lec9.tex

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\begin{document}

% Theory of functions and their correspondence to maths
% functions. Return values and None. print vs
% return. Parameters. What's the point of functions. Passing functions
% as parameters.

\begin{frame}
\frametitle{CS24420 \& MA25220 \& MT25220 \& MX35220 \& CSM0120}

\begin{center}
\begin{huge}
Lecture 9: Functions
\end{huge}
\bigskip

Amanda Clare (afc@aber.ac.uk)

\end{center}
\end{frame}

% Guido video on type hints


\begin{frame}[fragile]
\frametitle{Functions}
\begin{itemize}
\item Maths functions vs Python functions
\item Return values, None and print
\item Named parameters
\item Passing functions as arguments
\end{itemize}
\end{frame}


\begin{frame}[fragile]
\frametitle{Functions in maths}
Assume we have 2 sets, $A$ and $B$. For example: countries and capital cities
\bigskip

We could imagine all possible pairings of elements such that one
element in the pair is from $A$ and the other is from $B$. 
\bigskip

For example, if
$A = \{France, UK, Spain\}$ and $B = \{Paris, London, Madrid\}$ then
the set of all possible pairs is:
\begin{multline*}
\{ (France, London), (France, Paris), (France, Madrid), \\
(UK, London),
  (UK, Paris), (UK, Madrid), \\
(Spain, Paris), (Spain, London), (Spain,
  Madrid) \}
 \end{multline*}
\end{frame}


\begin{frame}[fragile]
\frametitle{Relations in maths}
The set of all possible pairs from $A$ and $B$ is known as the cartesian product of $A$
and $B$.

A subset of this (that is, some of the pairs but not necessarily all
of them) is called a {\em relation}.

For example: 
\begin{itemize} 
\item the capital cities of each country: $\{(France, Paris), (UK,
  London), (Spain, Madrid) \}$
\item popular holiday destinations: $\{ (France, Paris), (UK, Paris),
  (Spain, London) \}$
\item same number of letters in word: $\{ (France, London), (Spain,
  Paris) \}$
\item Continental flight destinations from UK: $\{ (UK, Paris), (UK, Madrid) \}$
\end{itemize}

\end{frame}

\begin{frame}[fragile]
\frametitle{Functions in maths}
\begin{itemize}
\item A {\em function} (in maths) is a relation where there's a single output for
any given input. \\
\item A single element of $B$ for every element of $A$ in the relation.
So we can pick any input element and know that an output
exists. \\
\item We also know it's the only output (there's no confusion). \\
\item We can think of a function as a mapping from elements in $A$
to elements in $B$.
\end{itemize}
\end{frame}


\begin{frame}[fragile]
\frametitle{Which of these are functions}

Which of these relations are functions?
 
\begin{itemize} 
\item the capital cities of each country: $\{(France, Paris), (UK,
  London), (Spain, Madrid) \}$
\item popular holiday destinations: $\{ (France, Paris), (UK, Paris),
  (Spain, London) \}$
\item same number of letters in word: $\{ (France, London), (Spain,
  Paris) \}$
\item Continental flight destinations from UK: $\{ (UK, Paris), (UK, Madrid) \}$
\end{itemize}
\end{frame}



\begin{frame}[fragile]
\frametitle{In Python}
\begin{code}
def capital_city(country):
   capital = None
   if country == "France":
      capital = "Paris"
   elif country == "UK":
      capital = "London"
   elif country == "Spain":
      capital = "Madrid"
   return capital
\end{code}
\end{frame}

\begin{frame}[fragile]
\frametitle{In Python}
\begin{code}
capital_dict = { 
  "France": "Paris", 
  "UK":     "London", 
  "Spain":  "Madrid" 
}

def capital_city(country):
   capital = None
   if country in capital_dict:
      capital = capital_dict[country]
   return capital
\end{code}
\end{frame}


\begin{frame}[fragile]
\frametitle{Return values}
When we return a value from a function, we can collect that value into
a variable:
\begin{code}
c = capital_city("UK")
print(c)

d = capital_city("Germany")
print(d) 
\end{code}
\end{frame}

\begin{frame}[fragile]
\frametitle{Functions in Python}
When talking about Python, we use the term ``function'' to refer to a block of
reusable code, defined with the \texttt{def} keyword. 

