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\section*{Handout 6 (Parser Combinators)}

This handout explains how \emph{parser combinators} work and how they

can be implemented in Scala. Their most distinguishing feature is that

they are very easy to implement (admittedly it is only easy in a

functional programming language).  Another good point of parser

combinators is that they can deal with any kind of input as long as

this input is of ``sequence-kind'', for example a string or a list of

tokens. The only two properties of the input we need is to be able to

test when it is empty and ``sequentially'' take it apart. Strings and

lists fit this bill. However, parser combinators also have their

drawbacks. For example they require that 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.

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

Given the extended effort we have spent in order to implement a lexer

in order to generate list of tokens, it might be surprising that in

what follows we shall often use strings as input. This is for making

the explanation more lucid. It does not make our previous work on

lexers obsolete (remember they transform a string into a list of

tokens). Lexers will still be needed for building a somewhat realistic

compiler.

But as said, parser combinators are relatively agnostic about what

kind of input they process. In my Scala code I use the following

polymorphic types for parser combinators:

\begin{center}

input:\;\; \texttt{I}  \qquad output:\;\; \texttt{T}

\end{center}

\noindent That is they take as input something of type \texttt{I} and

return a set of pairs of type \texttt{Set[(T, I)]}. Since the input needs

to be of ``sequence-kind'', I actually have to often write \texttt{I

  <\% Seq[\_]} for the input type. This ensures the input is a

subtype of Scala sequences. The first component of the generated pairs

corresponds to what the parser combinator was able to process from the

input and the second is the unprocessed part of the input (therefore

the type of this unprocessed part is the same as the input). As we

shall see shortly, a parser combinator might return more than one such

pair; the idea is that there are potentially several ways of how to

parse the input.  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 at all, then parser

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

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

parsed.

Also important to note is that the type \texttt{T} for the processed

part is different from the input type. In the example above is just

happens to be the same. The reason for the difference is that in

general we are interested in transforming our input into something

``different''\ldots for example into a tree; or if we implement the

grammar for arithmetic expressions, we might be interested in the

actual integer number the arithmetic expression, say \texttt{1 + 2 *

  3}, stands for. In this way we can use parser combinators to

implement relatively easily a calculator, for instance.

The main 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 states

that the function \texttt{parse} takes 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 It is the obligation in each instance (parser combinator) to

supply an implementation for \texttt{parse}.  From this function 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

single 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 return 

  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 is \texttt{Char}. This means

\texttt{parse} returns  \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(in: String) = 

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

}

\end{lstlisting}

\end{center}

\noindent You can see the \texttt{parse} tests whether the

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

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

\texttt{c} and the unprocessed part \texttt{in.tail}. In case

\texttt{in} does not start with \texttt{c} then the parser returns the

empty set (in Scala \texttt{Set()}). Since this parser recognises

characters and just returns characters as the processed part, the

output type of the parser is \texttt{Char}.

If we want to parse a list of tokens and interested in recognising a

number token, we could write something like this

\begin{center}

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

case object NumParser extends Parser[List[Token], Int] {

  def parse(ts: List[Token]) = ts match {

    case Num_token(s)::ts => Set((s.toInt, ts)) 

    case _ => Set ()

  }

}

\end{lstlisting}

\end{center}

\noindent

In this parser the input is of type \texttt{List[Token]}. The function

parse looks at the input \texttt{ts} and checks whether the first

token is a \texttt{Num\_token}. Let us assume our lexer generated

these tokens for numbers. But this parser does not just return this

token (and the rest of the list), like the \texttt{CharParser} above,

rather extracts the string \texttt{s} from the token and converts it

into an integer. The hope is that the lexer did its work well and this

conversion always succeeds. The consequence of this is that the output

type for this parser is \texttt{Int}, not \texttt{Token}. Such a

conversion would be needed if we want to implement a simple calculator

program.

These simple parsers that just look at the input and do a simple

transformation are often called \emph{atomic} parser combinators.

More interesting are the parser combinators that build larger parsers

out of smaller component parsers. There are three such parser

combinators that can be implemented generically. The \emph{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 of pairs):

\begin{center}

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

\end{center}

\noindent In Scala we can 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(in: I) = p.parse(in) ++ q.parse(in)

}

\end{lstlisting}

\end{center}

\noindent The types of this parser combinator are again generic (we

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} given as arguments.  Both parsers

need to be able to process input of type \texttt{I} and return in

\texttt{parse} 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 pairs) and then run the same input with \texttt{q} (producing

another set of pairs).  The result should be then just the union of

both sets, which is the operation \texttt{++} in Scala.

The alternative parser combinator allows us to construct a parser that

parses either a character \texttt{a} or \texttt{b} using the

\texttt{CharParser} shown above. For this we can write

\begin{center}

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

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

\end{lstlisting}

\end{center}

\noindent Later on we will use Scala mechanism for introducing some

more readable shorthand notation for this, like \texttt{"a" ||

  "b"}. Let us look in detail at what this parser combinator produces

with some sample strings

\begin{center}

\begin{tabular}{rcl}

input strings & & output\medskip\\

\texttt{\Grid{acde}} & $\rightarrow$ & $\left\{(\texttt{\Grid{a}}, \texttt{\Grid{cde}})\right\}$\\

\texttt{\Grid{bcde}} & $\rightarrow$ & $\left\{(\texttt{\Grid{b}}, \texttt{\Grid{cde}})\right\}$\\

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

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

parse anything with \pcode{ccde}, therefore the empty

set is returned.

