Abstract
Control sets on grammars are extended to depth-first derivations. It is proved that a context-free language is generated by the depth-first derivations of an arbitrary context-free grammar controlled by an arbitrary regular set. This result is sharpened to obtain a new characterization of the family of derivation-bounded languages: a languageL is derivation bounded if and only if it is generated by the depth-first derivations of a context-free grammarG controlled by a regular subsetR of the Szilard language ofG. The left-derivation-bounded languages are characterized analogously using leftmost derivations. It is proved that a grammarG is nonterminal bounded if and only if the Szilard language defined using only the depth-first derivations ofG is regular. Finally, it is proved that if a family of languagesC is a trio, a semi-AFL, an AFL, or an AFL closed under λ-free substitution, then the family of languages generated using arbitrary context-free grammars controlled by members ofC is full, is closed under reversal, and has the closure properties assumed ofC.
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Luker, M. Control sets on grammars using depth-first derivations. Math. Systems Theory 13, 349–359 (1979). https://doi.org/10.1007/BF01744305
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DOI: https://doi.org/10.1007/BF01744305