# Missouri State University Programming Worksheet

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1. Write a composition function that takes 2 functions which have the following signatures:
1. f = fn : ‘b -> ‘c option
2. g = fn : ‘a -> ‘b option
3. The composition is done in such a way that
1. If either f or g return NONE, the entire function returns NONE
2. if g returns SOME the returned value becomes input to f.
3. If the value returned by is SOME then is the return value of the
whole function.
2. Write a do_until function that takes a function (f) and a predicate (p) (a function that takes a
value and returns a boolean). It will apply the f repeatedly on the input variable until the
predicate applied to the value returned by f becomes false. eg.:
do_until (fn x => x + 1) (fn x => x 100 (as per predicate function)
3. Use the above function to implement factorial.
4. Write a map function for pairs; each value in a pair of type (‘a * ‘a) gets mapped to a pair of
type (‘b * ‘b) eg. If a doubles function used passed to the map function (2, 3) will return (4,
6).
5. Use builtin List.foldl, String.size functions to nd the longest string in a list of strings. If
there are more than 1 strings with a max length, return the string that is closer to the
beginning of the list. Eg: [“a”, “abc”, “defg”, “hi”, “jkl”, “mnop”, “qr”] should return
“defg” (not “mnop”) as it occurs earlier in the list.
6. Write a second version of the function above which returns the string that is closer to the
end of the list in case of a tie. Eg: [“a”, “abc”, “defg”, “hi”, “jkl”, “mnop”, “qr”] should return
“mnop” (not “defg”)
7. Use MLs o operator to derive a function that reverses a string and capitalizes the string.
Passing “Hello” returns “olleh”. Use builtin functions from the SML library.
Functional Programming
In languages like SML, functions are considered rst class citizens. This means functions can
be treated like any other variable. They can be passed as parameters to functions, returned by
functions. They can be elements in a tuple/record, computed, stored, etc. Functions that
accept functions as arguments and returns functions are known as higher order functions.
Functions can also refer to variables de ned outside it self, but the variables are de ned within
the scope of the function. This type of functions are called function closures. Functional
programming a style which involves the following:
1.
2.
3.
4.
5.
6.
Variables are immutable
Higher order functions and function closures
Recursive functions
Programming which is closer to math (i = i + 1 would not make sense in math)
Lazy evaluation
etc…
A functional programming language is then a language in which functional programming is
natural. Any language that enables immutable variables, higher order functions and function
closures can be considered “functional”. Most modern programming languages like Scala,
Java, Python, etc., support functional programming along with other paradigms like object
oriented programming.
Functions as Arguments
Consider the situation where a bunch of functions need to be implemented. The functions are
very similar in the sense that they all take the same input parameters, perform a similar
sequence of operations but only di er in how the return value is computed. Take a simple
example where the salary increment needs to be compiled: smallIncrement, mediumIncrement,
largeIncrement. See the initial implementations:
fun smallIncrement(ls: real list) =
if null ls then []
else
((hd ls) * 1.1) :: smallIncrement(tl ls);
fun mediumIncrement(ls: real list) =
if null ls then []
else
((hd ls) * Math.log10(hd ls)) :: mediumIncrement(tl ls);
fun largeIncrement(ls: real list) =
if null ls then []
else
(hd ls * hd ls) :: largeIncrement(tl ls);
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All the above functions accept a list of real values. They all check if the list is empty and
returns an empty list if true. If the list passed in is not empty, the increment for the rst element
in the list is calculated and prepended to the list of increments for the tail of the list computed
recursively. There is a lot of duplication of code. The caller is limited to 3 algorithms to calculate
the increment. If a new algorithms are needed by the caller, then additional functions will need
to be implemented, so this is not very scalable solution. So what if we could write a function
that not only takes the list but also an additional parameter that passes in the calculation to be
done? This is where the higher order functions come in. The above functions can be re-written
as:
fun increment(incrementer, ls) =
if null ls then []
else incrementer(hd ls) :: increment(incrementer, tl ls);
fun smallIncrementer(x: real) = x * 1.1;
fun mediumIncrementer(x: real) = x * Math.log10(x);
fun largeIncrementer(x: real) = x * x;
In the above implementation the incrementing calculation is captured in function
smallIncrementer, mediumIncrementer and largeIncrementer. The actual
increment function accepts an “incrementer” in addition to the actual list of salaries. Based on
the type of increment needed, caller can pass the appropriate incremented as below:
increment (smallIncrementer, [3000.0, 1000.0, 2200.0]);
increment (mediumIncrementer, [3000.0, 1000.0, 2200.0]);
increment (largeIncrementer, [3000.0, 1000.0, 2200.0]);
If needed aliases can be created for the above as below:
fun smallIncrement2(ls: real list) = increment(smallIncrementer, ls);
fun mediumIncrement2(ls: real list) = increment(mediumIncrementer,
ls);
fun largeIncrement2(ls: real list) = increment(largeIncrementer, ls);
The power in the above implementations is that the caller does not need to be limited to the
above incrementers. The caller can pass in custom incrementers to based on their need. Note
that the power in these implementations comes from being able to write a function that
accepts another function as a parameter (increment function) aka Higher Order functions.
