Context-free grammar

Informal languagetheory, acontext-free grammar(CFG) is aformal grammarwhoseproduction rules can be applied to anonterminal symbolregardless of its context. In particular, in a context-free grammar, each production rule is of the form

Simplified excerpt of the formal grammarfor theC programming language(left), and a derivation of a piece of C code (right) from the nonterminal symbol$\langle {\text{Stmt}}\rangle$.Nonterminal symbols are blue and terminal symbols are red.

$A\ \to \ \ Alpha$

with$A$asinglenonterminal symbol, and$\ Alpha$a string of terminals and/or nonterminals ($\ Alpha$can be empty). Regardless of which symbols surround it, the single nonterminal$A$on the left hand side can always be replaced by$\ Alpha$on the right hand side. This distinguishes it from acontext-sensitive grammar,which can have production rules in the form$\ Alpha A\beta \rightarrow \ Alpha \gamma \beta$with$A$a nonterminal symbol and$\ Alpha$,$\beta$,and$\gamma$strings of terminal and/or nonterminal symbols.

A formal grammar is essentially a set of production rules that describe all possible strings in a given formal language. Production rules are simple replacements. For example, the first rule in the picture,

$\langle {\text{Stmt}}\rangle \to \langle {\text{Id}}\rangle =\langle {\text{Expr}}\rangle;$

replaces$\langle {\text{Stmt}}\rangle$with$\langle {\text{Id}}\rangle =\langle {\text{Expr}}\rangle;$.There can be multiple replacement rules for a given nonterminal symbol. The language generated by a grammar is the set of all strings of terminal symbols that can be derived, by repeated rule applications, from some particular nonterminal symbol ( "start symbol" ). Nonterminal symbols are used during the derivation process, but do not appear in its final result string.

Languagesgenerated by context-free grammars are known ascontext-free languages(CFL). Different context-free grammars can generate the same context-free language. It is important to distinguish the properties of the language (intrinsic properties) from the properties of a particular grammar (extrinsic properties). Thelanguage equalityquestion (do two given context-free grammars generate the same language?) isundecidable.

Context-free grammars arise inlinguisticswhere they are used to describe the structure of sentences and words in anatural language,and they were invented by the linguistNoam Chomskyfor this purpose. By contrast, incomputer science,as the use of recursively-defined concepts increased, they were used more and more. In an early application, grammars are used to describe the structure ofprogramming languages.In a newer application, they are used in an essential part of theExtensible Markup Language(XML) called thedocument type definition.

In linguistics, some authors use the termphrase structure grammarto refer to context-free grammars, whereby phrase-structure grammars are distinct fromdependency grammars.In computer science, a popular notation for context-free grammars isBackus–Naur form,or BNF.

Background

Since at least the time of the ancient Indian scholarPāṇini,linguists have described thegrammarsof languages in terms of their block structure, and described how sentences arerecursivelybuilt up from smaller phrases, and eventually individual words or word elements. An essential property of these block structures is that logical units never overlap. For example, the sentence:

John, whose blue car was in the garage, walked to the grocery store.

can be logically parenthesized (with the logical metasymbols[ ]) as follows:

[John[,[whose[blue car]][was[in[the garage]]],]][walked[to[the[grocery store]]]].

A context-free grammar provides a simple and mathematically precise mechanism for describing the methods by which phrases in some natural language are built from smaller blocks, capturing the "block structure" of sentences in a natural way. Its simplicity makes the formalism amenable to rigorous mathematical study. Important features of natural language syntax such asagreementandreferenceare not part of the context-free grammar, but the basic recursive structure of sentences, the way in which clauses nest inside other clauses, and the way in which lists of adjectives and adverbs are swallowed by nouns and verbs, is described exactly.

Context-free grammars are a special form ofSemi-Thue systemsthat in their general form date back to the work ofAxel Thue.

