A famous theorem by Greibach (“The hardest context-free language”, SIAM J. Comp., 1973) states that there exists such a context-free language L 0 , that every context-free language over any alphabet is reducible to L 0 by a homomorphic reduction—in other words, is representable as its inverse homomorphic image h −1 (L 0 ), for a suitable homomorphism h. This paper establishes similar characterizations for conjunctive grammars, that is, for grammars extended with a conjunction operator, as well as for Boolean grammars, which are further equipped with a negation operator. At the same time, it is shown that no such characterization is possible for several subclasses of linear grammars.
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