@c -*-texinfo-*-
@c This is part of the GNU Guile Reference Manual.
@c Copyright (C) 2006, 2010, 2011
@c   Free Software Foundation, Inc.
@c See the file guile.texi for copying conditions.

@node PEG Parsing
@section PEG Parsing

Parsing Expression Grammars (PEGs) are a way of specifying formal
languages for text processing.  They can be used either for matching
(like regular expressions) or for building recursive descent parsers
(like lex/yacc).  Guile uses a superset of PEG syntax that allows more
control over what information is preserved during parsing.

Wikipedia has a clear and concise introduction to PEGs if you want to
familiarize yourself with the syntax:
@url{http://en.wikipedia.org/wiki/Parsing_expression_grammar}.

The module works by compiling PEGs down to lambda expressions.  These
can either be stored in variables at compile-time by the define macros
(@code{define-nonterm} and @code{define-grammar}) or calculated
explicitly at runtime with the compile functions
(@code{peg-sexp-compile} and @code{peg-string-compile}).

They can then be used for either parsing (@code{peg-parse}) or matching
(@code{peg-match}).  For convenience, @code{peg-match} also takes
pattern literals in case you want to inline a simple search (people
often use regular expressions this way).

The rest of this documentation consists of a syntax reference, an API
reference, and a tutorial.

@menu
* PEG Syntax Reference::
* PEG API Reference::
* PEG Tutorial::
* PEG Internals::
@end menu

@node PEG Syntax Reference
@subsection PEG Syntax Reference

@subsubheading Normal PEG Syntax:

@deftp {PEG Pattern} sequence a b
Parses @var{a}.  If this succeeds, continues to parse @var{b} from the
end of the text parsed as @var{a}.  Succeeds if both @var{a} and
@var{b} succeed.

@code{"a b"}

@code{(and a b)}
@end deftp

@deftp {PEG Pattern} {ordered choice} a b
Parses @var{a}.  If this fails, backtracks and parses @var{b}.
Succeeds if either @var{a} or @var{b} succeeds.

@code{"a/b"}

@code{(or a b)}
@end deftp

@deftp {PEG Pattern} {zero or more} a
Parses @var{a} as many times in a row as it can, starting each @var{a}
at the end of the text parsed by the previous @var{a}.  Always
succeeds.

@code{"a*"}

@code{(body lit a *)}
@end deftp

@deftp {PEG Pattern} {one or more} a
Parses @var{a} as many times in a row as it can, starting each @var{a}
at the end of the text parsed by the previous @var{a}.  Succeeds if at
least one @var{a} was parsed.

@code{"a+"}

@code{(body lit a +)}
@end deftp

@deftp {PEG Pattern} optional a
Tries to parse @var{a}.  Succeeds if @var{a} succeeds.

@code{"a?"}

@code{(body lit a ?)}
@end deftp

@deftp {PEG Pattern} {and predicate} a
Makes sure it is possible to parse @var{a}, but does not actually parse
it.  Succeeds if @var{a} would succeed.

@code{"&a"}

@code{(body & a 1)}
@end deftp

@deftp {PEG Pattern} {not predicate} a
Makes sure it is impossible to parse @var{a}, but does not actually
parse it.  Succeeds if @var{a} would fail.

@code{"!a"}

@code{(body ! a 1)}
@end deftp

@deftp {PEG Pattern} {string literal} ``abc''
Parses the string @var{"abc"}.  Succeeds if that parsing succeeds.

@code{"'abc'"}

@code{"abc"}
@end deftp

@deftp {PEG Pattern} {any character}
Parses any single character.  Succeeds unless there is no more text to
be parsed.

@code{"."}

@code{peg-any}
@end deftp

@deftp {PEG Pattern} {character class} a b
Alternative syntax for ``Ordered Choice @var{a} @var{b}'' if @var{a} and
@var{b} are characters.

@code{"[ab]"}

@code{(or "a" "b")}
@end deftp

@deftp {PEG Pattern} {range of characters} a z
Parses any character falling between @var{a} and @var{z}.

@code{"[a-z]"}

@code{(range #\a #\z)}
@end deftp

Example:

@example
"(a !b / c &d*) 'e'+"
@end example

Would be:

@lisp
(and
 (or
  (and a (body ! b 1))
  (and c (body & d *)))
 (body lit "e" +))
@end lisp

@subsubheading Extended Syntax

There is some extra syntax for S-expressions.

