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This paper introduces a class of formal grammars made up by augmenting the formalism of context-free grammars with an explicit set-theoretic intersection operation. It is shown that conjunctive grammars can generate some important non-context-free language constructs, including those not in the intersection closure of context-free languages, and that they(More)
A new generalization of context-free grammars is introduced: Boolean grammars allow the use of all set-theoretic operations as an integral part of the formalism of rules. Rigorous semantics for these grammars is defined by language equations in a way that allows to generalize some techniques from the theory of context-free grammars, including Chomsky normal(More)
It is proved that the set of scattered substrings of a language recognized by an n-state DFA requires a DFA with at least 2 n 2 −2 states (the known upper bound is 2 n), with witness languages given over an exponentially growing alphabet. For a 3-letter alphabet, scattered substrings are shown to require at least 2 √ 2n+30−6 states. A similar state(More)
It has recently been proved (Jeż, DLT 2007) that conjunctive grammars (that is, context-free grammars augmented by conjunction) generate some non-regular languages over a one-letter alphabet. The present paper improves this result by constructing conjunctive grammars for a larger class of unary languages. The results imply undecidability of a number of(More)
This paper studies systems of language equations that are resolved with respect to variables and contain the operations of concatenation, union and intersection. Every system of this kind is proved to have a least fixed point, and the equivalence of these systems to conjunctive grammars is established. This allows us to obtain an algebraic characterization(More)