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In [[computer science]], a '''parsing expression grammar''', or '''PEG''', is a type of [[formal grammar#Analytic grammars|analytic]] [[formal grammar]], i.e. it describes a [[formal language]] in terms of a set of rules for recognizing [[string (computer science)|strings]] in the language. The formalism was introduced by Bryan Ford in 2004<ref name="For04">
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{{cite conference
| ref = "pegs"
| first = Bryan p
| last = Ford
| title = Parsing Expression Grammars: A Recognition Based Syntactic Foundation
| booktitle = Proceedings of the 31st ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages
| pages =
| publisher = [[Association for Computing Machinery|ACM]]
| year = 2004
| url = http://portal.acm.org/citation.cfm?id=964001.964011
| doi = 10.1145/964001.964011
| isbn = 1-58113-729-X
| accessdate = 2010-07-30
}}</ref> and is closely related to the family of [[top-down parsing language]]s introduced in the early 1970s.
Syntactically, PEGs also look similar to [[context-free grammar]]s (CFGs), but they have a different interpretation: the choice operator selects the first match in PEG, while it is ambiguous in CFG. This is closer to how string recognition tends to be done in practice, e.g. by a [[recursive descent parser]].


Unlike CFGs, PEGs cannot be [[ambiguous grammar|ambiguous]]; if a string parses, it has exactly one valid [[parse tree]]. It is conjectured that there exist context-free languages that cannot be parsed by a PEG, but this is not yet proven.<ref name="For04" /> PEGs are well-suited to parsing computer languages, but not [[natural language]]s where their performance is comparable to general CFG algorithms such as the [[Earley algorithm]].<ref name="pegs">
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{{cite paper
 
| author = Bryan Ford
  <li>[http://www.think-group.cn/VK/blog/article.php?type=blog&cid=5&itemid=1728663 http://www.think-group.cn/VK/blog/article.php?type=blog&cid=5&itemid=1728663]</li>
| title = Functional Pearl: Packrat Parsing: Simple, Powerful, Lazy, Linear Time
 
| year = 2002
  <li>[http://clan.gamescraft.de/index.php?site=guestbook http://clan.gamescraft.de/index.php?site=guestbook]</li>
| url = http://pdos.csail.mit.edu/~baford/packrat/icfp02/packrat-icfp02.pdf
 
| format = [[Portable Document Format|PDF]]}}
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</ref>
 
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</ul>


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Formally, a parsing expression grammar consists of:
 
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</ul>


* A finite set ''N'' of ''[[nonterminal symbol]]s''.
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* A finite set Σ of ''[[terminal symbol]]s'' that is disjoint from ''N''.
* A finite set ''P'' of ''parsing rules''.
* An expression ''e<sub>S</sub>'' termed the ''starting expression''.


Each parsing rule in ''P'' has the form ''A'' ← ''e'', where ''A'' is a nonterminal symbol and ''e'' is a ''parsing expression''. A parsing expression is a hierarchical [[expression (mathematics)|expression]] similar to a [[regular expression]], which is constructed in the following fashion:
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# An ''atomic parsing expression'' consists of:
   <li>[http://jfeng.cn/forum.php?mod=viewthread&tid=160482 http://jfeng.cn/forum.php?mod=viewthread&tid=160482]</li>
#* any terminal symbol,
 
#* any nonterminal symbol, or
  <li>[http://www.jankloco.com/activity/p/52409/ http://www.jankloco.com/activity/p/52409/]</li>
#* the empty string ε.
 
# Given any existing parsing expressions ''e'', ''e''<sub>1</sub>, and ''e''<sub>2</sub>, a new parsing expression can be constructed using the following operators:
  <li>[http://colossuscorporation.net/appicker/index.php?option=com_kunena&func=view&catid=17&id=311449&Itemid=534#311449 http://colossuscorporation.net/appicker/index.php?option=com_kunena&func=view&catid=17&id=311449&Itemid=534#311449]</li>
#* ''Sequence'': ''e''<sub>1</sub> ''e''<sub>2</sub>
 
#* ''Ordered choice'': ''e''<sub>1</sub> / ''e''<sub>2</sub>
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#* ''Zero-or-more'': ''e''*
 
#* ''One-or-more'': ''e''+
  <li>[http://rybnik360.pl/spip.php?article1&archives=2013-03-01/ http://rybnik360.pl/spip.php?article1&archives=2013-03-01/]</li>
#* ''Optional'': ''e''?
 
