In what follows we explain \emph{parser combinators}, because

they are very easy to implement. However, they only work when

the grammar to be parsed is \emph{not} left-recursive and they

are efficient only when the grammar is unambiguous. It is the

responsibility of the grammar designer to ensure these two

properties.

Parser combinators can deal with any kind of input as long as

this input is a kind of sequence, for example a string or a

list of tokens. If the input are lists of tokens, then parser

combinators transform them into sets of pairs, like so

$\underbrace{\text{list of tokens}}_{\text{input}}$ 

$\Rightarrow$

$\underbrace{\text{set of (parsed input, unparsed input)}}_{\text{output}}$

\end{center} 

\noindent In Scala parser combinators will therefore be

functions of type

\noindent that is they take as input something of type

\texttt{I} and return a set of pairs. The first component of

these pairs corresponds to what the parser combinator was able

to process from the input and the second is the unprocessed

part of the input. As we shall see shortly, a parser

combinator might return more than one such pair, with the idea

that there are potentially several ways how to interpret the

input. To simplify matters we will first look at the case

where the input to the parser combinators is just strings. As

a concrete example, consider the case where the input is the

string

\noindent We might have a parser combinator which tries to

interpret this string as a keyword (\texttt{if}) or an

identifier (\texttt{iffoo}). Then the output will be the set

\noindent where the first pair means the parser could

recognise \texttt{if} from the input and leaves the rest as

`unprocessed' as the second component of the pair; in the

other case it could recognise \texttt{iffoo} and leaves

\texttt{\VS testbar} as unprocessed. If the parser cannot

recognise anything from the input then parser combinators just

return the empty set $\{\}$. This will indicate something

``went wrong''\ldots or more precisely, nothing could be

parsed.

The main attraction of parser combinators is that we can

easily build parser combinators out of smaller components

following very closely the structure of a grammar. In order to

implement this in an object-oriented programming language,

like Scala, we need to specify an abstract class for parser

combinators. This abstract class requires the implementation

of the function \texttt{parse} taking an argument of type

\texttt{I} and returns a set of type \mbox{\texttt{Set[(T,

I)]}}.

\begin{lstlisting}[language=Scala]

\noindent From the function \texttt{parse} we can then

``centrally'' derive the function \texttt{parse\_all}, which

just filters out all pairs whose second component is not empty

(that is has still some unprocessed part). The reason is that

at the end of parsing we are only interested in the results

where all the input has been consumed and no unprocessed part

is left.

One of the simplest parser combinators recognises just a

character, say $c$, from the beginning of strings. Its

behaviour can be described as follows:

	the set $\{(c, \textit{tail of}\; s)\}$, where \textit{tail of} 

	$s$ is the unprocessed part of the input string

\item otherwise it returns the empty set $\{\}$	

\noindent The \texttt{parse} function tests whether the first

character of the input string \texttt{sb} is equal to

\texttt{c}. If yes, then it splits the string into the

recognised part \texttt{c} and the unprocessed part

\texttt{sb.tail}. In case \texttt{sb} does not start with

\texttt{c} then the parser returns the empty set (in Scala

\texttt{Set()}).

More interesting are the parser combinators that build larger

parsers out of smaller component parsers. For example the

alternative parser combinator is as follows: given two

parsers, say, $p$ and $q$, then we apply both parsers to the

input (remember parsers are functions) and combine the input

\begin{center}

$p(\text{input}) \cup q(\text{input})$

\end{center}

In Scala we would implement alternative parsers as follows

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

\noindent The types of this parser combinator are polymorphic

(we just have \texttt{I} for the input type, and \texttt{T}

for the output type). The alternative parser builds a new

parser out of two existing parser combinator \texttt{p} and

\texttt{q}. Both need to be able to process input of type

\texttt{I} and return the same output type \texttt{Set[(T,

I)]}. (There is an interesting detail of Scala, namely the

\texttt{=>} in front of the types of \texttt{p} and

\texttt{q}. They will prevent the evaluation of the arguments

before they are used. This is often called \emph{lazy

evaluation} of the arguments.) The alternative parser should

run the input with the first parser \texttt{p} (producing a

set of outputs) and then run the same input with \texttt{q}.

The result should be then just the union of both sets, which

is the operation \texttt{++} in Scala.

This parser combinator already allows us to construct a parser

that either a character \texttt{a} or \texttt{b}, as

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

\noindent Scala allows us to introduce some more readable

shorthand notation for this, like \texttt{'a' || 'b'}. We can

call this parser combinator with the strings

input strings & & output\medskip\\

\texttt{\Grid{cc}} & $\rightarrow$ & $\{\}$

\end{tabular}

\end{center}

\noindent We receive in the first two cases a successful

output (that is a non-empty set). In each case, either

\pcode{a} or \pcode{b} are the processed parts, and \pcode{c}

is the unprocessed part. Clearly this parser cannot parse

anything with the string \pcode{cc}.