@deftp {PEG Pattern} ignore a
Ignore the text matching @var{a}
@end deftp

@deftp {PEG Pattern} capture a
Capture the text matching @var{a}.
@end deftp

@deftp {PEG Pattern} peg a
Embed the PEG pattern @var{a} using string syntax.
@end deftp

Example:

@example
"!a / 'b'"
@end example

Would be:

@lisp
(or (peg "!a") "b")
@end lisp

@node PEG API Reference
@subsection PEG API Reference

@subsubheading Define Macros

The most straightforward way to define a PEG is by using one of the
define macros (both of these macroexpand into @code{define}
expressions).  These macros bind parsing functions to variables.  These
parsing functions may be invoked by @code{peg-parse} or
@code{peg-match}, which return a PEG match record.  Raw data can be
retrieved from this record with the PEG match deconstructor functions.
More complicated (and perhaps enlightening) examples can be found in the
tutorial.

@deffn {Scheme Macro} define-grammar peg-string
Defines all the nonterminals in the PEG @var{peg-string}.  More
precisely, @code{define-grammar} takes a superset of PEGs.  A normal PEG
has a @code{<-} between the nonterminal and the pattern.
@code{define-grammar} uses this symbol to determine what information it
should propagate up the parse tree.  The normal @code{<-} propagates the
matched text up the parse tree, @code{<--} propagates the matched text
up the parse tree tagged with the name of the nonterminal, and @code{<}
discards that matched text and propagates nothing up the parse tree.
Also, nonterminals may consist of any alphanumeric character or a ``-''
character (in normal PEGs nonterminals can only be alphabetic).

For example, if we:
@lisp
(define-grammar 
  "as <- 'a'+
bs <- 'b'+
as-or-bs <- as/bs")
(define-grammar 
  "as-tag <-- 'a'+
bs-tag <-- 'b'+
as-or-bs-tag <-- as-tag/bs-tag")
@end lisp
Then:
@lisp
(peg-parse as-or-bs "aabbcc") @result{}
#<peg start: 0 end: 2 string: aabbcc tree: aa>
(peg-parse as-or-bs-tag "aabbcc") @result{}
#<peg start: 0 end: 2 string: aabbcc tree: (as-or-bs-tag (as-tag aa))>
@end lisp

Note that in doing this, we have bound 6 variables at the toplevel
(@var{as}, @var{bs}, @var{as-or-bs}, @var{as-tag}, @var{bs-tag}, and
@var{as-or-bs-tag}).
@end deffn

@deffn {Scheme Macro} define-nonterm name capture-type peg-sexp
Defines a single nonterminal @var{name}.  @var{capture-type} determines
how much information is passed up the parse tree.  @var{peg-sexp} is a
PEG in S-expression form.

Possible values for capture-type:

@table @code
@item all
passes the matched text up the parse tree tagged with the name of the
nonterminal.
@item body
passes the matched text up the parse tree.
@item none
passes nothing up the parse tree.
@end table

For Example, if we:
@lisp
(define-nonterm as body (body lit "a" +))
(define-nonterm bs body (body lit "b" +))
(define-nonterm as-or-bs body (or as bs))
(define-nonterm as-tag all (body lit "a" +))
(define-nonterm bs-tag all (body lit "b" +))
(define-nonterm as-or-bs-tag all (or as-tag bs-tag))
@end lisp
Then:
@lisp
(peg-parse as-or-bs "aabbcc") @result{} 
#<peg start: 0 end: 2 string: aabbcc tree: aa>
(peg-parse as-or-bs-tag "aabbcc") @result{} 
#<peg start: 0 end: 2 string: aabbcc tree: (as-or-bs-tag (as-tag aa))>
@end lisp

Note that in doing this, we have bound 6 variables at the toplevel
(@var{as}, @var{bs}, @var{as-or-bs}, @var{as-tag}, @var{bs-tag}, and
@var{as-or-bs-tag}).
@end deffn

These are macros, with all that entails.  If you've built up a list at
runtime and want to define a new PEG from it, you should e.g.:
@lisp
(define exp '(body lit "a" +))
(eval `(define-nonterm as body ,exp) (interaction-environment))
@end lisp
The @code{eval} function has a bad reputation with regard to efficiency,
but this is mostly because of the extra work that has to be done
compiling the expressions, which has to be done anyway when compiling
the PEGs at runtime.

