Editing Context-free language

{{Short description|Formal language generated by context-free grammar}}

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).

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 <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 language|regular]].
It is accepted by the [[pushdown automaton#Formal definition|pushdown automaton]] <math>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>

:<math>\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}</math>

Unambiguous CFLs are a proper subset of all CFLs: there are [[Inherently ambiguous language|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===

The [[Dyck language|language of all properly matched parentheses]] is generated by the grammar <math>S\to SS ~|~ (S) ~|~ \varepsilon</math>.

==Properties==

===Context-free parsing===
{{main|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 <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>).<ref>{{cite journal |first=Leslie G. |last=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 }}</ref><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.<ref>{{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=http://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 }}</ref>

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 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]].<ref>{{Cite journal | last1 = Knuth | first1 = 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 =  }}</ref>

See also [[parsing expression grammar]] as an alternative approach to grammar and parser.

===Closure properties===
The class of context-free languages is [[closure (mathematics)|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 (set theory)|union]] <math>L \cup P</math> of ''L'' and ''P''{{sfn|Hopcroft|Ullman|1979|p=131|loc=Corollary of Theorem 6.1}}
*the reversal of ''L''{{sfn|Hopcroft|Ullman|1979|p=142|loc=Exercise 6.4d}}
*the [[concatenation]] <math>L \cdot P</math> of ''L'' and ''P''{{sfn|Hopcroft|Ullman|1979|p=131|loc=Corollary of Theorem 6.1}}
*the [[Kleene star]] <math>L^*</math> of ''L''{{sfn|Hopcroft|Ullman|1979|p=131|loc=Corollary of Theorem 6.1}}
*the image <math>\varphi(L)</math> of ''L'' under a [[String operations#String homomorphism|homomorphism]] <math>\varphi</math>{{sfn|Hopcroft|Ullman|1979|p=131-132|loc=Corollary of Theorem 6.2}}
*the image <math>\varphi^{-1}(L)</math> of ''L'' under an [[String operations#String homomorphism|inverse homomorphism]] <math>\varphi^{-1}</math>{{sfn|Hopcroft|Ullman|1979|p=132|loc=Theorem 6.3}}
*the [[circular shift#Applications|circular shift]] of ''L'' (the language <math>\{vu : uv \in L \}</math>){{sfn|Hopcroft|Ullman|1979|p=142-144|loc=Exercise 6.4c}}
*the prefix closure of ''L'' (the set of all [[Prefix (computer science)|prefix]]es of strings from ''L''){{sfn|Hopcroft|Ullman|1979|p=142|loc=Exercise 6.4b}}
*the [[Quotient of a formal language|quotient]] ''L''/''R'' of ''L'' by a regular language ''R''{{sfn|Hopcroft|Ullman|1979|p=142|loc=Exercise 6.4a}}

====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>.<ref name="Scheinberg.1960">{{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 | author=Stephen Scheinberg | title=Note on the Boolean Properties of Context Free Languages | journal=Information and Control | volume=3 | pages=372&ndash;375 | year=1960 | issue=4 | doi=10.1016/s0019-9958(60)90965-7| doi-access=free }}</ref>

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.<ref>{{Cite web|last1=Beigel|first1=Richard|last2=Gasarch|first2=William|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=June 6, 2020|website=University of Maryland Department of Computer Science}}</ref>

===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 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)}}
*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,<ref>{{harvtxt|Salomaa|1973}}, p. 59, Theorem 6.7</ref>{{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)}}
*Universality: is <math>L(A)=\Sigma^*</math>?{{sfn|Hopcroft|Ullman|1979|p=203|loc=Theorem 8.11}}
*Regularity: is <math>L(A)</math> a regular language?{{sfn|Hopcroft|Ullman|1979|p=205|loc=Theorem 8.15}}
*Ambiguity: is every grammar for <math>L(A)</math> ambiguous?{{sfn|Hopcroft|Ullman|1979|p=206|loc=Theorem 8.16}}

The following problems are ''decidable'' for arbitrary context-free languages:
*Emptiness: Given a context-free grammar ''A'', is <math>L(A) = \emptyset</math> ?{{sfn|Hopcroft|Ullman|1979|p=137|loc=Theorem 6.6(a)}}
*Finiteness: Given a context-free grammar ''A'', is <math>L(A)</math> finite?{{sfn|Hopcroft|Ullman|1979|p=137|loc=Theorem 6.6(b)}}
*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 (2003),<ref>{{cite book|author1=John E. Hopcroft |author2=Rajeev Motwani |author3=Jeffrey D. Ullman | title=Introduction to Automata Theory, Languages, and Computation| year=2003| publisher=Addison Wesley}} Here: Sect.7.6, p.304, and Sect.9.7, p.411</ref> 
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<ref name="Bar-Hillel.Perles.Shamir.1961">{{cite journal|author1=Yehoshua Bar-Hillel |author2=Micha Asher Perles |author3=Eli 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}}</ref>

===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]]<ref name="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?}}</ref>

==Notes==
{{Reflist|group=note}}

==References==
{{Reflist}}

=== Works cited ===
{{Refbegin}}
*{{Hopcroft and Ullman 1979}}{{sfn whitelist|CITEREFHopcroftUllman1979}}
* {{cite book |first=Arto |last=Salomaa |title = Formal Languages |publisher = ACM Monograph Series |year= 1973}}
{{Refend}}

== Further reading ==
* {{cite book |first1=Jean-Michel |last1=Autebert |first2=Jean |last2=Berstel |first3=Luc |last3=Boasson |url=http://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}}

{{Formal languages and grammars}}
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