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\section*{A Crash-Course on Scala}

Scala is programming language that combines functional and
object-oriented programming-styles, and has received in the
last five years quite a bit of attention. One reason for this
attention is that, like the Java programming language, Scala
compiles to the Java Virtual Machine (JVM) and therefore can
run under MacOSX, Linux and Windows.\footnote{There are also
experimental backends for Android and JavaScript.} Unlike
Java, however, Scala often allows programmers to write concise
and elegant code. Some therefore say Scala is the much better
Java. If you want to try it out yourself, the Scala compiler
can be downloaded from

\begin{quote}
\url{http://www.scala-lang.org}
\end{quote}

Why do I use Scala in the AFL course? Actually, you can do
\emph{any} part of the programming coursework in \emph{any}
programming language you like. I use Scala for showing you
code during the lectures because its functional
programming-style allows me to implement the functions we will
discuss with very small code-snippets. Since the compiler is
free, you can download them and run every example I give. But
if you prefer, you can also easily translate the code-snippets
into any other functional language, for example Haskell, ML,
F\# and so on.

Writing programs in Scala can be done with the Eclipse IDE and
also with IntelliJ, but for the small programs I will develop
the good old Emacs-editor is adequate for me and I will run
the programs on the command line. One advantage of Scala over Java
is that it includes an interpreter (a REPL, or
Read-Eval-Print-Loop) with which you can run and test small
code-snippets without the need of the compiler. This helps a
lot with interactively developing programs. Once you installed
Scala correctly, you can start the interpreter by typing


\begin{alltt}
$ scala\small
Welcome to Scala version 2.11.2 (Java HotSpot(TM) 64-Bit Server VM).
Type in expressions to have them evaluated.
Type :help for more information.\normalsize

scala>
\end{alltt}

\noindent The precise response may vary due to the platform
where you installed Scala. At the scala prompt you can type
things like {\tt 2 + 3} \keys{Ret} and the output will be

\begin{alltt}
scala> 2 + 3
res0: Int = 5
\end{alltt}

\noindent indicating that the result of the addition is of
type {\tt Int} and the actual result is {\tt 5}. Another
classic example you can try out is

\begin{alltt}
scala> print ("hello world")
hello world
\end{alltt}

\noindent Note that in this case there is no result: the
reason is that {\tt print} does not actually produce a result
(there is no {\tt resXX}), rather it is a function that causes
the \emph{side-effect} of printing out a string. Once you are
more familiar with the functional programming-style, you will
know what the difference is between a function that returns a
result, like addition, and a function that causes a
side-effect, like {\tt print}. We shall come back to this
point in due course, but if you are curious now, the latter
kind of functions always have as return type {\tt Unit}.

If you want to write a stand-alone app, you can implement
an object that is an instance of {\tt App}, say

\begin{quote}
\begin{lstlisting}[language=Scala]
object Hello extends App {
    println ("hello world")
}
\end{lstlisting}
\end{quote}

\noindent save it in a file, say {\tt hellow-world.scala}, and
then run the compiler and runtime environment:

\begin{alltt}
$ scalac hello-world.scala
$ scala Hello
hello world
\end{alltt}

As mentioned above, Scala targets the JVM and
consequently Scala programs can also be executed by the
bog-standard Java runtime. This only requires the inclusion of
{\tt scala-library.jar}, which on my computer can be done as
follows:

\begin{alltt}
$ scalac hello-world.scala
$ java -cp /usr/local/src/scala/lib/scala-library.jar:. Hello
hello world
\end{alltt}


\subsection*{Inductive Datatypes}

The elegance and conciseness of Scala programs are often a
result of inductive datatypes that can be easily defined. For
example in ``every-day mathematics'' we would define regular
expressions simply by giving the grammar

\begin{center}
\begin{tabular}{r@{\hspace{2mm}}r@{\hspace{2mm}}l@{\hspace{13mm}}l}
  $r$ & $::=$ &   $\varnothing$         & null\\
        & $\mid$ & $\epsilon$           & empty string\\
        & $\mid$ & $c$                  & single character\\
        & $\mid$ & $r_1 \cdot r_2$      & sequence\\
        & $\mid$ & $r_1 + r_2$          & alternative / choice\\
        & $\mid$ & $r^*$                & star (zero or more)\\
  \end{tabular}
\end{center}