@subsubheading Compile Functions
It is sometimes useful to be able to compile anonymous PEG patterns at
runtime.  These functions let you do that using either syntax.

@deffn {Scheme Procedure} peg-string-compile peg-string capture-type
Compiles the PEG pattern in @var{peg-string} propagating according to
@var{capture-type} (capture-type can be any of the values from
@code{define-nonterm}).
@end deffn


@deffn {Scheme Procedure} peg-sexp-compile peg-sexp capture-type
Compiles the PEG pattern in @var{peg-sexp} propagating according to
@var{capture-type} (capture-type can be any of the values from
@code{define-nonterm}).
@end deffn


@subsubheading Parsing & Matching Functions

For our purposes, ``parsing'' means parsing a string into a tree
starting from the first character, while ``matching'' means searching
through the string for a substring.  In practice, the only difference
between the two functions is that @code{peg-parse} gives up if it can't
find a valid substring starting at index 0 and @code{peg-match} keeps
looking.  They are both equally capable of ``parsing'' and ``matching''
given those constraints.

@deffn {Scheme Procedure} peg-parse nonterm string 
Parses @var{string} using the PEG stored in @var{nonterm}.  If no match
was found, @code{peg-parse} returns false.  If a match was found, a PEG
match record is returned.

The @code{capture-type} argument to @code{define-nonterm} allows you to
choose what information to hold on to while parsing.  The options are:

@table @code
@item all
tag the matched text with the nonterminal
@item body
just the matched text
@item none
nothing
@end table

@lisp
(define-nonterm as all (body lit "a" +))
(peg-parse as "aabbcc") @result{} 
#<peg start: 0 end: 2 string: aabbcc tree: (as aa)>

(define-nonterm as body (body lit "a" +))
(peg-parse as "aabbcc") @result{} 
#<peg start: 0 end: 2 string: aabbcc tree: aa>

(define-nonterm as none (body lit "a" +))
(peg-parse as "aabbcc") @result{} 
#<peg start: 0 end: 2 string: aabbcc tree: ()>

(define-nonterm bs body (body lit "b" +))
(peg-parse bs "aabbcc") @result{} 
#f
@end lisp
@end deffn

@deffn {Scheme Macro} peg-match nonterm-or-peg string
Searches through @var{string} looking for a matching subexpression.
@var{nonterm-or-peg} can either be a nonterminal or a literal PEG
pattern.  When a literal PEG pattern is provided, @code{peg-match} works
very similarly to the regular expression searches many hackers are used
to.  If no match was found, @code{peg-match} returns false.  If a match
was found, a PEG match record is returned.

@lisp
(define-nonterm as body (body lit "a" +))
(peg-match as "aabbcc") @result{} 
#<peg start: 0 end: 2 string: aabbcc tree: aa>
(peg-match (body lit "a" +) "aabbcc") @result{} 
#<peg start: 0 end: 2 string: aabbcc tree: aa>
(peg-match "'a'+" "aabbcc") @result{} 
#<peg start: 0 end: 2 string: aabbcc tree: aa>

(define-nonterm as all (body lit "a" +))
(peg-match as "aabbcc") @result{} 
#<peg start: 0 end: 2 string: aabbcc tree: (as aa)>

(define-nonterm bs body (body lit "b" +))
(peg-match bs "aabbcc") @result{} 
#<peg start: 2 end: 4 string: aabbcc tree: bb>
(peg-match (body lit "b" +) "aabbcc") @result{} 
#<peg start: 2 end: 4 string: aabbcc tree: bb>
(peg-match "'b'+" "aabbcc") @result{} 
#<peg start: 2 end: 4 string: aabbcc tree: bb>

(define-nonterm zs body (body lit "z" +))
(peg-match zs "aabbcc") @result{} 
#f
(peg-match (body lit "z" +) "aabbcc") @result{} 
#f
(peg-match "'z'+" "aabbcc") @result{} 
#f
@end lisp
@end deffn

@subsubheading PEG Match Records
The @code{peg-parse} and @code{peg-match} functions both return PEG
match records.  Actual information can be extracted from these with the
following functions.

