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The pumping lemma is often used to prove that a particular language is non-regular: a proof by contradiction may consist of exhibiting a string (of the required length) in the language that lacks the property outlined in the pumping lemma. Example: The language = {:} over the alphabet = {,} can be shown to be non-regular as follows:
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.
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
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
Illustration of the pumping lemma for regular automata Chomsky and Miller (1957) [ 15 ] used the pumping lemma : they guess a part v of an input string uvw and try to build a corresponding cycle into the automaton to be learned; using membership queries they ask, for appropriate k , which of the strings uw , uvvw , uvvvw , ..., uv k w also ...
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. By changing the above grammar to S → aSb S → ε
Proof (1) If L {\displaystyle L} is regular, construct a minimal DFA to accept it. Clearly, if x , y {\displaystyle x,y} end up in the same state after running through the DFA, then x ∼ L y {\displaystyle x\sim _{L}y} , thus the number of equivalence classes of ∼ L {\displaystyle \sim _{L}} is at most the number of DFA states, which must be ...
For example, the regular language = ... (by the pumping lemma for regular ... (by the pumping lemma for context-free languages). A proof that this grammar ...