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.
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References
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)
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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
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DOI: https://doi.org/10.1007/978-3-319-19225-3_22
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