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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]]; a string has exactly one valid [[parse tree]] or none. It is conjectured that there exist context-free languages that cannot be recognized by a PEG, but this is not yet proven.<ref name="For04" /> PEGs are well-suited to parsing computer languages (and artificial human languages such as [[Lojban]])
{{cite journal
| first= Bryan |last=Ford
Line 29:
== Definition ==
{{No footnotes|section|date=December 2022}}
A '''parsing expression''' is a kind of pattern that each string may either '''match''' or '''not match'''. In case of a match, there is a unique prefix of the string (which may be the whole string, the empty string, or something in between) which has been ''consumed'' by the parsing expression; this prefix is what one would usually think of as having matched the expression. However, whether a string matches a parsing expression ''may'' (because of look-ahead predicates) depend on parts of it which come after the consumed part. A '''parsing expression language''' is a set of all strings that match some specific parsing expression.<ref name="For04" />{{rp|Sec.3.4}}
A '''parsing expression grammar''' is a collection of named parsing expressions, which may reference each other. The effect of one such reference in a parsing expression is as if the whole referenced parsing expression was given in place of the reference. A parsing expression grammar also has a designated '''starting expression'''; a string matches the grammar if it matches its starting expression.
An element of a string matched is called a ''[[terminal symbol]]'', or '''terminal''' for short. Likewise the names assigned to parsing expressions are called ''[[nonterminal symbol]]s'', or '''nonterminals''' for short. These terms would be descriptive for [[Chomsky hierarchy|generative grammars]], but in the case of parsing expression grammars they are merely terminology, kept mostly because of being near ubiquitous in discussions of [[parsing]] algorithms.
=== Syntax ===
Both ''abstract'' and ''concrete'' syntaxes of parsing expressions are seen in the literature, and in this article. The abstract syntax is essentially a [[expression (mathematics)|mathematical formula]] and primarily used in theoretical contexts, whereas concrete syntax parsing expressions could be used directly to control a [[parser]]. The primary concrete syntax is that defined by Ford,<ref name="For04"/>{{rp|Fig.1}} although many tools have their own dialect of this. Other tools<ref>{{cite web |last1=Sirthias |first1=Mathias |title=Parboiled: Rule Construction in Java |website=[[GitHub]] |url=https://github.com/sirthias/parboiled/wiki/Rule-Construction-in-Java |access-date=13 January 2024}}</ref> can be closer to using a programming-language native encoding of abstract syntax parsing expressions as their concrete syntax.
==== Atomic parsing expressions ====
The two main kinds of parsing expressions not containing another parsing expression are individual terminal symbols and nonterminal symbols. In concrete syntax, terminals are placed inside quotes (single or double), whereas identifiers not in quotes denote nonterminals:
<syntaxhighlight lang="peg">
"terminal" Nonterminal 'another terminal'
</syntaxhighlight>
In the abstract syntax there is no formalised distinction, instead each symbol is supposedly defined as either terminal or nonterminal, but a common convention is to use upper case for nonterminals and lower case for terminals.
The concrete syntax also has a number of forms for classes of terminals:
* A <code>.</code> (period) is a parsing expression matching any single terminal.
* Brackets around a list of characters <code>[abcde]</code> form a parsing expression matching one of the numerated characters. As in [[regular expression]]s, these classes may also include ranges <code>[0-9A-Za-z]</code> written as a hyphen with the range endpoints before and after it. (Unlike the case in regular expressions, bracket character classes do not have <code>^</code> for negation; that end can instead be had via not-predicates.)
* Some dialects have further notation for predefined classes of characters, such as letters, digits, punctuation marks, or spaces; this is again similar to the situation in regular expressions.
In abstract syntax, such forms are usually formalised as nonterminals whose exact definition is elided for brevity; in Unicode, there are tens of thousands of characters that are letters. Conversely, theoretical discussions sometimes introduce atomic abstract syntax for concepts that can alternatively be expressed using composite parsing expressions. Examples of this include:
* the empty string ε (as a parsing expression, it matches every string and consumes no characters),
* end of input ''E'' (concrete syntax equivalent is <code>!.</code>), and
* failure <math>\bot</math> (matches nothing).
