Context-free language: Difference between revisions
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{{Short description|Formal language generated by context-free grammar}} | {{Short description|Formal language generated by context-free grammar}} | ||
{{Use dmy dates|date=December 2025}} | |||
In [[formal language theory]], a '''context-free language''' ('''CFL'''), also called a '''[[Chomsky hierarchy|Chomsky]] type-2 language''', is a [[formal language|language]] generated by a [[context-free grammar]] (CFG). | In [[formal language theory]], a '''context-free language''' ('''CFL'''), also called a '''[[Chomsky hierarchy|Chomsky]] type-2 language''', is a [[formal language|language]] generated by a [[context-free grammar]] (CFG). | ||
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==Examples== | ==Examples== | ||
An example context-free language is <math>L = \{a^nb^n:n\geq1\}</math>, the language of all non-empty even-length strings, the entire first halves of which are {{mvar|a}}'s, and the entire second halves of which are {{mvar|b}}'s. {{mvar|L}} is generated by the grammar <math>S\to aSb ~|~ ab</math>. | An example context-free language is <math>L = \{a^nb^n:n\geq1\}</math>, the language of all non-empty even-length strings, the entire first halves of which are {{mvar|a}}'s, and the entire second halves of which are {{mvar|b}}'s. {{mvar|L}} is generated by the grammar <math>S\to aSb ~|~ ab</math>. This language is not regular. It is accepted by the pushdown automaton <math display="inline">M=(\{q_0,q_1,q_f\}, \{a,b\}, \{a,z\}, \delta, q_0, z, \{q_f\})</math> where <math>\delta</math> is defined as follows:<ref group="note">meaning of <math>\delta</math>'s arguments and results: <math>\delta(\mathrm{state}_1, \mathrm{read}, \mathrm{pop}) = (\mathrm{state}_2, \mathrm{push})</math></ref> | ||
This language is not | |||
It is accepted by the | |||
:<math>\begin{align} | :<math>\begin{align} | ||
\delta(q_0, a, z) &= (q_0, az) \\ | \delta(q_0, a, z) &= (q_0, az) \\ | ||
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\end{align}</math> | \end{align}</math> | ||
Unambiguous CFLs are a proper subset of all CFLs: there are | Unambiguous CFLs are a proper subset of all CFLs: there are inherently ambiguous CFLs. An example of an inherently ambiguous CFL is the union of <math>\{a^n b^m c^m d^n | n, m > 0\}</math> with <math>\{a^n b^n c^m d^m | n, m > 0\}</math>. This set is context-free, since the union of two context-free languages is always context-free. But there is no way to unambiguously parse strings in the (non-context-free) subset <math>\{a^n b^n c^n d^n | n > 0\}</math> which is the intersection of these two languages.{{sfn|Hopcroft|Ullman|1979|p=100|loc=Theorem 4.7}} | ||
===Dyck language=== | ===Dyck language=== | ||
The | The language of all properly matched parentheses is generated by the grammar <math>S\to SS ~|~ (S) ~|~ \varepsilon</math>. | ||
==Properties== | ==Properties== | ||
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The context-free nature of the language makes it simple to parse with a pushdown automaton. | The context-free nature of the language makes it simple to parse with a pushdown automaton. | ||
Determining an instance of the [[membership problem]]; i.e. given a string <math>w</math>, determine whether <math>w \in L(G)</math> where <math>L</math> is the language generated by a given grammar <math>G</math>; is also known as ''recognition''. Context-free recognition for [[Chomsky normal form]] grammars was shown by [[Leslie Valiant|Leslie G. Valiant]] to be reducible to Boolean [[matrix multiplication]], thus inheriting its complexity upper bound of [[Big O notation|''O'']](''n''<sup>2.3728596</sup>). | Determining an instance of the [[membership problem]]; i.e. given a string <math>w</math>, determine whether <math>w \in L(G)</math> where <math>L</math> is the language generated by a given grammar <math>G</math>; is also known as ''recognition''. Context-free recognition for [[Chomsky normal form]] grammars was shown by [[Leslie Valiant|Leslie G. Valiant]] to be reducible to Boolean [[matrix multiplication]], thus inheriting its complexity upper bound of [[Big O notation|''O'']](''n''<sup>2.3728596</sup>).