Introduction to Higher Order (Functional Programming) (Python)

John R. Woodward

Can Your Programming Language Do This? Motivating Example

• http://www.joelonsoftware.com/items/2006/08/01.html

You only need to write code for the “bit that changes”

What is repeated here? What is the abstraction?

The Abstraction

• We are

• - getting an “object” (lobster/chicken)

• - repeating an action twice

– Put in pot, put in pot

– Boom boom, boom boom.

• The first action is with the object.

• The second action is with another object (water, coconut)

We repeat the 2nd action TWICE

Higher Order Functions (HOF)

1. In most programming languages we pass around integers, Booleans, strings, as argument to function and return types

2. For example Boolean isPrime(Integer n)

3. Takes as input an integer and returns output true/false depending on if it is prime or not.

4. With functional languages we can pass around functions. Type signatures

Higher Order Function

• A higher order function is a function which

• - take another function as input

Or

• Returns a function as output.

Examples of higher order functions

1. At college you had to differentiate y=mx+c

2. Also integration (returns a function).

3. In physics…law of refraction/reflection.

4. Fermat's principle states that light takes a path that (locally) minimizes the optical length between its endpoints. (Lagrangian Mechanics)

5. Mechanics…hanging chain (a ball rolls down a slope – a chain takes on a shape).

6. Droplets minimize surface area, aerodynamics (wing shape).

7. CALCULUS OF VARIATION

Example: A linear function

def linear(gradient, intercept):

def result(x):

return gradient*x + intercept

return result #not result(x)

myLinearFunction = linear(4,3)

#makes a function 4*x+3

print myLinearFunction(8)

#evaluate the function at point 8

print linear(4,3)(8)

#or do directly

f(x) and f

1. In maths f and f(x) are different.

2. f is a function!

3.f(x) is the value of f at value x

4. Never say “the function f(x)”

5.Say - The function f that takes a variable x (i.e. f takes var x)

6. Or – the function f at the point x (i.e. f given x has the value ??)

Inputs, outputs and functions

• Input x, to a function f, outputs f(x)

• f is a “process”

1. x=5,

2. f(x) = x+2,

3. f(5)=7 5 7

f

• Typically inputs/outputs are basic data types (int, float, double, Boolean, char, String)

Maximum of two numbers def max(x,y):

if (x>y):

return x

else:

return y

#what is the input and output data-type

e.g. max(2,5)=???

2

5

?

Maximum of two functions def maxVal(f, g, x):

return max(f(x), g(x))

#what is the input and output data-type

example

def f1(x): return x+2

def f2(x): return 6

maxVal(f1,f2,3)=?

f2

f1

3

?

maxVal

Returning the Function

def maxFun(f, g):

def maximumFunction(x):

return max(f(x), g(x))

return maximumFunction

def f1(x): return x+2

def f2(x): return 6

maxFun(f1,f2)=?

f1

f2

?

maxFun

Returning the Function

def maxFun(f, g):

def maximumFunction(x):

return max(f(x), g(x))

return maximumFunction

def f1(x): return x+2

def f2(x): return 6

maxFun(f1,f2)=?

f1

f2

?

maxFun

Summation 1/2

Could you write this code for f(x) = 2x ???

How could you write this code for f(x) in general ???

1. In maths (a little like a for loop) 𝑓(𝑖) = 𝑓(1) + 𝑓(2) + ⋯+ 𝑓(10)𝑖=10𝑖=1

2. ∑ takes 3 arguments, and upper and lower bound and a function to sum over.

3. E.g. 2 ∗ 𝑖𝑖=10𝑖=1 = 2 + 4 +⋯+ 20 =

Summation 2/2

1. In maths 𝑓(𝑖) = 𝑓(1) + 𝑓(2) + ⋯+ 𝑓(10)𝑖=10𝑖=1

2. ∑ takes 3 arguments, and upper and lower bound and a function to sum over.

