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Context-free grammar

# Context-free grammar

Revision as of 06:31, 14 February 2008; view current revision

In formal language theory, a context-free grammar (CFG) is a grammar in which every production rule is of the form

Vw

where V is a single nonterminal symbol, and w is a string of terminals and/or nonterminals (possibly empty). The term "context-free" expresses the fact that nonterminals can be rewritten without regard to the context in which they occur. A formal language is context-free if some context-free grammar generates it.

Context-free grammars play a central role in the description and design of programming languages and compilers. They are also used for analyzing the syntax of natural languages.

## Background

Since the time of Pāṇini, at least, linguists have described the grammars of languages in terms of their block structure, and described how sentences are recursively built up from smaller phrases, and eventually individual words or word elements.

The context-free grammar (or "phrase-structure grammar" [1] as Chomsky called it) formalism developed by Noam Chomsky,[1] in the mid-1950s, took the manner in which linguistics had described this grammatical structure, and then turned it into rigorous mathematics. A context-free grammar provides a simple and precise mechanism for describing the methods by which phrases in some natural language are built from smaller blocks, capturing the "block structure" of sentences in a natural way. Its simplicity makes the formalism amenable to rigorous mathematical study, but it comes at a price: important features of natural language syntax such as agreement and reference cannot be expressed in a natural way, or not at all.

Block structure was introduced into computer programming languages by the Algol project, which, as a consequence, also featured a context-free grammar to describe the resulting Algol syntax. This became a standard feature of computer languages, and the notation for grammars used in concrete descriptions of computer languages came to be known as Backus-Naur Form, after two members of the Algol language design committee.

The "block structure" aspect that context-free grammars capture is so fundamental to grammar that the terms syntax and grammar are often identified with context-free grammar rules, especially in computer science. Formal constraints not captured by the grammar are then considered to be part of the "semantics" of the language.

Context-free grammars are simple enough to allow the construction of efficient parsing algorithms which, for a given string, determine whether and how it can be generated from the grammar. An Earley parser is an example of such an algorithm, while the widely used LR and LL parsers are more efficient algorithms that deal only with more restrictive subsets of context-free grammars.

## Formal definitions

A context-free grammar G is a 4-tuple:

$G = (V\,, \Sigma\,, R\,, S\,)$ where

1. $V\,$ is a finite set of non-terminal characters or variables. They represent different types of phrase or clause in the sentence.

2. $\Sigma\,$ is a finite set of terminals, disjoint with $V\,$, which make up the actual content of the sentence.

3. $S\,$ is the start variable, used to represent the whole sentence (or program). It must be an element of $V\,$.

4. $R\,$ is a relation from $V\,$ to $(V\cup\Sigma)^{*}$ such that $\exist\, w\in (V\cup\Sigma)^{*}: (S,w)\in R$.

In addition, $R\,$ is a finite set. The members of $R\,$ are called the rules or productions of the grammar. The asterisk represents the Kleene star operation.

For any strings $u, v\in (V\cup\Sigma)^{*}$, we say $u\,$ yields $v\,$, written as $u\Rightarrow v\,$, if $\exists (\alpha, \beta)\in R, u_{1}, u_{2}\in (V\cup\Sigma)^{*}$ such that $u\,=u_{1}\alpha u_{2}$ and $v\,=u_{1}\beta u_{2}$. Thus, $\! v$ is the result of applying the rule $\! (\alpha, \beta)$ to $\! u$.

For any $u, v\in (V\cup\Sigma)^{*}, u\stackrel{*}{\Rightarrow} v$ (or $u\Rightarrow\Rightarrow v\,$ in some textbooks) if $\exists u_{1}, u_{2}, \cdots u_{k}\in (V\cup\Sigma)^{*}, k\geq 0$ such that $u\Rightarrow u_{1}\Rightarrow u_{2}\cdots\Rightarrow u_{k}\Rightarrow v$

The language of a grammar $G = (V\,, \Sigma\,, R\,, S\,)$ is the set

$L(G) = \{ w\in\Sigma^{*} : S\stackrel{*}{\Rightarrow} w\}$

A language $L\,$ is said to be a context-free language (CFL) if there exists a CFG, $G\,$ such that $L\,=\,L(G)$.

