Introduction to Higher Order (Functional Programming) (Python) part 2 John R. Woodward

Higher Order Function A higher order function is a function which - take another function as input Or Returns a function as output.

What does the following print foo = [2, 18, 9, 22, 17, 24, 8, 12, 27] print filter(lambda x: x % 3 == 0, foo) print map(lambda x: x * 2 + 10, foo) print reduce(lambda x, y: x + y, foo)

output [18, 9, 24, 12, 27] [14, 46, 28, 54, 44, 58, 26, 34, 64] 139

What does the following print foo = [2, 18, 9, 22, 17, 24, 8, 12, 27] print type(filter(lambda x: x % 3 == 0, foo)) print type(map(lambda x: x * 2 + 10, foo)) print type(reduce(lambda x, y: x + y, foo))

output <type 'list'> <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 print filter(f1, range(2, 25)) print filter(f2, range(2, 25)) 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 divisible by 2 or 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 list1 = [0, 1, 0.0, 1.0, True, False, "True", "False", "", None, [True], [False]] #answer [1, 1.0, True, 'True', 'False', [True], [False]]

Reduce – try this reduce( (lambda x, y: x + y), [1,2,3,4] )

answers reduce( (lambda x, y: x + y), [1, 2, 3, 4] ) 10 -8 24

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 Map, reduce and filter all take a function as an argument. What type is the function in each case Map, Reduce, Filter

Data Types of Function Taken. Map: takes an argument of type T and returns a type S. It returns a list of S, denoted [S] Reduce: takes two arguments of type T, and returns an argument of type T Filter: take an argument of type T and returns a Boolean (True/False). It returns a list of T, denoted [T]

Partial Application Motivating example Addition (+) takes two arguments (arg1 + arg2) What if we only supply one argument? We cannot compute e.g. (1+) But it is still a function of one argument (1+ could be called what???)

Visualization ??? 3 ADD ADD 1 ??? 1 2 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 Haskell is a pure functional language referential transparency - the evaluation of an expression does not depend on context. 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. In the 2nd world war, Richard Feynman was in charge of making calculation for the atomic bomb project. These calculations were done by humans working in parallel. e.g. calculate exponential

Functional Programming Programming paradigm? Object oriented, imperative Immutable state (referential transparency) Higher order functions, partial functions, anonymous functions (lambda) Map, filter, reduce (fold) List data structures and recursion feature heavily Type inference (type signatures) Lazy and eager evaluation, list comprehension

State & Referential Transparency In maths, a function depends ONLY ON ITS INPUTS e.g. root(4) = +/-2 This is a “stateless function” A stateful function (method, procedure) stores some data which changes i.e. mutable data E.g. a “function” which returns the last argument it was given F(4) = -1, F(55) = 4, F(3) = 55, F(65) = 55, (side effects, e.g. accessing a file) 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 We can calculate the primes by first build a list of non-prime numbers, 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 You can nest list comprehensions (just like you can do with other python commands). In this case just substitute the “noprimes” variable with the python code. I would not recommend this, as it is harder to read 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] ['LAZY', 'lazy', 4] ['DOG', 'dog', 3]

Output 2 <type 'list'> 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'>

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

More examples vec = [-4, -2, 0, 2, 4] print [x*2 for x in vec] print [x for x in vec if x >= 0] print [abs(x) for x in vec]

More examples vec = [-4, -2, 0, 2, 4] print [x*2 for x in vec] print [x for x in vec if x >= 0] print [abs(x) for x in vec] [-8, -4, 0, 4, 8] [0, 2, 4] [4, 2, 0, 2, 4]

More examples fruits = [' banana', ' loganberry ', 'passion fruit '] 4. print [fruit.strip() for fruit in fruits] 5. print [(x, x**2) for x in range(6)]

More examples fruits = [' banana', ' loganberry ', 'passion fruit '] 4. print [fruit.strip() for fruit in fruits] 5. print [(x, x**2) for x in range(6)] ['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)