#* ''And-predicate'': &''e''
</ul>
#* ''Not-predicate'': !''e''
 
=== Semantics ===
 
The fundamental difference between [[context-free grammars]] and parsing expression grammars is that the PEG's choice operator is ''ordered''. If the first alternative succeeds, the second alternative is ignored. Thus ordered choice is not [[commutativity|commutative]], unlike unordered choice as in context-free grammars. Ordered choice is analogous to [[cut (logic programming)|soft cut]] operators available in some [[logic programming]] languages.
 
The consequence is that if a CFG is transliterated directly to a PEG, any ambiguity in the former is resolved by deterministically picking one parse tree from the possible parses. By carefully choosing the order in which the grammar alternatives are specified, a programmer has a great deal of control over which parse tree is selected.
 
Like [[Boolean grammar|boolean context-free grammars]], parsing expression grammars also add the and- and not- [[syntactic predicate]]s. Because they can use an arbitrarily complex sub-expression to "look ahead" into the input string without actually consuming it, they provide a powerful syntactic [[Parsing#Lookahead|lookahead]] and disambiguation facility, in particular when reordering the alternatives cannot specify the exact parse tree desired.
 
=== Operational interpretation of parsing expressions ===
Each nonterminal in a parsing expression grammar essentially represents a parsing [[function (mathematics)|function]] in a [[recursive descent parser]], and the corresponding parsing expression represents the "code" comprising the function.  Each parsing function conceptually takes an input string as its argument, and yields one of the following results:
 
* ''success'', in which the function may optionally move forward or ''consume'' one or more characters of the input string supplied to it, or
* ''failure'', in which case no input is consumed.
 
An atomic parsing expression consisting of a single '''terminal''' (i.e. literal) succeeds if the first character of the input string matches that terminal, and in that case consumes the input character; otherwise the expression yields a failure result.  An atomic parsing expression consisting of the empty string always trivially succeeds without consuming any input.
An atomic parsing expression consisting of a '''nonterminal''' ''A'' represents a [[recursion|recursive]] call to the nonterminal-function ''A''.  A nonterminal may succeed without actually consuming any input, and this is considered an outcome distinct from failure.
 
The '''sequence''' operator ''e''<sub>1</sub> ''e''<sub>2</sub> first invokes ''e''<sub>1</sub>, and if ''e''<sub>1</sub> succeeds, subsequently invokes ''e''<sub>2</sub> on the remainder of the input string left unconsumed by ''e''<sub>1</sub>, and returns the result. If either ''e''<sub>1</sub> or ''e''<sub>2</sub> fails, then the sequence expression ''e''<sub>1</sub> ''e''<sub>2</sub> fails.
 
The '''choice''' operator ''e''<sub>1</sub> / ''e''<sub>2</sub> first invokes ''e''<sub>1</sub>, and if ''e''<sub>1</sub> succeeds, returns its result immediately. Otherwise, if ''e''<sub>1</sub> fails, then the choice operator [[backtracking|backtracks]] to the original input position at which it invoked ''e''<sub>1</sub>, but then calls ''e''<sub>2</sub> instead, returning ''e''<sub>2</sub>'s result.
 
The '''zero-or-more''', '''one-or-more''', and '''optional''' operators consume zero or more, one or more, or zero or one consecutive repetitions of their sub-expression ''e'', respectively. Unlike in [[context-free grammar]]s and [[regular expression]]s, however, these operators ''always'' behave [[greedy algorithm|greedily]], consuming as much input as possible and never backtracking.  (Regular expression matchers may start by matching greedily, but will then backtrack and try shorter matches if they fail to match.)  For example, the expression a* will always consume as many a's as are consecutively available in the input string, and the expression (a* a) will always fail because the first part (a*) will never leave any a's for the second part to match.
 
The '''and-predicate''' expression &''e'' invokes the sub-expression ''e'', and then succeeds if ''e'' succeeds and fails if ''e'' fails, but in either case ''never consumes any input''.
 
The '''not-predicate''' expression !''e'' succeeds if ''e'' fails and fails if ''e'' succeeds, again consuming no input in either case.
 
