Unambiguous Conjunctive Grammars over a One-Letter Alphabet

  • Artur Jeż
  • Alexander Okhotin
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7907)


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.


Cellular Automaton Expressive Power Input Alphabet Nonterminal Symbol Start Symbol 
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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Artur Jeż
    • 1
    • 2
  • Alexander Okhotin
    • 3
  1. 1.Max-Planck-Institut für InformatikSaarbrückenGermany
  2. 2.Institute of Computer ScienceUniversity of WrocławPoland
  3. 3.Department of Mathematics and StatisticsUniversity of TurkuFinland

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