Theory of Computing Systems

, Volume 49, Issue 2, pp 319–342 | Cite as

One-Nonterminal Conjunctive Grammars over a Unary Alphabet

  • Artur Jeż
  • Alexander OkhotinEmail author


Conjunctive grammars over an alphabet Σ={a} are studied, with the focus on the special case with a unique nonterminal symbol. Such a grammar is equivalent to an equation X=ϕ(X) over sets of natural numbers, using union, intersection and addition. It is shown that every grammar with multiple nonterminals can be encoded into a grammar with a single nonterminal, with a slight modification of the language. Based on this construction, the compressed membership problem for one-nonterminal conjunctive grammars over {a} is proved to be EXPTIME-complete; the same problem for the context-free grammars is decidable in NLOGSPACE, but becomes NP-complete if the grammar is compressed as well. The equivalence problem for these grammars is shown to be co-r.e.-complete, both finiteness and co-finiteness are r.e.-complete, while equivalence to a fixed unary language with a regular positional notation is decidable.


Conjunctive grammars Language equations Unary languages Decision problems 


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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Institute of Computer ScienceUniversity of WrocławWrocławPoland
  2. 2.Department of MathematicsUniversity of TurkuTurkuFinland

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