Regulated Grammars and Automata by Alexander Meduna & Petr Zemek

Author:Alexander Meduna & Petr Zemek
Language: eng
Format: epub
Publisher: Springer New York, New York, NY

Then, . Clearly, (i)–(iii) hold. Since s′ℓ 2 → s ∈ R′ by the initialization part of the algorithm and , , so (iv) also holds. Thus, the basis holds.

Induction Hypothesis. Suppose that the claim holds for all derivations of length ℓ or less, where 1 ≤ ℓ ≤ n, for some n ≥ 1.

Induction Step. Consider any derivation of the form

where χ ∈ V ′+. Since n + 1 > 1, this derivation can be expressed as

for some x ∈ V ′+, β ∈ Ψ′∗, and t ∈ Ψ′. By the induction hypothesis, , x is of the form , where X i  ∈ V, for all i, 1 ≤ i ≤ h, for some h ≥ 1, , γ ∈ Ψ′∗, and , such that (i) through (iv) hold. Furthermore,

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