@deffn {Scheme Procedure} peg:string peg-match
Returns the original string that was parsed in the creation of
@code{peg-match}.
@end deffn

@deffn {Scheme Procedure} peg:start peg-match
Returns the index of the first parsed character in the original string
(from @code{peg:string}).  If this is the same as @code{peg:end},
nothing was parsed.
@end deffn

@deffn {Scheme Procedure} peg:end peg-match
Returns one more than the index of the last parsed character in the
original string (from @code{peg:string}).  If this is the same as
@code{peg:start}, nothing was parsed.
@end deffn

@deffn {Scheme Procedure} peg:substring peg-match
Returns the substring parsed by @code{peg-match}.  This is equivalent to
@code{(substring (peg:string peg-match) (peg:start peg-match) (peg:end
peg-match))}.
@end deffn

@deffn {Scheme Procedure} peg:tree peg-match
Returns the tree parsed by @code{peg-match}.
@end deffn

@deffn {Scheme Procedure} peg-record? peg-match
Returns true if @code{peg-match} is a PEG match record, or false
otherwise.
@end deffn

Example:
@lisp
(define-nonterm bs all (peg "'b'+"))

(peg-match bs "aabbcc") @result{}
#<peg start: 2 end: 4 string: aabbcc tree: (bs bb)>

(let ((pm (peg-match bs "aabbcc")))
   `((string ,(peg:string pm))
     (start ,(peg:start pm))
     (end ,(peg:end pm))
     (substring ,(peg:substring pm))
     (tree ,(peg:tree pm))
     (record? ,(peg-record? pm)))) @result{}
((string "aabbcc")
 (start 2)
 (end 4)
 (substring "bb")
 (tree (bs "bb"))
 (record? #t))
@end lisp

@subsubheading Miscellaneous

@deffn {Scheme Procedure} context-flatten tst lst
Takes a predicate @var{tst} and a list @var{lst}.  Flattens @var{lst}
until all elements are either atoms or satisfy @var{tst}.  If @var{lst}
itself satisfies @var{tst}, @code{(list lst)} is returned (this is a
flat list whose only element satisfies @var{tst}).

@lisp
(context-flatten (lambda (x) (and (number? (car x)) (= (car x) 1))) '(2 2 (1 1 (2 2)) (2 2 (1 1)))) @result{} 
(2 2 (1 1 (2 2)) 2 2 (1 1))
(context-flatten (lambda (x) (and (number? (car x)) (= (car x) 1))) '(1 1 (1 1 (2 2)) (2 2 (1 1)))) @result{} 
((1 1 (1 1 (2 2)) (2 2 (1 1))))
@end lisp

If you're wondering why this is here, take a look at the tutorial.
@end deffn

@deffn {Scheme Procedure} keyword-flatten terms lst
A less general form of @code{context-flatten}.  Takes a list of terminal
atoms @code{terms} and flattens @var{lst} until all elements are either
atoms, or lists which have an atom from @code{terms} as their first
element.
@lisp
(keyword-flatten '(a b) '(c a b (a c) (b c) (c (b a) (c a)))) @result{}
(c a b (a c) (b c) c (b a) c a)
@end lisp

If you're wondering why this is here, take a look at the tutorial.
@end deffn

@node PEG Tutorial
@subsection PEG Tutorial

@subsubheading Parsing /etc/passwd
This example will show how to parse /etc/passwd using PEGs.

First we define an example /etc/passwd file:

@lisp
(define *etc-passwd*
  "root:x:0:0:root:/root:/bin/bash
daemon:x:1:1:daemon:/usr/sbin:/bin/sh
bin:x:2:2:bin:/bin:/bin/sh
sys:x:3:3:sys:/dev:/bin/sh
nobody:x:65534:65534:nobody:/nonexistent:/bin/sh
messagebus:x:103:107::/var/run/dbus:/bin/false
")
@end lisp

As a first pass at this, we might want to have all the entries in
/etc/passwd in a list.

Doing this with string-based PEG syntax would look like this:
@lisp
(define-grammar
  "passwd <- entry* !.
entry <-- (! NL .)* NL*
NL < '\n'")
@end lisp

A @code{passwd} file is 0 or more entries (@code{entry*}) until the end
of the file (@code{!.} (@code{.} is any character, so @code{!.} means
``not anything'')).  We want to capture the data in the nonterminal
@code{passwd}, but not tag it with the name, so we use @code{<-}.

An entry is a series of 0 or more characters that aren't newlines
(@code{(! NL .)*}) followed by 0 or more newlines (@code{NL*}).  We want
to tag all the entries with @code{entry}, so we use @code{<--}.