## Examples

### Example 1

A simple context-free grammar is given as:

S → aSb | ab;

where | is used to separate multiple options for the same non-terminal; so this is the same as

S → aSb
S → ab

The terminals here are a and b, while the only non-terminal is S. This grammar generates the language $\{ a^n b^n : n \ge 1 \}$ which is not regular.

The special character ε stands for the empty string. By changing the above grammar to :S → aSb | ε we obtain a grammar generating the language $\{ a^n b^n : n \ge 0 \}$ instead. This differs only in that it contains the empty string while the original grammar did not.

### Example 2

Here is a context-free grammar for syntactically correct infix algebraic expressions in the variables x, y and z:

S → x | y | z | S + S | S - S | S * S | S/S | (S)

This grammar can, for example, generate the string "( x + y ) * x - z * y / ( x + x )" as follows: "S" is the initial string. "S - S" is the result of applying the fifth transformation [S → S - S] to the nonterminal S. "S * S - S / S" is the result of applying the sixth transform to the first S and the seventh one to the second S. "( S ) * S - S / ( S )" is the result of applying the final transform to certain of the nonterminals. "( S + S ) * S - S * S / ( S + S )" is the result of the fourth and fifth transforms to certain nonterminals. "( x + y ) * x - z * y / ( x + x )" is the final result, obtained by using the first three transformations to turn the S nonterminals into the terminals x, y, and z.

This grammar is ambiguous, meaning that one can generate the same string with more than one parse tree. For example, "x + y * z" might have either the + or the * parsed first; presumably these will produce different results.

### Example 3

A context-free grammar for the language consisting of all strings over {a,b} for which the number of a's and b's are different is

S → U | V
U → TaU | TaT
V → TbV | TbT
T → aTbT | bTaT | ε

Here, the nonterminal T can generate all strings with the same number of a's as b's, the nonterminal U generates all strings with more a's than b's and the nonterminal V generates all strings with fewer a's than b's.

### Example 4

Another example of a context-free language is $\{ b^n a^m b^{2n} : n \ge 0, m \ge 0 \}$. This is not a regular language, but it is context free as it can be generated by the following context-free grammar:

S → bSbb | A
A → aA | ε

### Other examples

Context-free grammars are not limited in application to mathematical ("formal") languages. For example, it has been suggested that a class of Tamil poetry called Venpa is governed by a context-free grammar.[2]

### Derivations and syntax trees

There are two common ways to describe how a given string can be derived from the start symbol of a given grammar. The simplest way is to list the consecutive strings of symbols, beginning with the start symbol and ending with the string, and the rules that have been applied. If we introduce a strategy such as "always replace the left-most nonterminal first" then for context-free grammars the list of applied grammar rules is by itself sufficient. This is called the leftmost derivation of a string. For example, if we take the following grammar:

(1) S → S + S
(2) S → 1
(3) S → a

and the string "1 + 1 + a" then a left derivation of this string is the list [ (1), (1), (2), (2), (3) ]. Analogously the rightmost derivation is defined as the list that we get if we always replace the rightmost nonterminal first. In this case this could be the list [ (1), (3), (1), (2), (2)].

The distinction between leftmost derivation and rightmost derivation is important because in most parsers the transformation of the input is defined by giving a piece of code for every grammar rule that is executed whenever the rule is applied. Therefore it is important to know whether the parser determines a leftmost or a rightmost derivation because this determines the order in which the pieces of code will be executed. See for an example LL parsers and LR parsers.

A derivation also imposes in some sense a hierarchical structure on the string that is derived. For example, if the string "1 + 1 + a" is derived according to the leftmost derivation:

S → S + S (1)
→ S + S + S (1)
→ 1 + S + S (2)
→ 1 + 1 + S (2)
→ 1 + 1 + a (3)

the structure of the string would be:

{ { { 1 }S + { 1 }S }S + { a }S }S

where { ... }S indicates a substring recognized as belonging to S. This hierarchy can also be seen as a tree:

           S
/|\
/ | \
/  |  \
S  '+'  S
/|\      |
/ | \     |
S '+' S   'a'
|     |
'1'   '1'