A bit more interesting is the \emph{sequence parser

combinator}. Given two parsers, say, $p$ and $q$,

apply first the input to $p$ producing a set of pairs;

then apply $q$ to the unparsed parts; then combine the 

results like

\begin{center}

\begin{tabular}{lcl}

$\{((\textit{output}_1, \textit{output}_2), u_2)$ & $\;|\;$ & 

$(\textit{output}_1, u_1) \in p(\text{input}) 

\;\wedge\;$\\

&& $(\textit{output}_2, u_2) \in q(u_1)\}$

This can be implemented in Scala as follows:

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

\noindent This parser takes as input two parsers, \texttt{p}

and \texttt{q}. It implements \texttt{parse} as follows: let

first run the parser \texttt{p} on the input producing a set

of pairs (\texttt{head1}, \texttt{tail1}). The \texttt{tail1}

stands for the unprocessed parts left over by \texttt{p}. Let

\texttt{q} run on these unprocessed parts producing again a

set of pairs. The output of the sequence parser combinator is

then a set containing pairs where the first components are

again pairs, namely what the first parser could parse together

with what the second parser could parse; the second component

is the unprocessed part left over after running the second

parser \texttt{q}. Therefore the input type of the sequence

parser combinator is as usual \texttt{I}, but the output type

is

Scala allows us to provide some shorthand notation for the

sequence parser combinator. So we can write for example

\texttt{'a' $\sim$ 'b'}, which is the parser combinator that

first consumes the character \texttt{a} from a string and then

\texttt{b}. Three examples of this parser combinator are as

follows:

\begin{center}

\begin{tabular}{rcl}

input strings & & output\medskip\\

\texttt{\Grid{abc}} & $\rightarrow$ & $\left\{((\texttt{\Grid{a}}, \texttt{\Grid{b}}), \texttt{\Grid{c}})\right\}$\\

\texttt{\Grid{bac}} & $\rightarrow$ & $\{\}$\\

\texttt{\Grid{ccc}} & $\rightarrow$ & $\{\}$

\end{tabular}

\end{center}

\noindent A slightly more complicated parser is \texttt{('a'

|| 'b') $\sim$ 'b'} which parses as first character either an

\texttt{a} or \texttt{b} followed by a \texttt{b}. This parser

produces the following results.

\begin{center}

\begin{tabular}{rcl}

input strings & & output\medskip\\

\texttt{\Grid{abc}} & $\rightarrow$ & $\left\{((\texttt{\Grid{a}}, \texttt{\Grid{b}}), \texttt{\Grid{c}})\right\}$\\

\texttt{\Grid{bbc}} & $\rightarrow$ & $\left\{((\texttt{\Grid{b}}, \texttt{\Grid{b}}), \texttt{\Grid{c}})\right\}$\\

\texttt{\Grid{aac}} & $\rightarrow$ & $\{\}$

\end{tabular}

\end{center}

\noindent Two more examples: first consider the parser \pcode{('a' ~

'a') ~ 'a'} and the input \pcode{aaaa}:

\texttt{\Grid{aaaa}} & $\rightarrow$ & 

$\left\{(((\texttt{\Grid{a}}, \texttt{\Grid{a}}), \texttt{\Grid{a}}), \texttt{\Grid{a}})\right\}$\\

\noindent Notice how the results nest deeper as pairs (the

last \pcode{a} is in the unprocessed part). To consume

everything we can use the parser \pcode{(('a' ~'a') ~ 'a') ~

'a'}. Then the output is as follows:

\texttt{\Grid{aaaa}} & $\rightarrow$ & 

$\left\{((((\texttt{\Grid{a}}, \texttt{\Grid{a}}), \texttt{\Grid{a}}), \texttt{\Grid{a}}), \texttt{""})\right\}$\\

\noindent This is an instance where the parser consumed

completely the input, meaning the unprocessed part is just the

empty string.

Note carefully that constructing a parser such \texttt{'a' ||

('a' $\sim$ 'b')} will result in a tying error. The first

parser has as output type a single character (recall the type

of \texttt{CharParser}), but the second parser produces a pair

of characters as output. The alternative parser is however

required to have both component parsers to have the same type.

We will see later how we can build this parser without the

typing error.

The next parser combinator does not actually combine smaller

parsers, but applies a function to the result of the parser.

It is implemented in Scala as follows