A newline is just a literal newline (@code{'\n'}).  We don't want a
bunch of newlines cluttering up the output, so we use @code{<} to throw
away the captured data.

Here is the same PEG defined using S-expressions:
@lisp
(define-nonterm passwd body (and (body lit entry *) (body ! peg-any 1)))
(define-nonterm entry all (and (body lit (and (body ! NL 1) peg-any) *)
			       (body lit NL *)))
(define-nonterm NL none "\n")
@end lisp

Obviously this is much more verbose.  On the other hand, it's more
explicit, and thus easier to build automatically.  However, there are
some tricks that make S-expressions easier to use in some cases.  One is
the @code{ignore} keyword; the string syntax has no way to say ``throw
away this text'' except breaking it out into a separate nonterminal.
For instance, to throw away the newlines we had to define @code{NL}.  In
the S-expression syntax, we could have simply written @code{(ignore
"\n")}.  Also, for the cases where string syntax is really much cleaner,
the @code{peg} keyword can be used to embed string syntax in
S-expression syntax.  For instance, we could have written:

@lisp
(define-nonterm passwd body (peg "entry* !."))
@end lisp

However we define it, parsing @code{*etc-passwd*} with the @code{passwd}
nonterminal yields the same results:

@lisp
(peg:tree (peg-parse passwd *etc-passwd*)) @result{}
((entry "root:x:0:0:root:/root:/bin/bash")
 (entry "daemon:x:1:1:daemon:/usr/sbin:/bin/sh")
 (entry "bin:x:2:2:bin:/bin:/bin/sh")
 (entry "sys:x:3:3:sys:/dev:/bin/sh")
 (entry "nobody:x:65534:65534:nobody:/nonexistent:/bin/sh")
 (entry "messagebus:x:103:107::/var/run/dbus:/bin/false"))
@end lisp

However, here is something to be wary of:

@lisp
(peg:tree (peg-parse passwd "one entry")) @result{}
(entry "one entry")
@end lisp

By default, the parse trees generated by PEGs are compressed as much as
possible without losing information.  It may not look like this is what
you want at first, but uncompressed parse trees are an enormous headache
(there's no easy way to predict how deep particular lists will nest,
there are empty lists littered everywhere, etc. etc.).  One side-effect
of this, however, is that sometimes the compressor is too aggressive.
No information is discarded when @code{((entry "one entry"))} is
compressed to @code{(entry "one entry")}, but in this particular case it
probably isn't what we want.

There are two functions for easily dealing with this:
@code{keyword-flatten} and @code{context-flatten}.  The
@code{keyword-flatten} function takes a list of keywords and a list to
flatten, then tries to coerce the list such that the first element of
all sublists is one of the keywords.  The @code{context-flatten}
function is similar, but instead of a list of keywords it takes a
predicate that should indicate whether a given sublist is good enough
(refer to the API reference for more details).

What we want here is @code{keyword-flatten}.
@lisp
(keyword-flatten '(entry) (peg:tree (peg-parse passwd *etc-passwd*))) @result{}
((entry "root:x:0:0:root:/root:/bin/bash")
 (entry "daemon:x:1:1:daemon:/usr/sbin:/bin/sh")
 (entry "bin:x:2:2:bin:/bin:/bin/sh")
 (entry "sys:x:3:3:sys:/dev:/bin/sh")
 (entry "nobody:x:65534:65534:nobody:/nonexistent:/bin/sh")
 (entry "messagebus:x:103:107::/var/run/dbus:/bin/false"))
(keyword-flatten '(entry) (peg:tree (peg-parse passwd "one entry"))) @result{}
((entry "one entry"))
@end lisp

Of course, this is a somewhat contrived example.  In practice we would
probably just tag the @code{passwd} nonterminal to remove the ambiguity
(using either the @code{all} keyword for S-expressions or the @code{<--}
symbol for strings)..

