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[clarification needed] lemma, is a lemma that gives a property shared by all context-free languages and generalizes the pumping lemma for regular languages.

The pumping lemma can be used to construct a proof by contradiction that a specific language is not context-free. Conversely, the pumping lemma does not suffice to guarantee that a language is context-free; there are other necessary conditions, such as Ogden's lemma, or the Interchange lemma.

Formal statementEdit

 
Proof idea: If   is sufficiently long, its derivation tree w.r.t. a Chomsky normal form grammar must contain some nonterminal   twice on some tree path (upper picture). Repeating   times the derivation part   ⇒...⇒   obtains a derivation for   (lower left and right picture for   and  , respectively).

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

 

with substrings   and  , such that

1.  ,
2.  , and
3.   for all  .

Below is a formal expression of the Pumping Lemma.

 

Informal statement and explanationEdit

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  , where   is a constant—called the pumping length—that varies between context-free languages.

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

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

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

Usage of the lemmaEdit

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   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   in L. The pumping lemma tells us that s can be written in the form  , where u, v, w, x, and y are substrings, such that  ,  , and   for every integer  . By the choice of s and the fact that  , 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:

  1.   for some  .
  2.   for some j and k with  
  3.   for some  .
  4.   for some j and k with  .
  5.   for some  .

For each case, it is easily verified that   does not contain equal numbers of each letter for any  . Thus,   does not have the form  . 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

 

for s=bjckdl with e.g. j≥1 choose vwx to consist only of b’s, for s=aibjcjdj choose vwx to consist only of a’s; in both cases all pumped strings are still in L.[2]

A precursor of the pumping lemma was used in 1960 by Scheinberg to prove that   is not context-free.[3]

ReferencesEdit

  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
  3. ^ Stephen Scheinberg (1960). "Note on the Boolean Properties of Context Free Languages" (PDF). Information and Control. 3: 372–375. doi:10.1016/s0019-9958(60)90965-7. Here: Lemma 3, and its use on p.374-375.