Context-free language: Difference between revisions

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+== art Louis Vuitton Luggage Tag ==
-In [[formal language theory]], a  '''context-free language''' (CFL) is a [[formal language|language]] generated by some [[context-free grammar]] (CFG). Different CF grammars can generate the same CF language, or conversely, a given CF language can be generated by different CF grammars. It is important to distinguish properties of the language (intrinsic properties) from properties of a particular grammar (extrinsic properties).
-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. Indeed, given a CFG, there is a direct way to produce a pushdown automaton for the grammar (and corresponding language), though going the other way (producing a grammar given an automaton) is not as direct.
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-Context-free languages have many applications in [[programming languages]]; for example, the [[Dyck language|language of all properly matched parentheses]] is generated by the grammar <math>S\to SS ~|~ (S) ~|~ \varepsilon</math>.  Also, most arithmetic expressions are generated by context-free grammars.
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-An archetypical 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 <math>a</math>'s, and the entire second halves of which are <math>b</math>'s. <math>L</math> is generated by the grammar <math>S\to aSb ~|~ ab</math>, and is accepted by the 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:
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-<math>\delta(q_0, a, z) = (q_0, az)</math><br>
-<math>\delta(q_0, a, a) = (q_0, aa)</math><br>
-<math>\delta(q_0, b, a) = (q_1, \varepsilon)</math><br>
-<math>\delta(q_1, b, a) = (q_1, \varepsilon)</math>
-<math>\delta(\mathrm{state}_1, \mathrm{read}, \mathrm{pop}) = (\mathrm{state}_2, \mathrm{push})</math>
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-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}}
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+ </ul>
-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 language]]s 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]] or a number of other methods, such as [[Ogden's lemma]] or [[Parikh's theorem]].<ref>[http://cs.stackexchange.com/questions/265/how-to-prove-that-a-language-is-not-context-free How to prove that a language is not context-free?]</ref>
+== I cherish them. Obama New Balance Australia Stores ==
-== Closure properties ==
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-Context-free languages are [[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:
+   <li>[http://bbs.duduqiu.com/forum.php?mod=viewthread&tid=24814 http://bbs.duduqiu.com/forum.php?mod=viewthread&tid=24814]</li>
-* the [[union (set theory)|union]] <math>L \cup P</math> of ''L'' and ''P''
+   <li>[http://dz.shuang.ca/forum.php?mod=viewthread&tid=32383&fromuid=1872 http://dz.shuang.ca/forum.php?mod=viewthread&tid=32383&fromuid=1872]</li>
-* the reversal of ''L''
-* the [[concatenation]] <math>L \cdot P</math> of ''L'' and ''P''
+   <li>[http://www.forestry.crs.gov.ng/index.php/forum/3-suggestion-box/226350-coaches-cheap-nike-shox-australia.html#227280 http://www.forestry.crs.gov.ng/index.php/forum/3-suggestion-box/226350-coaches-cheap-nike-shox-australia.html#227280]</li>
-* the [[Kleene star]] <math>L^*</math> of ''L''
-* the image <math>\varphi(L)</math> of ''L'' under a [[String operations#String homomorphism|homomorphism]] <math>\varphi</math>
+   <li>[http://124.128.87.5:8888/discuz/forum.php?mod=viewthread&tid=436427 http://124.128.87.5:8888/discuz/forum.php?mod=viewthread&tid=436427]</li>
-* the image <math>\varphi^{-1}(L)</math> of ''L'' under an [[String operations#String homomorphism|inverse homomorphism]] <math>\varphi^{-1}</math>
-* the [[cyclic shift]] of ''L'' (the language <math>\{vu : uv \in L \}</math>)
+ </ul>
-Context-free languages are not closed under [[complement (complexity)|complement]], [[intersection (set theory)|intersection]], or [[Complement (set theory)#Relative complement|difference]]. 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.
-===Nonclosure under intersection and complement===
-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>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]].
-Context-free languages are also not closed under complementation, as for any languages A and B:  <math>A \cap B = \overline{\overline{A} \cup \overline{B}} </math>.
-==Decidability properties==
-The following problems are [[Undecidable problem|undecidable]] for arbitrary [[context-free grammar]]s A and B:
-*Equivalence: Given two context-free grammars ''A'' and ''B'', is <math>L(A)=L(B)</math>?
-*Intersection Emptiness: Given two context-free grammars ''A'' and ''B'', is <math>L(A) \cap L(B) = \emptyset </math> ? 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> and the variant of the problem where ''B'' is a regular grammar is decidable.
-*Containment: Given a context-free grammar ''A'', is <math>L(A) \subseteq L(B)</math> ? Again, the variant of the problem where ''B'' is a regular grammar is decidable.
-*Universality: Given a context-free grammar ''A'', is <math>L(A)=\Sigma^*</math> ?
-The following problems are ''decidable'' for arbitrary context-free languages:
-*Emptiness: Given a context-free grammar ''A'', is <math>L(A)=\emptyset</math> ?
-*Finiteness: Given a context-free grammar ''A'', is <math>L(A)</math> finite?
-*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]].
-== Parsing ==
-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 some grammar <math>G</math>; is also known as ''[[parsing]]''.
-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 the [[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 doi|10.1016/S0019-9958(65)90426-2}}</ref>
-See also [[parsing expression grammar]] as an alternative approach to grammar and parser.
-== See also ==
-* [[Deterministic context-free language]]
-* [[Parsing]]
-== Notes ==
-{{Reflist}}
-== References ==
-{{refbegin}}
-* {{cite book| author = [[Seymour Ginsburg]] | title = The Mathematical Theory of Context-Free Languages | year = 1966 | publisher = McGraw-Hill, Inc. | location = New York, NY, USA}}
-*{{cite book
-  | last1 = Hopcroft
-  | first1 = John E.
-  | last2 = Ullman
-  | first2 = Jeffrey D.
-  | title = Introduction to Automata Theory, Languages, and Computation
-  | publisher = Addison-Wesley
-  | edition = 1st
-   | year = 1979
-  | ref = harv
-}}
-* {{cite book|author=Arto Salomaa|title = Formal Languages|publisher = ACM Monograph Series|year= 1973}}
-* {{cite book|author = [[Michael Sipser]] | year = 1997 | title = Introduction to the Theory of Computation | publisher = PWS Publishing | isbn = 0-534-94728-X}} Chapter 2: Context-Free Languages, pp.&nbsp;91–122.
-* Jean-Michel Autebert, Jean Berstel, Luc Boasson, [http://www-igm.univ-mlv.fr/~berstel/Articles/1997CFLPDA.pdf Context-Free Languages and Push-Down Automata], in: G. Rozenberg, A. Salomaa (eds.), Handbook of Formal Languages, Vol. 1, Springer-Verlag, 1997, 111-174.
-{{refend}}
-{{Formal languages and grammars}}
-[[Category:Formal languages]]
-[[Category:Languages]]
-[[Category:Linguistics]]

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