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. The pumping lemma can be used to construct a refutation 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 statement
If a language L {\displaystyle L} is context-free, then there exists some integer p ≥ 1 {\displaystyle p\geq 1} (called a "pumping length") such that every string s {\displaystyle s} in L {\displaystyle L} that has a length of p {\displaystyle p} or more symbols (i.e. with | s | ≥ p {\displaystyle |s|\geq p} ) can be written as
s = u v w x y {\displaystyle s=uvwxy}
with substrings u , v , w , x {\displaystyle u,v,w,x} and y {\displaystyle y} , such that
| v x | ≥ 1 {\displaystyle |vx|\geq 1} ,
| v w x | ≤ p {\displaystyle |vwx|\leq p} , and
u v n w x n y ∈ L {\displaystyle uv^{n}wx^{n}y\in L} for all n ≥ 0 {\displaystyle n\geq 0} . Below is a formal expression of the Pumping Lemma.
( ∀ L ⊆ Σ ∗ ) ( context free ( L ) ⇒ ( ( ∃ p ≥ 1 ) ( ( ∀ s ∈ L ) ( ( | s | ≥ p ) ⇒ ( ( ∃ u , v , w , x , y ∈ Σ ∗ ) ( s = u v w x y ∧ | v x | ≥ 1 ∧ | v w x | ≤ p ∧ ( ∀ n ≥ 0 ) ( u v n w x n y ∈ L ) ) ) ) ) ) ) {\displaystyle {\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 |vx|\geq 1\land |vwx|\leq p\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 holds for all strings in the language that are of length at least p {\displaystyle p} , where p {\displaystyle p} is a constant—called the pumping length—that varies between context-free languages. Say s {\displaystyle s} is a string of length at least p {\displaystyle p} that is in the language. The pumping lemma states that s {\displaystyle s} can be split into five substrings, s = u v w x y {\displaystyle s=uvwxy} , where v x {\displaystyle vx} is non-empty and the length of v w x {\displaystyle vwx} is at most p {\displaystyle p} , such that repeating v {\displaystyle v} and x {\displaystyle x} the same number of times ( n {\displaystyle n} ) in s {\displaystyle s} produces a string that is still in the language. It is often useful to repeat zero times, which removes v {\displaystyle v} and x {\displaystyle x} from the string (this is "pumping down"). This process of "pumping up" s {\displaystyle s} with additional copies of v {\displaystyle v} and x {\displaystyle 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 {\displaystyle p} equal to the maximum string length in L {\displaystyle L} plus one. As there are no strings of this length the pumping lemma holds vacuously.
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, if S ⊂ N {\displaystyle S\subset \mathbb {N} } is infinite but does not contain an (infinite) arithmetic progression, then L = { a n : n ∈ S } {\displaystyle L=\{a^{n}:n\in S\}} is not context-free. In particular, neither the prime numbers nor the square numbers are context-free. For example, the language L = { a n b n c n | n > 0 } {\displaystyle 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 {\displaystyle s=a^{p}b^{p}c^{p}} in L. The pumping lemma tells us that s can be written in the form s = u v w x y {\displaystyle s=uvwxy} , where u, v, w, x, and y are substrings, such that | v x | ≥ 1 {\displaystyle |vx|\geq 1} , | v w x | ≤ p {\displaystyle |vwx|\leq p} , and u v i w x i y ∈ L {\displaystyle uv^{i}wx^{i}y\in L} for every integer i ≥ 0 {\displaystyle i\geq 0} . By the choice of s and the fact that | v w x | ≤ p {\displaystyle |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:
v w x = a j {\displaystyle vwx=a^{j}} for some j ≤ p {\displaystyle j\leq p} .
v w x = a j b k {\displaystyle vwx=a^{j}b^{k}} for some j and k with j + k ≤ p {\displaystyle j+k\leq p}
v w x = b j {\displaystyle vwx=b^{j}} for some j ≤ p {\displaystyle j\leq p} .
v w x = b j c k {\displaystyle vwx=b^{j}c^{k}} for some j and k with j + k ≤ p {\displaystyle j+k\leq p} .
v w x = c j {\displaystyle vwx=c^{j}} for some j ≤ p {\displaystyle j\leq p} . In each case, u v i w x i y {\displaystyle uv^{i}wx^{i}y} does not contain equal numbers of each letter for any i ≠ 1 {\displaystyle i\neq 1} . Thus, u v 2 w x 2 y {\displaystyle uv^{2}wx^{2}y} does not have the form a i b i c i {\displaystyle a^{i}b^{i}c^{i}} . This contradicts the definition of L. Therefore, our initial assumption that L is context free must be false. In 1960, Scheinberg proved that L = { a n b n a n | n > 0 } {\displaystyle L=\{a^{n}b^{n}a^{n}|n>0\}} is not context-free using a precursor of the pumping lemma. While the pumping lemma is often a useful tool to prove that a given language is not context-free, 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 ∈ N } ∪ { a i b j c j d j | i , j ∈ N , i ≥ 1 } {\displaystyle L=\{b^{j}c^{k}d^{l}|j,k,l\in \mathbb {N} \}\cup \{a^{i}b^{j}c^{j}d^{j}|i,j\in \mathbb {N} ,i\geq 1\}}
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. To prove that a given language is context-free, it is sufficient to construct a pushdown automaton that accepts it.
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