In the concrete syntax, quoted and bracketed terminals have backslash escapes, so that "[[line feed]] or [[carriage return]]" may be written <code>[\n\r]</code>. The abstract syntax counterpart of a quoted terminal of length greater than one would be the sequence of those terminals; <code>"bar"</code> is the same as <code>"b" "a" "r"</code>. The primary concrete syntax assigns no distinct meaning to terminals depending on whether they use single or double quotes, but some dialects treat one as case-sensitive and the other as case-insensitive.
==== Composite parsing expressions ====
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:
* ''Sequence'': ''e''<sub>1</sub> ''e''<sub>2</sub>
* ''Ordered choice'': ''e''<sub>1</sub> / ''e''<sub>2</sub>
* ''Zero-or-more'': ''e''*
* ''One-or-more'': ''e''+
* ''Optional'': ''e''?
* ''And-predicate'': &''e''
* ''Not-predicate'': !''e''
* ''Group'': (''e'')
Operator priorities are as follows, based on Table 1 in:<ref name="For04" />
{| class="wikitable"
! Operator !! Priority
|-
| (''e'') || 5
|-
|
|-
| &''e
|-
| ''e''<sub>1</sub> ''e''<sub>2</sub> || 2
|-
| ''e''<sub>1</sub> / ''e''<sub>2</sub> || 1
|}
==== Grammars ====
In the concrete syntax, a parsing expression grammar is simply a sequence of nonterminal definitions, each of which has the form
<syntaxhighlight lang="peg">
Identifier LEFTARROW Expression
</syntaxhighlight>
The <code>Identifier</code> is the nonterminal being defined, and the <code>Expression</code> is the parsing expression it is defined as referencing. The <code>LEFTARROW</code> varies a bit between dialects, but is generally some left-pointing arrow or assignment symbol, such as <code><-</code>, <code>←</code>, <code>:=</code>, or <code>=</code>. One way to understand it is precisely as making an assignment or definition of the nonterminal. Another way to understand it is as a contrast to the right-pointing arrow → used in the rules of a [[context-free grammar]]; with parsing expressions the flow of information goes from expression to nonterminal, not nonterminal to expression.
As a mathematical object, a parsing expression grammar is a tuple <math>(N,\Sigma,P,e_S)</math>, where <math>N</math> is the set of nonterminal symbols, <math>\Sigma</math> is the set of terminal symbols, <math>P</math> is a [[Function (mathematics)|function]] from <math>N</math> to the set of parsing expressions on <math>N \cup \Sigma</math>, and <math>e_S</math> is the starting parsing expression. Some concrete syntax dialects give the starting expression explicitly,<ref name="ptKupries">{{cite web |last1=Kupries |first1=Andreas |title=pt::peg_language - PEG Language Tutorial |url=https://core.tcl-lang.org/tcllib/doc/tcllib-1-21/embedded/md/tcllib/files/modules/pt/pt_peg_language.md |website=Tcl Library Source Code |access-date=14 January 2024}}</ref> but the primary concrete syntax instead has the implicit rule that the first nonterminal defined is the starting expression.
It is worth noticing that the primary dialect of concrete syntax parsing expression grammars does not have an explicit definition terminator or separator between definitions, although it is customary to begin a new definition on a new line; the <code>LEFTARROW</code> of the next definition is sufficient for finding the boundary, if one adds the constraint that a nonterminal in an <code>Expression</code> must not be followed by a <code>LEFTARROW</code>. However, some dialects may allow an explicit terminator, or outright require<ref name="ptKupries"/> it.
=== Example ===
This is a PEG that recognizes mathematical formulas that apply the basic five operations to non-negative integers.
<syntaxhighlight lang="peg">
Expr ← Sum
Sum ← Product (('+' / '-') Product)*
Product ← Power (('*' / '/') Power)*
Power ← Value ('^' Power)?