{{sfn|Valiant|1975}}<ref group="note">In Valiant's paper, ''O''(''n''<sup>2.81</sup>) was the then-best known upper bound. See [[Matrix multiplication#Computational complexity]] for bound improvements since then.</ref> | ||
Conversely, [[Lillian Lee (computer scientist)|Lillian Lee]] has shown ''O''(''n''<sup>3−ε</sup>) Boolean matrix multiplication to be reducible to ''O''(''n''<sup>3−3ε</sup>) CFG parsing, thus establishing some kind of lower bound for the latter. | Conversely, [[Lillian Lee (computer scientist)|Lillian Lee]] has shown ''O''(''n''<sup>3−ε</sup>) Boolean matrix multiplication to be reducible to ''O''(''n''<sup>3−3ε</sup>) CFG parsing, thus establishing some kind of lower bound for the latter.{{sfn|Lee|2002}} | ||
Practical uses of context-free languages require also to produce a derivation tree that exhibits the structure that the grammar associates with the given string. The process of producing this tree is called ''[[parsing]]''. Known parsers have a time complexity that is cubic in the size of the string that is parsed. | Practical uses of context-free languages require also to produce a derivation tree that exhibits the structure that the grammar associates with the given string. The process of producing this tree is called ''[[parsing]]''. Known parsers have a time complexity that is cubic in the size of the string that is parsed. | ||
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Formally, the set of all context-free languages is identical to the set of languages accepted by pushdown automata (PDA). Parser algorithms for context-free languages include the [[CYK algorithm]] and [[Earley parser|Earley's Algorithm]]. | Formally, the set of all context-free languages is identical to the set of languages accepted by pushdown automata (PDA). Parser algorithms for context-free languages include the [[CYK algorithm]] and [[Earley parser|Earley's Algorithm]]. | ||
A special subclass of context-free languages are the [[deterministic context-free language]]s which are defined as the set of languages accepted by a [[deterministic pushdown automaton]] and can be parsed by a [[LR parser|LR(k) parser]]. | A special subclass of context-free languages are the [[deterministic context-free language]]s which are defined as the set of languages accepted by a [[deterministic pushdown automaton]] and can be parsed by a [[LR parser|LR(k) parser]].{{sfn|Knuth|1965}} | ||
See also [[parsing expression grammar]] as an alternative approach to grammar and parser. | See also [[parsing expression grammar]] as an alternative approach to grammar and parser. | ||
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====Nonclosure under intersection, complement, and difference==== | ====Nonclosure under intersection, complement, and difference==== | ||
