Abstract
The rational index \(\rho _L\) of a language L is an integer function, where \(\rho _L(n)\) is the maximum length of the shortest string in \(L \cap R\), over all regular languages R recognized by n-state nondeterministic finite automata (NFA). This paper investigates the rational index of languages defined by (context-free) grammars with bounded tree dimension, and shows that it is of polynomial in n. More precisely, it is proved that for a grammar with tree dimension bounded by d, its rational index is \(O(n^{2d})\), and that this estimation is asymptotically tight, as there exists a grammar with rational index \(\varTheta (n^{2d})\).
Keywords
Research supported by the Russian Science Foundation, project 18-11-00100.
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
Afrati, F., Papadimitriou, C.: The parallel complexity of simple chain queries. In: Proceedings of the Sixth ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems, PODS 1987, pp. 210–213. ACM, New York (1987). https://doi.org/10.1145/28659.28682
Bar-Hillel, Y., Perles, M., Shamir, E.: On formal properties of simple phreise structure grammars. STUF Lang. Typol. Univ. 14(1–4), 143–172 (1961). https://doi.org/10.1524/stuf.1961.14.14.143
Boasson, L., Courcelle, B., Nivat, M.: The rational index: a complexity measure for languages. SIAM J. Comput. 10(2), 284–296 (1981)
Brzozowski, J.A.: Regular-like expressions for some irregular languages. In: 9th Annual Symposium on Switching and Automata Theory (swat 1968), pp. 278–286 (1968). https://doi.org/10.1109/SWAT.1968.24
Chistikov, D., Czerwinski, W., Hofman, P., Pilipczuk, M., Wehar, M.: Shortest paths in one-counter systems. Log. Methods Comput. Sci. 15(1) (2019). https://doi.org/10.23638/LMCS-15(1:19)2019
Chytil, M.P., Monien, B.: Caterpillars and context-free languages. In: Choffrut, C., Lengauer, T. (eds.) STACS 1990. LNCS, vol. 415, pp. 70–81. Springer, Heidelberg (1990). https://doi.org/10.1007/3-540-52282-4_33
Esparza, J., Luttenberger, M., Schlund, M.: A brief history of strahler numbers. In: Dediu, A.-H., Martín-Vide, C., Sierra-Rodríguez, J.-L., Truthe, B. (eds.) LATA 2014. LNCS, vol. 8370, pp. 1–13. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-04921-2_1
Ganty, P., Valput, D.: Bounded-oscillation pushdown automata. Electron. Proc. Theor. Comput. Sci. 226, 178–197 (2016). https://doi.org/10.4204/eptcs.226.13
Greenlaw, R., Hoover, H.J., Ruzzo, W.L.: Limits to Parallel Computation: P-completeness Theory. Oxford University Press Inc., New York (1995)
Hellings, J.: Path results for context-free grammar queries on graphs. CoRR abs/1502.02242 (2015)
Holzer, M., Kutrib, M., Leiter, U.: Nodes connected by path languages. In: Mauri, G., Leporati, A. (eds.) DLT 2011. LNCS, vol. 6795, pp. 276–287. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-22321-1_24
Komarath, B., Sarma, J., Sunil, K.S.: On the complexity of L-reachability. In: Jürgensen, H., Karhumäki, J., Okhotin, A. (eds.) DCFS 2014. LNCS, vol. 8614, pp. 258–269. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-09704-6_23
Luttenberger, M., Schlund, M.: Convergence of newton’s method over commutative semirings. Inf. Comput. 246, 43–61 (2016). https://doi.org/10.1016/j.ic.2015.11.008
Pierre, L.: Rational indexes of generators of the cone of context-free languages. Theor. Comput. Sci. 95(2), 279–305 (1992). https://doi.org/10.1016/0304-3975(92)90269-L
Pierre, L., Farinone, J.M.: Context-free languages with rational index in \(\theta (n^\gamma )\) for algebraic numbers \(\gamma \). RAIRO - Theor. Inf. Appl. - Informatique Théorique et Appl. 24(3), 275–322 (1990)
Reps, T.W.: Program analysis via graph reachability. Inf. Softw. Technol. 40, 701–726 (1997)
Ullman, J.D., Van Gelder, A.: Parallel complexity of logical query programs. In: 27th Annual Symposium on Foundations of Computer Science (sfcs 1986), pp. 438–454 (Oct 1986). https://doi.org/10.1109/SFCS.1986.40
Wechsung, G.: The oscillation complexity and a hierarchy of context-free languages. In: Fundamentals of Computation Theory, FCT 1979, Proceedings of the Conference on Algebraic, Arthmetic, and Categorial Methods in Computation Theory, Berlin/Wendisch-Rietz, Germany, 17–21 September 1979, pp. 508–515 (1979)
Yannakakis, M.: Graph-theoretic methods in database theory. In: Rosenkrantz, D.J., Sagiv, Y. (eds.) Proceedings of the Ninth ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems, Nashville, Tennessee, USA, April 2–4, 1990, pp. 230–242. ACM Press (1990). https://doi.org/10.1145/298514.298576
Acknowledgment
The authors are grateful to the anonymous reviewers for numerous helpful remarks and suggestions, and particularly for alerting the authors of the work by Chytil and Monien [6].
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2022 Springer Nature Switzerland AG
About this paper
Cite this paper
Shemetova, E., Okhotin, A., Grigorev, S. (2022). Rational Index of Languages with Bounded Dimension of Parse Trees. In: Diekert, V., Volkov, M. (eds) Developments in Language Theory. DLT 2022. Lecture Notes in Computer Science, vol 13257. Springer, Cham. https://doi.org/10.1007/978-3-031-05578-2_21
Download citation
DOI: https://doi.org/10.1007/978-3-031-05578-2_21
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-05577-5
Online ISBN: 978-3-031-05578-2
eBook Packages: Computer ScienceComputer Science (R0)