each element of the list. Here is the complete
definition including the type
> import Prelude hiding (map, null, curry, uncurry)
> -- hide Prelude functions defined here
>
> map :: (a -> b) -> [a] -> [b]
> map f [] = []
> map f (x:xs) = f x : map f xs
Since a list can either be empty or of the form (x:xs) this is a complete defini-
tion. Also notice that pattern matching of variables is done here.
11

12 LECTURE 3. MORE ON PATTERN MATCHING.
3.2 Literate programming detour.
Before we proceed further, let us clarify the >’s at the beginning of the of
the lines. This is the literate programming convention that Haskell supports.
Literate programming is a style of programming championed by Knuth, where
comments are given more importance than the code. It is not restricted to
Haskell alone; in fact TeX and METAFONT were written first written by Knuth
in a literate version Pascal language called WEB and later on ported to CWEB,
a literate version of the C Programming language. The ghc compiler and the
ghci interpreter supports both the literate and non-literate version of Haskell.
Normally any line is treated as part of the program unless the commented. In
literate haskell all lines other than
1. Those that start with a ‘>’ or
2. A block of lines enclosed in a begin{code} end{code}
are treated as comments. We will use this literate style of programming; we will
use only the first form i.e. program lines start with a >. The advantage is that
one can download the notes directly and compile and execute them.
3.3 Wild card patterns.
We now illustrate the use of wild card patterns. Consider the function that tests
whether a list is empty. This can be defined as follows.
> null [] = True
> null _ = False
The pattern (under score) matches any expression just like a variable pattern.
However, unlike a variable pattern where the matched value is bound to the
variable, a wild card discards the value. This can be used when we do not care
about the value in the RHS of the equation.
3.4 Tuples and pattern matching.
Besides lists Haskell supports the tuple type. Tuple types corresponds to taking
set theoretic products. For example the tuple (1,"Hello") is an ordered pair
consisting of the integer 1 and the string "Hello". Its type is (Int,String) or
equivalently (Int,[Char]) as String is nothing but [Char]. We illustrate the
pattern matching of tuples by giving the definition of the standard functions
curry and uncurry.

3.5. A BRIEF DETOUR ON CURRYING 13
3.5 A brief detour on currying
In haskell functions are univariate functions unlike other languages. Multi-
parameter functions are captured using the process called currying. A function
taking two arguments a and b and returning c can be seen as a function taking
the a and returning a function that takes b and returning c. This kind of
function is called a curried function. Another way in which we can represent a
function taking 2 arguments is to think of the function as taking a tuple. This
is its uncurried form. We now deﬁne the higher order functions that transforms
between these two forms.
> curry :: ((a,b) -> c) -> a -> b -> c
> uncurry :: (a -> b -> c) -> (a,b) -> c
>
> curry f a b = f (a,b)
> uncurry f (a,b) = f a b
The above code clearly illustrates the power of Haskell when it comes to ma-
nipulating functions. Use of higher order functions is one of the features that
we will ﬁnd quite a bit of use.
3.6 Summary
In this lecture we saw
1. Pattern matching for lists,
2. Tuples and pattern matching on them,
3. Literate haskell
4. Higher order functions.

14 LECTURE 3. MORE ON PATTERN MATCHING.

Lecture 4
The Sieve of Eratosthenes.
In this lecture, we look at an algorithm to ﬁnd primes which you might have
learned in school. It is called the Sieve of Eratosthenes. The basic idea is the
following.
1. Enumerate all positive integers starting from 2.
2. Forever do the following
3. Take the smallest of the remaining uncrossed integer say p and circle it.
4. Cross out all numbers that are the factors of the circled integer p.
All the circled integers are the list of primes. For an animated description see
the wiki link http://en.wikipedia.org/Sieve_of_Eratosthenes.
We want to convert this algorithm to haskell code. Notices that the Sieve
seperates the primes from the composites. The primes are those that are circled
and composites are those that are crossed. So let us start by deﬁning
> primes = circledIntegers
i.e. the primes are precisely the list of circled integers.
An integer gets circled if and only if it is not crossed at any stage of the sieving
process. Furthermore, a integer gets crossed in the stage when its least prime
factor is circled. So to check whether an integer is crossed all we need to do is
to check whether there is a prime which divides it.
15

16 LECTURE 4. THE SIEVE OF ERATOSTHENES.
> divides x n = n ‘mod‘ x == 0
>
> check (x:xs) n | x * x > n = True -- the rest are bigger.
> | x ‘divides‘ n = False -- We hit a factor
> | otherwise = check xs n -- need to do more work
>
> isCircled = check primes
>
The function isCircle x checks whether x will eventually be circled. One
need not go over all primes as the smallest non-trivial prime p that divides a
composite number n should always be less than or equal to
√
n. This explains
the first guard statement of the check function.
4.1 Guards detour.
We now explain another feature of Haskell namely guards. Look at the definition
of the function check. Recall that a function is defined by giving a set of
equations. For each such equation, we can have a set of guards. The syntax of
these guarded equation looks like
f p_1 ... p_n | g_1 = e_1
| g_2 = e_2
| g_3 = e_3
| ...
| g_m = e_m
Each of the guards g i is a boolean expression. You should read this as “if f’s
arguments match the patterns p1 ... pn then its value is e 1 if g 1 is true,
e 2 if g 2 is true, etc e m if g m is true”. If multiple guards are true then the
guard listed first has priority. For example consider the following function
>
> f x | x >= 0 = "non-negative"
> | x <= 0 = "negative"
Then f 0 is the string "non-negative". If you want to add a default guard,
then use the keyword otherwise. The keyword otherwise is nothing but the
boolean value True. However, in guards it is a convention to write otherwise
instead of True.

4.2. HOW IS THE CIRCULARITY OF PRIMES HANDLED? 17
Finally, we want to define the list of circledInteger. Clearly the first integer
to be circled is 2. The rest are those integers on which the function isCircled
returns true. And here is the Haskell code.
>
> circledIntegers = 2 : filter isCircled [3..]
>
Here filter is a function that does what you expect it to do. Its first argument
is a predicate, i.e it take an element and returns a boolean, and its second
argument is a list. It returns all those elements in the list that satisfies the
predicate. The type of filter is given by filter :: (a -> Bool) -> [a]
-> [a]. This completes the program for sieving primes. Now load the program
into the interperter
$ ghci Sieve-of-Eratosthense.lhs
[1 of 1] Compiling Main ( src/lectures/Sieve-of-Eratosthenes.lhs, interpreted )
*Main> :type take
take :: Int -> [a] -> [a]
*Main> take 10 primes
[2,3,5,7,11,13,17,19,23,29]
The take n xs returns the first n elements of the list xs.
4.2 How is the circularity of primes handled?
One thing you might have noticed is that the list primes has a circular definition.
The compiler is able to grok this circularity due to the fact that Haskell is a lazy
language. No expression is evaluated unless it is required. For each integer in
the list primes to decide that it is circled, we need to consider only the primes
that are less than its square root. As a result the definition of primes does not
blow up into an infinitary computation.