\noindent This grammar specifies what regular expressions are
(essentially a kind of tree-structure with three kinds of
inner nodes---sequence, alternative and star---and three kinds
of leave nodes---null, empty and character). If you are
familiar with Java, it might be an instructive exercise to
define this kind of inductive datatypes in
Java.\footnote{Happy programming! ;o)}

Implementing the regular expressions from above in Scala is
very simple: It first requires an \emph{abstract class}, say,
{\tt Rexp}. This will act as the type for regular expressions.
Second, it requires some instances. The cases for
$\varnothing$ and $\epsilon$ do not have any arguments, while
in all the other cases we do have arguments. For example the
character regular expression needs to take as an argument the
character it is supposed to recognise. In Scala, the cases
without arguments are called \emph{case objects}, while the
ones with arguments are \emph{case classes}. The corresponding
code is as follows:

\begin{quote}
\begin{lstlisting}[language=Scala]
abstract class Rexp 
case object NULL extends Rexp
case object EMPTY extends Rexp
case class CHAR (c: Char) extends Rexp
case class SEQ (r1: Rexp, r2: Rexp) extends Rexp 
case class ALT (r1: Rexp, r2: Rexp) extends Rexp 
case class STAR (r: Rexp) extends Rexp 
\end{lstlisting}
\end{quote}

\noindent In order to be an instance of {\tt Rexp}, each case
object and case class needs to extend {\tt Rexp}. Given the
grammar above, I hope you can see the underlying pattern. If
you want to play further with such definitions, feel free to
define for example binary trees.

Once you make a definition like the one above, you can
represent, for example, the regular expression for $a + b$ in
Scala as {\tt ALT(CHAR('a'), CHAR('b'))}. Expressions such as
{\tt 'a'} stand for ASCII characters. If you want to assign
this regular expression to a variable, you can use the keyword
{\tt val} and type

\begin{alltt}
scala> val r = ALT(CHAR('a'), CHAR('b'))
r: ALT = ALT(CHAR(a),CHAR(b))
\end{alltt}

\noindent As you can see, in order to make such assignments,
no constructor is required in the class (as in Java). However,
if there is the need for some non-standard initialisation, you
can of course define such a constructor in Scala. But we omit
this here. 

Note that Scala in its response says the variable {\tt r} is
of type {\tt ALT}, not {\tt Rexp}. This might be a bit
unexpected, but can be explained as follows: Scala always
tries to find the most general type that is needed for a
variable or expression, but does not ``over-generalise''. In
this case there is no need to give {\tt r} the more general
type of {\tt Rexp}. This is different if you want to form a
list of regular expressions, for example

\begin{alltt}
scala> val ls = List(ALT(CHAR('a'), CHAR('b')), NULL)
ls: List[Rexp] = List(ALT(CHAR(a),CHAR(b)), NULL)
\end{alltt}

\noindent In this case Scala needs to assign a common type to
the regular expressions, so that it is compatible with the
fact that lists can only contain elements of a single type, in
this case the type is {\tt Rexp}.\footnote{If you type in this
example, you will notice that the type contains some further
information, but lets ignore this for the moment.} 

For types like {\tt List[...]} the general rule is that when a
type takes another type as argument, then this is written in
angle-brackets. This can also contain nested types as in {\tt
List[Set[Rexp]]}, which is a list of sets each of which
contains regular expressions.

\subsection*{Functions and Pattern-Matching}

I mentioned above that Scala is a very elegant programming
language for the code we will write in this module. This
elegance mainly stems from the fact that functions can be
implemented very easily in Scala. Lets first consider a
problem from number theory, called the \emph{Collatz-series},
which corresponds to a famous unsolved problem in
mathematics.\footnote{See for example
\url{http://mathworld.wolfram.com/CollatzProblem.html}.}
Mathematician define this series as:

\[
collatz_{n + 1} \dn 
\begin{cases}
\frac{1}{2} * collatz_n & \text{if $collatz_n$ is even}\\
3 * collatz_n + 1 & \text{if $collatz_n$ is odd}
\end{cases}
\]

\noindent The famous unsolved question is whether this
series started with any $n > 0$ as $collaz_0$ will always
return to $1$. This is obvious when started with $1$, and also
with $2$, but already needs a bit of head-scratching for the
case of $3$.