=== Examples ===
This is a PEG that recognizes mathematical formulas that apply the basic four operations to non-negative integers.
<pre>
Value  &larr; [0-9]+ / '(' Expr ')'
Product &larr; Value (('*' / '/') Value)*
Sum    &larr; Product (('+' / '-') Product)*
Expr    &larr; Sum
</pre>
In the above example, the terminal symbols are characters of text, represented by characters in single quotes, such as <code>'('</code> and <code>')'</code>. The range '''<code>[0-9]</code>''' is also a shortcut for ten characters, indicating any one of the digits 0 through 9. (This range syntax is the same as the syntax used by [[regular expression]]s.) The nonterminal symbols are the ones that expand to other rules: ''Value'', ''Product'', ''Sum'', and ''Expr''.
 
The parsing expression '''('a'/'b')*''' matches and consumes an arbitrary-length sequence of a's and b's. The [[Formal grammar#The syntax of grammars|production rule]] {{nowrap|'''''S'' ← 'a' ''S''? 'b''''}} describes the simple [[context-free language|context-free]] "matching language" <math> \{ a^n b^n : n \ge 1 \} </math>. 
The following parsing expression grammar describes the classic non-context-free language <math> \{ a^n b^n c^n : n \ge 1 \} </math>:
<pre>
S &larr; &(A 'c') 'a'+ B !('a'/'b'/'c')
A &larr; 'a' A? 'b'
B &larr; 'b' B? 'c'
</pre>
The following recursive rule matches standard C-style if/then/else statements in such a way that the optional "else" clause always binds to the innermost "if", because of the implicit prioritization of the '/' operator.  (In a [[context-free grammar]], this construct yields the classic [[dangling else]] [[ambiguous|ambiguity]].)
<pre>
S &larr; 'if' C 'then' S 'else' S / 'if' C 'then' S
</pre>
The parsing expression '''foo &(bar)''' matches and consumes the text "foo" but only if it is followed by the text "bar". The parsing expression '''foo !(bar)''' matches the text "foo" but only if it is ''not'' followed by the text "bar". The expression '''!(a+ b) a''' matches a single "a" but only if it is not part of an arbitrarily long sequence of a's followed by a b.
 
The following recursive rule matches Pascal-style nested comment syntax, (* which can (* nest *) like this *). The comment symbols appear in single quotes to distinguish them from PEG operators.
<pre>
Begin &larr; '(*'
End  &larr; '*)'
C    &larr; Begin N* End
N    &larr; C / (!Begin !End Z)
Z    &larr; any single character
</pre>
 
== Implementing parsers from parsing expression grammars ==
Any parsing expression grammar can be converted directly into a [[recursive descent parser]].<ref name="ford02">
{{cite web
|url=http://pdos.csail.mit.edu/~baford/packrat/thesis
|title=Packrat Parsing: a Practical Linear-Time Algorithm with Backtracking
|accessdate=2007-07-27
|last=Ford
|first=Bryan
|date=September 2002
|publisher=Massachusetts Institute of Technology
}}
<!-- I'm not sure how to incorporate all the info from this old ref into the new one above.  If you can do so, or can verify that it has already been done, please delete this.  --~~~~
 
Ford, Bryan, ''Packrat Parsing: a Practical Linear-Time Algorithm with Backtracking'', Master’s thesis, Massachusetts Institute of Technology, September, 2002, http://pdos.csail.mit.edu/~baford/packrat/thesis/
 
-->
</ref>  Due to the unlimited [[Parsing#Lookahead|lookahead]] capability that the grammar formalism provides, however, the resulting parser could exhibit [[exponential time]] performance in the worst case.
 
It is possible to obtain better performance for any parsing expression grammar by converting its recursive descent parser into a ''packrat parser'', which always runs in [[linear time]], at the cost of substantially greater storage space requirements.  A packrat parser<ref name="ford02" />
is a form of [[parsing|parser]] similar to a recursive descent parser in construction, except that during the parsing process it [[memoization|memoizes]] the intermediate results of all invocations of the [[mutual recursion|mutually recursive]] parsing functions, ensuring that each parsing function is only invoked at most once at a given input position. Because of this memoization, a packrat parser has the ability to parse many [[context-free grammar]]s and ''any'' parsing expression grammar (including some that do not represent context-free languages) in linear time.  Examples of memoized recursive descent parsers are known from at least as early as 1993.<ref name="merritt01">
{{cite web
|url=http://compilers.iecc.com/comparch/article/93-11-012
|title=Transparent Recursive Descent
|accessdate=2009-09-04
|last=Merritt
|first=Doug
|date=November 1993
|publisher=Usenet group comp.compilers
}}
</ref>
Note that this analysis of the performance of a packrat parser assumes that enough memory is available to hold all of the memoized results; in practice, if there were not enough memory, some parsing functions might have to be invoked more than once at the same input position, and consequently the parser could take more than linear time.
 
