Ogden's lemma

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In the theory of formal languages, Ogden's lemma (named after William F. Ogden) is a generalization of the pumping lemma for context-free languages.

Ogden's lemma states that if a language is context-free, then there exists some number (where may or may not be a pumping length) such that for any string of length at least in and every way of "marking" or more of the positions in , can be written as

with strings and , such that

  1. has at most marked positions,
  2. has at least one marked position, and
  3. for all .

In the special case where every position is marked, Ogden's lemma is equivalent to the pumping lemma for context-free languages. Ogden's lemma can be used to show that certain languages are not context-free in cases where the pumping lemma is not sufficient. An example is the language . Ogden's lemma can also be used to prove the inherent ambiguity of some languages[citation needed].

See also[edit]


  • Ogden, W. (1968). "A helpful result for proving inherent ambiguity". Mathematical Systems Theory. 2 (3): 191–194. doi:10.1007/BF01694004.
  • Hopcroft, Motwani and Ullman (1979). Automata Theory, Languages, and Computation. Addison-Wesley. ISBN 81-7808-347-7.