Some Python functions would not satisfy the mathematical definition of
a function. 
\end{frame}

\begin{frame}[fragile]
\frametitle{Return values and None}
Some functions in Python don't return a value. They are written
solely for their side effects (often printing to a screen or writing
to a file)
\begin{code}
def print_hashes(num):
   for i in range(num):
      print('#' * i)
\end{code}

This is still known as a {\em function} in Python, even though it seems
that it doesn't return a value. In fact it does actually return a
value: it returns the value \texttt{None}. 

\bigskip

N.B. Sometimes when we write a function just for its side effects rather than
its return value we might instead choose to return a boolean,
as an indicator of success or failure.
\end{frame}


\begin{frame}[fragile]
\frametitle{Returning multiple values}
We can return a tuple of values. For example:

\begin{itemize}
\item a pair of latitude and longitude
\item the longest element in a list, and its length
\item an RGB (red-green-blue) colour triple
\end{itemize}

\begin{code}
(lat, lon) = convert_postcode("SY23 3DB")
\end{code}
\end{frame}

\begin{frame}[fragile]
\frametitle{Returning multiple values}
We can also return a dictionary of values. For example:

\begin{code}
def convert_colour(hex_colour):
   red = int(hex_colour[:2], 16)
   green = int(hex_colour[2:4], 16)
   blue = int(hex_colour[4:], 16)
   return { "r": red, "g": green, "b": blue }

rgb = convert_colour("FFA583")
print(rgb["r"])
\end{code}
\end{frame}


\begin{frame}[fragile]
\frametitle{Partial functions}
A partial function is not defined for all elements of the input set
$A$, only some
of them. For some elements there is no output. 

From our previous example of two sets, which of these are partial
functions over these sets, rather than total functions?
 
\begin{itemize} 
\item the capital cities of each country: $\{(France, Paris), (UK,
  London), (Spain, Madrid) \}$
\item popular holiday destinations: $\{ (France, Paris), (UK, Paris),
  (Spain, London) \}$
\item same number of letters in word: $\{ (France, London), (Spain,
  Paris) \}$
\item Continental flight destinations from UK: $\{ (UK, Paris), (UK, Madrid) \}$
\end{itemize}
\end{frame}


\begin{frame}[fragile]
\frametitle{Partial functions}
This matters when coding. Think about which inputs your function
should provide an answer for, and which inputs can't be handled. What
do you want to happen for other inputs? 

\begin{code}
>>> import math
>>> n = math.sqrt(9)
>>> print(n)
3.0
>>> n = math.sqrt(-9)
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
ValueError: math domain error
\end{code}
\end{frame}

\begin{frame}[fragile]
\frametitle{Parameters}
If I define a function that takes two parameters, then I must call it
with two parameters:
\begin{code}
def perimeter(width, height):
   return (width*2) + (height*2)

print( perimeter(4, 5) )
\end{code}

\end{frame}

\begin{frame}[fragile]
\frametitle{Named parameters}
If we name the parameters when I supply the values then we can put
them in any order. However this is not a good idea, for code readability.
\begin{code}
def perimeter(width, height):
   return (width*2) + (height*2)

print( perimeter(4, 5) )
print( perimeter(width=4, height=5) )
print( perimeter(height=5, width=4) )
\end{code}
\end{frame}


\begin{frame}[fragile]
\frametitle{Named parameters}
However we can give the parameters default values and then the names
and ability to use them in any order becomes more useful.
\begin{code}
def perimeter(width=7, height=6):
   return (width*2) + (height*2)

print( perimeter(4, 5) )  
print( perimeter(width=4) )  
print( perimeter(height=5) ) 
print( perimeter() )  
\end{code}
\end{frame}






\begin{frame}[fragile]
\frametitle{Functions as arguments}
We can pass functions as arguments. Here I could pass the function \texttt{len}, or
I could pass the function \texttt{first\_char}.
\begin{code}
def is_ok_text( word, f ):
   score = f(word)
   return score < 10

def first_char(word):
   return (ord (word[0]) )

x = is_ok_text("hello", len)
y = is_ok_text("hello", first_char)
\end{code}
\end{frame}

\begin{frame}[fragile]
\frametitle{Functions as arguments}
Suppose we're playing Scrabble. We want to define rules for which
words are acceptable. 
\begin{code}
def tough(word):
   return ( len(word) > 3
            and not word.endswith('s')
            and word[0].upper() != word[0] )

def easy(word):
   return True
\end{code}
Both of these functions will take a word as argument, and return a boolean.
\end{frame}


\begin{frame}[fragile]
\frametitle{Functions as arguments}
\begin{code}
def playScrabble(num_players, rules_func):
   # some code to set up and play
   # at some point we can test a word
   word = get_word_from_player(n)
   if rules_func(word):
      # play carries on
   else:
      # choose another word

