Regular Realizability Problems and Context-Free Languages

  • A. RubtsovEmail author
  • M. Vyalyi
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9118)


We investigate regular realizability (RR) problems, which are the problems of verifying whether the intersection of a regular language – the input of the problem – and a fixed language, called a filter, is non-empty. In this paper we focus on the case of context-free filters. The algorithmic complexity of the RR problem is a very coarse measure of the complexity of context-free languages. This characteristic respects the rational dominance relation. We show that a RR problem for a maximal filter under the rational dominance relation is \(\mathbf {P}\)-complete. On the other hand, we present an example of a \(\mathbf {P}\)-complete RR problem for a non-maximal filter. We show that RR problems for Greibach languages belong to the class \(\mathbf {NL}\). We also discuss RR problems with context-free filters that might have intermediate complexity. Possible candidates are the languages with polynomially-bounded rational indices. We show that RR problems for these filters lie in the class \(\mathbf {NSPACE}(\log ^2 n)\).



We are acknowledged to Abuzer Yakaryilmaz for pointing on the result of Lemma 5 and for reference to a lemma similar to Lemma 6.


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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Computing Centre of RASMoscowRussia
  2. 2.Moscow Institute of Physics and TechnologyMoscowRussia
  3. 3.National Research University Higher School of EconomicsMoscowRussia

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