Context-free language#Decidability properties

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

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 $L = \{a^nb^n:n\geq1\}$ , 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 $S\to aSb ~|~ ab$ .

This language is not regular.

It is accepted by the pushdown automaton $M=(\{q_0,q_1,q_f\}, \{a,b\}, \{a,z\}, \delta, q_0, z, \{q_f\})$ where $\delta$ is defined as follows:meaning of $\delta$ 's arguments and results: $\delta(\mathrm{state}_1, \mathrm{read}, \mathrm{pop}) = (\mathrm{state}_2, \mathrm{push})$

: $\begin{align}
\delta(q_0, a, z) &= (q_0, az) \\
\delta(q_0, a, a) &= (q_0, aa) \\
\delta(q_0, b, a) &= (q_1, \varepsilon) \\
\delta(q_1, b, a) &= (q_1, \varepsilon) \\
\delta(q_1, \varepsilon, z) &= (q_f, \varepsilon)
\end{align}$

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 $\{a^n b^m c^m d^n | n, m > 0\}$ 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.{{sfn|Hopcroft|Ullman|1979|p=100|loc=Theorem 4.7}}

=Dyck language=

The language of all properly matched parentheses is generated by the grammar $S\to SS ~|~ (S) ~|~ \varepsilon$ .

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 $w$ , determine whether $w \in L(G)$ where $L$ is the language generated by a given grammar $G$ ; 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(n^2.3728596).{{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 }}In Valiant's paper, O(n^2.81) was the then-best known upper bound. See Matrix multiplication#Computational complexity for bound improvements since then.

Conversely, Lillian Lee has shown O(n^3−ε) Boolean matrix multiplication to be reducible to O(n^3−3ε) CFG parsing, thus establishing some kind of lower bound for the latter.{{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 }}

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.{{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 = }}

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 $L \cup P$ 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 $L \cdot P$ of L and P{{sfn|Hopcroft|Ullman|1979|p=131|loc=Corollary of Theorem 6.1}}
the Kleene star $L^*$ of L{{sfn|Hopcroft|Ullman|1979|p=131|loc=Corollary of Theorem 6.1}}
the image $\varphi(L)$ of L under a homomorphism $\varphi$ {{sfn|Hopcroft|Ullman|1979|p=131-132|loc=Corollary of Theorem 6.2}}
the image $\varphi^{-1}(L)$ of L under an inverse homomorphism $\varphi^{-1}$ {{sfn|Hopcroft|Ullman|1979|p=132|loc=Theorem 6.3}}
the circular shift of L (the language $\{vu : uv \in L \}$ ){{sfn|Hopcroft|Ullman|1979|p=142-144|loc=Exercise 6.4c}}
the prefix closure of L (the set of all prefixes of strings from L){{sfn|Hopcroft|Ullman|1979|p=142|loc=Exercise 6.4b}}
the 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 $A = \{a^n b^n c^m \mid m, n \geq 0 \}$ and $B = \{a^m b^n c^n \mid m,n \geq 0\}$ , which are both context-free.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. 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 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: $A \cap B = \overline{\overline{A} \cup \overline{B}}$ . In particular, context-free language cannot be closed under difference, since complement can be expressed by difference: $\overline{L} = \Sigma^* \setminus L$ .{{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=372–375 | year=1960 | issue=4 | doi=10.1016/s0019-9958(60)90965-7| doi-access=free }}

However, if L is a context-free language and D is a regular language then both their intersection $L\cap D$ and their difference $L\setminus D$ are context-free languages.{{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}}

=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 $L(A)=L(B)$ ?{{sfn|Hopcroft|Ullman|1979|p=203|loc=Theorem 8.12(1)}}
Disjointness: is $L(A) \cap L(B) = \emptyset$ ?{{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,{{harvtxt|Salomaa|1973}}, p. 59, 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 $L(A) \subseteq L(B)$ ?{{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 $L(A)=\Sigma^*$ ?{{sfn|Hopcroft|Ullman|1979|p=203|loc=Theorem 8.11}}
Regularity: is $L(A)$ a regular language?{{sfn|Hopcroft|Ullman|1979|p=205|loc=Theorem 8.15}}
Ambiguity: is every grammar for $L(A)$ 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 $L(A) = \emptyset$ ?{{sfn|Hopcroft|Ullman|1979|p=137|loc=Theorem 6.6(a)}}
Finiteness: Given a context-free grammar A, is $L(A)$ finite?{{sfn|Hopcroft|Ullman|1979|p=137|loc=Theorem 6.6(b)}}
Membership: Given a context-free grammar G, and a word $w$ , does $w \in L(G)$ ? Efficient polynomial-time algorithms for the membership problem are the CYK algorithm and Earley's Algorithm.

According to Hopcroft, Motwani, Ullman (2003),{{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

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{{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}}

=Languages that are not context-free=

The set $\{a^n b^n c^n d^n | n > 0\}$ is a context-sensitive language, but there does not exist a context-free grammar generating this language.{{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 or a number of other methods, such as Ogden's lemma or Parikh's theorem.{{cite web| url = https://cs.stackexchange.com/q/265| title = How to prove that a language is not context-free?}}

Notes

References

= Works cited =

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