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.

Statement

Ogden's lemma states that if a language L is context-free, then there exists some number ${\displaystyle p\geq 1}$ (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

${\displaystyle s=uvwxy}$

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

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

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

Generalized condition

Bader and Moura have generalized the lemma[1] to allow marking some positions that are not to be included in vx. Their dependence of the parameters was later improved by Dömösi and Kudlek. If we denote the number of such excluded positions by e, then the number d of distinguished positions of which we want to include some in vx must satisfy ${\displaystyle d\geq p(e+1)}$, where p is some constant that depends only on the language. The statement becomes that every s can be written as

${\displaystyle s=uvwxy}$

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

1. vx has at least one distinguished position and no excluded position,
2. vwx has at most ${\displaystyle p^{(e+1)}}$ distinguished positions, and
3. ${\displaystyle uv^{n}wx^{n}y\in L}$ for all ${\displaystyle n\geq 0}$.

Moreover, either each of u,v,w has a distinguished position, or each of ${\displaystyle w,x,y}$ has a distinguished position.

References

1. Bader, Christopher; Moura, Arnaldo (April 1982). "A Generalization of Ogden's Lemma". Applied Mathematics and Computation. 29 (2): 404–407. doi:10.1145/322307.322315. S2CID 33988796.

Further reading

• Dömösi, P.; Kudlek, M. (1999). "Strong Iteration Lemmata for Regular, Linear, Context-Free, and Linear Indexed Languages". FCT'99, LNCS. Lecture Notes in Computer Science. 1684: 226–233. doi:10.1007/3-540-48321-7_18. ISBN 978-3-540-66412-3.
• Ogden, W. (1968). "A helpful result for proving inherent ambiguity". Mathematical Systems Theory. 2 (3): 191–194. doi:10.1007/BF01694004. S2CID 13197551.
• Hopcroft; Motwani; Ullman (1979). Automata Theory, Languages, and Computation. Addison-Wesley. ISBN 81-7808-347-7.