Conjunctive Grammars Can Generate Non-regular Unary Languages

@inproceedings{Je2008ConjunctiveGC,
  title={Conjunctive Grammars Can Generate Non-regular Unary Languages},
  author={Artur Jeż},
  booktitle={International Conference on Developments in Language Theory},
  year={2008}
}
  • Artur Jeż
  • Published in
    International Conference on…
    1 June 2008
  • Linguistics
Conjunctive grammars were introduced by A. Okhotin in [1] as a natural extension of context-free grammars with an additional operation of intersection in the body of any production of the grammar. Several theorems and algorithms for context-free grammars generalize to the conjunctive case. Still some questions remained open. A. Okhotin posed nine problems concerning those grammars. One of them was a question, whether a conjunctive grammar over unary alphabet can generate only regular languages… 

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References

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Conjunctive Grammars were introduced in 2000 as a generalization of context-free grammars that allows the use of an explicit intersection oper-ation in rules and several theoretical results on their properties have been obtained and numerous open problems are proposed.

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It is proved that conjunctive grammars can still be parsed in cubic time and that the notion of the derivation tree is retained, which gives reasonable hope for their practical applicability.

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This note is aimed to report on the two solved problems, as well as to correct a couple of small errors, in the survey of conjunctive and Boolean grammars.

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A new class of formal grammars, which allow the use of all set-theoretic operations as an integral part of the formalism of rules, are introduced, which allows to conjecture the practical applicability of the new concept.

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Every system of this kind of language equations that are resolved with respect to variables and contain the operations of concatenation, union and intersection is proved to have a least fixed point, and the equivalence of these systems to conjunctive grammars is established.

An overview of conjunctive grammars, Formal Language Theory Column