3. E.g. 2 ∗ 𝑖𝑖=10𝑖=1 = 2 + 4 +⋯+ 20 =

def sum(f, a, b):

total = 0

for i in range(a, b+1):

total += f(i)

return total

Product Symbol in Maths

1. In mathematics we often use the product

symbol . 𝑓(𝑖)𝑖=3𝑖=1 means f(1)*f(2)*f(3).

2. You can do this for the labs

3. Hint: is its almost “cut and paste”, but you need to think a little.

Your turn…in the lab

1. Write a function f which returns a function which is the minimum of functions f1, f2

2. i.e. f(x)=min {f1(x) and f2(x)}

3. Write a function which returns a function which is the average of two functions

4. i.e. f(x) = {f1(x) + f2(x)}/2

5. Can you generalize this to a list of functions?

1 Simple example: Higher order Function

• #Takes a function f and value x and evaluates f(x), returning the result.

def apply(f, x):

return f(x)

• #f is a function, x is input to f

2 Composition as HOF

def compositionValue(f, g, x):

return f(g(x))

1. This take two functions f and g.

2. And a value x

3. And calculates the values f(g(x))

4. What restrictions are there?

f

g

x

f(g(x))

g(x)

3 Returning a Function fg def compositionFunction(f, g): def result(x): return f(g(x)) return result • myFunction = compositionFunction(timesThree, timesTwo)

• print myFunction(4) • print apply(compositionFunction(timesThree, timesTwo) , 4)

• print (lambda x, y :compositionFunction(x,y)) (timesThree, timesTwo) (4)

Variable as Strings, and “literal strings”

name = "john"

print name

#Or we can just print the name directly

print "john"

1. If we only use “John” once – we could use a string literal (a one-off use).

2. If we use “John” multiple times – define a variable – name – and use that.

Anonymous Functions (lambda)

def f (x): return x**2

print f(8)

#Or we can do

g = lambda x: x**2

print g(8)

#Or just do it directly

print (lambda x: x+2) (4)

y = (lambda x: x*2) (4)

print y # the value is stored in y

What do the following print

def inc(x): return x+1

1. print inc(5) 2. print type(inc) 3. print type(inc(5)) 4. print type("inc(5)") 5. print type(eval ("inc(5)" )) 6. print type((lambda x: x*2)) 7. print type((lambda x: x*2) (4))

And the types are…

1. 6

2. <type 'function'>

3. <type 'int'>

4. <type 'str'>

5. <type 'int'>

6. <type 'function'>

7. <type 'int'>

What does the following do 1

def f(a, b): return a+b

def inc(x): return x+1

def double(x): return x*2

What does the following do 2

def f(a, b): return a+b

def inc(x): return x+1

def double(x): return x*2

1. print f(1,2)

2. print f("cat","dog")

3. print f(inc, double)

output

1. 3 2. catdog 3. Traceback (most recent call last): File "C:\Users\jrw\workspace\Python1\addition1.py", line 6, in <module> • print f(inc, double) • File

"C:\Users\jrw\workspace\Python1\addition1.py", line 1, in f

• def f(a, b): return a+b • TypeError: unsupported operand type(s) for +:

'function' and 'function'

What does the following print

foo = [2, 18, 9, 22, 17, 24, 8, 12, 27]

1. print filter(lambda x: x % 3 == 0, foo)

2. print map(lambda x: x * 2 + 10, foo)

3. print reduce(lambda x, y: x + y, foo)

output

1. [18, 9, 24, 12, 27]

2. [14, 46, 28, 54, 44, 58, 26, 34, 64]

3. 139

What does the following print

foo = [2, 18, 9, 22, 17, 24, 8, 12, 27]

1. print type(filter(lambda x: x % 3 == 0, foo))

2. print type(map(lambda x: x * 2 + 10, foo))

3. print type(reduce(lambda x, y: x + y, foo))

output

1. <type 'list'>

2. <type 'list'>

3. <type 'int'>

Lambda – filter, map, reduce

foo = [2, 18, 9, 22, 17, 24, 8, 12, 27]

print filter(lambda x: x % 3 == 0, foo)

