Unambiguous conjunctive grammars over a one-symbol alphabet

It is demonstrated that unambiguous conjunctive grammars over a unary alphabet Σ={a} have non-trivial expressive power, and that their basic properties are undecidable. The key result is that for every base of positional notation, k⩾11, and for every one-way real-time cellular automaton operating over the alphabet of base-k digits between ⌊k+94⌋ and ⌊k+12⌋, the language of all strings an with the base-k representation of the form 1w1, where w is accepted by the automaton, is described by an unambiguous conjunctive grammar. Another encoding is used to simulate a cellular automaton in a unary language containing almost all strings. These constructions are used to show that for every fixed unambiguous conjunctive language L0, testing whether a given unambiguous conjunctive grammar generates L0 is undecidable.