Value ← [0-9]+ / '(' Expr ')'
</syntaxhighlight>
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 a shortcut for the ten characters from <code>'0'</code> to <code>'9'</code>. (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'', ''Power'', ''Product'', ''Sum'', and ''Expr''. Note that rules ''Sum'' and ''Product'' don't lead to desired left-associativity of these operations (they don't deal with associativity at all, and it has to be handled in post-processing step after parsing), and the ''Power'' rule (by referring to itself on the right) results in desired right-associativity of exponent. Also note that a rule like {{code|Sum ← Sum (('+' / '-') Product)?|peg}} (with intention to achieve left-associativity) would cause infinite recursion, so it cannot be used in practice even though it can be expressed in the grammar.
=== Semantics ===
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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.
=== Operational interpretation of parsing expressions ===
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The '''not-predicate''' expression !''e'' succeeds if ''e'' fails and fails if ''e'' succeeds, again consuming no input in either case.
===
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|dangling else ambiguity]].)
<syntaxhighlight lang="peg">
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</syntaxhighlight>
The following recursive rule matches Pascal-style nested comment syntax, {{code|(* which can (* nest *) like this *)}}.
<syntaxhighlight lang="peg">
C ← Begin N* End
Begin ← '(*'
End ← '*)'
</syntaxhighlight>
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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 is 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,{{fact|date=December 2023}} with better worst-case performance than a recursive descent parser without memoization, 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.
=== Bottom-up PEG parsing ===
A pika parser<ref name="Hut20">{{cite arXiv
| first= Luke A. D. |last=Hutchison
| title = Pika parsing: parsing in reverse solves the left recursion and error recovery problems
| year = 2020
|class=cs.PL
| eprint = 2005.06444
}}</ref> uses dynamic programming to apply PEG rules bottom-up and right to left, which is the inverse of the normal recursive descent order of top-down, left to right. Parsing in reverse order solves the left recursion problem, allowing left-recursive rules to be used directly in the grammar without being rewritten into non-left-recursive form, and also confers optimal error recovery capabilities upon the parser, which historically proved difficult to achieve for recursive descent parsers.
== Advantages ==
=== No compilation required ===
Many parsing algorithms require a preprocessing step where the grammar is first compiled into an opaque executable form, often some sort of automaton. Parsing expressions can be executed directly (even if it is typically still advisable to transform the human-readable PEGs shown in this article into a more native format, such as [[Lisp (programming language)#Symbolic_expressions_(S-expressions)|S-expressions]], before evaluating them).
=== Compared to regular expressions ===
Compared to pure [[regular expressions]] (i.e., describing a language recognisable using a [[finite automaton]]), PEGs are vastly more powerful. In particular they can handle unbounded recursion, and so match parentheses down to an arbitrary nesting depth; regular expressions can at best keep track of nesting down to some fixed depth, because a finite automaton (having a finite set of internal states) can only distinguish finitely many different nesting depths. In more theoretical terms, <math> \{a^n b^n\}_{n \geqslant 0} </math> (the language of all strings of zero or more <math>a</math>'s, followed by an ''equal number'' of <math>b</math>s) is not a regular language, but it is easily seen to be a parsing expression language, matched by the grammar
<syntaxhighlight lang="peg">
start ← AB !.
AB ← ('a' AB 'b')?
</syntaxhighlight>
Here <code>AB !.</code> is the starting expression. The <code>!.</code> part enforces that the input ends after the <code>AB</code>, by saying “there is no next character”; unlike regular expressions, which have magic constraints <code>$</code> or <code>\Z</code> for this, parsing expressions can express the end of input using only the basic primitives.
The <code>*</code>, <code>+</code>, and <code>?</code> of parsing expressions are similar to those in regular expressions, but a difference is that these operate strictly in a greedy mode. This is ultimately due to <code>/</code> being an ordered choice. A consequence is that something can match as a regular expression which does not match as parsing expression:
: <code>[ab]?[bc][cd]</code>
is both a valid regular expression and a valid parsing expression. As regular expression, it matches <code>bc</code>, but as parsing expression it does not match, because the <code>[ab]?</code> will match the <code>b</code>, then <code>[bc]</code> will match the <code>c</code>, leaving nothing for the <code>[cd]</code>, so at that point matching the sequence fails. "Trying again" with having <code>[ab]?</code> match the empty string is explicitly against the semantics of parsing expressions; this is not an edge case of a particular matching algorithm, instead it is the sought behaviour.