The formalism of context-free grammars was developed in the mid-1950s byNoam Chomsky,and also theirclassification as a special typeofformal grammar(which he calledphrase-structure grammars).Some authors, however, reserve the term for more restricted grammars in the Chomsky hierarchy: context-sensitive grammars or context-free grammars. In a broader sense,phrase structure grammarsare also known as constituency grammars. The defining trait of phrase structure grammars is thus their adherence to the constituency relation, as opposed to the dependency relation ofdependency grammars.In Chomsky'sgenerative grammarframework, the syntax of natural language was described by context-free rules combined with transformation rules.

Block structure was introduced into computerprogramming languagesby theAlgolproject (1957–1960), which, as a consequence, also featured a context-free grammarto describe the resulting Algol syntax. This became a standard feature of computer languages, and the notation for grammars used in concrete descriptions of computer languages came to be known asBackus–Naur form,after two members of the Algol language design committee.The "block structure" aspect that context-free grammars capture is so fundamental to grammar that the terms syntax and grammar are often identified with context-free grammar rules, especially in computer science. Formal constraints not captured by the grammar are then considered to be part of the "semantics" of the language.

Context-free grammars are simple enough to allow the construction of efficientparsing algorithmsthat, for a given string, determine whether and how it can be generated from the grammar. AnEarley parseris an example of such an algorithm, while the widely usedLRandLL parsersare simpler algorithms that deal only with more restrictive subsets of context-free grammars.

Formal definitions

A context-free grammarGis defined by the 4-tuple$G=(V,\Sigma,R,S)$, where

1. Vis a finite set; each element$v\in V$is calleda nonterminal characteror avariable.Each variable represents a different type of phrase or clause in the sentence. Variables are also sometimes called syntactic categories. Each variable defines a sub-language of the language defined byG.
2. Σis a finite set ofterminals, disjoint fromV,which make up the actual content of the sentence. The set of terminals is the Alpha bet of the language defined by the grammarG.
3. Ris a finiterelationin$V\times (V\cup \Sigma )^{*}$,where the asterisk represents theKleene staroperation. The members ofRare called the(rewrite) rules orproductions of the grammar. (also commonly symbolized by aP)
4. Sis the start variable (or start symbol), used to represent the whole sentence (or program). It must be an element ofV.

Production rule notation

Aproduction ruleinRis formalized mathematically as a pair$(\ Alpha,\beta )\in R$,where$\ Alpha \in V$is a nonterminal and$\beta \in (V\cup \Sigma )^{*}$is astringof variables and/or terminals; rather than usingordered pairnotation, production rules are usually written using an arrow operator with$\ Alpha$as its left hand side andβas its right hand side: $\ Alpha \rightarrow \beta$.

It is allowed forβto be theempty string,and in this case it is customary to denote it by ε. The form$\ Alpha \rightarrow \varepsilon$is called anε-production.

It is common to list all right-hand sides for the same left-hand side on the same line, using | (thevertical bar) to separate them. Rules$\ Alpha \rightarrow \beta _{1}$and$\ Alpha \rightarrow \beta _{2}$can hence be written as$\ Alpha \rightarrow \beta _{1}\mid \beta _{2}$.In this case,$\beta _{1}$and$\beta _{2}$are called the first and second alternative, respectively.

Rule application

For any strings$u,v\in (V\cup \Sigma )^{*}$,we sayudirectly yieldsv,written as$u\Rightarrow v\,$,if$\exists (\ Alpha,\beta )\in R$with$\ Alpha \in V$and$u_{1},u_{2}\in (V\cup \Sigma )^{*}$such that$u\,=u_{1}\ Alpha u_{2}$and$v\,=u_{1}\beta u_{2}$.Thus,vis a result of applying the rule$(\ Alpha,\beta )$tou.

Repetitive rule application

For any strings$u,v\in (V\cup \Sigma )^{*},$we sayuyieldsvorvisderivedfromuif there is a positive integerkand strings$u_{1},\ldots,u_{k}\in (V\cup \Sigma )^{*}$such that$u=u_{1}\Rightarrow u_{2}\Rightarrow \cdots \Rightarrow u_{k}=v$.This relation is denoted$u{\stackrel {*}{\Rightarrow }}v$,or$u\Rightarrow \Rightarrow v$in some textbooks. If$k\geq 2$,the relation$u{\stackrel {+}{\Rightarrow }}v$holds. In other words,$({\stackrel {*}{\Rightarrow }})$and$({\stackrel {+}{\Rightarrow }})$are thereflexive transitive closure(allowing a string to yield itself) and thetransitive closure(requiring at least one step) of$(\Rightarrow )$,respectively.

