DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-09-12

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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Marked positions and admissible Ogden decompositions

Definition

Let $z = a_{1} \dots a_{n}$ be a word over $Σ$ . A set $M \subseteq {1, \dots, n}$ is a set of marked positions of $z$ .

Fix an integer $p \geq 1$ . A decomposition $z = u v w x y$ is $p$ -admissible with respect to $M$ when:

the concatenated substring $v x$ contains at least one marked position;
the substring $v w x$ contains at most $p$ marked positions.

Remarks

Ogden's lemma strengthens the ordinary pumping lemma by controlling where the pumped material $v x$ must lie relative to the marked positions.
The definition does not itself assert that pumping preserves the language; it only records the combinatorial shape of the decomposition.

Depends on

Computation alphabets, words, the empty word, and $\Sigma^*$

Used by

Ogden's lemma Theorem

Dependency tree · two levels

4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Jean Gallier and Jocelyn Quaintance, Introduction to the Theory of Computation: Some Notes for CIS511 (standard reference, not scraped)