# Which rules are we playing?
playScrabble(4, tough)
\end{code}
\end{frame}

\begin{frame}[fragile]
\frametitle{Functions have a type}
Functions have the type 'function'.
\begin{code}
>>> type(perimeter)
<type 'function'>
>>> type(len)
<type 'builtin_function_or_method'>
>>> type(max)
<type 'builtin_function_or_method'>
\end{code}
When we just use the name, e.g. \texttt{perimeter} we are referring
to the function itself. When we add brackets, we are calling the
function instead, e.g.  \texttt{perimeter(3,4)}.
\end{frame}


\begin{frame}[fragile]
\frametitle{Functions and types}
\begin{itemize}
\item There's no way to declare the input types or output types of
  functions in Python 
\item We have to do this in the documentation that we provide
  instead. For example, see the documentation for numpy.loadtxt:
  \url{https://docs.scipy.org/doc/numpy/reference/generated/numpy.loadtxt.html}
\item Mypy: Optional type checking for Python \url{http://mypy-lang.org/}
\item Type hints, Guido van Rossum, PyCon 2015: \url{https://www.youtube.com/watch?v=2wDvzy6Hgxg}
\end{itemize}
\begin{code}
def greeting(name: str) -> str:
    return 'Hello ' + name
\end{code}
\end{frame}

% \begin{frame}[fragile]
% \frametitle{Lambda functions [Optional extra]}
% We can also define simple functions using the lambda keyword. 
% \begin{code}
% number_of_ls = lambda s : return s.count('l')
% say_hello = lambda name: return "hello " + name 

% print (say_hello("Amanda"))
% print (number_of_ls(say_hello("Amanda")))
% \end{code}
% \end{frame}

% \begin{frame}[fragile]
% \frametitle{Lambda functions}
% Lambda functions are a convenient notation but not necessary. 

% \bigskip

% These functions are equivalent:
% \begin{code}
% number_of_ls = lambda s : s.count('l')

% def number_of_ls(s):
%    return s.count('l')
% \end{code}
% \end{frame}

% \begin{frame}[fragile]
% \frametitle{Lambda functions}
% These functions are equivalent:
% \begin{code}
% say_hello = lambda name: "hello " + name 

% def say_hello(name):
%    return "hello " + name 
% \end{code}
% \end{frame}

% \begin{frame}[fragile]
% \frametitle{Lambda functions}
% Lambda functions can only be used for simple, single line functions.

% \bigskip

% After the colon must be an expression which produces the returned
% value.

% \bigskip

% Lambda functions can take more than one argument:

% \begin{code}
% perimeter = lambda height,width : width*2 + height*2 
% \end{code}
% \end{frame}

\begin{frame}
\frametitle{Why use functions}
\begin{itemize}
\item To group code into reusable units.
\item To make small, testable units of code, that are separate from
  other pieces of code.
\item To structure your code so that the flow of code is readable at a high level
  (details hidden by the functions).
\item Abstraction: once it's written and tested you don't need to know
  how a function works inside. Just use it. 
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{Writing a function}
When writing a function:
\begin{itemize}
\item Decide what this function should do. Keep this simple. This is
  the most difficult part. 
\item Decide what parameters it should take.
\item Decide what output it should return
\item Write the code and test it.
\end{itemize}
\end{frame}

\begin{frame}[fragile]
\frametitle{Summary}
\begin{itemize}
\item Functions in maths and Python
\item Return values, None and print
\item Named parameters
\item Passing functions as arguments
\end{itemize}

\bigskip

\begin{code}
def my_function( num, modifier_func, extra=5):
   x = modifier_func(num) + extra
   return x

y = my_function( 4, math.sqrt, extra=8)
\end{code}
\end{frame}

\end{document}