The context-free languages are not closed under intersection. This can be seen by taking the languages <math>A = \{a^n b^n c^m \mid m, n \geq 0 \}</math> and <math>B = \{a^m b^n c^n \mid m,n \geq 0\}</math>, which are both context-free.<ref group=note>A context-free grammar for the language ''A'' is given by the following production rules, taking ''S'' as the start symbol: ''S'' → ''Sc'' | ''aTb'' | ''ε''; ''T'' → ''aTb'' | ''ε''. The grammar for ''B'' is analogous.</ref> Their intersection is <math>A \cap B = \{ a^n b^n c^n \mid n \geq 0\}</math>, which can be shown to be non-context-free by the [[pumping lemma for context-free languages]]. As a consequence, context-free languages cannot be closed under complementation, as for any languages ''A'' and ''B'', their intersection can be expressed by union and complement: <math>A \cap B = \overline{\overline{A} \cup \overline{B}} </math>. In particular, context-free language cannot be closed under difference, since complement can be expressed by difference: <math>\overline{L} = \Sigma^* \setminus L</math>. | The context-free languages are not closed under intersection. This can be seen by taking the languages <math>A = \{a^n b^n c^m \mid m, n \geq 0 \}</math> and <math>B = \{a^m b^n c^n \mid m,n \geq 0\}</math>, which are both context-free.<ref group="note">A context-free grammar for the language ''A'' is given by the following production rules, taking ''S'' as the start symbol: ''S'' → ''Sc'' | ''aTb'' | ''ε''; ''T'' → ''aTb'' | ''ε''. The grammar for ''B'' is analogous.</ref> Their intersection is <math>A \cap B = \{ a^n b^n c^n \mid n \geq 0\}</math>, which can be shown to be non-context-free by the [[pumping lemma for context-free languages]]. As a consequence, context-free languages cannot be closed under complementation, as for any languages ''A'' and ''B'', their intersection can be expressed by union and complement: <math>A \cap B = \overline{\overline{A} \cup \overline{B}} </math>. In particular, context-free language cannot be closed under difference, since complement can be expressed by difference: <math>\overline{L} = \Sigma^* \setminus L</math>.{{sfn|Scheinberg|1960}} | ||
However, if ''L'' is a context-free language and ''D'' is a regular language then both their intersection <math>L\cap D</math> and their difference <math>L\setminus D</math> are context-free languages. | However, if ''L'' is a context-free language and ''D'' is a regular language then both their intersection <math>L\cap D</math> and their difference <math>L\setminus D</math> are context-free languages.{{sfn|Beigel|Gasarch}} | ||
===Decidability=== | ===Decidability=== | ||
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The following problems are [[Undecidable problem|undecidable]] for arbitrarily given [[context-free grammar]]s A and B: | The following problems are [[Undecidable problem|undecidable]] for arbitrarily given [[context-free grammar]]s A and B: | ||
*Equivalence: is <math>L(A)=L(B)</math>?{{sfn|Hopcroft|Ullman|1979|p=203|loc=Theorem 8.12(1)}} | *Equivalence: is <math>L(A)=L(B)</math>?{{sfn|Hopcroft|Ullman|1979|p=203|loc=Theorem 8.12(1)}} | ||
*Disjointness: is <math>L(A) \cap L(B) = \emptyset </math> ?{{sfn|Hopcroft|Ullman|1979|p=202|loc=Theorem 8.10}} However, the intersection of a context-free language and a ''regular'' language is context-free, | *Disjointness: is <math>L(A) \cap L(B) = \emptyset </math> ?{{sfn|Hopcroft|Ullman|1979|p=202|loc=Theorem 8.10}} However, the intersection of a context-free language and a ''regular'' language is context-free,{{sfn|Salomaa|1973|p=59|loc=Theorem 6.7}}{{sfn|Hopcroft|Ullman|1979|p=135|loc=Theorem 6.5}} hence the variant of the problem where ''B'' is a regular grammar is decidable (see "Emptiness" below). | ||
*Containment: is <math>L(A) \subseteq L(B)</math> ?{{sfn|Hopcroft|Ullman|1979|p=203|loc=Theorem 8.12(2)}} Again, the variant of the problem where ''B'' is a regular grammar is decidable,{{citation needed|date=December 2015}} while that where ''A'' is regular is generally not.{{sfn|Hopcroft|Ullman|1979|p=203|loc=Theorem 8.12(4)}} | *Containment: is <math>L(A) \subseteq L(B)</math> ?{{sfn|Hopcroft|Ullman|1979|p=203|loc=Theorem 8.12(2)}} Again, the variant of the problem where ''B'' is a regular grammar is decidable,{{citation needed|date=December 2015}} while that where ''A'' is regular is generally not.{{sfn|Hopcroft|Ullman|1979|p=203|loc=Theorem 8.12(4)}} | ||
