Abstract
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 \(\mathbf {NSPACE}(\log ^2 n)\).
A. Rubtsov—Supported in part by RFBR grant 14–01–00641.
M. Vyalyi—Supported in part RFBR grant 14–01–93107 and the scientific school grant NSh4652.2012.1.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Anderson, T., Loftus, J., Rampersad, N., Santean, N., Shallit, J.: Detecting palindromes, patterns and borders in regular languages. Inf. Comput. 207, 1096–1118 (2009)
Berstel, J.: Transductions and Context-Free Languages. Teubner Verlag, Stuttgart/Leipzig/Wiesbaden (1979)
Berstel, J., Boasson, L.: Context-free languages. In: Leeuwen, J. (ed.) Handbook of Theoretical Computer Science, vol. B, pp. 59–102. Elsevier, Amsterdam (1990)
Boasson, L.: Non-générateurs algébriques et substitution. RAIRO Informatique théorique 19, 125–136 (1985)
Boasson, L., Courcelle, B., Nivat, M.: The rational index, a complexity measure for languages. SIAM J. Comput. 10(2), 284–296 (1981)
Greenlaw, R., Hoover, H.J., Ruzzo, L.: Limits to Parallel Computation: P-Completeness Theory. Oxford University Press, Oxford (1995)
Greibach, S.A.: An infinite hierarchy of context-free languages. J. ACM 16, 91–106 (1969)
Lewis, P.M., Stearns, R.E., Hartmanis, J.: Memory bounds for recognition of context-free and context-sensitive languages. In: Switching Circuit Theory and Logical Design, pp. 191–202. IEEE, New York (1965)
Pierre, L.: Rational indexes of generators of the cone of context-free languages. Theor. Comput. Sci. 95, 279–305 (1992)
Pierre, L., Farinone, J.M.: Rational index of context-free languages with rational index in \(\Theta (n^\gamma )\) for algebraic numbers \(\gamma \). Informatique théorique et applications 24(3), 275–322 (1990)
Vyalyi, M.N.: On regular realizability problems. Probl. Inf. Transm. 47(4), 342–352 (2011)
Vyalyi, M.N.: Universality of regular realizability problems. In: Bulatov, A.A., Shur, A.M. (eds.) CSR 2013. LNCS, vol. 7913, pp. 271–282. Springer, Heidelberg (2013)
Yakaryılmaz, A.: One-counter verifiers for decidable languages. In: Bulatov, A.A., Shur, A.M. (eds.) CSR 2013. LNCS, vol. 7913, pp. 366–377. Springer, Heidelberg (2013)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Rubtsov, A., Vyalyi, M. (2015). Regular Realizability Problems and Context-Free Languages. In: Shallit, J., Okhotin, A. (eds) Descriptional Complexity of Formal Systems. DCFS 2015. Lecture Notes in Computer Science(), vol 9118. Springer, Cham. https://doi.org/10.1007/978-3-319-19225-3_22
Download citation
DOI: https://doi.org/10.1007/978-3-319-19225-3_22
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-19224-6
Online ISBN: 978-3-319-19225-3
eBook Packages: Computer ScienceComputer Science (R0)