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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.
For example, a paper in 1999 described how JFLAP now allowed one to experiment with construction type proofs, such as converting an NFA to a DFA to a minimal state DFA, and as another example, converting NPDA to CFG and vice versa. [6] In 2002 JFLAP was converted to Swing. In 2005-2007 a study was run with fourteen institutions using JFLAP.
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
Abstract syntax tree of the arithmetic expression "a^2+4*b" wrt. the example grammar (top) and its Chomsky normal form (bottom) The following grammar, with start symbol Expr , describes a simplified version of the set of all syntactical valid arithmetic expressions in programming languages like C or Algol60 .
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
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.