*Universality: is <math>L(A)=\Sigma^*</math>?{{sfn|Hopcroft|Ullman|1979|p=203|loc=Theorem 8.11}} | *Universality: is <math>L(A)=\Sigma^*</math>?{{sfn|Hopcroft|Ullman|1979|p=203|loc=Theorem 8.11}} | ||
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*Membership: Given a context-free grammar ''G'', and a word <math>w</math>, does <math>w \in L(G)</math> ? Efficient polynomial-time algorithms for the membership problem are the [[CYK algorithm]] and [[Earley parser|Earley's Algorithm]]. | *Membership: Given a context-free grammar ''G'', and a word <math>w</math>, does <math>w \in L(G)</math> ? Efficient polynomial-time algorithms for the membership problem are the [[CYK algorithm]] and [[Earley parser|Earley's Algorithm]]. | ||
According to Hopcroft, Motwani, Ullman ( | According to [[John Hopcroft|Hopcroft]], [[Rajeev Motwani|Motwani]], [[Jeffrey Ullman|Ullman]] (2006),{{sfn|Hopcroft|Motwani|Ullman|2006|loc=See Section 7.6 for properties of context-free languages and Section 9.7 for related exercises}} many of the fundamental closure and (un)decidability properties of context-free languages were shown in the 1961 paper of [[Yehoshua Bar-Hillel|Bar-Hillel]], Perles, and Shamir.{{sfn|Bar-Hillel|Perles|Shamir|1961}} | ||
many of the fundamental closure and (un)decidability properties of context-free languages were shown in the 1961 paper of [[Yehoshua Bar-Hillel|Bar-Hillel]], Perles, and | |||
===Languages that are not context-free=== | ===Languages that are not context-free=== | ||
The set <math>\{a^n b^n c^n d^n | n > 0\}</math> is a [[context-sensitive language]], but there does not exist a context-free grammar generating this language.{{sfn|Hopcroft|Ullman|1979}} So there exist context-sensitive languages which are not context-free. To prove that a given language is not context-free, one may employ the [[pumping lemma for context-free languages]] | The set <math>\{a^n b^n c^n d^n | n > 0\}</math> is a [[context-sensitive language]], but there does not exist a context-free grammar generating this language.{{sfn|Hopcroft|Ullman|1979}} So there exist context-sensitive languages which are not context-free. To prove that a given language is not context-free, one may employ the [[pumping lemma for context-free languages]]{{sfn|Bar-Hillel|Perles|Shamir|1961}} or a number of other methods, such as [[Ogden's lemma]] or [[Parikh's theorem]].<ref>{{cite web| url = https://cs.stackexchange.com/q/265| title = How to prove that a language is not context-free?|author=Stack Exchange}}</ref> | ||
==Notes== | ==Notes== | ||
{{Reflist|group=note}} | {{Reflist|group=note|2}} | ||
==References== | ==References== | ||
{{Reflist}} | {{Reflist|2}} | ||
=== Works cited === | === Works cited === | ||
{{Refbegin}} | {{Refbegin}} | ||
*{{Hopcroft and Ullman 1979}}{{sfn whitelist|CITEREFHopcroftUllman1979}} | * {{cite journal|first1=Yehoshua|last1=Bar-Hillel |author1-link=Yehoshua Bar-Hillel|first2=Micha Asher |last2=Perles |first3=Eli |last3=Shamir | title=On Formal Properties of Simple Phrase-Structure Grammars| journal=Zeitschrift für Phonetik, Sprachwissenschaft und Kommunikationsforschung| year=1961| volume=14| number=2| pages=143–172}} | ||
* {{cite book |first=Arto | | * {{Cite web|last1=Beigel|first1=Richard|last2=Gasarch|first2=William|author2-link =William Gasarch|title=A Proof that if L = L1 ∩ L2 where L1 is CFL and L2 is Regular then L is Context Free Which Does Not use PDA's|url=http://www.cs.umd.edu/~gasarch/BLOGPAPERS/cfg.pdf |archive-url=https://web.archive.org/web/20141212060332/http://www.cs.umd.edu/~gasarch/BLOGPAPERS/cfg.pdf |archive-date=2014-12-12 |url-status=live|access-date=2020-06-06|website=University of Maryland Department of Computer Science}} | ||