#[18, 9, 24, 12, 27]

print map(lambda x: x * 2 + 10, foo)

#[14, 46, 28, 54, 44, 58, 26, 34, 64]

print reduce(lambda x, y: x + y, foo)

#139

#(more detail in a few slides)

Lists

• Functional programming uses LISTs as its primary data structure. E.g. [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]

• listNumbers = [1,2,3] #[1,2,3]

• listNumbers.append(4)#[1, 2, 3, 4]

• listNumbers.insert(2, 55)#[1, 2, 55, 3, 4]

• listNumbers.remove(55)#[1, 2, 3, 4]

• listNumbers.index(4)#3

• listNumbers.count(2)#1

Map – (a transformation)

• Map takes a function and a list and applies the function to each elements in the list

• def cube(x): return x*x*x

• print map(cube, range(1, 11))

• print map(lambda x :x*x*x, range(1, 11))

• print map(lambda x :x*x*x, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10])

• # [1, 8, 27, 64, 125, 216, 343, 512, 729, 1000]

• #what is the type signature

filter

• Filter takes a function (what type???) and a list, and returns items which pass the test

• def f1(x): return x % 2 != 0

• def f2(x): return x % 3 != 0

• def f3(x): return x % 2 != 0 and x % 3 != 0

1. print filter(f1, range(2, 25))

2. print filter(f2, range(2, 25))

3. print filter(f3, range(2, 25))

filter 2

• def f1(x): return x % 2 != 0 • def f2(x): return x % 3 != 0 • def f3(x): return x % 2 != 0 and x % 3 != 0

• print filter(f1, range(2, 25)) • # [3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23]#odd • print filter(f2, range(2, 25)) • # [2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23] • #not divisible by 3 • print filter(f3, range(2, 25)) • # [5, 7, 11, 13, 17, 19, 23]#not divisiable by 2 and 3

A list can contain different datatypes

• list1 = [0, 1, 0.0, 1.0, True, False, "True", "False", "", None, [True], [False]]

• def isTrue(x):

• return x

• print filter(isTrue, list1)

output

• [1, 1.0, True, 'True', 'False', [True], [False]]

• list1 = [0, 1, 0.0, 1.0, True, False, "True", "False", "", None, [True], [False]]

Reduce – try this

• reduce( (lambda x, y: x + y), [1,2,3,4] )

• reduce( (lambda x, y: x - y), [1,2,3,4] )

• reduce( (lambda x, y: x * y), [1,2,3,4] )

• reduce( (lambda x, y: x / y), [1,2,3,4] )

answers

• reduce( (lambda x, y: x + y), [1, 2, 3, 4] ) • reduce( (lambda x, y: x - y), [1, 2, 3, 4] ) • reduce( (lambda x, y: x * y), [1, 2, 3, 4] ) • reduce( (lambda x, y: x / y), [1, 2, 3, 4] ) • 10 • -8 • 24 • 0

The last one

• print reduce( (lambda x, y: x / y), [1.0, 2.0, 3.0, 4.0] )

• ??? answer

The last one

• print reduce( (lambda x, y: x / y), [1.0, 2.0, 3.0, 4.0] )

• 0.0416666666667

• # what is the type signature?

Map, Reduce, Filter

1. Map, reduce and filter all take a function as an argument.

2. What type is the function in each case

3. Map,

4. Reduce,

5. Filter

Data Types of Function Taken.

1. Map: takes an argument of type T and returns a type S. It returns a list of S, denoted [S]

2. Reduce: takes two arguments of type T, and returns an argument of type T

3. Filter: take an argument of type T and returns a Boolean (True/False). It returns a list of T, denoted [T]

Partial Application

1. Motivating example

2. Addition (+) takes two arguments

3. (arg1 + arg2)

4. What if we only supply one argument?

5. We cannot compute e.g. (1+)

6. But it is still a function of one argument

7. (1+ could be called what???)