18 LECTURE 4. THE SIEVE OF ERATOSTHENES.
4.3 Infix application of function and prefix ap-
plication of operators.
We now explain another feature of a Haskell. Notice the use of mod in the
expression n ‘mod‘ x == 0 in the definition of the function divides and in the
guard x ‘divides‘ n in the definition of the function check. We have used a two
argument function (i.e. its type is a -> b -> c) like an operator. For any such
function foo we can convert it into a binary operator by back quoting it, i.e.
‘foo‘.
The converse is also possible, i.e. we can convert an operator into the corre-
sponding function by enclosing it in a bracket. For example the expression (+)
(there is no space in side the bracket otherwise it is an error) is a function Int
-> Int -> Int.
> incr1 = map ((+) 1)
> incr2 = map (+1)
The two function incr1 and incr2 both does the same thing; increments all
the elements in a list of integers by 1.

Lecture 5
Fibonacci Series
In continuation with the theme of the last lecture we define another infinite
series — the fibonacci series. Ofcourse all of you know the Fibonacci Series. It
is defined by the linear recurrence relation Fi+2 = Fi + Fi+1. We assume that
F0 = 1 and F1 = 1 to begin with.
The defintion is straight forward; it is just a one liner, but we use this as an
excuse to introduce two standard list functions
5.1 Ziping a list
The zip of the list x0, ..., xn and y0, . . . , ym is the list of tuples (x0, y0), . . . , (xk, yk)
where k is the minimum of n and m.
> zip :: [a] -> [b] -> [(a,b)]
> zip [] _ = []
> zip _ [] = []
> zip (x:xs) (y:ys) = (x,y) : zip xs ys
The function zipWith is a general form of ziping which instead of tupling com-
bines the values with an input functions.
> zipWith :: (a -> b -> c) -> [a] -> [b] -> [c]
> zipWith f xs ys = map (uncurry f) $ zip xs ys
We can now give the code for fibonacci numbers.
19

20 LECTURE 5. FIBONACCI SERIES
> fib = 1:1:zipWith (+) fib (tail fib)
Notice that the zip of fib and tail fib gives a set of tuples whose caluse are
consecutive ﬁbonacci numbers. Therefore, once the intial values are set all that
is required is zipping through with a (+) operation.

Lecture 6
Folding Lists
Will be up soon.
21

Lecture 7
Data types
We have already seen an example of a compound data type namely list. Recall
that, a list is either an empty list or a list with a head element and rest of the
list. We begin by defining a list data type. Haskell already provides a list data
type so we do not need to define a user defined data type. However, we do this
for illustration
> import Prelude hiding (sum) -- hide the standard sum function
> data List a = EmptyList
> | Cons a (List a)
One reads this as follows “List of a is either EmptyList or a Cons of a and List
of a”. Here the variable a is a type variable. The result of this is that List
is now a polymorphic data type. We can instatiate this with any other Haskell
data types. A list of integers is then List Integer.
The identifiers EmptyList and Cons are the two constructors of our new data
type List. The constructors can now be used in Haskell expressions. For
example EmptyList is a valid Haskell expression. So is Cons 2 EmptyList and
Cons 1 (Cons 2 EmptyList). The standard list actually has two constructors,
namely [] and (:).
7.1 Pattern Matching
We can now define functions on our new data type List using pattern matching
in the most obvious way. Here is a version of sum that works with List Int
instead of [Int]
23

24 LECTURE 7. DATA TYPES
> sum :: List Int -> Int
> sum EmptyList = 0
> sum (Cons x xs) = x + sum xs
As the next example, we two functions to convert from our list type to the
standard list type.
> toStdList :: List a -> [a]
> fromStdList :: [a] -> List a
> toStdList EmptyList = []
> toStdList (Cons x xs) = x : toStdList xs
> fromStdList [] = EmptyList
> fromStdList (x:xs) = Cons x (fromStdList xs)
1. Exercise: Define the functions map foldr and foldl for our new list
type.
7.2 Syntax of a data type
We now give the general syntax for defining data types.
data Typename tv_1 tv_2 tv_n = C1 te_11 te_12 ... te_1r1
| C2 te_21 te_22 ... te_2r2
| . . .
| Cm te_m1 te_m2 ... te_mrm
Here data is a key word that tells the complier that the next equation is a
data type definition. This gives a polymorphic data type with n type arguments
tv 1,...,tv n. The te ij’s are arbitrary type expressions and the identifiers
C1 to Cm are the constructors of the type. Recall that in Haskell there is a
constraint that each variable, or for that matter type variable, should be an
identifer which starts with a lower case alphabet. In the case of type names and
constructors, they should start with upper case alphabet.
7.3 Constructors
Constructors of a data type play a dual role. In expressions they behave like
functions. For example in the List data type that we defined the EmptyList

7.4. THE BINARY TREE 25
constructor is a constant List (which is the same as 0-argument function) and
Cons has type a -> List a -> List a. On the other hand constructors can
be used in pattern matching when deﬁning functions.
7.4 The Binary tree
We now look at another example the binary tree. Recall that a binary tree
is either an empty tree or has root and two children. In haskell this can be
captured as follows
>
> data Tree a = EmptyTree
> | Node (Tree a) a (Tree a)
>
To illustrate function on tree let us deﬁne the depth function
> depth :: Tree a -> Int
> depth EmptyTree = 0
> depth (Node left _ right) | dLeft <= dRight = dRight + 1
> | otherwise = dLeft + 1
> where dLeft = depth left
> dRight = depth right

Lecture 8
An expression evaluator
Our goal is to build a simple calculator. We will not worry about the parsing as
that requires input output for which we are not ready yet. We will only build
the expression evaluator. We start by deﬁning a data type that captures the
syntax of expressions. Our expressions are simple and do not have variables.
> data Expr = Const Double
> | Plus Expr Expr
> | Minus Expr Expr
> | Mul Expr Expr
> | Div Expr Expr
Now the evaluator is easy.
> eval :: Expr -> Double
> eval (Const d) = d
> eval (Plus a b) = eval a + eval b
> eval (Minus a b) = eval a - eval b
> eval (Mul a b) = eval a * eval b
> eval (Div a b) = eval a / eval b
8.1 The Either data type
You might have noticed that that we do not handle the division by zero errors
So we want to capture functions that evaluates to a value or returns an error.
27