* {{Hopcroft and Ullman 1979}}{{sfn whitelist|CITEREFHopcroftUllman1979}} | |||
* {{Hopcroft, Motwani, and Ullman 2006}}{{sfn whitelist|CITEREFHopcroftMotwaniUllman2006}} | |||
* {{Cite journal | last = Knuth | first = D. E. | author-link = Donald Knuth | title = On the translation of languages from left to right | doi = 10.1016/S0019-9958(65)90426-2 | journal = Information and Control | volume = 8 | issue = 6 | pages = 607–639 | date = July 1965 | doi-access = }} | |||
* {{cite journal |first=Lillian |last=Lee |author-link=Lillian Lee (computer scientist) |title=Fast Context-Free Grammar Parsing Requires Fast Boolean Matrix Multiplication |journal=J ACM |date=January 2002 |volume=49 |number=1 |pages=1–15 |url=https://www.cs.cornell.edu/home/llee/papers/bmmcfl-jacm.pdf |archive-url=https://web.archive.org/web/20030427152836/http://www.cs.cornell.edu/home/llee/papers/bmmcfl-jacm.pdf |archive-date=2003-04-27 |url-status=live |doi=10.1145/505241.505242 |arxiv=cs/0112018|s2cid=1243491 }} | |||
* {{cite book | last=Salomaa | first=Arto | author-link=Arto Salomaa | title=Formal Languages | series=ACM Monograph Series | publisher=Academic Press | location=New York | year=1973 | isbn=978-0126157505}} | |||
* {{cite journal | url=https://core.ac.uk/download/pdf/82210847.pdf |archive-url=https://web.archive.org/web/20181126005901/https://core.ac.uk/download/pdf/82210847.pdf |archive-date=2018-11-26 |url-status=live | first=Stephen |last=Scheinberg | title=Note on the Boolean Properties of Context Free Languages | journal=Information and Control | volume=3 | pages=372-375 | year=1960 | issue=4 | doi=10.1016/s0019-9958(60)90965-7| doi-access=free }} | |||
* {{cite journal |first=Leslie G. |last=Valiant |author-link=Leslie Valiant|title=General context-free recognition in less than cubic time |journal=Journal of Computer and System Sciences |date=April 1975 |volume=10 |number=2 |pages=308–315 |doi=10.1016/s0022-0000(75)80046-8 |doi-access=free |url=https://figshare.com/articles/journal_contribution/General_context-free_recognition_in_less_than_cubic_time/6605915/1/files/12096398.pdf }} | |||
{{Refend}} | {{Refend}} | ||
== Further reading == | == Further reading == | ||
* {{cite book |first1=Jean-Michel |last1=Autebert |first2=Jean |last2=Berstel |first3=Luc |last3=Boasson |url= | {{Refbegin}} | ||
* {{cite book|first=Seymour |last=Ginsburg |author-link=Seymour Ginsburg | title = The Mathematical Theory of Context-Free Languages | year = 1966 | publisher = McGraw-Hill | location = New York, NY, USA}} | * {{cite book |first1=Jean-Michel |last1=Autebert |first2=Jean |last2=Berstel |first3=Luc |last3=Boasson |url=https://www-igm.univ-mlv.fr/~berstel/Articles/1997CFLPDA.pdf |archive-url=https://web.archive.org/web/20110516030515/http://www-igm.univ-mlv.fr/%7Eberstel/Articles/1997CFLPDA.pdf |archive-date=2011-05-16 |url-status=live |chapter=Context-Free Languages and Push-Down Automata |editor1=G. Rozenberg |editor2=A. Salomaa |title=Handbook of Formal Languages |volume=1 |publisher=Springer-Verlag |date=1997 |pages=111–174}} | ||
* {{cite book |first=Seymour |last=Ginsburg |author-link=Seymour Ginsburg |title=The Mathematical Theory of Context-Free Languages |year=1966 |publisher=McGraw-Hill |location=New York, NY, USA}} | |||
* {{Sipser 1997|chapter='''2''': Context-Free Languages |pages=91-122}} | * {{Sipser 1997|chapter='''2''': Context-Free Languages |pages=91-122}} | ||
{{Refend}} | |||
{{Formal languages and grammars}} | {{Formal languages and grammars}} | ||
Latest revision as of 17:03, 9 May 2026
In formal language theory, a context-free language (CFL), also called a Chomsky type-2 language, is a language generated by a context-free grammar (CFG).