Visualization

ADD

3

2 1

ADD

3

??? 1

This is how we normally think of the add function

Inc as partial add

• #add (2 args), inc(x)=add(1,x)

• #inc is a special case of add

• from functools import partial

• def add(a,b):

• return a+b

• inc = partial(add, 1)

• print inc(4)

Inc defined with add

• #add takes 2 arguments

• def add(x,y):

• return x+y

• #we can define a new function by hardcoding one variable

• def inc(x):

• return add(1,x)

• print inc(88)

double as partial mul

• #mul(2 args), double(x)=mul(2,x)

• #double is a special case of mul

• from functools import partial

• def mul(a,b):

• return a*b

• double = partial(mul, 2)

• print double(10)

Functions as special cases

• #square and cube are special cases of the power function

• def power(base, exponent):

• return base ** exponent

• def square(base):

• return power(base, 2)

• def cube(base):

• return power(base, 3)

Equivalent Code

• from functools import partial

• square = partial(power, exponent=2)

• cube = partial(power, exponent=3)

referential transparency

1. Haskell is a pure functional language referential transparency - the evaluation of an expression does not depend on context.

2. The value of an expression can be evaluated in any order (all sequences that terminate return the same value) (1+2)+(3+4) we could reduce this in any order.

3. In the 2nd world war, Richard Feynman was in charge of making calculation for the atomic bomb project.

4. These calculations were done by humans working in parallel. e.g. calculate exponential

Functional Programming

1. Programming paradigm? Object oriented, imperative

2. Immutable state (referential transparency)

3. Higher order functions, partial functions, anonymous functions (lambda)

4. Map, filter, reduce (fold)

5. List data structures and recursion feature heavily

6. Type inference (type signatures)

7. Lazy and eager evaluation, list comprehension

State & Referential Transparency

1. In maths, a function depends ONLY ON ITS INPUTS e.g. root(4) = +/-2

2. This is a “stateless function” 3. A stateful function (method, procedure) stores some

data which changes i.e. mutable data 4. E.g. a “function” which returns the last argument it

was given 5. F(4) = -1, F(55) = 4, F(3) = 55, F(65) = 55, 6. (side effects, e.g. accessing a file) 7. Harder to debug stateful function than stateless

(why).

List Comprehensions

• List comprehensions provide a concise way to create lists.

• To make new lists, where each element is the result of some operation applied to each member of another sequence.

List Comprehensions

• The following are common ways to describe lists (or sets, or tuples, or vectors) in mathematics.

S = {x² : x in {0 ... 9}} V = (1, 2, 4, 8, ..., 2¹²) M = {x | x in S and x even}

Create a list of squares

For example, assume we want to create a

list of squares, like:

• squares = []

• for x in range(10):

• squares.append(x**2)

• print squares

• #[0, 1, 4, 9, 16, 25, 36, 49, 64, 81]

As a list comprehension

We can obtain the same result with:

• squares = [x**2 for x in range(10)]

This is also equivalent to

• squares = map(lambda x: x**2, range(10)),

However the list comprehension is more

concise and readable.

(choose the way that works for you, but do try different techniques)

In Python

• S = [x**2 for x in range(10)]

• V = [2**i for i in range(13)]

• M = [x for x in S if x % 2 == 0]

• print S;

• print V;

• print M

In Python

• S = [x**2 for x in range(10)]

• V = [2**i for i in range(13)]

• M = [x for x in S if x % 2 == 0]

• print S;

• print V;

• print M

[0, 1, 4, 9, 16, 25, 36, 49, 64, 81] [1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096] [0, 4, 16, 36, 64]

Calculating Primes

1. We can calculate the primes by

2. first build a list of non-prime numbers,

3. then use another list comprehension to get the "inverse" of the list,

Calculating Primes

noprimes = [j for i in range(2, 8) for j in range(i*2, 50, i)]

print noprimes

primes = [x for x in range(2, 50) if x not in noprimes]

print primes

output

• noprimes = [j for i in range(2, 8) for j in range(i*2, 50, i)]