Even regular expressions that depend on nondeterminism ''can'' be compiled into a parsing expression grammar, by having a separate nonterminal for every state of the corresponding [[deterministic finite automaton|DFA]] and encoding its transition function into the definitions of these nonterminals —
<syntaxhighlight lang="peg">
A ← 'x' B / 'y' C
</syntaxhighlight>
is effectively saying "from state A transition to state B if the next character is x, but to state C if the next character is y" — but this works because nondeterminism can be eliminated within the realm of regular languages. It would not make use of the parsing expression variants of the repetition operations.
=== Compared to context-free grammars ===
PEGs can comfortably be given in terms of characters, whereas [[context-free grammar]]s (CFGs) are usually given in terms of tokens, thus requiring an extra step of tokenisation in front of parsing proper.<ref>CFGs ''can'' be used to describe the syntax of common programming languages down to the character level, but doing so is rather cumbersome, because the standard tokenisation rule that a token consists of the longest consecutive sequence of characters of the same kind does not mesh well with the nondeterministic side of CFGs. To formalise that whitespace between two adjacent tokens is mandatory if the characters on both sides of the token boundary are letters, but optional if they are non-letters, a CFG needs multiple variants of most nonterminals, to keep track of what kind of character has to be at the boundary. If there are <math>3</math> different kinds of non-whitespace characters, that adds up to <math>3^2 = 9</math> possible variants per nonterminal — significantly bloating the grammar.</ref> An advantage of not having a separate tokeniser is that different parts of the language (for example embedded [[Domain-specific language|mini-language]]s) can easily have different tokenisation rules.
In the strict formal sense, PEGs are likely incomparable to CFGs, but practically there are many things that PEGs can do which pure CFGs cannot, whereas it is difficult to come up with examples of the contrary. In particular PEGs can be crafted to natively resolve ambiguities, such as the "[[dangling else]]" problem in C, C++, and Java, whereas CFG-based parsing often needs a rule outside of the grammar to resolve them. Moreover any PEG can be parsed in linear time by using a packrat parser, as described above, whereas parsing according to a general CFG is asymptotically equivalent<ref>{{cite journal |last1=Lee |first1=Lillian |title=Fast Context-free Grammar Parsing Requires Fast Boolean Matrix Multiplication |journal=J. ACM |date=January 2002 |volume=49 |issue=1 |page=1–15 |doi=10.1145/505241.505242|arxiv=cs/0112018 }}</ref> to [[logical matrix|boolean matrix]] multiplication (thus likely between quadratic and cubic time).
One classical example of a formal language which is provably not context-free is the language <math> \{a^n b^n c^n\}_{n \geqslant 0} </math>: an arbitrary number of <math>a</math>s are followed by an equal number of <math>b</math>s, which in turn are followed by an equal number of <math>c</math>s. This, too, is a parsing expression language, matched by the grammar
<syntaxhighlight lang="peg">
start ← AB 'c'*
AB ← 'a' AB 'b' / &(BC !.)
BC ← ('b' BC 'c')?
</syntaxhighlight>
For <code>AB</code> to match, the first stretch of <math>a</math>s must be followed by an equal number of <math>b</math>s, and in addition <code>BC</code> has to match where the <math>a</math>s switch to <math>b</math>s, which means those <math>b</math>s are followed by an equal number of <math>c</math>s.
