The paper introduces the concept of derivation trees for general grammars and defines context-dependent pairs of nodes within these trees. It then proves that:
If a linear core general grammar generates each of its sentences through a slow-branching derivation tree where any two neighboring nonterminal paths contain at most a constant number of context-dependent node pairs, then the generated language is k-linear.
If a general grammar generates each of its sentences through a derivation tree where any two neighboring nonterminal paths contain at most a constant number of context-dependent node pairs, and all node pairs in non-neighboring paths are context-independent, then the generated language is regular.
The paper explains that these results provide a powerful tool for demonstrating that certain languages are k-linear or regular. It also discusses the limitations of deciding whether a given general grammar satisfies the required conditions.
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