Unambiguous conjunctive grammars over a one-symbol alphabet

Abstract 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 + 9 4 ⌋ and ⌊ k + 1 2 ⌋ , the language of all strings a n with the base-k representation of the form 1 w 1 , 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 L 0 , testing whether a given unambiguous conjunctive grammar generates L 0 is undecidable.