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\section*{A Crash-Course on Scala}

Scala is a programming language that combines functional and

object-oriented programming-styles, and has received in the

\begin{quote}

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

\end{quote}

Why do I use Scala in the AFL module? 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,

Standard ML, F\#, Ocaml and so on.

Developing programs in Scala can be done with the Eclipse IDE

and also with IntelliJ IDE, 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

\begin{quote}

\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}

\end{quote}

\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{quote}

\begin{alltt}

scala> 2 + 3

res0: Int = 5

\end{alltt}

\end{quote}

\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{quote}

\begin{alltt}

scala> print ("hello world")

hello world

\end{alltt}

\end{quote}

\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 later, 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 in Scala, you can

implement an object that is an instance of {\tt App}, say

\begin{quote}

\begin{lstlisting}[language=Scala,numbers=none]

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{quote}

\begin{alltt}

$ scalac hello-world.scala

$ scala Hello

hello world

\end{alltt}

\end{quote}

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{quote}

\begin{alltt}

$ scalac hello-world.scala

$ java -cp /usr/local/src/scala/lib/scala-library.jar:. Hello

hello world

\end{alltt}

\end{quote}

\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! \Smiley} and then compare it how 

it can be defined in Scala.

Implementing the regular expressions from above in Scala is

actually very simple: It first requires an \emph{abstract

class}, say, {\tt Rexp}. This will act as the type for regular

expressions. Second, it requires a case for each clause in the

grammar. 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 Scala code is as

follows:

\begin{quote}

\begin{lstlisting}[language=Scala,numbers=none]

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 of inductive

datatypes, 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, though in the output

syntax the quotes are omitted. If you want to assign this

regular expression to a variable, you can use the keyword {\tt

val} and type

\begin{quote}

\begin{alltt}

scala> val r = ALT(CHAR('a'), CHAR('b'))

r: ALT = ALT(CHAR(a),CHAR(b))

\end{alltt}

\end{quote}

\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 too. But we

omit such ``tricks'' 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

our definition the type {\tt Rexp} is more general than {\tt

ALT}, since it is the abstract class. But 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{quote}

\begin{alltt}

scala> val ls = List(ALT(CHAR('a'), CHAR('b')), NULL)

ls: List[Rexp] = List(ALT(CHAR(a),CHAR(b)), NULL)

\end{alltt}

\end{quote}

\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 first common 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 compound types like {\tt List[...]}, the general rule is

that when a type takes another type as argument, then this

argument type 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 in addition to

inductive datatypes, also functions can be implemented very

easily in Scala. To show you this, 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,numbers=none]{../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 in Scala for arbitrary precision

integers (in case you want to try out the function on really

big numbers). 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 Java, for example, which does not

have pattern-matching, the resulting code would be just

awkward.

Mathematicians already use the power of pattern-matching 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{2mm}}c@{\hspace{2mm}}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 This function is defined by recursion analysing each

pattern of what the regular expression could look like. 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

keyword can be another pattern, but often as in the case

above, it is just a variable you want to pattern-match

(the {\tt r} after {\tt =} in Line 1).

Each case in this definition 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 $ba + ca$, 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{quote}

\begin{alltt}

scala> (1 to 10).toList

res1: List[Int] = List(1, 2, 3, 4, 5, 6, 7, 8, 9, 10)

\end{alltt}

\end{quote}

\noindent Generating from this list the list of squares in a

non-functional programming 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 such 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{quote}

\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}

\end{quote}

\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}. As you can see, 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 in Scala by using directly {\tt map} as follows:

\begin{quote}

\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}

\end{quote}

\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 of 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 {\tt

for}-constructions 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{quote}

\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}

\end{quote}

\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{quote}

\begin{alltt}

scala> (1 to 10).toSet[Int].map(n => n \% 3)

res5 = Set(2, 1, 0)

\end{alltt}

\end{quote}

\noindent This is the correct result for sets, as there are

only three equivalence classes of integers modulo 3. Note that

in this example we need to ``help'' Scala to transform the

numbers into a set of integers by explicitly annotating the

type {\tt Int}. Since maps and {\tt for}-constructions are

just syntactic variants of each other, the latter can also be

written as

\begin{quote}

\begin{alltt}

scala> for (n <- (1 to 10).toSet[Int]) yield n \% 3

res5 = Set(2, 1, 0)

\end{alltt}

\end{quote}