If we want to avoid the head-scratching, we could implement
this as the following function in Scala:

\begin{quote}
\lstinputlisting[language=Scala]{../progs/collatz.scala}
\end{quote}

\noindent The keyword for function definitions is {\tt def}
followed by the name of the function. After that you have a
list of arguments (enclosed in parentheses and separated by
commas). Each argument in this list needs its type annotated.
In this case we only have one argument, which is of type {\tt
BigInt}. This type stands for arbitrary precision integers.
After the arguments comes the type of what the function
returns---a Boolean in this case for indicating that the
function has reached {\tt 1}. Finally, after the {\tt =} comes
the \emph{body} of the function implementing what the function
is supposed to do. What the {\tt collatz} function does should
be pretty self-explanatory: the function first tests whether
{\tt n} is equal to $1$ in which case it returns {\tt true}
and so on.

Notice a quirk in Scala's syntax for {\tt if}s: The condition
needs to be enclosed in parentheses and the then-case comes
right after the condition---there is no {\tt then} keyword in
Scala.

The real power of Scala comes, however, from the ability to
define functions by \emph{pattern matching}. In the {\tt
collatz} function above we need to test each case using a
sequence of {\tt if}s. This can be very cumbersome and brittle
if there are many cases. If we wanted to define a function
over regular expressions in say Java, which does not have
pattern-matching, the resulting code would just be awkward.

Mathematicians already use the power of pattern-matching, for
example, when they define the function that takes a regular
expression and produces another regular expression that can
recognise the reversed strings. The resulting recursive
function is often defined as follows:

\begin{center}
\begin{tabular}{r@{\hspace{1mm}}c@{\hspace{1mm}}l}
$rev(\varnothing)$   & $\dn$ & $\varnothing$\\
$rev(\epsilon)$      & $\dn$ & $\epsilon$\\
$rev(c)$             & $\dn$ & $c$\\
$rev(r_1 + r_2)$     & $\dn$ & $rev(r_1) + rev(r_2)$\\
$rev(r_1 \cdot r_2)$ & $\dn$ & $rev(r_2) \cdot rev(r_1)$\\
$rev(r^*)$                   & $\dn$ & $rev(r)^*$\\
\end{tabular}
\end{center}

\noindent The corresponding Scala code looks very similar
to this definition, thanks to pattern-matching.


\begin{quote}
\lstinputlisting[language=Scala]{../progs/rev.scala}
\end{quote}

\noindent The keyword for starting a pattern-match is {\tt
match} followed by a list of {\tt case}s. Before the match
can be another pattern, but often as in the case above, 
it is just a variable you want to pattern-match.

In the {\tt rev}-function above, each case follows the
structure of how we defined regular expressions as inductive
datatype. For example the case in Line 3 you can read as: if
the regular expression {\tt r} is of the form {\tt EMPTY} then
do whatever follows the {\tt =>} (in this case just return
{\tt EMPTY}). Line 5 reads as: if the regular expression
{\tt r} is of the form {\tt ALT(r1, r2)}, where the
left-branch of the alternative is matched by the variable {\tt
r1} and the right-branch by {\tt r2} then do ``something''.
The ``something'' can now use the variables {\tt r1} and {\tt
r2} from the match. 

If you want to play with this function, call, it for 
example, with the regular expression $ab + ac$. This regular
expression can recognise the strings $ab$ and $ac$. The 
function {\tt rev} produces the result $ba + ca$, which
can recognise the reversed strings $ba$ and $ca$.

In Scala each pattern-match can also be guarded as in

\begin{quote}
\begin{lstlisting}[language=Scala, numbers=none]
case Pattern if Condition => Do_Something
\end{lstlisting}
\end{quote}

\noindent This allows us, for example, to re-write the {\tt
collatz}-function from above as follows:
 
\begin{quote}
\lstinputlisting[language=Scala]{../progs/collatz2.scala}
\end{quote}

\noindent Although in this case the pattern-match does not
improve the code in any way. The underscore in the last case
indicates that we do not care what the pattern looks like. 
Thus Line 4 acts like a default case whenever the cases above 
did not match. Cases are always tried out from top to bottom.
 
