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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 ...
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 {\displaystyle p} , where p {\displaystyle p} is a constant—called the ...
The set of all context-free languages is identical to the set of languages accepted by pushdown automata, which makes these languages amenable to parsing.Further, for a given CFG, there is a direct way to produce a pushdown automaton for the grammar (and thereby the corresponding language), though going the other way (producing a grammar given an automaton) is not as direct.
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
This allows natural language features such as agreement and reference, and programming language analogs 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.
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).
Burnside's lemma also known as the Cauchy–Frobenius lemma; Frattini's lemma (finite groups) Goursat's lemma; Mautner's lemma (representation theory) Ping-pong lemma (geometric group theory) Schreier's subgroup lemma; Schur's lemma (representation theory) Zassenhaus lemma
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