28 LECTURE 8. AN EXPRESSION EVALUATOR
The Haskell library exports a data type called Either which is useful in such a
situation. For completeness, we give the definition of Either. You can use the
standard type instead.
data Either a b = Left a
| Right b
The data type Either is used in haskell often is situation where a computation
can go wrong like for example expression evaluation. It has two constructors
Left and Right. By convention Right is used for the correct value and Left
for the error message: A way to remeber this convention is to remember that
“Right is always right”.
We would want our evaluator to return either an error message or a double. For
convenience, we capture this type as Result.
> type Result a = Either String a
The above expression is a type alias. This means that Result a is the same type
as Either String a. As far as the compiler is concerned, they are both same.
We have already seen a type aliasing in action namely String and [Char].
We are now ready to define the new evaluator. We first give its type
> eval’ :: Expr -> Result Double
The definition of eval’ is similar to eval for constructors Const, Plus, Minus
and Mul except that it has to ensure.
1. Ensure each of the sub expressions have not resulted a division by zero
error.
2. If the previous condition is met has to wrap the result into a Result data
type using the Right constructor.
Instead of explicit pattern matching we define a helper function for this calle
app that simplify this.
> zeroDivision = Left "Division by zero"
> app op (Right a) (Right b) = Right (op a b)
> app _ _ _ = zeroDivision

8.1. THE EITHER DATA TYPE 29
The constructor Div has to be handled seperately as it is the only operator that
generates an error. Again we write a helper here.
> divOp (Right a) (Right b) | b == 0 = zeroDivision
> | otherwise = Right (a / b)
> divOp _ _ = zeroDivision -- problem any way.
Now we are ready to deﬁne eval’.
> eval’ (Const d) = Right d
> eval’ (Plus a b) = app (+) (eval’ a) (eval’ b)
> eval’ (Minus a b) = app (-) (eval’ a) (eval’ b)
> eval’ (Mul a b) = app (*) (eval’ a) (eval’ b)
> eval’ (Div a b) = divOp (eval’ a) (eval’ b)
Let us load the code in the interpreter
$ ghci src/lectures/An-expression-evaluator.lhs
[1 of 1] Compiling Main ( src/lectures/Expression.lhs, interpreted )
*Main> let [zero,one,two,three,four,five] = map Const [0..5]
*Main> eval’ (Div five zero)
Left "Division by zero"
*Main> eval’ (Plus (Div five zero) two)
Left "Division by zero"
*Main> eval’ (Plus (Div five two) two)
Right 4.5
*Main>

30 LECTURE 8. AN EXPRESSION EVALUATOR

Lecture 9
Lambda Calculus
We now look at lambda calculus, the theoretical stuff that underlies functional
programming. It was introduced by Alonzo Church to formalise two key con-
cepts when dealing with functions in mathematics and logic namely: function
definition and function application. In this lecture, we build enough stuff to get
a lambda calculus evaluator.
> module Lambda where
> import Data.List -- I need some standard list functions
We start with an example. Consider the squaring function, i.e. the function
that maps x to x2
. In the notation of lambda calculus this is denoted as λx.x2
.
This is called a lambda abstraction. Apply a function f on an expression N is
written as fN. The key rule in expression evaluation is the β-reduction: the
expression (λx.M)N reduces under β-reduction to the expression M with N
substituted in it. We now look at lambda calculus formally.
The goal of this lecture is to introduce basics of lambda calculus and on the way
implement a small lambda calculus interpreter.
9.1 Abstract syntax
As with any other formal system we first define its abstract syntax. A lambda
calculus expression is defined via the grammar
e := v|e1e2|λx.e
Here e1 and e2 expressions themselves. We now capture this abstract syntax as
a Haskell data type
31

32 LECTURE 9. LAMBDA CALCULUS
> -- | The datatype that captures the lambda calculus expressions.
> data Expr = V String -- ^ A variable
> | A Expr Expr -- ^ functional application
> | L String Expr -- ^ lambda abstraction
> deriving Show
9.2 Free and bound variables
The notion of free and bound variables are fundamental to whole of mathemat-
ics. For example in the integral sin xydy, the variable x occurs free where as
the variables y occurs bound (to the corresponding dy). Clearly the value of
the expression does not depend on the bound variable; in fact we can write the
same integral as sin xtdt.
In lambda calculus we say a variable occurs bound if it can be linked to a lambda
abstraction. For example in the expression λx.xy the variable x is bound where
as y occurs free. A variable can occur free as well as bound in an expression —
consider x in λy.x(λx.x).
Formally we can deﬁne the free variables of a lambda expression as follows.
FV (v) = {v}
FV (e1e2) = FV (e1) ∪ FV (e2)
FV (λx.e) = FV (e) {x}
We turn this into haskell code
> freeVar :: Expr -> [String]
> freeVar (V x ) = [x]
> freeVar (A f e) = freeVar f ‘union‘ freeVar e
> freeVar (L x e) = delete x $ freeVar e
9.3 Variable substitution
We often need to substitute variables with other expressions. Since it is so
frequent we give a notation for this. By M[x := e], we mean the expression
obtained by replacing all free occurrence of x in M by e. Let us code this up in
haskell.

9.4. CHANGE OF BOUND VARIABLES (α-REDUCTION) 33
> subst :: Expr -> String -> Expr -> Expr
> subst var@(V y) x e | y == x = e
> | otherwise = var
> subst (A f a) x e = A (subst f x e) (subst a x e)
> subst lam@(L y a) x e | y == x = lam
> | otherwise = L y (subst a x e)
9.4 Change of bound variables (α-reduction)
You are already familiar with this in mathematics. If we have an integral of the
kind xtdt we can rewrite it as xydy by a change of variable. The same is
true for lambda calculus. We say call the “reduction” λx.M ← λt.M[x := t] as
the α-reduction. However care should be taken when the new variable is chosen.
Two pitfalls to avoid when performing α-reduction of the expression λx.M to
λt.M[x := t] is
1. The variable t should not be free in M for otherwise by changing from x
to t we have bound an otherwise free variable. For example if M = t then
λt.M[x = t] becomes λt.t which is clearly wrong.
2. If M has a free occurrence of x in the scope of a bound occurrence of t
then we cannot perform change the bound variable x to t. For example
consider M = λt.xt. Then λt.M[x = t] will become λt.λt.tt which is
clearly wrong.
Clearly, one safe way to do α-reduction on λx.M is to use a fresh variable t,
i.e. a variable that is neither free nor bound in M. We write a helper function
to compute all the variables of a lambda calculus expression.
> varsOf :: Expr -> [String]
> varsOf (V x) = [x]
> varsOf (A f e) = varsOf f ‘union‘ varsOf e
> varsOf (L x e) = varsOf e ‘union‘ [x]
We now give the code to perform a safe change of bound variables.
> alpha :: Expr -> [String] -> Expr
> alpha (A f e) vars = A (alpha f vars) (alpha e vars)
> alpha (L x e) vars | x ‘elem‘ vars = L t $ alpha e’ vars
> | otherwise = L x $ alpha e vars