== Disadvantages ==
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|archive-date= 28 July 2011
|date=10 March 2010
}}</ref> to eliminate redundant parsing steps. Packrat parsing requires internal storage proportional to the total input size, rather than to the depth of the parse tree as with LR parsers. Whether this is a significant difference depends on circumstances; if parsing is a service provided as a [[Function (computer programming)|function]] then the parser will have stored the full parse tree up until returning it, and already that parse tree will typically be of size proportional to the total input size. If parsing is instead provided as a [[Generator (computer programming)|generator]] then one might get away with only keeping parts of the parse tree in memory, but the feasibility of this depends on the grammar. A parsing expression grammar can be designed so that only after consuming the full input will the parser discover that it needs to backtrack to the beginning,<ref>For example, there could at the very end of input be a directive to the effect that “in this file, comma is a [[decimal separator]], so all those function calls f(3,14*r) you thought had two arguments? They don't. Now go back to the start of input and parse it all again.” Arguably that would be a poor design of the input language, but the point is that parsing expression grammars are powerful enough to handle this, purely as a matter of syntax.</ref> which again could require storage proportional to total input size.
For recursive 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. However in many domains, for example hand-written source code, the expression nesting depth has an effectively constant bound quite independent of the length of the program, because expressions nested beyond a certain depth tend to get [[refactor]]ed. When it is not necessary to keep the full parse tree, a more accurate analysis would take the depth of the parse tree into account separately from the input size.<ref>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 quadratic for sparse graphs with <math>|E| \in O(|V|)</math>.</ref>
=== Computational model ===
In order to attain linear overall complexity, the storage used for memoization must furthermore provide [[amortized analysis|amortized constant time]] access to individual data items memoized. In practice that is no problem — for example a dynamically sized [[hash table]] attains this – but that makes use of [[Pointer (computer programming)|pointer]] arithmetic, so it presumes having a [[random-access machine]]. Theoretical discussions of data structures and algorithms have an unspoken tendency to presume a more restricted model (possibly that of [[lambda calculus]], possibly that of [[Scheme (programming language)|Scheme]]), where a sparse table rather has to be built using trees, and data item access is not constant time. Traditional parsing algorithms such as the [[LL parser]] are not affected by this, but it becomes a penalty for the reputation of packrat parsers: they rely on operations of seemingly ill repute.
Viewed the other way around, this says packrat parsers tap into computational power readily available in real life systems, that older parsing algorithms do not understand to employ.
=== Indirect left recursion ===
A PEG is called ''well-formed''<ref name="For04"/> if it contains no ''[[left recursion|left-recursive]]'' rules, i.e., rules that allow a nonterminal to expand to an expression in which the same nonterminal occurs as the leftmost symbol. For a left-to-right top-down parser, such rules cause infinite regress: parsing will continually expand the same nonterminal without moving forward in the string. Therefore, to allow packrat parsing, left recursion must be eliminated.
==== Practical significance ====
Direct recursion, be that left or right, is important in context-free grammars, because there recursion is the only way to describe repetition:
<syntaxhighlight lang="peg">
Sum → Term | Sum '+' Term | Sum '-' Term
Args → Arg | Arg ',' Args
</syntaxhighlight>
People trained in using context-free grammars often come to PEGs expecting to use the same idioms, but parsing expressions can do repetition without recursion:
<syntaxhighlight lang="peg">
Sum ← Term ( '+' Term / '-' Term )*
Args ← Arg ( ',' Arg )*
</syntaxhighlight>
A difference lies in the [[abstract syntax tree]]s generated: with recursion each <code>Sum</code> or <code>Args</code> can have at most two children, but with repetition there can be arbitrarily many. If later stages of processing require that such lists of children are recast as trees with bounded [[degree (graph theory)|degree]], for example microprocessor instructions for addition typically only allow two operands, then properties such as [[left-associative|left-associativity]] would be imposed after the PEG-directed parsing stage.
Therefore left-recursion is practically less likely to trouble a PEG packrat parser than, say, an LL(k) context-free parser, unless one insists on using context-free idioms. However, not all cases of recursion are about repetition.