Context-free language

The language of a grammar$G=(V,\Sigma,R,S)$is the set

$L(G)=\{w\in \Sigma ^{*}:S{\stackrel {*}{\Rightarrow }}w\}$

of all terminal-symbol strings derivable from the start symbol.

A languageLis said to be a context-free language (CFL), if there exists a CFGG,such that$L\,=\,L(G)$.

Non-deterministic pushdown automatarecognize exactly the context-free languages.

Examples

Words concatenated with their reverse

The grammar$G=(\{S\},\{a,b\},P,S)$,with productions

S→aSa,

S→bSb,

S→ ε,

is context-free. It is not proper since it includes an ε-production. A typical derivation in this grammar is

S→aSa→aaSaa→aabSbaa→aabbaa.

This makes it clear that $L(G)=\{ww^{R}:w\in \{a,b\}^{*}\}$. The language is context-free, however, it can be proved that it is notregular.

If the productions

S→a,

S→b,

are added, a context-free grammar for the set of allpalindromesover the Alpha bet{a,b}is obtained.

Well-formed parentheses

The canonical example of a context-free grammar is parenthesis matching, which is representative of the general case. There are two terminal symbols "(" and ")" and one nonterminal symbol S. The production rules are

S→SS,

S→ (S),

S→ ()

The first rule allows the S symbol to multiply; the second rule allows the S symbol to become enclosed by matching parentheses; and the third rule terminates the recursion.

Well-formed nested parentheses and square brackets

A second canonical example is two different kinds of matching nested parentheses, described by the productions:

S→SS

S→ ()

S→ (S)

S→ []

S→ [S]

with terminal symbols [ ] ( ) and nonterminal S.

The following sequence can be derived in that grammar:

([ [ [ ()() [ ][ ] ] ]([ ]) ])

Matching pairs

In a context-free grammar, we can pair up characters the way we do withbrackets.The simplest example:

S → aSb

S → ab

This grammar generates the language$\{a^{n}b^{n}:n\geq 1\}$,which is notregular(according to thepumping lemma for regular languages).

The special character ε stands for the empty string. By changing the above grammar to

S → aSb

S → ε

we obtain a grammar generating the language$\{a^{n}b^{n}:n\geq 0\}$instead. This differs only in that it contains the empty string while the original grammar did not.

Distinct number of a's and b's

A context-free grammar for the language consisting of all strings over {a,b} containing an unequal number of a's and b's:

S → T | U

T → VaT | VaV | TaV

U → VbU | VbV | UbV

V → aVbV | bVaV | ε

Here, the nonterminal T can generate all strings with more a's than b's, the nonterminal U generates all strings with more b's than a's and the nonterminal V generates all strings with an equal number of a's and b's. Omitting the third alternative in the rules for T and U does not restrict the grammar's language.

Second block of b's of double size

Another example of a non-regular language is$\{{\text{b}}^{n}{\text{a}}^{m}{\text{b}}^{2n}:n\geq 0,m\geq 0\}$.It is context-free as it can be generated by the following context-free grammar:

S→bSbb|A

A→aA|ε

First-order logic formulas

Theformation rulesfor the terms and formulas of formal logic fit the definition of context-free grammar, except that the set of symbols may be infinite and there may be more than one start symbol.

Examples of languages that are not context free

In contrast to well-formed nested parentheses and square brackets in the previous section, there is no context-free grammar for generating all sequences of two different types of parentheses, each separately balanceddisregarding the other,where the two types need not nest inside one another, for example:

[ ( ] )

or

[ [ [ [(((( ] ] ] ]))))(([ ))(([ ))([ )( ])( ])( ])

The fact that this language is not context free can be proven usingpumping lemma for context-free languagesand a proof by contradiction, observing that all words of the form ${(}^{n}{[}^{n}{)}^{n}{]}^{n}$ should belong to the language. This language belongs instead to a more general class and can be described by aconjunctive grammar,which in turn also includes other non-context-free languages, such as the language of all words of the form ${\text{a}}^{n}{\text{b}}^{n}{\text{c}}^{n}$.