It is also possible to build [[LL parser]]s and [[LR parser]]s from parsing expression grammars, with better worst-case performance than a recursive descent parser, but the unlimited lookahead capability of the grammar formalism is then lost.  Therefore, not all languages that can be expressed using parsing expression grammars can be parsed by LL or LR parsers.
 
== Advantages ==
Compared to pure [[regular expressions]] (i.e. without back-references), PEGs are strictly more powerful, but require significantly more memory. For example, a regular expression inherently cannot find an arbitrary number of matched pairs of parentheses, because it is not recursive, but a PEG can. However, a PEG will require an amount of memory proportional to the length of the input, while a regular expression matcher will require only a constant amount of memory.
 
Any PEG can be parsed in linear time by using a packrat parser, as described above.
 
Parsers for languages expressed as a CFG, such as LR parsers, require a separate [[tokenization]] step to be done first, which breaks up the input based on the location of spaces, punctuation, etc. The tokenization is necessary because of the way these parsers use ''lookahead'' to parse CFGs that meet certain requirements in linear time.{{Citation needed|date=November 2011}} PEGs do not require tokenization to be a separate step, and tokenization rules can be written in the same way as any other grammar rule.
 
Many CFGs contain ambiguities, even when they're intended to describe unambiguous languages. The "[[dangling else]]" problem in C, C++, and Java is one example. These problems are often resolved by applying a rule outside of the grammar. In a PEG, these ambiguities never arise, because of prioritization.
 
== Disadvantages ==
 
=== Memory consumption ===
PEG parsing is typically carried out via ''packrat parsing'', which uses [[memoization]]<ref>{{cite web
| title = The Packrat Parsing and Parsing Expression Grammars Page
| author = Bryan Ford
| url = http://pdos.csail.mit.edu/~baford/packrat/
| accessdate = 23 Nov 2010
}}</ref><ref>{{cite web
| title = What is a Packrat Parser? What are Brzozowski Derivatives?
| author = Rick Jelliffe
| url = http://broadcast.oreilly.com/2010/03/what-is-a-packrat-parser-what.html
| accessdate = 23 Nov 2010
}}</ref> to eliminate redundant parsing steps.  Packrat parsing requires storage proportional to the total input size, rather than the depth of the parse tree as with LR parsers. This is a significant difference in many domains: for example, hand-written source code has an effectively constant expression nesting depth independent of the length of the program—expressions nested beyond a certain depth tend to get refactored.
 
For some grammars and some inputs, the depth of the parse tree can be proportional to the input size,<ref>for example, the LISP expression (x (x (x (x ....))))</ref>
so both an LR parser and a packrat parser will appear to have the same worst-case asymptotic performance.  A more accurate analysis would take the depth of the parse tree into account separately from the input size. This is similar to a situation which arises in [[graph algorithms]]: the [[Bellman–Ford algorithm]] and [[Floyd–Warshall algorithm]] appear to have the same running time (<math>O(|V|^3)</math>) if only the number of vertices is considered.  However, a more precise analysis which accounts for the number of edges as a separate parameter assigns the [[Bellman–Ford algorithm]] a time of <math>O(|V|*|E|)</math>, which is only quadratic in the size of the input (rather than cubic).
 