And functions that can reference bindings that are outside themselves as in (smallIncrement2,
etc referencing smallIncrementer in their body (function closures!)
Polymorphic Types:
Try out the above examples in SML and notice the types associated with each function.
Particularly the type of function increment will be shown as fn : (‘a -> ‘b) * ‘a list
-> ‘b list. The function is pretty generic and is not even aware that it’s being used to do
calculations on real numbers. So in theory it would be passed a list of strings and return a list
of numbers! We don’t even have to write an “incrementer” we can just pass in the builtin
Real.fromString as below:
increment (Real.fromString, [“3000.0”, “1000.0”, “2200.0”]);
which will return a real option list:
[SOME 3000.0,SOME 1000.0,SOME 2200.0]
Such functions are called polymorphic. Higher order functions need not be polymorphic and
not all polymorphic functions are higher order functions! The function below (used for
illustrative purpose) is higher order but not polymorphic. The REPL will show that the type for
this function is fn : (int -> int) * int -> int.
fun do_something(f, i) =
if i = 0 then 0
else 1 + do_something(f, f i);
Anonymous Functions
In the above example, the helper functions smallIncrementer, mediumIncrementer,
and largeIncrementer do not need to be written as stand alone functions. Instead
anonymous functions can be used to implement smallIncrement2, mediumIncrement2 and
largeIncrement2 as below:
fun smallIncrement3(ls: real list) = increment(fn x => x * 1.1, ls);
fun mediumIncrement3(ls: real list) = increment(fn x => x *
Math.log10(x), ls);
fun largeIncrement3(ls: real list) = increment(fn x => x * x, ls);
Take smallIncrement3 further analyze the anonymous function. An anonymous function is a
function without a name, aka a lambda. In SML the syntax for declaring an anonymous
function is as:
fn () =>
e.g.:
val a = fn (i, j) => if i = 0 then 0 else i + j;
When using anonymous functions sometimes there is a tendency to unnecessarily use
anonymous function to wrap another function. As in:
increment (fn x => Real.fromString(x), [“3000.0”, “1000.0”, “2200.0″]);
While the above code will work as expected, there is no need to wrap call to
Real.fromString. Just call in the function from the builtin library as is. Functions are rst
class citizens in SML.
Maps and Filters
Map and lter are the most popular use cases for higher order functions. The increment
function above is essentially a map function. It essentially “maps” each value in a list to another
value and creates a new list of the mapped values. The map function can be re-written using
case expression as below:
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fun map(f, l) =
case l of
[] => []
| ls::ls’ => f ls :: map(f, ls’)
Note: It’s common convention to use a variable and it’s “prime” to represent head and tail of a
list in case expressions, as above with ls and ls’ (read as “ls prime”)
The above map function can be used to map a list of integers to their doubles and squares
using anonymous functions as below:
val doubles = map (fn x => x + x, [12, 31, 55, 16, 613]);
val squares = map (fn x => x * x, [12, 31, 55, 16, 613]);
You can also nest calls to the map function to apply multiple mappings:
val logs = map (Math.log10, map (
Real.fromInt,
[12, 31, 55, 16, 613]
))
In the code above the inner call to map function (in bold) converts the integers to real
numbers. The outer map in turn maps each real value to the logs of their numbers. As will be
seen later this can be made even more concise by using function composition, given below for
an initial look:
val logs = map (Math.log10 o Real.fromInt,
[12, 31, 55, 16, 613]
)
The other popular use of higher order functions is the lter function. A lter function is used to
select only elements in a list that meet a certain requirement. For instance there might be a
need to extract only even numbers from a list of integers. Given below is such a lter
implemented in SML:
fun filter(f, l) =
case l of
[] => []
| l::l’ => if f l then l :: filter(f, l’)
else filter(f, l’);
The type for the function lter above is fn : (‘a -> bool) * ‘a list -> ‘a list.