34 LECTURE 9. LAMBDA CALCULUS
> where t = fresh (varsOf e ‘union‘ vars)
> e’ = subst e x (V t)
> alpha e _ = e
9.5 Function evaluation (β-reduction)
The way lambda calculus captures computation is through β reduction. We
already saw what is β reduction. Under beta reduction, an expression (λx.M)N
reduces to M[x := N], where M[x := N] denotes substitution of free occurrences
of x by N. However, there is a chance that a free variable of N could become
bound suddenly. For example consider N to be just y and M to be λy.xy. Then
reducing (λx.M)N to M[x := N] will bind the free variable y in N.
We now give the code for β reduction. It performs one step of beta reduction
that too if and only if the expression is of the form (λx.M)N.
> beta :: Expr -> Expr
> beta (A (L x m) n) = carefulSubst m x n
> carefulSubst m x n = subst (alpha m $ freeVar n) x n
>
9.6 Generating Fresh variables
We saw that for our evaluator we needed a source of fresh variables. Our function
fresh is given a set of variables and its task is to compute a variable name that
does not belong to the list. We use diagonalisation, a trick ﬁrst used by Cantor
to prove that Real numbers are of strictly higher cardinality than integers.
> fresh :: [String] -> String
> fresh = foldl diagonalise "a"
>
> diagonalise [] [] = "a" -- diagonalised any way
> diagonalise [] (y:ys) | y == ’a’ = "b" -- diagonalise on this character
> | otherwise = "a"
> diagonalise s [] = s -- anyway s is differnt from t
> diagonalise s@(x:xs) (y:ys) | x /= y = s -- differs at this position anyway
> | otherwise = x : diagonalise xs ys
>

9.7. EXERCISE 35
9.7 Exercise
1. Read up about β-normal forms. Write a function that converts a lambda
calculus expression to its normal form if it exists. There are diﬀerent
evaluation strategies to get to β-normal form. Code them all up.
2. The use of varOf in α-reduction is an overkill. See if it can be improved.
3. Read about η-reduction and write code for it.

Lecture 10
Modules
Will be up soon.
37

Lecture 11
Towards type inference
A powerful feature of Haskell is the automatic type inference of expressions. In
the next few lectures, we will attempt to give an idea of how the type inference
algorithm works. Ofcourse giving the type inference algorithm for the entire
Haskell language is beyond the scope of this lecture so we take a toy example.
Our aim is to give a complete type inference algorithm for an enriched version
of lambda calculus, that has facilities to do integer arithmetic. Therefore, our
lambda calulus expressions have, besides the other stuﬀ we have seen, integer
constants and the built in function ‘+’. We limit ourselves to only one opera-
tor because it is straight forward to extend our algorithm to work with other
operation
11.1 Syntax of our Enriched Lambda calculus.
The syntax is given below. Here v and x stands for arbitrary variable and e1,e2
stands for arbitrary expressions.
e = ...| − 1|0|1|...| + |v|e1e2|λx.e
We will write the lambda calculus expression +e1e2 in its inﬁx form e1 + e2 for
ease of readability.
The haskell datatype that captures our enriched lambda calculus expression is
the following
> module Lambda where
>
> -- | The enriched lambda calculus expression.
39

40 LECTURE 11. TOWARDS TYPE INFERENCE
> data Expr = C Integer -- ^ A constant
> | P -- ^ The plus operator
> | V String -- ^ The variable
> | A Expr Expr -- ^ function application
> | L String Expr -- ^ lambda abstraction
> deriving (Show, Eq, Ord)
Clearly stuﬀ like A (C 2) (C 3) are invalid expressions but we ignore this for
time being. One thing that the type checker can do for us is to catch such stuﬀ.
11.2 Types
We now want to assign types to the enriched lambda calculus that we have. As
far as we are concerned the types for us are
t = Z|α|t1 → t2
Here α is an arbitrary type variable. Again, we capture it in a Haskell datatype.
> data Type = INTEGER
> | TV String
> | TA Type Type deriving (Show, Eq, Ord)
11.3 Conventions
We will follow the following convention when dealing with type inference. The
lambda calculus expressions will be denoted by Latin letters e, f, g etc with
appropriate subscripts. We will reserve the Latin letters x, y, z and t for lambda
calculus variables. Types will be represented by the Greek letter τ and σ with
the letters α and β reserved for type variables.
11.4 Type specialisation
The notion of type specialisation is intuitivly clear. The type α → β is a more
general type than α → α. We use σ ≤ τ to denote the fact that σ is specialisation
of τ. How do we formalise this notion of specialisation ? Firstly note that any
constant type like for example integer cannot be specialised further. Secondly
notice that a variable α can be specialised to a type τ as long as τ does not
have an occurance of α in it. We will denote a variable specialisation by α ← τ.

11.5. TYPE ENVIRONMENT 41
When we have a set of variable specialisation we have to ensure that there
is no cyclicity indirectly. We doe this as follows. We say a sequence Σ =
{α1 ← τ1, . . . , αn ← τn} is a consistent set of specialisation if for each i, τi
does not contain any of the variables αj, 1 ≤ j ≤ i. Now we can deﬁne what a
specialisation is. Given a consistent sequence of specialisation Σ let τ[Σ] denote
the type obtained by substituting for variables in τ with their specialisations in
Σ. Then we say that σ ≤ τ if there is a specialisation Σ such that τ[Σ] = σ. The
specialisation order gives a way to compare two types. It is not a partial order
but can be converted to one by appropriate quotienting. We say two types τ σ
are isomorphic, denoted by σ ≡ τ if σ ≤ τ and τ ≤ σ. It can be shown that
≡ forms an equivalence relation on types. Let τ denote the equivalence class
associated with τ then, it can be show that ≤ is a partial order on τ .
11.5 Type environment
Recall that the value of a closed lambda calculus expression, i.e. a lambda calcu-
lus expression with no free variables, is completely determined. More generally,
given an expression M, its value depends only on the free variables in it. Simi-
lary the type of an expression M is completely speciﬁed once all its free variables
are assigned types. A type environment is an assignment of types to variables.
So the general task is to infer the type of a lambda calculus expression M in a
given type environment Γ where all the free varaibles of M have been assigned
types. We will denote the type environments with with capital Greek letter
Γ with appropriate subscripts if required. Some notations that we use is the
following.
1. We write x :: τ to denote that the variable x has been assigned the type
τ.
2. For a variable x, we use Γ(x) to denote the type that the type environment
Γ assigns to x.
3. We write x ∈ Γ if Γ assigned a type for the variable x.
4. The type environment Γ1 ∪ Γ2 denotes the the type environment Γ such
that Γ(x) = Γ2(x) if x ∈ Γ2 and Γ1(x) otherwise, i.e. the second type
environment has a precedence.
As we described before, given a type environment Γ, the types of any lambda
calculus expression whose free variables are assigned types in Γ can be infered.
We use the notation Γ e :: τ to say that under the type environment Γ one
can infer the type τ for e.
The type inference is like theorem proving: Think of infering e :: τ as proving
that the expression e has type τ. Such an inference requires a set of rules which