Regular grammars

Main article:Regular grammar

Everyregular grammaris context-free, but not all context-free grammars are regular.The following context-free grammar, for example, is also regular.

S→a

S→aS

S→bS

The terminals here areaandb,while the only nonterminal isS. The language described is all nonempty strings of$a$s and$b$s that end in$a$.

This grammar isregular:no rule has more than one nonterminal in its right-hand side, and each of these nonterminals is at the same end of the right-hand side.

Every regular grammar corresponds directly to anondeterministic finite automaton,so we know that this is aregular language.

Using vertical bars, the grammar above can be described more tersely as follows:

S→a|aS|bS

Derivations and syntax trees

Aderivationof a string for a grammar is a sequence of grammar rule applications that transform the start symbol into the string. A derivation proves that the string belongs to the grammar's language.

A derivation is fully determined by giving, for each step:

• the rule applied in that step
• the occurrence of its left-hand side to which it is applied

For clarity, the intermediate string is usually given as well.

For instance, with the grammar:

1. S→S+S
2. S→ 1
3. S→a

the string

1 + 1 +a

can be derived from the start symbolSwith the following derivation:

S

→S+S(by rule 1. onS)

→S+S+S(by rule 1. on the secondS)

→ 1 +S+S(by rule 2. on the firstS)

→ 1 + 1 +S(by rule 2. on the secondS)

→ 1 + 1 +a(by rule 3. on the thirdS)

Often, a strategy is followed that deterministically chooses the next nonterminal to rewrite:

• in aleftmost derivation,it is always the leftmost nonterminal;
• in arightmost derivation,it is always the rightmost nonterminal.

Given such a strategy, a derivation is completely determined by the sequence of rules applied. For instance, one leftmost derivation of the same string is

S

→S+S(by rule 1 on the leftmostS)

→ 1 +S(by rule 2 on the leftmostS)

→ 1 +S+S(by rule 1 on the leftmostS)

→ 1 + 1 +S(by rule 2 on the leftmostS)

→ 1 + 1 +a(by rule 3 on the leftmostS),

which can be summarized as

rule 1

rule 2

rule 1

rule 2

rule 3.

One rightmost derivation is:

S

→S+S(by rule 1 on the rightmostS)

→S+S+S(by rule 1 on the rightmostS)

→S+S+a(by rule 3 on the rightmostS)

→S+ 1 +a(by rule 2 on the rightmostS)

→ 1 + 1 +a(by rule 2 on the rightmostS),

which can be summarized as

rule 1

rule 1

rule 3

rule 2

rule 2.

The distinction between leftmost derivation and rightmost derivation is important because in mostparsersthe transformation of the input is defined by giving a piece of code for every grammar rule that is executed whenever the rule is applied. Therefore, it is important to know whether the parser determines a leftmost or a rightmost derivation because this determines the order in which the pieces of code will be executed. See for an exampleLL parsersandLR parsers.

A derivation also imposes in some sense a hierarchical structure on the string that is derived. For example, if the string "1 + 1 + a" is derived according to the leftmost derivation outlined above, the structure of the string would be:

{{1}_S+ {{1}_S+ {a}_S}_S}_S

where{...}_Sindicates a substring recognized as belonging toS.This hierarchy can also be seen as a tree:

This tree is called aparse treeor "concrete syntax tree" of the string, by contrast with theabstract syntax tree.In this case the presented leftmost and the rightmost derivations define the same parse tree; however, there is another rightmost derivation of the same string

S

→S+S(by rule 1 on the rightmostS)

→S+a(by rule 3 on the rightmostS)

→S+S+a(by rule 1 on the rightmostS)

→S+ 1 +a(by rule 2 on the rightmostS)

→ 1 + 1 +a(by rule 2 on the rightmostS),

which defines a string with a different structure

{{{1}_S+ {1}_S}_S+ {a}_S}_S

and a different parse tree:

