Abstract
A criterion to measure derivational complexity of formal grammars and languages is proposed and discussed. That is, the associate language and the L -associate language are defined for a grammar such that the former represents all the valid derivations and the latter represents all the valid leftmost derivations. It is shown that for any phrase\3-structure grammar, the associate language is a contex\3-sensitive language and the L \3-associate language is a context\3-free language. Necessary and sufficient conditions for an associate language to be a regular set and to be a context\3-free language are found. The idea in the above necessary and sufficient conditions is extended to the notion of “rank≓ for a measure of derivational complexity of context\3-free grammars and languages. It is shown that for each nonnegative integer k , there exists a context\3-free language whose rank is k . The paper also includes a few solvable decision problems concerning derivational complexity of grammars.
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