for us will be the type inference rules. We express this inference rules in the
following notation
Premise 1, . . . , Premise n
conclusion
The type inference rules that we have are the following
Rule Const :
Γ n :: Z
where n is an arbitrary integer.
Rule Plus :
Γ + :: Z → Z → Z
Rule Var :
Γ ∪ {x :: τ} x :: τ
Rule Apply :
Γ f :: σ → τ, Γ e :: σ
Γ fe :: τ
Rule Lambda :
Γ ∪ {x :: σ} e :: τ
Γ λx.e :: σ → τ
Rule Specialise :
Γ e :: τ, σ ≤ τ
Γ e :: σ
The goal of the type inference algorithm is to infer the most general type,
i.e. Given an type environment Γ and an expression e find the type τ that
satisfies the following two conditions
1. Γ e :: τ and,
2. If Γ e :: σ then σ ≤ τ.
11.6 Exercises
1. A pre-order is a relation that is both reflexive and transitive.
• Show that the specialisation order ≤ defined on types is a pre-order.

11.6. EXERCISES 43
• Given any pre-oder deﬁne the associated relation as a b if
a b and b a. Prove that is an equivalence class. Show that
can be converted into a natural partial order on the equivalence class
of .
2. Prove that if σ and τ are two types such that σ ≡ τ then prove that
there is a bijection between the set V ar(σ) and V ar(τ) given by αi → βi
such that σ[Σ] = τ where Σ is a specialisation {αi ← βi|1 ≤ i ≤ n}. In
particular isomorphic types have same number of variables. (Hint: use
induction on the number of variables that occur in σ and τ).

Lecture 12
Unification algorithm
On of the key steps involved in type inference is the unification algorithm. Given
two type τ1 and τ2 a unifier if it exists is a type τ such that τ is a specialisation
of both τ1 and τ2. The most general unifier of τ1 and τ2, if it exists, is the
unifier which is most general, i.e. it is the unifier τ∗
of τ1 and τ2 such that all
other unifiers τ is a specialisation of τ∗
. In this lecture we develop an algorithm
to compute the most general unifier of two types τ1 and τ2.
> {- | Module defining the unification algorithm -}
>
> module Unification where
> import Lambda -- See the last lecture
> import qualified Data.Map as M -- To define specialisation.
> import Data.Either
> import Data.Maybe
>
> type Map = M.Map
Although we defined the unifier as a type, it is convenient when computing the
unifier of τ1 and τ2 to compute the type specialisation Σ that unifies them to
their most general unifier. We will therefore abuse the term most general unifier
to also mean this specialisation. Also for simplicity we drop the adjective “most
general” and just use unifier to mean the most general unifier.
> type Specialisation = Map String Type -- ^ A type specialisation
> type Result = Either String -- ^ The error message if it failed
> Specialisation -- ^ The unifier
>
> unify :: Type -> Type -> Result
45

46 LECTURE 12. UNIFICATION ALGORITHM
A type specialisation is captured by a Map from type variable names to the
corresponding specialisation. We given an function to compute τ[Σ] given a
specialisation Σ.
> specialise :: Specialisation -> Type -> Type
> specialise sp t@(TV x) = maybe t (specialise sp) $ M.lookup x sp
> specialise sp (TA s t) = TA (specialise sp s) (specialise sp t)
> specialise sp i = i
We generalise the unification in two ways:
1. We consider unification of types under a particular specialisation Σ. The
unifier of τ1 and τ2 under the specialisation Σ is nothing but the unifier
of τ1[Σ] and τ2[Σ]. The unification of two types can then be seen as
unification under empty specialisation.
> genUnify :: Specialisation -> Type -> Type -> Result
> unify = genUnify M.empty
2. Instead of considering unifiers of pairs of types τ1 and τ2, we consider
the simultaneous unifier of a sequence of pairs {(τ1, σ1), . . . , (τn, σn)}. A
unifier of such a sequence is a specialisation Σ such that τi[Σ] = σi[Σ] for
all 1 ≤ i ≤ n. Of course we want to find the most general of such unifier.
We call this function genUnify’. Given, genUnify it is easy to define
genUnify’.
> genUnify’ :: Specialisation -- ^ pair of types to unify
> -> [(Type,Type)] -- ^ the specialisation to unify under
> -> Result -- ^ the resulting unifier
> genUnify’ = foldl fld . Right
> where fld (Right sp) (tau,sigma) = genUnify sp tau sigma
> fld err _ = err
What is left is the definition of genUnify.
> genUnify sp (TV x) t = unifyV sp x t
> genUnify sp t (TV x) = unifyV sp x t
> genUnify sp INTEGER t = unifyI sp t
> genUnify sp t INTEGER = unifyI sp t
> genUnify sp ap1 ap2 = unifyA sp ap1 ap2
>

12.1. UNIFYING A VARIABLE AND A TYPE 47
The order of the pattern matches are important. For example notice that in
line 6, both ap1 and ap2 have to be an arrow type (why?) Also in lines 3 and
4, t can either be INTEGER or an arrow type.
So our rules of unification can are split into unifying a variable α with a type
τ, a constant types (here Integer) with a type τ and unifying two arrow types.
We capture these rules with functions unifyV, unifyI and unifyA respectively.
> unifyV :: Specialisation -> String -> Type -> Result
> unifyI :: Specialisation -> Type -> Result
> unifyA :: Specialisation -> Type -> Type -> Result
12.1 Unifying a variable and a type
Unification of variables can be tricky since specialisations are involved. Firstly,
notice that if α occurs in a type τ, we cannot unify α with τ unless τ is α itself.
We first give a function to check this.
> occurs :: Specialisation -> String -> Type -> Bool
> occurs sp x (TV y) | x == y = True
> | otherwise = maybe False (occurs sp x) $ M.lookup y sp
> occurs sp x (TA s t) = occurs sp x s || occurs sp x t
> occurs sp x _ = False
The rules of unification of a variable α and a type τ are thus
1. If α has a specialisation σ, unify σ and tau,
2. If τ is α or can be specialised to α then we already have the specialisation.
3. If α is unspecialised then specialise it with τ provided there occurs no
recursion either directly or indirectly.
> unifyV sp x t = maybe (unifyV’ sp x t) (genUnify sp t) $ M.lookup x sp
>
> unifyV’ sp x INTEGER = Right $ M.insert x INTEGER sp
> unifyV’ sp x var@(TV y) | x == y = Right $ sp
> | otherwise = maybe (Right $ M.insert x var sp)
> (unifyV’ sp x)
> $ M.lookup y sp
> unifyV’ sp x ap@(TA s t) | occurs sp x s = failV sp x ap