Note however that both parse trees can be obtained by both leftmost and rightmost derivations. For example, the last tree can be obtained with the leftmost derivation as follows:

S

→S+S(by rule 1 on the leftmostS)

→S+S+S(by rule 1 on the leftmostS)

→ 1 +S+S(by rule 2 on the leftmostS)

→ 1 + 1 +S(by rule 2 on the leftmostS)

→ 1 + 1 +a(by rule 3 on the leftmostS),

If a string in the language of the grammar has more than one parsing tree, then the grammar is said to be anambiguous grammar.Such grammars are usually hard to parse because the parser cannot always decide which grammar rule it has to apply. Usually, ambiguity is a feature of the grammar, not the language, and an unambiguous grammar can be found that generates the same context-free language. However, there are certain languages that can only be generated by ambiguous grammars; such languages are calledinherently ambiguous languages.

Normal forms

Every context-free grammar with no ε-production has an equivalent grammar inChomsky normal form,and a grammar inGreibach normal form."Equivalent" here means that the two grammars generate the same language.

The especially simple form of production rules in Chomsky normal form grammars has both theoretical and practical implications. For instance, given a context-free grammar, one can use the Chomsky normal form to construct apolynomial-timealgorithm that decides whether a given string is in the language represented by that grammar or not (theCYK algorithm).

Closure properties

Context-free languages areclosedunder the various operations, that is, if the languagesKandLare context-free, so is the result of the following operations:

• unionK∪L;concatenationK∘L;Kleene starL^*
• substitution(in particularhomomorphism)
• inverse homomorphism
• intersectionwith aregular language

They are not closed under general intersection (hence neither undercomplementation) and set difference.

Decidable problems

The following are some decidable problems about context-free grammars.

Parsing

The parsing problem, checking whether a given word belongs to the language given by a context-free grammar, is decidable, using one of the general-purpose parsing algorithms:

• CYK algorithm(for grammars inChomsky normal form)
• Earley parser
• GLR parser
• LL parser(only for the proper subclass of LL(k) grammars)

Context-free parsing forChomsky normal formgrammars was shown byLeslie G. Valiantto be reducible to Booleanmatrix multiplication,thus inheriting its complexity upper bound ofO(n^2.3728639).Conversely,Lillian Leehas shownO(n^3−ε) Boolean matrix multiplication to be reducible toO(n^3−3ε) CFG parsing, thus establishing some kind of lower bound for the latter.

Reachability, productiveness, nullability

Example grammar:
S→Bb|Cc|Ee
B→Bb|b
C→C
D→Bd|Cd|d
E→Ee

A nonterminal symbol$X$is calledproductive,orgenerating,if there is a derivation$X{\stackrel {*}{\Rightarrow }}w$for some string$w$of terminal symbols.$X$is calledreachableif there is a derivation$S{\stackrel {*}{\Rightarrow }}\ Alpha X\beta$for some strings$\ Alpha,\beta$of nonterminal and terminal symbols from the start symbol.$X$is calleduselessif it is unreachable or unproductive.$X$is callednullableif there is a derivation$X{\stackrel {*}{\Rightarrow }}\varepsilon$.A rule$X\rightarrow \varepsilon$is called anε-production.A derivation$X{\stackrel {+}{\Rightarrow }}X$is called acycle.

Algorithms are known to eliminate from a given grammar, without changing its generated language,

• unproductive symbols,
• unreachable symbols,
• ε-productions, with one possible exception,and
• cycles.

In particular, an alternative containing a useless nonterminal symbol can be deleted from the right-hand side of a rule. Such rules and alternatives are calleduseless.

In the depicted example grammar, the nonterminalDis unreachable, andEis unproductive, whileC→Ccauses a cycle. Hence, omitting the last three rules does not change the language generated by the grammar, nor does omitting the alternatives "|Cc|Ee"from the right-hand side of the rule forS.

A context-free grammar is said to beproperif it has neither useless symbols nor ε-productions nor cycles. Combining the above algorithms, every context-free grammar not generating ε can be transformed into aweakly equivalentproper one.