=== Indirect left recursion ===
 
PEGs cannot express ''[[left recursion|left-recursive]]'' rules where a rule refers to itself without moving forward in the string. For example, in the arithmetic grammar above, it would be tempting to move some rules around so that the precedence order of products and sums could be expressed in one line:
 
<pre>
Value   &larr; [0-9.]+ / '(' Expr ')'
Product &larr; Expr (('*' / '/') Expr)*
Sum    &larr; Expr (('+' / '-') Expr)*
Expr   &larr; Product / Sum / Value
</pre>
 
In this new grammar matching an <tt>Expr</tt> requires testing if a <tt>Product</tt> matches while matching a <tt>Product</tt> requires testing if an <tt>Expr</tt> matches. This [[circular definition]] cannot be resolved. However, left-recursive rules can always be rewritten to eliminate left-recursion. For example, the following left-recursive CFG rule:
 
<pre>
string-of-a &larr; string-of-a 'a' | 'a'
</pre>
 
can be rewritten in a PEG using the plus operator:
 
<pre>
string-of-a &larr; 'a'+
</pre>
 
The process of rewriting [[Left recursion#Indirect left recursion|''indirectly'' left-recursive]] rules is complex in some packrat parsers, especially when semantic actions are involved.
 
With some modification, traditional packrat parsing can support direct left recursion,<ref name="ford02"/><ref name="warth08">
{{cite paper
| author = Alessandro Warth, James R. Douglass, Todd Millstein
| title = Packrat Parsers Can Support Left Recursion
| date = January 2008
| url = http://www.vpri.org/pdf/tr2007002_packrat.pdf
| format = [[Portable Document Format|PDF]]
| publisher = Viewpoints Research Institute
| accessdate = 2008-10-02 }}
</ref><ref name="ruediste09">
{{cite paper
| author = Ruedi Steinmann
| title = Handling Left Recursion in Packrat Parsers
| date = March 2009
| url = http://n.ethz.ch/~ruediste/packrat.pdf
| format = [[Portable Document Format|PDF]]}}
</ref> but doing so results in a loss of the linear-time parsing property<ref name="warth08"/> which is generally the justification for using PEGs and packrat parsing in the first place.  Only the [[OMeta]] parsing algorithm<ref name="ometa08">
{{cite web
|url=http://www.vpri.org/pdf/tr2007002_packrat.pdf
|title=Packrat Parsers Can Support Left Recursion
|accessdate=2009-08-04
|date=January 2008
|publisher=PEPM '08
}}
</ref> supports full direct and indirect left recursion without additional attendant complexity (but again, at a loss of the linear time complexity), whereas all [[GLR parser]]s support left recursion.
 
=== Expressive power ===
 
PEG packrat parsers cannot recognize some unambiguous nondeterministic CFG rules, such as the following:<ref name="pegs"/>
 
<pre>
S &larr; 'x' S 'x' | 'x'
</pre>
Neither LL(k) nor LR(k) parsing algorithms are capable of recognizing this example. However, this language is parsable by a general CFG parser like the [[CYK algorithm]].
 
== See also ==
* [[Formal grammar]]
* [[Regular expressions]]
* [[Top-down parsing language]]
* [[Comparison of parser generators]]
* [[Parser combinators]]
 
== References ==
<references />
* {{cite conference
| first = Sérgio
| last = Medeiros
| authorlink =  
| coauthors = [[Roberto Ierusalimschy|Ierusalimschy, Roberto]]
| title = A parsing machine for PEGs
| booktitle = Proc. of the 2008 symposium on Dynamic languages
| pages = article #2
| publisher = [[Association for Computing Machinery|ACM]]
| year = 2008
| url =
| doi = 10.1145/1408681.1408683
| isbn = 978-1-60558-270-2
| accessdate =
}}
 
== External links ==
{{external links|date=September 2011}}
* [http://www.brynosaurus.com/pub/lang/peg-slides.pdf Parsing Expression Grammars: A Recognition-Based Syntactic Foundation] (PDF slides)
* [http://mathosproject.com/updates/convert-a-string-expression-into-a-lambda-expression/ Converting a string expression into a lambda expression using an expression parser]
* [http://bford.info/packrat/ The Packrat Parsing and Parsing Expression Grammars Page]
* [http://pdos.csail.mit.edu/~baford/packrat/thesis/ Packrat Parsing: a Practical Linear-Time Algorithm with Backtracking]
* The [[constructed language]] [[Lojban]] has a [http://www.digitalkingdom.org/~rlpowell/hobbies/lojban/grammar/ fairly large PEG grammar] allowing unambiguous parsing of Lojban text.
 
{{DEFAULTSORT:Parsing Expression Grammar}}
[[Category:Formal languages]]
[[Category:Parsing algorithms]]

Latest revision as of 05:10, 29 July 2014

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