> | occurs sp x t = failV sp x ap
> | otherwise = Right $ M.insert x ap sp
>
> failV sp x ap = Left $ unwords [ "Fail to unify", x, "with" , ppSp sp ap
> , ": recursion detected."
> ]
>
Notice here that M.lookup x sp returns Nothing if x is not specialised un-
der the specialisation sp. The function unifyV uniﬁes only an unspecialised
variable with τ.
12.2 Unifying an integer and type
Notice here that the type τ is not a variable. Then it can only be Integer or an
arrow type.
> unifyI sp INTEGER = Right sp
> unifyI sp t = Left $ unwords [ "Failed to unify Integer with type"
> , ppSp sp t
> ]
12.3 Unifying two arrow types
> unifyA sp ap1@(TA a b) ap2@(TA c d) = either errmsg Right
> $ genUnify’ sp [(a,c),(b,d)]
> where errmsg str = Left $ unwords [ "while unifying"
> , ppSp sp ap1, "and"
> , ppSp sp ap2
> ] ++ "n" ++ str
12.4 Helper functions.
We now document the helper functions used in the uniﬁcation algorithm. First
we give an algorithm to pretty print types. This makes our error messages more
readable. Notice that this version does not put unnecessary brackets.

12.5. TESTING THIS CODE USING GHCI 49
> -- | Pretty print a type
>
> pp :: Type -> String
> pp INTEGER = "Integer"
> pp (TV x) = x
> pp (TA s t) = bracket s ++ " -> " ++ pp t
> where bracket t@(TA r s) = "(" ++ pp t ++ ")"
> bracket s = pp s
>
> -- | Pretty print a specialised type
> ppSp :: Specialisation -> Type -> String
> ppSp sp = pp . specialise sp
12.5 Testing this code using ghci
Since the lecture used the module of the previous lecture you need to give the
following commandline arguments.
$ ghci src/lectures/Unification-algorithm.lhs src/lectures/Towards-type-inference.lhs
[1 of 2] Compiling Lambda ( src/lectures/Towards-type-inference.lhs, interpreted )
[2 of 2] Compiling Unification ( src/lectures/Unification-algorithm.lhs, interpreted )
Ok, modules loaded: Unification, Lambda.
*Unification> let [a,b,c,d,e] = map TV $ words "a b c d e"
*Unification> unify (TA INTEGER a) b
Loading package array-0.3.0.2 ... linking ... done.
Loading package containers-0.4.0.0 ... linking ... done.
Right (fromList [("b",TA INTEGER (TV "a"))])
*Unification> unify (TA INTEGER a) (TA b (TA b c))
Right (fromList [("a",TA (TV "b") (TV "c")),("b",INTEGER)])
*Unification>
*Unification> let t = (TA a b)
*Unification> unify (TA INTEGER a) (TA c t)
Left "while unifying Integer -> a and c -> a -> bnFail to unify a with a -> b : recursion detect
*Unification> let Left l = unify (TA INTEGER a) (TA c t)
*Unification> putStrLn l
while unifying Integer -> a and c -> a -> b
Fail to unify a with a -> b : recursion detected.

*Unification>

Lecture 13
The type inference
algorithm
Will be up some day.
51

52 LECTURE 13. THE TYPE INFERENCE ALGORITHM

Lecture 14
Type classes
Consider the following interaction with the ghci
Prelude> (+) 1
<interactive>:1:1:
No instance for (Show (a0 -> a0))
arising from a use of ‘print’
Possible fix: add an instance declaration for (Show (a0 -> a0))
In a stmt of an interactive GHCi command: print it
Prelude> (+) 1 2
3
Prelude>
Here the interpreter is able to display the value 3 but not the function (+) 1.
The error message is instructive. It says that there is no instance of Show (a0
-> a0) for use with print. Let us see what Show and print are by using the
:info command of the interpreter.
$ ghci
53

54 LECTURE 14. TYPE CLASSES
Prelude> :info Show
class Show a where
showsPrec :: Int -> a -> ShowS
show :: a -> String
showList :: [a] -> ShowS
-- Defined in GHC.Show
instance (Show a, Show b) => Show (Either a b)
-- Defined in Data.Either
instance Show a => Show [a] -- Defined in GHC.Show
instance Show Ordering -- Defined in GHC.Show
instance Show a => Show (Maybe a) -- Defined in GHC.Show
instance Show Int -- Defined in GHC.Show
instance Show Char -- Defined in GHC.Show
instance Show Bool -- Defined in GHC.Show
[lots of similar lines delete]
Prelude> :info print
print :: Show a => a -> IO () -- Defined in System.IO
Prelude>
To understand what it means we need to know how the interpreter behaves.
When even you enter an expression on the command line the interpreter eval-
uates and tries to print it. However not all haskell types are printable. The
function print has the type Show a => a -> IO () which means print ar-
gument type a is not a general type variable but is constranined to take types
which are an instance of Show. Here Show is a type class.
Note that from :info Show it is clear that the standard types like Int, Bool
etc are all instances of the class Show. Therefore the interpreter could print any
value of those type. However, a function type is not an instance of Show and
hence could not be printed.

