Conjunctive Grammars Can Generate Non-regular Unary Languages

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. We give a negative answer, contrary to the conjectured positive one, by constructing a conjunctive grammar for the language {a n : n ∈ N}. We then generalise this result—for every set of natural numbers L, such that their representation in some k-ary system is a regular set, we show that {a : n ∈ L} is generated by a conjunctive grammar over unary alphabet and that this grammar can be efficiently computed.