@lisp
(define-nonterm tag-passwd all (peg "entry* !."))
(peg:tree (peg-parse tag-passwd *etc-passwd*)) @result{}
(tag-passwd
  (entry "root:x:0:0:root:/root:/bin/bash")
  (entry "daemon:x:1:1:daemon:/usr/sbin:/bin/sh")
  (entry "bin:x:2:2:bin:/bin:/bin/sh")
  (entry "sys:x:3:3:sys:/dev:/bin/sh")
  (entry "nobody:x:65534:65534:nobody:/nonexistent:/bin/sh")
  (entry "messagebus:x:103:107::/var/run/dbus:/bin/false"))
(peg:tree (peg-parse tag-passwd "one entry"))
(tag-passwd 
  (entry "one entry"))
@end lisp

If you're ever uncertain about the potential results of parsing
something, remember the two absolute rules:
@enumerate
@item
No parsing information will ever be discarded.
@item
There will never be any lists with fewer than 2 elements.
@end enumerate

For the purposes of (1), "parsing information" means things tagged with
the @code{any} keyword or the @code{<--} symbol.  Plain strings will be
concatenated.

Let's extend this example a bit more and actually pull some useful
information out of the passwd file:

@lisp
(define-grammar
  "passwd <-- entry* !.
entry <-- login C pass C uid C gid C nameORcomment C homedir C shell NL*
login <-- text
pass <-- text
uid <-- [0-9]*
gid <-- [0-9]*
nameORcomment <-- text
homedir <-- path
shell <-- path
path <-- (SLASH pathELEMENT)*
pathELEMENT <-- (!NL !C  !'/' .)*
text <- (!NL !C  .)*
C < ':'
NL < '\n'
SLASH < '/'")
@end lisp

This produces rather pretty parse trees:
@lisp
(passwd
  (entry (login "root")
         (pass "x")
         (uid "0")
         (gid "0")
         (nameORcomment "root")
         (homedir (path (pathELEMENT "root")))
         (shell (path (pathELEMENT "bin") (pathELEMENT "bash"))))
  (entry (login "daemon")
         (pass "x")
         (uid "1")
         (gid "1")
         (nameORcomment "daemon")
         (homedir
           (path (pathELEMENT "usr") (pathELEMENT "sbin")))
         (shell (path (pathELEMENT "bin") (pathELEMENT "sh"))))
  (entry (login "bin")
         (pass "x")
         (uid "2")
         (gid "2")
         (nameORcomment "bin")
         (homedir (path (pathELEMENT "bin")))
         (shell (path (pathELEMENT "bin") (pathELEMENT "sh"))))
  (entry (login "sys")
         (pass "x")
         (uid "3")
         (gid "3")
         (nameORcomment "sys")
         (homedir (path (pathELEMENT "dev")))
         (shell (path (pathELEMENT "bin") (pathELEMENT "sh"))))
  (entry (login "nobody")
         (pass "x")
         (uid "65534")
         (gid "65534")
         (nameORcomment "nobody")
         (homedir (path (pathELEMENT "nonexistent")))
         (shell (path (pathELEMENT "bin") (pathELEMENT "sh"))))
  (entry (login "messagebus")
         (pass "x")
         (uid "103")
         (gid "107")
         nameORcomment
         (homedir
           (path (pathELEMENT "var")
                 (pathELEMENT "run")
                 (pathELEMENT "dbus")))
         (shell (path (pathELEMENT "bin") (pathELEMENT "false")))))
@end lisp

Notice that when there's no entry in a field (e.g. @code{nameORcomment}
for messagebus) the symbol is inserted.  This is the ``don't throw away
any information'' rule---we succesfully matched a @code{nameORcomment}
of 0 characters (since we used @code{*} when defining it).  This is
usually what you want, because it allows you to e.g. use @code{list-ref}
to pull out elements (since they all have known offsets).

If you'd prefer not to have symbols for empty matches, you can replace
the @code{*} with a @code{+} and add a @code{?} after the
@code{nameORcomment} in @code{entry}.  Then it will try to parse 1 or
more characters, fail (inserting nothing into the parse tree), but
continue because it didn't have to match the nameORcomment to continue.


@subsubheading Embedding Arithmetic Expressions

We can parse simple mathematical expressions with the following PEG:

@lisp
(define-grammar
  "expr <- sum
sum <-- (product ('+' / '-') sum) / product
product <-- (value ('*' / '/') product) / value
value <-- number / '(' expr ')'
number <-- [0-9]+")
@end lisp

Then:
@lisp
(peg:tree (peg-parse expr "1+1/2*3+(1+1)/2")) @result{}
(sum (product (value (number "1")))
     "+"
     (sum (product
            (value (number "1"))
            "/"
            (product
              (value (number "2"))
              "*"
              (product (value (number "3")))))
          "+"
          (sum (product
                 (value "("
                        (sum (product (value (number "1")))
                             "+"
                             (sum (product (value (number "1")))))
                        ")")
                 "/"
                 (product (value (number "2")))))))
@end lisp

There is very little wasted effort in this PEG.  The @code{number}
nonterminal has to be tagged because otherwise the numbers might run
together with the arithmetic expressions during the string concatenation
stage of parse-tree compression (the parser will see ``1'' followed by
``/'' and decide to call it ``1/'').  When in doubt, tag.

