This is a series of notes taken from the Coursera course: “Programming Language” by professor Dan Grossman. I plan to chain these notes up with the catchy terms that I learnt from the course.

In programming language, higher-order functions and closures are often two closely related concepts that needs to be discussed together.

Funcion as first class objects

In Python, functions are first class objects, in other words, they are expressions that can be evaluated, stored and passed as function arguments, just like other primitive types such as int, float etc. When we start passing a function to another function as an argument, we call the argument as higher-order function.

def apply(x, f):
    return f(x)

def concat_myname(s):
    return s + 'Michael'
apply('Hello', concat_myname)

def double(d):
    return d * d
apply(3, double)

In this somewhat naive example, we see that function as first class objects made function more generic than before - that we can not only define routines dependent on primitive types, but also dependent on function behaviors. Frequent use cases are map, reduce and filter . Their behaviors are:

def mymap(l, f):
    if len(l) == 0:
        return []
        return [f(l[0])] + mymap(l[1:], f)

def myreduce(l, f, acc):
    if len(l) == 0:
        return acc
        return myreduce(l[1:], f, f(acc, l[0]))

def myfilter(l, f):
    if len(l) == 0:
        return []
        return ([l[0]] if f(l[0]) else []) + myfilter(l[1:], f)

Using these functions can be greately simplified with lambda keyword, which creates an anonymous function upon evaluation.

l = [1, 2, 3, 4]
mymap(l, lambda x: x*2)             # doubles list
myreduce(l, lambda a,b: a + b, 0)   # summing list
myfilter(l, lambda x: x % 2 == 0)   # drop non-even numbers

Lexical scope and closure

With higher order functions, behaviors can be passed around like variables. However, as they gets passed around, the environment where the function expresion is evaluated also changes. Take at look at the example below:

class Dog:
    def __init__(self, name): = name
    def bark(self):
        print(f'Bark! My name is {}')

a_dog = Dog('Charlie')

# Say we want to adopt the Charlie dog
def adopt_charlie():

# However, the pet center has other dogs.
# Specifically, variable `a_dog` is shadowed
# by another dog named 'Milo'
def in_pet_care_center(my_action):    
    a_dog = Dog('Milo')
    b_dog = Dog('Oscar')
    # When we execute our action, which dog will we adopt?

# Bark! My name is Charlie

Yay! We still have Charlie! But let us look closer what happened there. Upon defining adopt_charlie, a_dog is bound to “Charlie”. Inside care center, a_dog is shadowed by another Dog instance. When my_action is evaluated, the current environment of evaluation has a_dog bound to “Milo”. However, the actual evaluation of my_action used the environment at which the function is defined. Evaluating a funtion under the environment where it is defined, is called Lexical scope rules. Most modern programming languages follow this rule.

It has many benefits, one of them is that one can easily understands the behavior of a function, only required to read the function body and all the codes before it, without having to worry about where it is called.

As an implementation detail of lexical scope, when one defines a function, the compiler/interpreter will need to save not only the function body, but all (free) variable bindings used by the function. Putting the function body and the bindings together creates a closure. Thus, when we defines a function inside our code, we are actually adding a new binding of the function name to its closure into the environment.

In languages that does not support higher order functions such as C, we could simulate the behavior with function pointers, except, we need to also pass the bindings as an extra argument. For example:

#include "stdio.h"
/* A simplified environment structure */
struct Env
    char **names;
    int **values;

/* Same apply as above */
void apply(void* x, void (*f)(void*, struct Env*), struct Env* f_env)
    f(x, f_env);

/* Can retrieve free variables from my_env here */
void power_of_2(void *x, struct Env* my_env)
    int *i_x = (int*)x;
    *i_x = (*i_x) * (*i_x);

int main()
    struct Env some_env; // Empty env upon definition
    int n = 4;
    apply((void*)&n, power_of_2, &some_env);
    printf("%d\n", n); // 16
    return 0;