Context-free language

Informal language theory,acontext-free language(CFL), also called aChomskytype-2 language,is alanguagegenerated by acontext-free grammar(CFG).

Context-free languages have many applications inprogramming languages,in particular, most arithmetic expressions are generated by context-free grammars.

Background[edit]

Context-free grammar[edit]

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[edit]

The set of all context-free languages is identical to the set of languages accepted bypushdown 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[edit]

An example context-free language is $L=\{a^{n}b^{n}:n\geq 1\}$ ,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 $S\to aSb~|~ab$ . This language is notregular. It is accepted by thepushdown automaton $M=(\{q_{0},q_{1},q_{f}\},\{a,b\},\{a,z\},\delta,q_{0},z,\{q_{f}\})$ where $\delta$ is defined as follows:^{[note 1]}

{\begin{aligned}\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{aligned}}

Unambiguous CFLs are a proper subset of all CFLs: there areinherently ambiguousCFLs. An example of an inherently ambiguous CFL is the union of $\{a^{n}b^{m}c^{m}d^{n}|n,m>0\}$ with $\{a^{n}b^{n}c^{m}d^{m}|n,m>0\}$ .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 $\{a^{n}b^{n}c^{n}d^{n}|n>0\}$ which is the intersection of these two languages.^[1]

Dyck language[edit]

Thelanguage of all properly matched parenthesesis generated by the grammar $S\to SS~|~(S)~|~\varepsilon$ .

Properties[edit]

Context-free parsing[edit]

The context-free nature of the language makes it simple to parse with a pushdown automaton.

Determining an instance of themembership problem;i.e. given a string $w$ ,determine whether $w\in L(G)$ where $L$ is the language generated by a given grammar $G$ ;is also known asrecognition.Context-free recognition forChomsky normal formgrammars was shown byLeslie G. Valiantto be reducible to booleanmatrix multiplication,thus inheriting its complexity upper bound ofO(n^2.3728596).^[2]^{[note 2]} 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.^[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 calledparsing.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 theCYK algorithmandEarley's Algorithm.

A special subclass of context-free languages are thedeterministic context-free languageswhich are defined as the set of languages accepted by adeterministic pushdown automatonand can be parsed by aLR(k) parser.^[4]

See alsoparsing expression grammaras an alternative approach to grammar and parser.

Closure properties[edit]

The class of context-free languages isclosedunder the following operations. That is, ifLandPare context-free languages, the following languages are context-free as well:

theunion $L\cup P$ ofLandP^[5]
the reversal ofL^[6]
theconcatenation $L\cdot P$ ofLandP^[5]
theKleene star $L^{*}$ ofL^[5]
the image $\varphi (L)$ ofLunder ahomomorphism $\varphi$ ^[7]
the image $\varphi ^{-1}(L)$ ofLunder aninverse homomorphism $\varphi ^{-1}$ ^[8]
thecircular shiftofL(the language $\{vu:uv\in L\}$ )^[9]
the prefix closure ofL(the set of allprefixesof strings fromL)^[10]
thequotientL/RofLby a regular languageR^[11]

Nonclosure under intersection, complement, and difference[edit]

The context-free languages are not closed under intersection. This can be seen by taking the languages $A=\{a^{n}b^{n}c^{m}\mid m,n\geq 0\}$ and $B=\{a^{m}b^{n}c^{n}\mid m,n\geq 0\}$ ,which are both context-free.^{[note 3]}Their intersection is $A\cap B=\{a^{n}b^{n}c^{n}\mid n\geq 0\}$ ,which can be shown to be non-context-free by thepumping lemma for context-free languages.As a consequence, context-free languages cannot be closed under complementation, as for any languagesAandB,their intersection can be expressed by union and complement: $A\cap B={\overline {{\overline {A}}\cup {\overline {B}}}}$ .In particular, context-free language cannot be closed under difference, since complement can be expressed by difference: ${\overline {L}}=\Sigma ^{*}\setminus L$ .^[12]

However, ifLis a context-free language andDis a regular language then both their intersection $L\cap D$ and their difference $L\setminus D$ are context-free languages.^[13]

Decidability[edit]

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 areundecidablefor arbitrarily givencontext-free grammarsA and B:

Equivalence: is $L(A)=L(B)$ ?^[14]
Disjointness: is $L(A)\cap L(B)=\emptyset$ ?^[15]However, the intersection of a context-free language and aregularlanguage is context-free,^[16]^[17]hence the variant of the problem whereBis a regular grammar is decidable (see "Emptiness" below).
Containment: is $L(A)\subseteq L(B)$ ?^[18]Again, the variant of the problem whereBis a regular grammar is decidable,^{[citation needed]}while that whereAis regular is generally not.^[19]
Universality: is $L(A)=\Sigma ^{*}$ ?^[20]
Regularity: is $L(A)$ a regular language?^[21]
Ambiguity: is every grammar for $L(A)$ ambiguous?^[22]

The following problems aredecidablefor arbitrary context-free languages:

Emptiness: Given a context-free grammarA,is $L(A)=\emptyset$ ?^[23]
Finiteness: Given a context-free grammarA,is $L(A)$ finite?^[24]
Membership: Given a context-free grammarG,and a word $w$ ,does $w\in L(G)$ ?Efficient polynomial-time algorithms for the membership problem are theCYK algorithmandEarley's Algorithm.

According to Hopcroft, Motwani, Ullman (2003),^[25] many of the fundamental closure and (un)decidability properties of context-free languages were shown in the 1961 paper ofBar-Hillel,Perles, and Shamir^[26]

Languages that are not context-free[edit]

The set $\{a^{n}b^{n}c^{n}d^{n}|n>0\}$ is acontext-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 thepumping lemma for context-free languages^[26]or a number of other methods, such asOgden's lemmaorParikh's theorem.^[28]

Notes[edit]

^meaning of $\delta$ 's arguments and results: $\delta (\mathrm {state} _{1},\mathrm {read},\mathrm {pop} )=(\mathrm {state} _{2},\mathrm {push} )$
^In Valiant's paper,O(n^2.81) was the then-best known upper bound. SeeMatrix multiplication#Computational complexityfor bound improvements since then.
^A context-free grammar for the languageAis given by the following production rules, takingSas the start symbol:S→Sc|aTb|ε;T→aTb|ε.The grammar forBis analogous.

References[edit]

^Hopcroft & Ullman 1979,p. 100, Theorem 4.7.
^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.
^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 2003-04-27.
^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.
^^a ^b ^cHopcroft & 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.
^Stephen Scheinberg (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 2018-11-26.
^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 2014-12-12.RetrievedJune 6,2020.
^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).
^John E. Hopcroft; Rajeev Motwani; Jeffrey D. Ullman (2003).Introduction to Automata Theory, Languages, and Computation.Addison Wesley.Here: Sect.7.6, p.304, and Sect.9.7, p.411
^^a ^bYehoshua Bar-Hillel; Micha Asher Perles; Eli Shamir (1961). "On Formal Properties of Simple Phrase-Structure Grammars".Zeitschrift für Phonetik, Sprachwissenschaft und Kommunikationsforschung.14(2): 143–172.
^Hopcroft & Ullman 1979.
^"How to prove that a language is not context-free?".

Works cited[edit]

Hopcroft, John E.; Ullman, Jeffrey D. (1979).Introduction to Automata Theory, Languages, and Computation(1st ed.). Addison-Wesley.ISBN 9780201029888.
Salomaa, Arto (1973).Formal Languages.ACM Monograph Series.