This function is polymorphic as we can see from the presence of the generic type ‘a above.
The rst argument is a function that take a value of type ‘a and returns a boolean value. The
second parameter is an ‘a list. and the return value is also an ‘a list. As this function is
generic, it can be used to lter lists of any type. Given below is an example where the lter
function is used to lter for even numbers:
val evens = filter (fn x => x mod 2 = 0, [12, 31, 55, 16, 613]);
The anonymous function contains the logic used for the lter. x mod 2 gives the remainder
from the integer division with 2. If the remainder is 0 then the number is even and hence is a
good check for evenness of the number. To show the generic nature of the lter function we
can lter strings using the same lter:
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val a_words = filter (fn s => String.isPrefix “a” s orelse
String.isPrefix “A” s,
[“Fox”, “Asterdam”, “Atlas”, “Box”, “Fall”, “Beetle”]);
Functions that return functions: If functions can accept functions as arguments then
functions can also return functions. Given below is a bit of a contrived example for the
illustration purpose:
exception InvalidArgument of real;
fun multiplier(r) =
if r 1.0 then
fn x => x * r
else
fn x => x / r;
The function accepts a real number. If the value of the real number is > 1.0 then the function
returns a function that accepts one parameter (x) and return the product of it’s own argument
and the argument that is passed to the wrapper function. The multiplier can be used as below:
val res1 = multiplier(2.0)(3.0); (* return value of 6.0 *)
val res2 = multiplier(0.2)(3.0); (* return value of 15.0 *)
Note: The map and lter functions are the most popular uses of higher order functions but
these can be used in any situation where part of the algorithm needs to be either passed into
the function or returned from the function.
Lexical Scope
Most of programming languages use what is known as “Lexical Scope”. To understand this,
consider the code below:
val x = 1
fun f y = x + y
val x = 2
val y = 3
val z = f (x + y)
What would be the value of z be? It might appear that the answer should be 7 as below
val z = f (x + y)
(* becomes f (2 + 3) -> 5 gets passed to function and in the
body of the function x is added to the argument hence
should be evaluated to 7, as x = 2 (?) *)
Contrary to what one might expect, the value assigned to z is 6. This is because though the
variable x de ned in line 1 is shadowed by the binding in line 3, the function f still keeps track
of the initial binding of x, and uses that binding when evaluating the body. So as far as the
body of the function f is concerned, the variable x is still bound to the value 1! This is the
essence of “Lexical Scope”. Formally it might be stated as:
The body of a function is evaluated in the environment where the function is de ned, not the
environment where the function is called.
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Most modern programming use this type of scoping, which is also called static scoping.
“Dynamic scoping” reverses this rule. Most modern languages shun “Dynamic scoping” as this
means that any function should account for all possible environments that this function can be
called from! In the code above the value returned by function f can’t be predicted, as binding of
x might be changed in any way before the function is called. Lexical scoping makes it easier to
reason about the function. We can say exactly that the function increments the value of y by 1
and returns the incremented value, because value of x is 1 for the function.
Environments and Closures
For the all the functions de ned so far, only the type of the function has been discussed. The
“value” of the functions is not discussed. The “value” of a function consists of the code and
also the environment where the function is de ned. These 2 aspects of the “value” of a
function is called a Closure or function closure.
So for the function f above:
1. The code part of the function is fn y => x + y and
2. The environment has a binding of x to the value 1.
Passing Closures:
Consider a situation where you want to use the leverage the lter function above to lter
numbers greater than a number speci ed by the caller, as below:
fun allGreaterThan (xs, n) = filter (fn x => x > n, xs)
The above function works because of lexical scoping. This is because though the variable n is
not present in the lter function, it gets passed as a closure via the anonymous function which
is bold above.
Fold
Another popular uses of higher functions is the fold function. This is also known as a reduce
function. The map/ lter/reduce functions go together in a lot of applications related to
processing data, where these functions can be used to clean data, do aggregations, etc before
passing on to other stages in the application. Given below is an implementation of the fold
function:
fun fold (f, acc, ls) =
case ls of
[] => acc
| ls::ls’ => fold(f, f(ls, acc), ls’);
The function above takes a function (f), and initial value (acc) that is built on during the process
of recursion, and the list to apply the fold on. The function f is used to combine the head value
with acc.