Context-free languages have many applications in programming languages, in particular, most arithmetic expressions are generated by context-free grammars.
Background
Context-free grammar
Different context-free grammars can generate the same context-free language. Intrinsic properties of the language can be distinguished from extrinsic properties of a particular grammar by comparing multiple grammars that describe the language.
Automata
The set of all context-free languages is identical to the set of languages accepted by pushdown automata, which makes these languages amenable to parsing. Further, for a given CFG, there is a direct way to produce a pushdown automaton for the grammar (and thereby the corresponding language), though going the other way (producing a grammar given an automaton) is not as direct.
Examples
An example context-free language is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L = \{a^nb^n:n\geq1\}} , the language of all non-empty even-length strings, the entire first halves of which are a's, and the entire second halves of which are b's. L is generated by the grammar Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S\to aSb ~|~ ab} . This language is not regular. It is accepted by the pushdown automaton Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle M=(\{q_0,q_1,q_f\}, \{a,b\}, \{a,z\}, \delta, q_0, z, \{q_f\})} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta} is defined as follows:[note 1]
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \delta(q_0, a, z) &= (q_0, az) \\ \delta(q_0, a, a) &= (q_0, aa) \\ \delta(q_0, b, a) &= (q_1, \varepsilon) \\ \delta(q_1, b, a) &= (q_1, \varepsilon) \\ \delta(q_1, \varepsilon, z) &= (q_f, \varepsilon) \end{align}}
Unambiguous CFLs are a proper subset of all CFLs: there are inherently ambiguous CFLs. An example of an inherently ambiguous CFL is the union of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{a^n b^m c^m d^n | n, m > 0\}} with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{a^n b^n c^m d^m | n, m > 0\}} . This set is context-free, since the union of two context-free languages is always context-free. But there is no way to unambiguously parse strings in the (non-context-free) subset Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{a^n b^n c^n d^n | n > 0\}} which is the intersection of these two languages.[1]
Dyck language
The language of all properly matched parentheses is generated by the grammar Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S\to SS ~|~ (S) ~|~ \varepsilon} .
Properties
Context-free parsing
The context-free nature of the language makes it simple to parse with a pushdown automaton.
Determining an instance of the membership problem; i.e. given a string Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w} , determine whether Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w \in L(G)} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L} is the language generated by a given grammar Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} ; is also known as recognition. Context-free recognition for Chomsky normal form grammars was shown by Leslie G. Valiant to be reducible to Boolean matrix multiplication, thus inheriting its complexity upper bound of O(n2.3728596).[2][note 2] Conversely, Lillian Lee has shown O(n3−ε) Boolean matrix multiplication to be reducible to O(n3−3ε) CFG parsing, thus establishing some kind of lower bound for the latter.[3]
Practical uses of context-free languages require also to produce a derivation tree that exhibits the structure that the grammar associates with the given string. The process of producing this tree is called parsing. Known parsers have a time complexity that is cubic in the size of the string that is parsed.
Formally, the set of all context-free languages is identical to the set of languages accepted by pushdown automata (PDA). Parser algorithms for context-free languages include the CYK algorithm and Earley's Algorithm.
A special subclass of context-free languages are the deterministic context-free languages which are defined as the set of languages accepted by a deterministic pushdown automaton and can be parsed by a LR(k) parser.[4]
See also parsing expression grammar as an alternative approach to grammar and parser.