A bit more interesting is the \emph{sequence parser combinator}. Given

two parsers, say again, $p$ and $q$, we want to apply first the input

to $p$ producing a set of pairs; then apply $q$ to all the unparsed

parts in the pairs; and then combine the results. Mathematically we would

write something like this for the expected set of pairs:

\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 Notice that the $p$ wil first be run on the input, producing

pairs of the form $(\textit{output}_1, u_1)$ where the $u_1$ stands

for the unprocessed, or left-over, parts. We want that $q$ runs on all

these unprocessed parts $u_1$. This again will produce some

processed part , $p$ and

$q$, we apply both parsers to the input (remember parsers are

functions) and combine the output (remember they are sets of pairs)

\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(in: I) = p.parse(in) ++ q.parse(in)

}

\end{lstlisting}

\end{center}

\noindent The types of this parser combinator are again generic (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.}  Therefore the output

type of this parser is \texttt{T}. The alternative parser should run

the input with the first parser \texttt{p} (producing a set of pairs)

and then run the same input with \texttt{q} (producing another set of

pairs).  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 Later on we will use again Scala mechanism for introducing some

more readable shorthand notation for this, like \texttt{"a" || "b"}. Let us look in detail at what this parser combinator produces with some sample strings 

\begin{center}

\begin{tabular}{rcl}

input strings & & output\medskip\\

\texttt{\Grid{acde}} & $\rightarrow$ & $\left\{(\texttt{\Grid{a}}, \texttt{\Grid{cde}})\right\}$\\

\texttt{\Grid{bcde}} & $\rightarrow$ & $\left\{(\texttt{\Grid{b}}, \texttt{\Grid{cde}})\right\}$\\

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

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

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

set.

A bit more interesting is the \emph{sequence parser combinator}. Given

two parsers, say again, $p$ and $q$, we want to apply first the input

to $p$ producing a set of pairs; then apply $q$ to all the unparsed

parts in the pairs; and 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 Notice that the $p$ will first be run on the input,

producing pairs of the form $\textit{output}_1$ and unprocessed part

$u_1$. This unprocessed part is fed into the second parser $q$. The

overall result of the sequence parser combinator is pairs of the form

$((\textit{output}_1, \textit{output}_2), u_2)$. This means the

unprocessed part of both parsers is the unprocessed part the second

parser $q$ produces leaves as left-over. The processed parts of both

parsers is a pair consisting of the outputs of $p$ and $q$, namely

$(\textit{output}_1, \textit{output}_2)$. This behaviour 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(in: I) = 

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

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

            yield ((output1, output2), u2)

}

\end{lstlisting}

\end{center}

\noindent This parser takes again as arguments 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 then \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}

\noindent

If any of the runs of \textit{p} and \textit{q} fail, that is produce

the empty set, then \texttt{parse} will also produce the empty set.

Notice that we have now two output types for the sequence parser

combinator, because in general \textit{p} and \textit{q} might produce

different things (for example first we recognise a number and then a

string corresponding to an operator).

With the shorthand notation we shall introduce later for the sequence

parser combinator, we can write for example \pcode{"a" ~ "b"}, which

is the parser combinator that first recognises the character

\texttt{a} from a string and then \texttt{b}. Let us look again at

three examples of how this parser combinator processes some strings:

\begin{center}

\begin{tabular}{rcl}

input strings & & output\medskip\\

\texttt{\Grid{abcde}} & $\rightarrow$ & $\left\{((\texttt{\Grid{a}}, \texttt{\Grid{b}}), \texttt{\Grid{cde}})\right\}$\\

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

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

\end{tabular}

\end{center}

\noindent In the first line we have a successful parse, because the

string starts with \texttt{ab}, which is the prefix we are looking

for. But since the parsing combinator is constructed as sequence of

the two simple (atomic) parsers for \texttt{a} and \texttt{b}, the

result is a nested pair of the form \texttt{((a, b), cde)}. It is

\emph{not} a simple pair \texttt{(ab, cde)} as one might erroneously

expect.  The parser returns the empty set in the other examples,

because they do not fit with what the parser is supposed to parse.

A slightly more complicated parser is \pcode{("a" || "b") ~ "c"} which

parses as first character either an \texttt{a} or \texttt{b}, followed

by a \texttt{c}. This parser produces the following outputs.

\begin{center}

\begin{tabular}{rcl}

input strings & & output\medskip\\

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

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

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

\end{tabular}

\end{center}

\noindent

Now consider the parser \pcode{("a" ~ "b") ~ "c"} which parses

\texttt{a}, \texttt{b}, \texttt{c} in sequence. This parser produces

the following outputs.

\begin{center}

\begin{tabular}{rcl}

input strings & & output\medskip\\

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

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

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

\end{tabular}

\end{center}

\noindent The second and third example fail, because something is

``missing'' in the sequence we are looking for. Also notice how the

results nest with sequences: the parsed part is a nested pair of the

form \pcode{((a, b), c)}. 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 again 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}