It is very easy to turn these parse trees into lisp expressions:

@lisp
(define (parse-sum sum left . rest)
  (if (null? rest)
      (apply parse-product left)
      (list (string->symbol (car rest))
	    (apply parse-product left)
	    (apply parse-sum (cadr rest)))))

(define (parse-product product left . rest)
  (if (null? rest)
      (apply parse-value left)
      (list (string->symbol (car rest))
	    (apply parse-value left)
	    (apply parse-product (cadr rest)))))

(define (parse-value value first . rest)
  (if (null? rest)
      (string->number (cadr first))
      (apply parse-sum (car rest))))

(define parse-expr parse-sum)
@end lisp

(Notice all these functions look very similar; for a more complicated
PEG, it would be worth abstracting.)

Then:
@lisp
(apply parse-expr (peg:tree (peg-parse expr "1+1/2*3+(1+1)/2"))) @result{}
(+ 1 (+ (/ 1 (* 2 3)) (/ (+ 1 1) 2)))
@end lisp

But wait!  The associativity is wrong!  Where it says @code{(/ 1 (* 2
3))}, it should say @code{(* (/ 1 2) 3)}.

It's tempting to try replacing e.g. @code{"sum <-- (product ('+' / '-')
sum) / product"} with @code{"sum <-- (sum ('+' / '-') product) /
product"}, but this is a Bad Idea.  PEGs don't support left recursion.
To see why, imagine what the parser will do here.  When it tries to
parse @code{sum}, it first has to try and parse @code{sum}.  But to do
that, it first has to try and parse @code{sum}.  This will continue
until the stack gets blown off.

So how does one parse left-associative binary operators with PEGs?
Honestly, this is one of their major shortcomings.  There's no
general-purpose way of doing this, but here the repetition operators are
a good choice:

@lisp
(use-modules (srfi srfi-1))

(define-grammar
  "expr <- sum
sum <-- (product ('+' / '-'))* product
product <-- (value ('*' / '/'))* value
value <-- number / '(' expr ')'
number <-- [0-9]+")

;; take a deep breath...
(define (make-left-parser next-func)
  (lambda (sum first . rest) ;; general form, comments below assume
    ;; that we're dealing with a sum expression
    (if (null? rest) ;; form (sum (product ...))
      (apply next-func first)
      (if (string? (cadr first));; form (sum ((product ...) "+") (product ...))
	  (list (string->symbol (cadr first))
		(apply next-func (car first))
		(apply next-func (car rest)))
          ;; form (sum (((product ...) "+") ((product ...) "+")) (product ...))
	  (car 
	   (reduce ;; walk through the list and build a left-associative tree
	    (lambda (l r)
	      (list (list (cadr r) (car r) (apply next-func (car l)))
		    (string->symbol (cadr l))))
	    'ignore
	    (append ;; make a list of all the products
             ;; the first one should be pre-parsed
	     (list (list (apply next-func (caar first))
			 (string->symbol (cadar first))))
	     (cdr first)
             ;; the last one has to be added in
	     (list (append rest '("done"))))))))))

(define (parse-value value first . rest)
  (if (null? rest)
      (string->number (cadr first))
      (apply parse-sum (car rest))))
(define parse-product (make-left-parser parse-value))
(define parse-sum (make-left-parser parse-product))
(define parse-expr parse-sum)
@end lisp

Then:
@lisp
(apply parse-expr (peg:tree (peg-parse expr "1+1/2*3+(1+1)/2"))) @result{}
(+ (+ 1 (* (/ 1 2) 3)) (/ (+ 1 1) 2))
@end lisp

As you can see, this is much uglier (it could be made prettier by using
@code{context-flatten}, but the way it's written above makes it clear
how we deal with the three ways the zero-or-more @code{*} expression can
parse).  Fortunately, most of the time we can get away with only using
right-associativity.