Closure properties
The class of context-free languages is closed under the following operations. That is, if L and P are context-free languages, the following languages are context-free as well:
- the union Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L \cup P} of L and P[5]
- the reversal of L[6]
- the concatenation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L \cdot P} of L and P[5]
- the Kleene star Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L^*} of L[5]
- the image Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi(L)} of L under a homomorphism Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi} [7]
- the image Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi^{-1}(L)} of L under an inverse homomorphism Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi^{-1}} [8]
- the circular shift of L (the language Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{vu : uv \in L \}} )[9]
- the prefix closure of L (the set of all prefixes of strings from L)[10]
- the quotient L/R of L by a regular language R[11]
Nonclosure under intersection, complement, and difference
The context-free languages are not closed under intersection. This can be seen by taking the languages Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A = \{a^n b^n c^m \mid m, n \geq 0 \}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B = \{a^m b^n c^n \mid m,n \geq 0\}} , which are both context-free.[note 3] Their intersection is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \cap B = \{ a^n b^n c^n \mid n \geq 0\}} , which can be shown to be non-context-free by the pumping lemma for context-free languages. As a consequence, context-free languages cannot be closed under complementation, as for any languages A and B, their intersection can be expressed by union and complement: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \cap B = \overline{\overline{A} \cup \overline{B}} } . In particular, context-free language cannot be closed under difference, since complement can be expressed by difference: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{L} = \Sigma^* \setminus L} .[12]
However, if L is a context-free language and D is a regular language then both their intersection Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L\cap D} and their difference Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L\setminus D} are context-free languages.[13]
Decidability
In formal language theory, questions about regular languages are usually decidable, but ones about context-free languages are often not. It is decidable whether such a language is finite, but not whether it contains every possible string, is regular, is unambiguous, or is equivalent to a language with a different grammar.
The following problems are undecidable for arbitrarily given context-free grammars A and B:
- Equivalence: is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L(A)=L(B)} ?[14]
- Disjointness: is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L(A) \cap L(B) = \emptyset } ?[15] However, the intersection of a context-free language and a regular language is context-free,[16][17] hence the variant of the problem where B is a regular grammar is decidable (see "Emptiness" below).
- Containment: is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L(A) \subseteq L(B)} ?[18] Again, the variant of the problem where B is a regular grammar is decidable,[citation needed] while that where A is regular is generally not.[19]
- Universality: is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L(A)=\Sigma^*} ?[20]
- Regularity: is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L(A)} a regular language?[21]
- Ambiguity: is every grammar for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L(A)} ambiguous?[22]
The following problems are decidable for arbitrary context-free languages:
- Emptiness: Given a context-free grammar A, is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L(A) = \emptyset} ?[23]
- Finiteness: Given a context-free grammar A, is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L(A)} finite?[24]
- Membership: Given a context-free grammar G, and a word Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w} , does Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w \in L(G)} ? Efficient polynomial-time algorithms for the membership problem are the CYK algorithm and Earley's Algorithm.
According to Hopcroft, Motwani, Ullman (2006),[25] many of the fundamental closure and (un)decidability properties of context-free languages were shown in the 1961 paper of Bar-Hillel, Perles, and Shamir.[26]
Languages that are not context-free
The set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{a^n b^n c^n d^n | n > 0\}} is a context-sensitive language, but there does not exist a context-free grammar generating this language.[27] So there exist context-sensitive languages which are not context-free. To prove that a given language is not context-free, one may employ the pumping lemma for context-free languages[26] or a number of other methods, such as Ogden's lemma or Parikh's theorem.[28]
Notes
- ↑ meaning of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta} 's arguments and results: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta(\mathrm{state}_1, \mathrm{read}, \mathrm{pop}) = (\mathrm{state}_2, \mathrm{push})}
- ↑ In Valiant's paper, O(n2.81) was the then-best known upper bound. See Matrix multiplication#Computational complexity for bound improvements since then.
- ↑ A context-free grammar for the language A is given by the following production rules, taking S as the start symbol: S → Sc | aTb | ε; T → aTb | ε. The grammar for B is analogous.
References
- ↑ Hopcroft & Ullman 1979, p. 100, Theorem 4.7.
- ↑ Valiant 1975.
- ↑ Lee 2002.
- ↑ Knuth 1965.
- ↑ 5.0 5.1 5.2 Hopcroft & Ullman 1979, p. 131, Corollary of Theorem 6.1.
- ↑ Hopcroft & Ullman 1979, p. 142, Exercise 6.4d.
- ↑ Hopcroft & Ullman 1979, p. 131-132, Corollary of Theorem 6.2.