\subsection*{Loops}

Coming from Java or C, you might be surprised that Scala does
not really have loops. It has instead, what is in functional
programming called \emph{maps}. To illustrate how they work,
lets assume you have a list of numbers from 1 to 10 and want to
build the list of squares. The list of numbers from 1 to 10 
can be constructed in Scala as follows:

\begin{alltt}
scala> (1 to 10).toList
res1: List[Int] = List(1, 2, 3, 4, 5, 6, 7, 8, 9, 10)
\end{alltt}

\noindent Generating from this list the list of squares in a
non-functional language (e.g.~Java), you would assume the list
is given as a kind of array. You would then iterate, or loop,
an index over this array and replace each entry in the array
by the square. Right? In Scala, and in other functional
programming languages, you use maps to achieve the same. 

Maps essentially take a function that describes how each
element is transformed (for example squaring) and a list over
which this function should work. There are two forms to
express maps in Scala. The first way is in a {\tt
for}-construction. Squaring the numbers from 1 to 10 would
look in this form as follows:

\begin{alltt}
scala> for (n <- (1 to 10).toList) yield n * n
res2: List[Int] = List(1, 4, 9, 16, 25, 36, 49, 64, 81, 100)
\end{alltt}

\noindent The keywords are {\tt for} and {\tt yield}. This
{\tt for}-construction roughly says that from the list of
numbers we draw {\tt n}s and compute the result of {\tt n *
n}. In consequence we specified the list where each {\tt n}
comes from, namely {\tt (1 to 10).toList}, and how each
element needs to be transformed. This can also be expressed
in a second way by using directly {\tt map} as follows:

\begin{alltt}
scala> (1 to 10).toList.map(n => n * n)
res3 = List(1, 4, 9, 16, 25, 36, 49, 64, 81, 100)
\end{alltt}

\noindent In this way the expression {\tt n => n * n} stands
for the function that calculates the square (this is how the
{\tt n}s are transformed). This expression for functions might
remind you on your lessons about the lambda-calculus where
this would have been written as $\lambda n.\,n * n$.

It might not be obvious, but {\tt for}-constructions are just
syntactic sugar: when compiling, Scala translates them into
equivalent maps.


The very charming feature of Scala is that such maps or {\tt
for}-constructions can be written for any kind of data
collection, such as lists, sets, vectors and so on. For
example if we instead compute the reminders modulo $3$ of this
list, we can write

\begin{alltt}
scala> (1 to 10).toList.map(n => n \% 3)
res4 = List(1, 2, 0, 1, 2, 0, 1, 2, 0, 1)
\end{alltt}

\noindent If we, however, transform the numbers 1 to 10 not
into a list, but into a set, and then compute the reminders
modulo $3$ we obtain

\begin{alltt}
scala> (1 to 10).toSet[Int].map(n => n \% 3)
res5 = Set(2, 1, 0)
\end{alltt}

\noindent This is the correct result for sets, as there are
only 3 equivalence classes of integers modulo 3. Note that we
need to ``help'' Scala in this example to transform the
numbers into a set of integers by explicitly annotating the
type {\tt Int}. Since maps and {\tt for}-construction are just
syntactic variants of each other, the latter can also be
written as

\begin{alltt}
scala> for (n <- (1 to 10).toSet[Int]) yield n \% 3
res5 = Set(2, 1, 0)
\end{alltt}

While hopefully this all looks reasonable, there is one
complication: In the examples above we always wanted to
transform one list into another list (e.g.~list of squares),
or one set into another set (set of numbers into set of
reminders modulo 3). What happens if we just want to print out
a list of integers? Then actually the {\tt for}-construction
needs to be modified. The reason is that {\tt print}, you
guessed it, does not produce any result, but only produces, in
the functional-programming-lingo, a side-effect. Printing out
the list of numbers from 1 to 5 would look as follows

\begin{alltt}
scala> for (n <- (1 to 5).toList) println(n)
1
2
3
4
5
\end{alltt}

\noindent
where you need to omit the keyword {\tt yield}. You can
also do more elaborate calculations such as

\begin{alltt}
scala> for (n <- (1 to 5).toList) \{
  val square_n = n * n
  println(s"$n * $n = $square_n") 
\}
1 * 1 = 1
2 * 2 = 4
3 * 3 = 9
4 * 4 = 16
5 * 5 = 25
\end{alltt}