@subsubheading Simplified Functions

For a more tantalizing example, consider the following grammar that
parses (highly) simplified C functions:

@lisp
(define-grammar
  "cfunc <-- cSP ctype cSP cname cSP cargs cLB cSP cbody cRB
ctype <-- cidentifier
cname <-- cidentifier
cargs <-- cLP (! (cSP cRP) carg cSP (cCOMMA / cRP) cSP)* cSP
carg <-- cSP ctype cSP cname
cbody <-- cstatement *
cidentifier <- [a-zA-z][a-zA-Z0-9_]*
cstatement <-- (!';'.)*cSC cSP
cSC < ';'
cCOMMA < ','
cLP < '('
cRP < ')'
cLB < '@{'
cRB < '@}'
cSP < [ \t\n]*")
@end lisp

Then:
@lisp
(peg-parse cfunc "int square(int a) @{ return a*a;@}") @result{}
(32
 (cfunc (ctype "int")
        (cname "square")
        (cargs (carg (ctype "int") (cname "a")))
        (cbody (cstatement "return a*a"))))
@end lisp

And:
@lisp
(peg-parse cfunc "int mod(int a, int b) @{ int c = a/b;return a-b*c; @}") @result{}
(52
 (cfunc (ctype "int")
        (cname "mod")
        (cargs (carg (ctype "int") (cname "a"))
               (carg (ctype "int") (cname "b")))
        (cbody (cstatement "int c = a/b")
               (cstatement "return a- b*c"))))
@end lisp

By wrapping all the @code{carg} nonterminals in a @code{cargs}
nonterminal, we were able to remove any ambiguity in the parsing
structure and avoid having to call @code{context-flatten} on the output
of @code{peg-parse}.  We used the same trick with the @code{cstatement}
nonterminals, wrapping them in a @code{cbody} nonterminal.

The whitespace nonterminal @code{cSP} used here is a (very) useful
instantiation of a common pattern for matching syntactically irrelevant
information.  Since it's tagged with @code{<} and ends with @code{*} it
won't clutter up the parse trees (all the empty lists will be discarded
during the compression step) and it will never cause parsing to fail.

@node PEG Internals
@subsection PEG Internals

A PEG parser takes a string as input and attempts to parse it as a given
nonterminal. The key idea of the PEG implementation is that every
nonterminal is just a function that takes a string as an argument and
attempts to parse that string as its nonterminal. The functions always
start from the beginning, but a parse is considered successful if there
is material left over at the end.

This makes it easy to model different PEG parsing operations. For
instance, consider the PEG grammar @code{"ab"}, which could also be
written @code{(and "a" "b")}. It matches the string ``ab''. Here's how
that might be implemented in the PEG style:

@lisp
(define (match-and-a-b str)
  (match-a str)
  (match-b str))
@end lisp

As you can see, the use of functions provides an easy way to model
sequencing. In a similar way, one could model @code{(or a b)} with
something like the following:

@lisp
(define (match-or-a-b str)
  (or (match-a str) (match-b str)))
@end lisp

Here the semantics of a PEG @code{or} expression map naturally onto
Scheme's @code{or} operator. This function will attempt to run
@code{(match-a str)}, and return its result if it succeeds. Otherwise it
will run @code{(match-b str)}.

Of course, the code above wouldn't quite work. We need some way for the
parsing functions to communicate. The actual interface used is below.

@subsubheading Parsing Function Interface

A parsing function takes three arguments - a string, the length of that
string, and the position in that string it should start parsing at. In
effect, the parsing functions pass around substrings in pieces - the
first argument is a buffer of characters, and the second two give a
range within that buffer that the parsing function should look at.

Parsing functions return either #f, if they failed to match their
nonterminal, or a list whose first element must be an integer
representing the final position in the string they matched and whose cdr
can be any other data the function wishes to return, or '() if it
doesn't have any more data.

The one caveat is that if the extra data it returns is a list, any
adjacent strings in that list will be appended by @code{peg-parse}. For
instance, if a parsing function returns @code{(13 ("a" "b" "c"))},
@code{peg-parse} will take @code{(13 ("abc"))} as its value.

For example, here is a function to match ``ab'' using the actual
interface.

@lisp
(define (match-a-b str len pos)
   (and (<= (+ pos 2) len)
        (string= str "ab" pos (+ pos 2))
        (list (+ pos 2) '()))) ; we return no extra information
@end lisp

The above function can be used to match a string by running
@code{(peg-parse match-a-b "ab")}.