- ↑ Hopcroft & Ullman 1979, p. 132, Theorem 6.3.
- ↑ Hopcroft & Ullman 1979, p. 142-144, Exercise 6.4c.
- ↑ Hopcroft & Ullman 1979, p. 142, Exercise 6.4b.
- ↑ Hopcroft & Ullman 1979, p. 142, Exercise 6.4a.
- ↑ Scheinberg 1960.
- ↑ Beigel & Gasarch.
- ↑ Hopcroft & Ullman 1979, p. 203, Theorem 8.12(1).
- ↑ Hopcroft & Ullman 1979, p. 202, Theorem 8.10.
- ↑ Salomaa 1973, p. 59, Theorem 6.7.
- ↑ Hopcroft & Ullman 1979, p. 135, Theorem 6.5.
- ↑ Hopcroft & Ullman 1979, p. 203, Theorem 8.12(2).
- ↑ Hopcroft & Ullman 1979, p. 203, Theorem 8.12(4).
- ↑ Hopcroft & Ullman 1979, p. 203, Theorem 8.11.
- ↑ Hopcroft & Ullman 1979, p. 205, Theorem 8.15.
- ↑ Hopcroft & Ullman 1979, p. 206, Theorem 8.16.
- ↑ Hopcroft & Ullman 1979, p. 137, Theorem 6.6(a).
- ↑ Hopcroft & Ullman 1979, p. 137, Theorem 6.6(b).
- ↑ 26.0 26.1 Bar-Hillel, Perles & Shamir 1961.
- ↑ Hopcroft & Ullman 1979.
- ↑ Stack Exchange. "How to prove that a language is not context-free?".
Works cited
- Bar-Hillel, Yehoshua; Perles, Micha Asher; Shamir, Eli (1961). "On Formal Properties of Simple Phrase-Structure Grammars". Zeitschrift für Phonetik, Sprachwissenschaft und Kommunikationsforschung. 14 (2): 143–172.
- Beigel, Richard; Gasarch, William. "A Proof that if L = L1 ∩ L2 where L1 is CFL and L2 is Regular then L is Context Free Which Does Not use PDA's" (PDF). University of Maryland Department of Computer Science. Archived (PDF) from the original on 12 December 2014. Retrieved 6 June 2020.
- Template:Hopcroft and Ullman 1979Template:Sfn whitelist
- Template:Hopcroft, Motwani, and Ullman 2006Template:Sfn whitelist
- Knuth, D. E. (July 1965). "On the translation of languages from left to right". Information and Control. 8 (6): 607–639. doi:10.1016/S0019-9958(65)90426-2.
- Lee, Lillian (January 2002). "Fast Context-Free Grammar Parsing Requires Fast Boolean Matrix Multiplication" (PDF). J ACM. 49 (1): 1–15. arXiv:cs/0112018. doi:10.1145/505241.505242. S2CID 1243491. Archived (PDF) from the original on 27 April 2003.
- Salomaa, Arto (1973). Formal Languages. ACM Monograph Series. New York: Academic Press. ISBN 978-0126157505.
- Scheinberg, Stephen (1960). "Note on the Boolean Properties of Context Free Languages" (PDF). Information and Control. 3 (4): 372–375. doi:10.1016/s0019-9958(60)90965-7. Archived (PDF) from the original on 26 November 2018.
- Valiant, Leslie G. (April 1975). "General context-free recognition in less than cubic time" (PDF). Journal of Computer and System Sciences. 10 (2): 308–315. doi:10.1016/s0022-0000(75)80046-8.
Further reading
- Autebert, Jean-Michel; Berstel, Jean; Boasson, Luc (1997). "Context-Free Languages and Push-Down Automata". In G. Rozenberg; A. Salomaa (eds.). Handbook of Formal Languages (PDF). 1. Springer-Verlag. pp. 111–174. Archived (PDF) from the original on 16 May 2011.
- Ginsburg, Seymour (1966). The Mathematical Theory of Context-Free Languages. New York, NY, USA: McGraw-Hill.
- Template:Sipser 1997