• print noprimes • #[4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28,

30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 10, 15, 20, 25, 30, 35, 40, 45, 12, 18, 24, 30, 36, 42, 48, 14, 21, 28, 35, 42, 49]

• primes = [x for x in range(2, 50) if x not in noprimes]

• print primes • #[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41,

43, 47]

Nested List Comprehensions

1. You can nest list comprehensions (just like you can do with other python commands).

2. In this case just substitute the “noprimes” variable with the python code.

3. I would not recommend this, as it is harder to read

4. It is also harder to debug (where would you insert a print statement)

Nested List Comprehensions

• primes = [x for x in range(2, 50) if x not in [j for i in range(2, 8) for j in range(i*2, 50, i)]]

• print primes

• #[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47]

List Comprehension with Strings

words = 'The quick brown fox jumps over the lazy dog'.split()

print words

stuff = [[w.upper(), w.lower(), len(w)] for w in words]

for i in stuff:

print i

List Comprehension with Strings

words = 'The quick brown fox jumps over the lazy dog'.split()

print words

stuff = [[w.upper(), w.lower(), len(w)] for w in words]

for i in stuff:

print i

['THE', 'the', 3] ['QUICK', 'quick', 5] ['BROWN', 'brown', 5] ['FOX', 'fox', 3] ['JUMPS', 'jumps', 5] ['OVER', 'over', 4] ['THE', 'the', 3] ['LAZY', 'lazy', 4] ['DOG', 'dog', 3]

Output 2

• words = 'The quick brown fox jumps over the lazy dog'.split()

• stuff = [[w.upper(), w.lower(), len(w)] for w in words]

• for i in stuff:

• print i

• print type(words)

• print type(stuff)

<type 'list'> <type 'list'>

3 EXAMPLES

print [y for y in [3,1,4]]

print [(x, y) for x in [1,2,3] for y in [3,1,4]]

print [(x, y) for x in [1,2,3] for y in [3,1,4] if x != y]

OUTPUT

• [3, 1, 4]

• [(1, 3), (1, 1), (1, 4), (2, 3), (2, 1), (2, 4), (3, 3), (3, 1), (3, 4)]

• [(1, 3), (1, 4), (2, 3), (2, 1), (2, 4), (3, 1), (3, 4)]

Written as nested for loops

• [(x, y) for x in [1,2,3] for y in [3,1,4] if x != y]

Is equivalent to

• combs = []

• for x in [1,2,3]:

• for y in [3,1,4]:

• if x != y:

• combs.append((x, y))

the order of the for and if statements is the same in both

(choose the way that works for you, but do try different techniques)

More examples

vec = [-4, -2, 0, 2, 4]

1. print [x*2 for x in vec]

2. print [x for x in vec if x >= 0]

3. print [abs(x) for x in vec]

• fruits = [' banana', ' loganberry ', 'passion fruit ']

4. print [fruit.strip() for fruit in fruits]

5. print [(x, x**2) for x in range(6)]

output

• [-8, -4, 0, 4, 8]

• [0, 2, 4]

• [4, 2, 0, 2, 4]

• ['banana', 'loganberry', 'passion fruit']

• [(0, 0), (1, 1), (2, 4), (3, 9), (4, 16), (5, 25)]

examples

matrix = [ [1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12],] print [row[i] for row in matrix for i in range(4)] print [[row[i]] for row in matrix for i in range(4)] print [[row[i] for row in matrix] for i in range(4)]

output

• [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]

• [[1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12]]

• [[1, 5, 9], [2, 6, 10], [3, 7, 11], [4, 8, 12]]

Equivalent to …

transposed = []

for i in range(4):

# the following 3 lines implement the nested listcomp

transposed_row = []

for row in matrix:

transposed_row.append(row[i])

transposed.append(transposed_row)