==== Non-repetition left-recursion ====
For example, in the arithmetic grammar above, it could seem tempting to express operator precedence as a matter of ordered choice — <code>Sum / Product / Value</code> would mean first try viewing as <code>Sum</code> (since we parse top–down), second try viewing as <code>Product</code>, and only third try viewing as <code>Value</code> — rather than via nesting of definitions. This (non-well-formed) grammar seeks to keep precedence order only in one line:
<syntaxhighlight lang="peg">
Value ← [0-9.]+ / '(' Expr ')'
Product ← Expr (('*' / '/') Expr)
Sum ← Expr (('+' / '-') Expr)
Expr ←
</syntaxhighlight>
<syntaxhighlight lang="peg">
string-of-a ← string-of-a 'a' | 'a'
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</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="warth08"/> 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.
A common first impression of PEGs is that they look like CFGs with certain convenience features — repetition operators <code>*+?</code> as in regular expressions and lookahead predicates <code>&!</code> — plus ordered choice for disambiguation. This understanding can be sufficient when one's goal is to create a parser for a language, but it is not sufficient for more theoretical discussions of the computational power of parsing expressions. In particular the [[Nondeterministic Turing machine|nondeterminism]] inherent in the unordered choice <code>|</code> of context-free grammars makes it very different from the deterministic ordered choice <code>/</code>.
=== The midpoint problem ===
PEG packrat parsers cannot recognize some unambiguous nondeterministic CFG rules, such as the following:<ref name="pegs"/>
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Neither [[LL(k)]] nor LR(k) parsing algorithms are capable of recognizing this example. However, this grammar can be used by a general CFG parser like the [[CYK algorithm]]. However, the ''language'' in question can be recognised by all these types of parser, since it is in fact a regular language (that of strings of an odd number of x's).
It is instructive to work out exactly what a PEG parser does when attempting to match
<syntaxhighlight lang="peg">
S ← 'x' S 'x' / 'x'
</syntaxhighlight>
against the string <code>xxxxxq</code>. As expected, it recursively tries to match the nonterminal <code>S</code> at increasing positions in this string, until failing the match against the <code>q</code>, and after that begins to backtrack. This goes as follows:
Position: 123456
String: xxxxxq
Results: ↑ Pos.6: Neither branch of S matches
↑ Pos.5: First branch of S fails, second branch succeeds, yielding match of length 1.
↑ Pos.4: First branch of S fails, second branch succeeds, yielding match of length 1.
↑ Pos.3: First branch of S succeeds, yielding match of length 3.
↑ Pos.2: First branch of S fails, because after the S match at 3 comes a q.
Second branch succeeds, yielding match of length 1.
↑ Pos.1: First branch of S succeeds, yielding match of length 3.
Matching against a parsing expression is [[greedy algorithm|greedy]], in the sense that the first success encountered is the only one considered. Even if locally the choices are ordered longest first, there is no guarantee that this greedy matching will find the globally longest match.
=== Ambiguity detection and influence of rule order on language that is matched ===
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</syntaxhighlight>
==
It is an open problem to give a concrete example of a context-free language which cannot be recognized by a parsing expression grammar.<ref name="For04" /> In particular, it is open whether a parsing expression grammar can recognize the language of palindromes.<ref>{{cite arXiv |last1=Loff |first1=Bruno |last2=Moreira |first2=Nelma |last3=Reis |first3=Rogério |date=2020-02-14 |title=The computational power of parsing expression grammars |class=cs.FL |eprint=1902.08272 }}</ref>
The class of parsing expression languages is closed under set intersection and complement, thus also under set union.<ref name="For04" />{{rp|Sec.3.4}}
=== Undecidability of emptiness ===
In stark contrast to the case for context-free grammars, it is not possible to generate elements of a parsing expression language from its grammar. Indeed, it is algorithmically [[undecidable problem|undecidable]] whether the language recognised by a parsing expression grammar is empty! One reason for this is that any instance of the [[Post correspondence problem]] reduces to an instance of the problem of deciding whether a parsing expression language is empty.
