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\section*{Handout 6 (Parser Combinators)}

In what follows we explain \emph{parser combinators}. Their

distinguishing feature is that 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. The general idea behind parser combinators is

to transform the input into sets of pairs, like so

\begin{center}

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

$\Rightarrow$

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

\end{center} 

\noindent In Scala parser combinators are functions of type

\begin{center}

\texttt{I $\Rightarrow$ Set[(T, I)]}

\end{center}

\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, the idea

being that there are potentially several ways of 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 string

\begin{center}

\tt\Grid{iffoo\VS testbar}

\end{center}

\noindent We might have a parser combinator which tries to

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

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

\begin{center}

$\left\{ \left(\texttt{\Grid{if}}\;,\; \texttt{\Grid{foo\VS testbar}}\right), 

           \left(\texttt{\Grid{iffoo}}\;,\; \texttt{\Grid{\VS testbar}}\right) \right\}$

\end{center}

\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 idea 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{center}

\begin{lstlisting}[language=Scala]

abstract class Parser[I, T] {

  def parse(ts: I): Set[(T, I)]

  def parse_all(ts: I): Set[T] =

    for ((head, tail) <- parse(ts); if (tail.isEmpty)) 

      yield head

}

\end{lstlisting}

\end{center}

\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 the parsing we are only interested in the results

where all the input has been consumed and no unprocessed part

is left over.

One of the simplest parser combinators recognises just a

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

behaviour can be described as follows:

\begin{itemize}

\item if the head of the input string starts with a $c$, then returns 

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

	$s$ is the unprocessed part of the input string

\item otherwise return the empty set $\{\}$	

\end{itemize}

\noindent

The input type of this simple parser combinator for characters is

\texttt{String} and the output type \mbox{\texttt{Set[(Char, String)]}}. 

The code in Scala is as follows:

\begin{center}

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

case class CharParser(c: Char) extends Parser[String, Char] {

  def parse(sb: String) = 

    if (sb.head == c) Set((c, sb.tail)) else Set()

}

\end{lstlisting}

\end{center}

\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 $q$, we apply both parsers to the

input (remember parsers are functions) and combine the output

(remember they are sets)

\begin{center}

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

\end{center}

\noindent In Scala we would implement alternative parser

combinator as follows

\begin{center}

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

class AltParser[I, T]

       (p: => Parser[I, T], 

        q: => Parser[I, T]) extends Parser[I, T] {

  def parse(sb: I) = p.parse(sb) ++ q.parse(sb)

}

\end{lstlisting}

\end{center}

\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 parsers \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)]}.\footnote{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. We will explain this later.} 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.

The alternative parser combinator already allows us to

construct a parser that parses either a character \texttt{a}

or \texttt{b}, as

\begin{center}

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

new AltParser(CharParser('a'), CharParser('b'))

\end{lstlisting}

\end{center}

\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

\begin{center}

\begin{tabular}{rcl}

input strings & & output\medskip\\

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

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

\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} is in the processed part, and

\pcode{c} in the unprocessed part. Clearly this parser cannot

parse anything in the string \pcode{cc}, therefore the empty

set.

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

all 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\;$\\

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

\end{tabular}

\end{center}

\noindent

This can be implemented in Scala as follows:

\begin{center}

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

class SeqParser[I, T, S]

       (p: => Parser[I, T], 

        q: => Parser[I, S]) extends Parser[I, (T, S)] {

  def parse(sb: I) = 

    for ((output1, u1) <- p.parse(sb); 

         (output2, u2) <- q.parse(u1)) 

            yield ((output1, output2), u2)

}

\end{lstlisting}

\end{center}

\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{output1}, \texttt{u1}). The \texttt{u1}

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

\begin{center}

\texttt{Set[((T, S), I)]}

\end{center}

Scala allows us to provide some shorthand notation for the

sequence parser combinator. We can write for example

\pcode{'a' ~ '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 \pcode{('a'

|| 'b') ~ 'b'} which parses as first character either an

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

produces the following outputs.

\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}:

\begin{center}

\begin{tabular}{rcl}

input string & & output\medskip\\

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

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

\end{tabular}

\end{center}

\noindent Notice how the results nest deeper and deeper as

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

consume everything of this string we can use the parser

\pcode{(('a' ~'a') ~ 'a') ~ 'a'}. Then the output is as

follows:

\begin{center}

\begin{tabular}{rcl}

input string & & output\medskip\\

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

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

\end{tabular}

\end{center}

\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 \pcode{'a' ||

('a' ~ 'b')} will result in a typing 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 a parser.

It is implemented in Scala as follows

\begin{center}

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

class FunParser[I, T, S]

         (p: => Parser[I, T], 

          f: T => S) extends Parser[I, S] {

  def parse(sb: I) = 

    for ((head, tail) <- p.parse(sb)) yield (f(head), tail)

}

\end{lstlisting}

\end{center}

\noindent This parser combinator takes a parser \texttt{p}

with output type \texttt{T} as one argument as well as a

function \texttt{f} with type \texttt{T => S}. The parser

\texttt{p} produces sets of type \texttt{(T, I)}. The

\texttt{FunParser} combinator then applies the function

\texttt{f} to all the parser outputs. Since this function is of

type \texttt{T => S}, we obtain a parser with output type

\texttt{S}. Again Scala lets us introduce some shorthand

notation for this parser combinator. Therefore we will write

\texttt{p ==> f} for it.

\subsubsection*{How to build parsers using parser combinators?}

\subsubsection*{Implementing an Interpreter}

%\bigskip

%takes advantage of the full generality---have a look

%what it produces if we call it with the string \texttt{abc}

%

%\begin{center}

%\begin{tabular}{rcl}

%input string & & 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}} & $\rightarrow$ & $\varnothing$

%\end{tabular}

%\end{center}

\end{document}

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