The above fold (or reduce) function can be used to implement a function that sums up all the
numbers in a list as below:
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fun sum(ls) = fold (fn (x, y) => x + y, 0, ls);
Calling the sum function as below gives the result 45:
val s = sum([1, 2, 3, 4, 5, 6, 7, 8, 9]);
Function Composition
Function composition is a way to combine 2 or more functions. Given two function f() and
g() and there is a need to call these in a nested manner, as in f(g()). Using composition it’s
possible to create a composite function that does this, as in:
fun compose (f, g) = fn x => f (g x)
SML provides a builtin in x operator o to do function composition. The code below:
fun sqrt_of_abs i = Math.sqrt(Real.fromInt (abs i))
can be re-written as below:
fun sqrt_of_abs i = (Math.sqrt o Real.fromInt o abs) i
The above can be written more concisely as:
val sqrt_of_abs = Math.sqrt o Real.fromInt o abs
There is another way to compose functions, which is using using pipeline operators. In SML it’s
possible to de ne our own in x operator, as below:
infix |>
fun x |> f = f x
Once the in x operator is de ned, sqrt_of_abs can be rede ned in a more intuitive way as
below.
fun sqrt_of_abs i = i |> abs |> Real.fromInt |> Math.sqrt;
This way of composing functions is more intuitive because it’s easier think of passing results of
one function to the next function and so on.
Creating fallback functions:
Consider a situation where 2 functions f, and g are used as below:
1. Call g with the argument passed
2. If it returned SOME value then return the value
3. If it returns NONE, then return the value returned by f
This may be implemented as below:
fun fallback(f, g) = fn x => case g x of
NONE => f x
| SOME y => y;
To illustrate the usage of the above fallback function, see the code below:
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fun sqrt_options x = if x > 0.0 then
SOME (Math.sqrt x)
else NONE;
val v1 = fallback (abs, sqrt_options) (2.0);
val v2 = fallback (abs, sqrt_options) (~2.0);
v1 will get the value of the square root of 2.0. But as negative numbers don’t have a square
root, the function falls back to returning the absolute value of number.
Function Currying and Partial Functions
The term currying is named after Haskell Curry who was a mathematician who studied
combinatory logic. Function currying comes from the idea of function that return functions.
SML accepts multiple arguments as a tuple. There is however another elegant way to
implement functions that accept multiple arguments not as tuples but multiple arguments. This
is implemented by implementing functions that accept only one parameter but returns a
function that accepts second parameter and so on. Look at the following 2 implementations of
a function that multiplies 2 numbers:
fun multiplier1 (i, j) = i * j;
fun multiplier2 i = fn j => i * j;
Both return the same result but the way they are called is di erent. The types for both the
functions are respectively given below:
fn : int * int -> int
fn : int -> int -> int
The code in bold highlights the di erence. The function multiplier2 does not result a numeric
result, but rather it returns an anonymous function which accepts the second parameter.
Because of this, they are called di erently as below:
val val1 = multiplier1 (3, 4);
val val2 = multiplier2 3 4;
SML provides syntactic sugar to make writing curried functions even more concise. The
multiplier can re-written as:
fun multiplier2 i j = i * j;
In SML both the ways of writing the multiplier2 are equivalent.
The advantage of the curried function is that they can be used to write partial functions, as
below:
val doubler = multiplier2 2;
val d = doubler 4;
The function doubler can be also written as:
fun doubler x = multiplier2 2 x;
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But this is considered as “Unnecessary function wrapping”. This style is not preferred.
The advantages of currying can be summarized as below:
1. The parameters passed to a function can be logically grouped
2. The secondary (partial) functions can be re-used to create specialized functions
3. Functions can be re-used as above, without using lambdas
Note: SML provide built in map and lter functions. They are List.map and List.filter.
They work just like the map and lter functions implemented above, but these use currying.
In most situations the function wrapping is not needed but sometimes SML might not be able
to determine the types of the input parameters. e.g.:
val pair_with_one = List.map (fn x => (x, 1));
The above function will give a warning but is unusable. The work around to this is either
1. Use wrapper functions! as below:
fun pair_with_one_fun xs = List.map (fn x => (x, 1)) xs;
2. Or specify the function type explicitly:
val pair_with_one: int list -> (int * int) list = List.map (fn x =>
(x, 1));
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The rst work around will keep the generic nature of the function. The second work around ties
the function to a speci c type of list.

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