Ogden's lemma

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 $L$ is context-free, then there exists some number $p\geq 1$ (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

s=uvwxy

with strings $u,v,w,x,$ and $y$ , such that

$vwx$ has at most $p$ marked positions,
$vx$ has at least one marked position, and
$uv^{n}wx^{n}y\in L$ for all $n\geq 0$ .

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 $\{a^{i}b^{j}c^{k}d^{l}:i=0{\text{ or }}j=k=l\}$ . Ogden's lemma can also be used to prove the inherent ambiguity of some languages.

References

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.

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

Ogden's lemma

See also

References