Unambiguous Conjunctive Grammars over a One-Letter 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 k > 11 and for every one-way real-time cellular automaton operating over the alphabet of base-k digits { d k+9 4 e, . . . , b k+1 2 c } , the language of all strings a with the base-k notation of the form 1w1, where w is accepted by the automaton, is generated 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.