\noindent
In this code I use a \emph{string interpolation},
written {\tt s"..."} in order to print out an equation.
This string interpolation allows me to refer to the integer 
values {\tt n} and {\tt square\_n} inside a string. This
is very convenient for printing out ``things''. 

The corresponding map construction for functions with 
side-effexts is in Scala called {\tt foreach}. So you 
could also write

\begin{alltt}
scala> (1 to 5).toList.foreach(n => println(n))
1
2
3
4
5
\end{alltt}

\noindent or even just

\begin{alltt}
scala> (1 to 5).toList.foreach(println)
1
2
3
4
5
\end{alltt}

\noindent If you want to find out more maps and functions
with side-efffects, you can ponder about the response Scala
gives if you replace {\tt foreach} by {\tt map} in the
expression above. Scala will still allow {\tt map}, but
then reacts with an interesting result.

\subsection*{Types}

In most functional programming languages types play an
important role. Scala is such a language. You have already
seen built-in types, like {\tt Int}, {\tt Boolean}, {\tt
String} and {\tt BigInt}, but also user-defined ones, like {\tt Rexp}.
Unfortunately, types can be a thorny subject, especially in
Scala. For example, why do we need to give the type to {\tt
toSet[Int]} but not to {\tt toList}? The reason is the power
of Scala, which sometimes means it cannot infer all necessary
typing information. At the beginning while getting familiar
with Scala, I recommend a ``play-it-by-ear-approach'' to
types. Fully understanding type-systems, especially compicated
ones like in Scala, can take a module on their
own.\footnote{Still, such a study can be a rewarding training:
If you are in the business of designing new programming
languages, you will not be able to turn a blind eye to types.
They essentially help programmers to avoid common programming
errors and help with maintaining code.}

In Scala, types are needed whenever you define an inductive
datatype and also whenever you define functions (their
arguments and their results need a type). Base types are types
that do not take any (type)arguments, for example {\tt Int}
and {\tt String}. Compound types take one or more arguments,
which need to be given in angle-brackets, for example {\tt
List[Int]} or {\tt Set[List[String]]} or {\tt Map[Int, Int]}.

There are e few special type-constructors. One is for tuples,
which is written with parentheses. For example {\tt (Int, Int,
String)} for a triple consisting of two integers and a string.
Tuples are helpful if you want to define a function with
multiple results, say the function returning the quotient and
reminder of two numbers. For this you might define:

\begin{alltt}
def quo_rem(m: Int, n: Int) : (Int, Int) = (m \verb|/| n, m \% n)
\end{alltt}

\noindent
Since this function returns a pair of integers, its type
needs to be {\tt (Int, Int)}. 

Another special type-constructor is for functions, written
{\tt =>}. For example, the type {\tt Int => String} stands 
for a function that takes an integer as argument and produces 
a string. A function of this type is for instance

\begin{quote}
\begin{lstlisting}[language=Scala]
def mk_string(n: Int) : String = n match {
  case 0 => "zero"
  case 1 => "one"
  case 2 => "two"
  case _ => "many" 
} 
\end{lstlisting}
\end{quote}

\noindent Unlike other functional programming languages, there
is no easy way to find out the types of existing functions
except for looking into the documentation

\begin{quote}
\url{http://www.scala-lang.org/api/current/}
\end{quote}

\noindent The function arrow can also be iterated, as in {\tt
Int => String => Boolean}. This is the type for a function
taking an integer as first argument and a string as second,
and the result of the function is a boolean. Though silly, a
function of this type is

\begin{quote}
\begin{lstlisting}[language=Scala]
def chk_string(n: Int, s: String) : Boolean = 
  mk_string(n) == s
\end{lstlisting}
\end{quote}

\noindent

\subsection*{Cool Stuff}

\subsection*{More Info}

\end{document}

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