Regularity and LL(k) checks

It is decidable whether a givengrammaris aregular grammar,as well as whether it is anLL(k) grammarfor a givenk≥0.Ifkis not given, the latter problem is undecidable.

Given a context-free grammar, it is not decidable whether its language is regular,nor whether it is an LL(k) language for a givenk.

Emptiness and finiteness

There are algorithms to decide whether the language of a given context-free grammar is empty, as well as whether it is finite.

Undecidable problems

Some questions that are undecidable for wider classes of grammars become decidable for context-free grammars; e.g. theemptiness problem(whether the grammar generates any terminal strings at all), is undecidable forcontext-sensitive grammars,but decidable for context-free grammars.

However, many problems areundecidableeven for context-free grammars; the most prominent ones are handled in the following.

Universality

Given a CFG, does it generate the language of all strings over the Alpha bet of terminal symbols used in its rules?

A reduction can be demonstrated to this problem from the well-known undecidable problem of determining whether aTuring machineaccepts a particular input (thehalting problem). The reduction uses the concept of acomputation history,a string describing an entire computation of aTuring machine.A CFG can be constructed that generates all strings that are not accepting computation histories for a particular Turing machine on a particular input, and thus it will accept all strings only if the machine does not accept that input.

Language equality

Given two CFGs, do they generate the same language?

The undecidability of this problem is a direct consequence of the previous: it is impossible to even decide whether a CFG is equivalent to the trivial CFG defining the language of all strings.

Language inclusion

Given two CFGs, can the first one generate all strings that the second one can generate?

If this problem was decidable, then language equality could be decided too: two CFGs$G_{1}$and$G_{2}$generate the same language if$L(G_{1})$is a subset of$L(G_{2})$and$L(G_{2})$is a subset of$L(G_{1})$.

Being in a lower or higher level of the Chomsky hierarchy

UsingGreibach's theorem,it can be shown that the two following problems are undecidable:

• Given acontext-sensitive grammar,does it describe a context-free language?
• Given a context-free grammar, does it describe aregular language?

Grammar ambiguity

Given a CFG, is itambiguous?

The undecidability of this problem follows from the fact that if an algorithm to determine ambiguity existed, thePost correspondence problemcould be decided, which is known to be undecidable.This may be proved byOgden's lemma.

Language disjointness

Given two CFGs, is there any string derivable from both grammars?

If this problem was decidable, the undecidablePost correspondence problem(PCP) could be decided, too: given strings$\ Alpha _{1},\ldots,\ Alpha _{N},\beta _{1},\ldots,\beta _{N}$over some Alpha bet$\{a_{1},\ldots,a_{k}\}$,let the grammar⁠$G_{1}$⁠consist of the rule

$S\to \ Alpha _{1}S\beta _{1}^{rev}|\cdots |\ Alpha _{N}S\beta _{N}^{rev}|b$;

where$\beta _{i}^{rev}$denotes thereversedstring$\beta _{i}$and$b$does not occur among the$a_{i}$;and let grammar⁠$G_{2}$⁠consist of the rule

$T\to a_{1}Ta_{1}^{rev}|\cdots |a_{k}Ta_{k}^{rev}|b$;

Then the PCP instance given by$\ Alpha _{1},\ldots,\ Alpha _{N},\beta _{1},\ldots,\beta _{N}$has a solution if and only if⁠$L(G_{1})$⁠and⁠$L(G_{2})$⁠share a derivable string. The left of the string (before the$b$) will represent the top of the solution for the PCP instance while the right side will be the bottom in reverse.

Extensions

An obvious way to extend the context-free grammar formalism is to allow nonterminals to have arguments, the values of which are passed along within the rules. This allows natural language features such asagreementandreference,and programming language analogs such as the correct use and definition of identifiers, to be expressed in a natural way. E.g. we can now easily express that in English sentences, the subject and verb must agree in number. In computer science, examples of this approach includeaffix grammars,attribute grammars,indexed grammars,and Van Wijngaardentwo-level grammars.Similar extensions exist in linguistics.