Lecture 15
Monads and IO
Will be up soon
55

Lecture 16
State monads
Will be up soon
57

Lecture 17
Monad transformes
Will be up soon
59

60 LECTURE 17. MONAD TRANSFORMES

Lecture 18
Functions with varible
number of arguments.
Consider the printf function in C. The number of arguments it take depends
on the format string that is provided to it. Can one have similar functions in
Haskell ?
Let us analyse the type of printf in various expressions. In the expression
printf "Hello", printf should have type String -> IO () where as in an
expression like printf "Hello %s" "world", it should have the type String
-> String -> IO(). This chamelon like behaviour is what we need to define
printf.
In this lecture we show how to hack Haskell’s type class mechanism to simulate
such variable argument functions. The stuff we will do is beyond Haskell 98 and
hence we need to enable certain extensison. We do it via the following compiler
pragmas. The very first set of lines in the source code if they are comments
that start with {-# and end with #-} are treated as compiler pragmas. In this
case we want to allow the extensions FlexibleInstances OverlappingInstances
and InvoherentInstances. One can also give these extensions at compile time;
to enable the extension Foo use the flag -XFoo at compile time.
Ofcourse it is difficult to remember all the extensions that you need for this
particular code to work. The easiest way to know what extensions are required
is to go ahead and use it in your file. GHC will warn you with an appropriate
error message if it expects an extension.
> {-# LANGUAGE FlexibleInstances #-}
> {-# LANGUAGE OverlappingInstances #-}
> {-# LANGUAGE IncoherentInstances #-}
61

62LECTURE 18. FUNCTIONS WITH VARIBLE NUMBER OF ARGUMENTS.
>
> module Printf where
Recall that an expression using printf would look something like the one below.
printf fmt e1 e2 ... em
We want our type system to infer IO () for this expression. The type of printf
in such a case should be String -> t1 -> ... -> tm -> IO () where ti
is the type of ei. The main idea is to define a type classes say Printf whose
instances are precisely those types that are of the form t1 -> ... -> tm
-> IO (). As a base case we will define an instance for IO (). We will then
inductively define it for other types.
The definition of the type class Printf and therefore printf will be easier if
we first declare a data type for formating.
>
> data Format = L String -- ^ A literal string
> | S -- ^ %s
> | G -- ^ %g for instances of Show
>
>
We would rather work with the more convenient [Format] instead of the format
string. Writing a function to convert from format string to [Format] is not too
difficult.
Now for the actual definition of the Printf class.
> class Printf s where
> printH :: IO [Format] -> s
>
The member function printH of the class Printf is a helper function that will
lead us to definition of printf. Intutively it takes as argument an IO action
which prints whatever arguments it has seen so far and returns the rest of the
formating action required to carry out s.
We will define the Printf instances for each of the type t1 -> ... tm ->
I0 () inductively. The base case is when there are no arguments.

63
> instance Printf (IO ()) where
> printH fmtIO = do fmt <- printLit fmtIO
> if null fmt then return ()
> else fail "Too few arguments provided"
>
Now the inductive instance declaration.
> instance (Printf s, Show a) => Printf (a -> s) where
> printH fmtIO a = printH action
> where action = do rfmt <- printLit fmtIO
> case rfmt of
> [] -> fail "Too many argument"
> (_:xs) -> do putStr $ show a; return xs
We give a specialised instance for string which depending on whether the for-
mating character is %s or %g uses putStr or print.
>
> instance Printf s => Printf (String -> s) where
> printH fmtIO s = printH action
> where action = do rfmt <- printLit fmtIO
> case rfmt of
> [] -> fail "Too many arguments"
> (G:xs) -> do putStr $ show s; return xs
> (S:xs) -> do putStr s; return xs
>
What is remaining is to deﬁne printLit
>
> printLit fmtIO = do fmt <- fmtIO
> let (lits,rest) = span isLit fmt
> in do sequence_ [putStr x | L x <- lits]
> return rest
> isLit (L _) = True
> isLit _ = False
>
We can now deﬁne printf

64LECTURE 18. FUNCTIONS WITH VARIBLE NUMBER OF ARGUMENTS.
> printf :: Printf s => String -> s
> printf = printH . format
>
> format :: String -> IO [Format]
> format "" = return []
> format "%" = fail "incomplete format string"
> format (’%’:x:xs) = do fmts <- format xs
> case x of
> ’s’ -> return (S:fmts)
> ’g’ -> return (G:fmts)
> ’%’ -> return (L "%" :fmts)
> _ -> fail ("bad formating character %" ++ [x])
> format zs = let (lit,xs) = span (/=’%’) zs
> in do fmts <- format xs
> return (L lit:fmts)
>

Lecture 19
Template Haskell based
implementation of printf
In the last lecture we saw a type class based solution to create functions that
take variable number of arguments. Here we give a template haskell based
solution.
19.1 What is Template Haskell ?
Template haskell is the Haskell way of doing Meta programming. At the very
least one can use it like a macro substitution but it can be used to do much
more. The idea is to process the Haskell code at compile time using Haskell
itself. A programmer can write Haskell functions to manipulate Haskell code
and using special syntax arrange the compiler to manipulate code at compile
time. In this lecture we will see how to define a version of printf using template
haskell.
Template Haskell consists of two important steps.
1. Quoting: To allow user defined function to manipulate the Haskell code
one needs to represent the program as a value in some suitable data type.
The data types defined in the module Language.Haskell.TH.Syntax is
used for this purpose. For example the type Exp defined in the above
module is used to represent a valid Haskell expression. Have a look into
the documentation of that module.
2. Splicing. Once the Haskell language fragment is processed using various
function defined by the user, it needs to be compiled by the compiler. This
processing is called splicing.
65

66LECTURE 19. TEMPLATE HASKELL BASED IMPLEMENTATION OF PRINTF
One point to be noted though is that template haskell does not splice the code
directly but only those that are expressions that are inside the quoting monad
Q. This monad is required because while generate code various side effects are
created. For example a variable x used in a fragment of the code has a different
meaning if there is a local binding defined on it. Besides one would want to read
in data from files (think of config files) to perform compile time operations.
There are two syantactic extensions to Haskell that makes template Haskell
possible. If a haskell expression is written between [| and |], the compiler will re-
place it with the corresponding representation in Language.Haskell.TH.Syntax.
For example, the expression [| "Hello" |] is of type Q Exp. The corresponding
Exp value is LitE (StringL "Hello").
The following are the extra syntactic conventions used.
1. [e| ... |] or just [| . . . ‘|]’ for quoting expressions. This has type Q
Exp.
2. [d| ... |] for quoting declarations. This has type Q [Decl]
3. [t| ... |] for quoting types. This has type Q Type.
4. [p| ... |] for quoting patterns. This has type Q Pat.
The splicing is done using the syntax $(...) (no space between $ and ()
19.2 Some convenience types and combinators
The types Q Exp and Q Pat etc occur so often that there are aliases for them
ExpQ and PatQ respectively. As an exercise guess the types aliases for Q Type
and Q Decl.
Whenever possible it is better to make use of the [| ... |] notation to build
quoted expressions. However sometimes it is better to use the constructors of
Exp directly. Recall that we can splice only quoted expressions, i.e values of
type Q Expr (or equivalently ExpQ). Say you have a quoted expressions qe and
qf which correspondes to the haskell expression e and f respectively. If one
wants to obtaine the quoted expression which correspondes to the application
of the function f on the expression e, we would have to do something like the
following
qfe = do e <- qe
f <- qf
return $ AppE f e.