This tree is called a concrete syntax tree (see also abstract syntax tree) of the string. In this case the presented leftmost and the rightmost derivations define the same syntax tree; however, there is another (leftmost) derivation of the same string

S → S + S (1)
→ 1 + S (2)
→ 1 + S + S (1)
→ 1 + 1 + S (2)
→ 1 + 1 + a (3)

and this defines the following syntax tree:

           S
/|\
/ | \
/  |  \
S  '+'  S
|      /|\
|     / | \
'1'   S '+' S
|     |
'1'   'a'


If, for certain strings in the language of the grammar, there is more than one parsing tree, then the grammar is said to be an ambiguous grammar. Ambiguity is the feature of grammars rather than the language because language is derived from grammar so;ambiguity does not depend upon language. Such grammars are usually hard to parse because the parser cannot always decide which grammar rule it has to apply.

## Normal forms

Every context-free grammar that does not generate the empty string can be transformed into one in which no rule has the empty string as a product [a rule with ε as a product is called an ε-production]. If it does generate the empty string, it will be necessary to include the rule $S \rarr \epsilon$, but there need be no other ε-rule. Every context-free grammar with no ε-production has an equivalent grammar in Chomsky normal form or Greibach normal form. "Equivalent" here means that the two grammars generate the same language.

Because of the especially simple form of production rules in Chomsky Normal Form grammars, this normal form has both theoretical and practical implications. For instance, given a context-free grammar, one can use the Chomsky Normal Form to construct a polynomial-time algorithm which decides whether a given string is in the language represented by that grammar or not (the CYK algorithm).

## Undecidable problems

Although some operations on context-free grammars are decidable due to their limited power, CFGs do have interesting undecidable problems. One of the simplest and most cited is the problem of deciding whether a CFG accepts the language of all strings. A reduction can be demonstrated to this problem from the well-known undecidable problem of determining whether a Turing machine accepts a particular input. The reduction uses the concept of a computation history, a string describing an entire computation of a Turing machine. We can construct a CFG that generates all strings that are not accepting computation histories for a particular Turing machine on a particular input, and thus it will accept all strings only if the machine does not accept that input.

As a consequence of this, it is also undecidable whether two CFGs describe the same language, since we can't even decide whether a CFG is equivalent to the trivial CFG deciding the language of all strings.

Another point worth mentioning is that the problem of determining if a context-sensitive grammar describes a context-free language is undecidable.

## Extensions

An obvious way to extend the context-free grammar formalism is to allow nonterminals to have arguments, the values of which are passed along within the rules. This allows natural language features such as agreement and reference, and programming language analogons such as the correct use and definition of identifiers, to be expressed in a natural way. E.g. we can now easily express that in English sentences, the subject and verb must agree in number.

In computer science, examples of this approach include affix grammars, attribute grammars, indexed grammars, and Van Wijngaarden two-level grammars.

Similar extensions exist in linguistics.

Another extension is to allow additional symbols to appear at the left hand side of rules, constraining their application. This produces the formalism of context-sensitive grammars.

## Linguistic applications

Chomsky initially hoped to overcome the limitations of context-free grammars by adding transformation rules.[1]

Such rules are another standard device in traditional linguistics; e.g. passivization in English. However, arbitrary transformations must be disallowed, since they are much too powerful (Turing complete). Much of generative grammar has been devoted to finding ways of refining the descriptive mechanisms of phrase-structure grammar and transformation rules such that exactly the kinds of things can be expressed that natural language actually allows.

His general position regarding the non-context-freeness of natural language has held up since then[3], although his specific examples regarding the inadequacy of CFGs in terms of their weak generative capacity were later disproved.[4] Gerald Gazdar and Geoffrey Pullum have argued that despite a few non-context-free constructions in natural language (such as cross-serial dependencies in Swiss German[3] and reduplication in Bambara[5]), the vast majority of forms in natural language are indeed context-free.[4]

## References

• Chomsky, Noam (Sept. 1956). "Three models for the description of language". Information Theory, IEEE Transactions 2 (3).

• Template:Cite book Section 2.1: Context-Free Grammars, pp.91–101. Section 4.1.2: Decidable problems concerning context-free languages, pp.156–159. Section 5.1.1: Reductions via computation histories: pp.176–183.

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