Anextended context-free grammar(orregular right part grammar) is one in which the right-hand side of the production rules is allowed to be aregular expressionover the grammar's terminals and nonterminals. Extended context-free grammars describe exactly the context-free languages.

Another extension is to allow additional terminal symbols to appear at the left-hand side of rules, constraining their application. This produces the formalism ofcontext-sensitive grammars.

Subclasses

There are a number of important subclasses of the context-free grammars:

• LR(k)grammars (also known asdeterministic context-free grammars) allowparsing(string recognition) withdeterministic pushdown automata(PDA), but they can only describedeterministic context-free languages.
• Simple LR,Look-Ahead LRgrammars are subclasses that allow further simplification of parsing. SLR and LALR are recognized using the same PDA as LR, but with simpler tables, in most cases.
• LL(k) and LL(*)grammars allow parsing by direct construction of a leftmost derivation as described above, and describe even fewer languages.
• Simple grammarsare a subclass of the LL(1) grammars mostly interesting for its theoretical property that language equality of simple grammars is decidable, while language inclusion is not.
• Bracketed grammarshave the property that the terminal symbols are divided into left and right bracket pairs that always match up in rules.
• Linear grammarshave no rules with more than one nonterminal on the right-hand side.
• Regular grammarsare a subclass of the linear grammars and describe theregularlanguages, i.e. they correspond tofinite automataandregular expressions.

LR parsing extends LL parsing to support a larger range of grammars; in turn,generalized LR parsingextends LR parsing to support arbitrary context-free grammars. On LL grammars and LR grammars, it essentially performs LL parsing and LR parsing, respectively, while onnondeterministic grammars,it is as efficient as can be expected. Although GLR parsing was developed in the 1980s, many new language definitions andparser generatorscontinue to be based on LL, LALR or LR parsing up to the present day.

Linguistic applications

Chomskyinitially hoped to overcome the limitations of context-free grammars by addingtransformation rules.

Such rules are another standard device in traditional linguistics; e.g.passivizationin English. Much ofgenerative grammarhas been devoted to finding ways of refining the descriptive mechanisms of phrase-structure grammar and transformation rules such that exactly the kinds of things can be expressed that natural language actually allows. Allowing arbitrary transformations does not meet that goal: they are much too powerful, beingTuring completeunless significant restrictions are added (e.g. no transformations that introduce and then rewrite symbols in a context-free fashion).

Chomsky's general position regarding the non-context-freeness of natural language has held up since then,although his specific examples regarding the inadequacy of context-free grammars in terms of their weak generative capacity were later disproved.Gerald GazdarandGeoffrey Pullumhave argued that despite a few non-context-free constructions in natural language (such ascross-serial dependenciesinSwiss GermanandreduplicationinBambara), the vast majority of forms in natural language are indeed context-free.

See also

• Parsing expression grammar
• Stochastic context-free grammar
• Algorithms for context-free grammar generation
• Pumping lemma for context-free languages

Further reading

• Hopcroft, John E.;Ullman, Jeffrey D.(1979),Introduction to Automata Theory, Languages, and Computation,Addison-Wesley.Chapter 4: Context-Free Grammars, pp. 77–106; Chapter 6: Properties of Context-Free Languages, pp. 125–137.
• Hopcroft; Motwani, Rajeev; Ullman, Jeffrey D. (2003).Introduction to automata theory, languages, and computation(2nd ed.). Upper Saddle River: Pearson Education International.ISBN978-0321210296.
• Sipser, Michael(1997),Introduction to the Theory of Computation,PWS Publishing,ISBN978-0-534-94728-6.Chapter 2: Context-Free Grammars, pp. 91–122; Section 4.1.2: Decidable problems concerning context-free languages, pp. 156–159; Section 5.1.1: Reductions via computation histories: pp. 176–183.
• J. Berstel, L. Boasson (1990). Jan van Leeuwen (ed.).Context-Free Languages.Handbook of Theoretical Computer Science. Vol. B. Elsevier. pp. 59–102.

External links

• Computer programmers may find thestack exchange answerto be useful.
• CFG Developercreated by Christopher Wong at Stanford University in 2014; modified by Kevin Gibbons in 2015.