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In the theory of formal languages, the pumping lemma for regular languages is a lemma that describes an essential property of all regular languages. Informally, it says that all sufficiently long strings in a regular language may be pumped —that is, have a middle section of the string repeated an arbitrary number of times—to produce a new ...
In computer science, in particular in formal language theory, the pumping lemma for context-free languages, also known as the Bar-Hillel lemma, [1] 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 ...
Pumping lemma for context-free languages, the fact that all sufficiently long strings in such a language have a pair of substrings that can be repeated arbitrarily many times, usually used to prove that certain languages are not context-free; Pumping lemma for indexed languages; Pumping lemma for regular tree languages
In a context-free grammar, we can pair up characters the way we do with brackets. The simplest example: S → aSb S → ab. This grammar generates the language {:}, which is not regular (according to the pumping lemma for regular languages). The special character ε stands for the empty string.
Ogden's lemma is often stated in the following form, which can be obtained by "forgetting about" the grammar, and concentrating on the language itself: If a language L is context-free, then there exists some number (where p may or may not be a pumping length) such that for any string s of length at least p in L and every way of "marking" p or more of the positions in s, s can be written as
To convert a grammar to Chomsky normal form, a sequence of simple transformations is applied in a certain order; this is described in most textbooks on automata theory. [4]: 87–94 [5] [6] [7] The presentation here follows Hopcroft, Ullman (1979), but is adapted to use the transformation names from Lange, Leiß (2009).
The proof is essentially the same as the standard pumping lemma: use the pigeonhole principle to find copies of some nonterminal symbol in the longest path in the shortest derivation tree. Now we prove the first part of Parikh's theorem, making use of the above lemma.
A grammar formalism is said to have polynomial parsing if its membership problem can be solved in deterministic polynomial time. This is the problem to decide, given a grammar G written in the formalism and a string w, whether w is generated by G – that is, whether w is "grammatical" according to G.
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