Pumping lemma for context-free languages

In computer science, in particular in formal language theory, the pumping lemma for context-free languages, also known as the Bar-Hillel lemma, is a lemma that gives a property shared by all context-free languages and generalizes the pumping lemma for regular languages. As the pumping lemma does not suffice to guarantee that a language is context-free there are more stringent necessary conditions, such as Ogden's lemma.

Formal statement

Proof idea: If

s

is sufficiently long, its derivation tree w.r.t. a Chomsky normal form grammar must contain some nonterminal

N

twice on some tree path (upper picture). Repeating

n

times the derivation part

N

⇒...⇒

vNx

obtains a derivation for

uv^{n}wx^{n}y

(lower left and right picture for

n=0

and

2

, respectively).

If a language $L$ is context-free, then there exists some integer $p\geq 1$ (called a "pumping length"^[1]) such that every string $s$ in $L$ that has a length of $p$ or more symbols (i.e. with $|s|\geq p$ ) can be written as

s=uvwxy

with substrings $u,v,w,x$ and $y$ , such that

1.

|vwx|\leq p

,

2.

|vx|\geq 1

, and

3.

uv^{n}wx^{n}y\in L

for all

n\geq 0

.

Below is a formal expression of the Pumping Lemma.

${\begin{array}{l}(\forall L\subseteq \Sigma ^{*})\\\quad ({\mbox{context free}}(L)\Rightarrow \\\quad ((\exists p\geq 1)((\forall s\in L)((|s|\geq p)\Rightarrow \\\quad ((\exists u,v,w,x,y\in \Sigma ^{*})(s=uvwxy\land |vwx|\leq p\land |vx|\geq 1\land (\forall n\geq 0)(uv^{n}wx^{n}y\in L)))))))\end{array}}$

Informal statement and explanation

The pumping lemma for context-free languages (called just "the pumping lemma" for the rest of this article) describes a property that all context-free languages are guaranteed to have.

The property is a property of all strings in the language that are of length at least $p$ , where $p$ is a constant—called the pumping length—that varies between context-free languages.

Say $s$ is a string of length at least $p$ that is in the language.

The pumping lemma states that $s$ can be split into five substrings, $s=uvwxy$ , where $vx$ is non-empty and the length of $vwx$ is at most $p$ , such that repeating $v$ and $x$ any (and the same) number of times in $s$ produces a string that is still in the language (it is possible and often useful to repeat zero times, which removes $v$ and $x$ from the string). This process of "pumping up" additional copies of $v$ and $x$ is what gives the pumping lemma its name.

Finite languages (which are regular and hence context-free) obey the pumping lemma trivially by having $p$ equal to the maximum string length in $L$ plus one. As there are no strings of this length the pumping lemma is not violated.

Usage of the lemma

The pumping lemma is often used to prove that a given language $L$ is non-context-free, by showing that arbitrarily long strings $s$ are in $L$ that cannot be "pumped" without producing strings outside $L$ .

For example, the language $L=\{a^{n}b^{n}c^{n}|n>0\}$ can be shown to be non-context-free by using the pumping lemma in a proof by contradiction. First, assume that $L$ is context free. By the pumping lemma, there exists an integer $p$ which is the pumping length of language $L$ . Consider the string $s=a^{p}b^{p}c^{p}$ in $L$ . The pumping lemma tells us that $s$ can be written in the form $s=uvwxy$ , where $u,v,w,x$ , and $y$ are substrings, such that $|vwx|\leq p$ , $|vx|\geq 1$ , and $uv^{i}wx^{i}y\in L$ for every integer $i\geq 0$ . By the choice of $s$ and the fact that $|vwx|\leq p$ , it is easily seen that the substring $vwx$ can contain no more than two distinct symbols. That is, we have one of five possibilities for $vwx$ :

$vwx=a^{j}$ for some $j\leq p$ .
$vwx=a^{j}b^{k}$ for some $j$ and $k$ with $j+k\leq p$
$vwx=b^{j}$ for some $j\leq p$ .
$vwx=b^{j}c^{k}$ for some $j$ and $k$ with $j+k\leq p$ .
$vwx=c^{j}$ for some $j\leq p$ .

For each case, it is easily verified that $uv^{i}wx^{i}y$ does not contain equal numbers of each letter for any $i\neq 1$ . Thus, $uv^{2}wx^{2}y$ does not have the form $a^{i}b^{i}c^{i}$ . This contradicts the definition of $L$ . Therefore, our initial assumption that $L$ is context free must be false.

While the pumping lemma is often a useful tool to prove that a given language is not context-free, it does not give a complete characterization of the context-free languages. If a language does not satisfy the condition given by the pumping lemma, we have established that it is not context-free.

On the other hand, there are languages that are not context-free, but still satisfy the condition given by the pumping lemma, for example $L=\{b^{j}c^{k}d^{l}|j,k,l\in \mathbb {N} \}$ ∪ { aⁱb^jc^jd^j | i, j ∈ ℕ, i≥1 }: for s=b^jc^kd^l with e.g. j≥1 choose vwx to consist only of b’s, for s=aⁱb^jc^jd^j choose vwx to consist only of a’s; in both cases all pumped strings are still in L.^[2]

References

↑ Berstel, Jean; Lauve, Aaron; Reutenauer, Christophe; Saliola, Franco V. (2009). Combinatorics on words. Christoffel words and repetitions in words (PDF). CRM Monograph Series. 27. Providence, RI: American Mathematical Society. p. 90. ISBN 978-0-8218-4480-9. Zbl 1161.68043. (Also see [www-igm.univ-mlv.fr/~berstel/ Aaron Berstel's website)
↑ John E. Hopcroft, Jeffrey D. Ullman (1979). Introduction to Automata Theory, Languages, and Computation. Addison-Wesley. ISBN 0-201-02988-X. Here: sect.6.1, p.129

Bar-Hillel, Y.; Micha Perles; Eli Shamir (1961). "On formal properties of simple phrase-structure grammars". Zeitschrift für Phonetik, Sprachwissenschaft, und Kommunikationsforschung. 14 (2): 143–172. — Reprinted in: Y. Bar-Hillel (1964). Language and Information: Selected Essays on their Theory and Application. Addison-Wesley series in logic. Addison-Wesley. pp. 116–150. ISBN 0201003732. OCLC 783543642.
Michael Sipser (1997). Introduction to the Theory of Computation. PWS Publishing. ISBN 0-534-94728-X. Section 1.4: Nonregular Languages, pp. 77–83. Section 2.3: Non-context-free Languages, pp. 115–119.

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[BLRS90-1] Berstel, Jean; Lauve, Aaron; Reutenauer, Christophe; Saliola, Franco V. (2009). Combinatorics on words. Christoffel words and repetitions in words (PDF). CRM Monograph Series. 27. Providence, RI: American Mathematical Society. p. 90. ISBN 978-0-8218-4480-9. Zbl 1161.68043. (Also see [www-igm.univ-mlv.fr/~berstel/ Aaron Berstel's website)

[2] John E. Hopcroft, Jeffrey D. Ullman (1979). Introduction to Automata Theory, Languages, and Computation. Addison-Wesley. ISBN 0-201-02988-X. Here: sect.6.1, p.129