Recall that an instance of the Post correspondence problem consists of a list <math> (\alpha_1,\beta_1), (\alpha_2,\beta_2), \dotsc, (\alpha_n,\beta_n) </math> of pairs of strings (of terminal symbols). The problem is to determine whether there exists a sequence <math>\{k_i\}_{i=1}^m</math> of indices in the range <math>\{1,\dotsc,n\}</math> such that <math> \alpha_{k_1} \alpha_{k_2} \dotsb \alpha_{k_m} = \beta_{k_1} \beta_{k_2} \dotsb \beta_{k_m} </math>. To [[reduction (complexity)|reduce]] this to a parsing expression grammar, let <math> \gamma_0, \gamma_1, \dotsc, \gamma_n </math> be arbitrary pairwise distinct equally long strings of terminal symbols (already with <math>2</math> distinct symbols in the terminal symbol alphabet, length <math>\lceil \log_2(n+1) \rceil</math> suffices) and consider the parsing expression grammar
<math display="block">
\begin{aligned}
S &\leftarrow \&(A \, !.) \&(B \, !.) (\gamma_1/\dotsb/\gamma_n)^+ \gamma_0 \\
A &\leftarrow \gamma_0 / \gamma_1 A \alpha_1 / \dotsb / \gamma_n A \alpha_n \\
B &\leftarrow \gamma_0 / \gamma_1 B \beta_1 / \dotsb / \gamma_n B \beta_n
\end{aligned}
</math>
Any string matched by the nonterminal <math>A</math> has the form <math>\gamma_{k_m} \dotsb \gamma_{k_2} \gamma_{k_1} \gamma_0 \alpha_{k_1} \alpha_{k_2} \dotsb \alpha_{k_m}</math> for some indices <math>k_1,k_2,\dotsc,k_m</math>. Likewise any string matched by the nonterminal <math>B</math> has the form <math>\gamma_{k_m} \dotsb \gamma_{k_2} \gamma_{k_1} \gamma_0 \beta_{k_1} \beta_{k_2} \dotsb \beta_{k_m}</math>. Thus any string matched by <math>S</math> will have the form <math>\gamma_{k_m} \dotsb \gamma_{k_2} \gamma_{k_1} \gamma_0 \rho</math> where <math> \rho = \alpha_{k_1} \alpha_{k_2} \dotsb \alpha_{k_m} = \beta_{k_1} \beta_{k_2} \dotsb \beta_{k_m}</math>.
== Practical use ==
* [[Python (programming language)|Python]] reference implementation [[CPython]] introduced a PEG parser in version 3.9 as an alternative to the [[LL parser|LL(1) parser]] and uses just PEG from version 3.10.<ref>{{Cite web |title=PEP 617 – New PEG parser for CPython {{!}} peps.python.org |url=https://peps.python.org/pep-0617/ |access-date=2023-01-16 |website=peps.python.org}}</ref>
* The [[Jq_(programming_language)#Parsing_Expression_Grammars|jq programming language]] uses a formalism closely related to PEG.
* The [[Lua (programming language)|Lua]] authors created [https://www.inf.puc-rio.br/~roberto/lpeg/ LPeg], a [[pattern-matching]] library that uses PEG instead of [[regular expressions]],<ref>{{cite journal |last1=Ierusalimschy |first1=Roberto |title=A text pattern-matching tool based on Parsing Expression Grammars |journal=Software: Practice and Experience |date=10 March 2009 |volume=39 |issue=3 |pages=221–258 |doi=10.1002/spe.892 |url=https://onlinelibrary.wiley.com/doi/10.1002/spe.892 |language=en |issn=0038-0644|url-access=subscription }}</ref> as well as the [https://www.inf.puc-rio.br/~roberto/lpeg/re.html re] module which implements a regular-expression-like syntax utilizing the LPeg library.<ref>{{cite web |last1=Ierusalimschy |first1=Roberto |title=LPeg.re - Regex syntax for LPEG |url=https://www.inf.puc-rio.br/~roberto/lpeg/re.html |website=www.inf.puc-rio.br}}</ref>
== See also ==
* [[Boolean grammar|Boolean context-free grammar]]
* [[Compiler Description Language]] (CDL)
* [[Formal grammar]]
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