Conjunctive Grammars Can Generate Non-regular Unary Languages

Abstract

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 $\{ a^{4^{n}} : n \in \mathbb{N} \}$. We then generalise this result—for every set of numbers L such that their representation in some k-ary system is regular set we show that $\{ a^{k^{n}} : n \in L \}$ is generated by some conjunctive grammar over unary alphabet.