19.3. PRINTF 67
To make this efficient there is a combinator called appE which does essentially
what the constructor AppE does but works on Q Exp rather than Exp.
appE :: ExpQ -> ExpQ -> ExpQ
The above code will then look like appE qe qf. There are such monadic version
of all the constructors of Exp available. Make use of it.
19.3 Printf
First note that we have enabled template Haskell using the compiler pragma
given above. It should be the very first line of your source code. A unrecognised
pragma is ignored by the compiler but sometimes a warning is issued.
> {-# LANGUAGE TemplateHaskell #-}
> module Printf where
> import Data.List
> import Language.Haskell.TH
Few things about enabling the template haskell. Strictly speaking this module
does not need TemplateHaskell, rather it can be written without using template
Haskell. This is because all it does is define functions that process objects of
type Expr or ExprQ. I have enabled it so as to write appE [|show|] instead of
the more complicated. appE (varE ’show)
First let us capture the formating via a data type
> data Format = L String -- ^ literal string
> | S -- ^ %s
> | G -- ^ %g generic type
> deriving Show
We need a function that will parse a string and the give the corresponding list
for format. The exact details are not really of interest as far as template haskell
is concerned. See the end of the file for an implementation.
> format :: String -> [Format]
The printf function can then be defined as.

> printf :: String -> ExpQ
> printfP :: [Format] -> ExpQ
> printf = printfP . format
We would rather implement the prinfP function. Let the list of formatting
instructions be [f 1,..,f n], then we want prinfP [f 1,...,f n] when spliced
to return the code.
x0 ... xm -> concat [e1,...,e_n]
Here e i depends on the ith format instruction f i. If f i is a literal then it will
just be a literal string. Otherwise it would be an appropriate variable. In our
case it should be xj where j is the number of non-literal, i.e. S or G, formating
instructions to the left of e i. The number m is the total number of non-literal
formatting instructions in the list [f 1,. . . ,f n] and should be less than or equal
to n.
Suppose we know the number of variables to the left of a format instruction f i,
how do we generate the expression e i ? The following function does that
> toExpQ :: Int -- ^ formatting instructions to the left
> -> Format -- ^ The current spec.
> -> (Int,ExpQ) -- ^ The total number of non-literal instruction
> -- to the left and the resulting expression.
> toExpQ i (L s) = (i,string s)
> toExpQ i S = (i+1,varExp i)
> toExpQ i G = (i+1,showE $ varExp i)
Here we make use of the following helper Template haskell functions which we
have deﬁned subsequently.
> string :: String -> ExpQ -- ^ quote the string
> showE :: ExpQ -> ExpQ -- ^ quoted showing
> varExp :: Int -> ExpQ -- ^ quoted variable xi
> varPat :: Int -> PatQ -- ^ quoted pattern xi
The printfP function then is simple. Recall that when spliced it should generate
the expression
x0 ... xm -> concat [e1,...,e_n]‘

19.3. PRINTF 69
The complete definition is given below.
> printfP fmts = lamE args . appE conc $ listE es
> where (nvars,es) = mapAccumL toExpQ 0 fmts
> args = map varPat [0 .. (nvars-1)]
> conc = [|concat|]
Here are the definition of the helper functions
> string = litE . StringL
> varExp i = varE $ mkName ("x" ++ show i)
> varPat i = varP $ mkName ("x" ++ show i)
> showE = appE [|show|]
We now come to the parsing of the format string. For simplicity of imple-
mentation if % occurs before an unknow format string it is treated as literal
occurance.
> format "" = []
> format [’%’] = [L "%"]
> format (’%’:x:xs)
> | x == ’s’ = S : format xs
> | x == ’g’ = G : format xs
> | x == ’%’ = L "%" : format xs
> | otherwise = L [’%’,x] : format xs
> format zs = L x : format xs
> where (x,xs) = span (/=’%’) zs
To try it out load it with ghci using the option -XTemplateHaskell. You need
this option to tell ghci that it better expect template haskell stuff like Oxford
bracket [| |] and splicing $(...)
$ ghci -XTemplateHaskell src/lectures/Template-Haskell-based-printf.lhs
[1 of 1] Compiling Printf ( src/lectures/Template-Haskell-based-printf.lhs, interpreted
Ok, modules loaded: Printf.

*Printf> $(printf "%s is a string") "Hello"
"Hello is a string"
*Printf> $(printf "%s is a string %g is an int") "Hello" 10
"Hello is a string 10 is an int"
*Printf>
19.4 Exercise
1. Write a function that will optimise the format instructions by merging
consecutive literal strings. Is there any point in optimising the format
instructions?
2. Rewrite the printf module to not use any template haskell itself.

Lecture 20
Concurrent programming in
Haskell
Haskell has a very good support for concurrent programming. In this lecture
we see a very basic introduction to concurrent programming.
20.1 Threads
The Haskell runtime implements threads which are really lightweight. This
means that on a decent machine you can open say 10K threads and still be have
decent performance. This makes Haskell a great platform to implement high
connectivity servers like http/ftp servers etc.
However you need to be carefull about FFI calls to C code. If one of your
thread makes a call to a C function that blocks then all the threads will block.
Sometimes the call to C functions might be indirect, you might use a library
that uses Haskells FFI to talk to an already implemented C library. Then you
can be in trouble if you are not careful.
One creates threads in Haskell using the forkIO :: IO () -> IO ThreadId
function. The threads created using forkIO are really local to you ghc process
and will not be visible in the rest of the OS. There is a similar forkOS function
that creates an OS level thread. As long as the code uses pure haskell functions
forkIO is all that you need.
We give here a basic function that forks a lot of worker process. The module to
import is Control.Concurrent.
> import Control.Concurrent -- Concurrency related module
71

72 LECTURE 20. CONCURRENT PROGRAMMING IN HASKELL
> import System.Environment -- For command line args in main
This is the worker process. In real life program this is were stuff happen.
> worker :: Int -> IO ()
> worker inp = do tId <- myThreadId
> let say x = putStrLn (show tId ++ ": " ++ x)
> in do say ("My input is " ++ show inp)
> say "Oh no I am dying."
>
Here is where the process is created. Note that forkIO . worker takes as input
an integer and runs the worker action on it in a seperate thread
> runThreads :: [Int] -> IO ()
> runThreads = sequence_ . map (forkIO . worker)
>
And finally this is the main function where all the command line arguments are
parsed and things done.
> main = do args <- getArgs
> case args of
> [a] -> let nthreads = read a
> in runThreads [1..nthreads]
> _ -> putStrLn "Bad arguments"
>
20.2 Daemonic thread termination
There is a big bug in the code you just saw. All threads are terminated as soon
as the main thread terminates. For real world applications one wants main
thread to hang around till the others are done. We will see one way to handle
this in the next lecture.
Last modified on Monday (31 October 2011 10:14:13 UTC)

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