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 \(k \geqslant 11\) and for every one-way real-time cellular automaton operating over the alphabet of base-k digits \(\big\{\lfloor\frac{k+9}{4}\rfloor, \ldots, \lfloor\frac{k+1}{2}\rfloor\big\}\), the language of all strings a n with the base-k notation of the form 1 w 1, 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 L 0, testing whether a given unambiguous conjunctive grammar generates L 0 is undecidable.
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Jeż, A., Okhotin, A. (2013). Unambiguous Conjunctive Grammars over a One-Letter Alphabet. In: Béal, MP., Carton, O. (eds) Developments in Language Theory. DLT 2013. Lecture Notes in Computer Science, vol 7907. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38771-5_25
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