Abstract
We answer two open questions by (Gruber, Holzer, Kutrib, 2009) on the state-complexity of representing sub- or superword closures of context-free grammars (CFGs): (1) We prove a (tight) upper bound of \(2^{\mathcal {O}(n)}\) on the size of nondeterministic finite automata (NFAs) representing the subword closure of a CFG of size \(n\). (2) We present a family of CFGs for which the minimal deterministic finite automata representing their subword closure matches the upper-bound of \(2^{2^{\mathcal {O}(n)}}\) following from (1). Furthermore, we prove that the inequivalence problem for NFAs representing sub- or superword-closed languages is only NP-complete as opposed to PSPACE-complete for general NFAs. Finally, we extend our results into an approximation method to attack inequivalence problems for CFGs.
This work was partially funded by the DFG project “Polynomial Systems on Semirings: Foundations, Algorithms, Applications”.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Atig, M.F., Bouajjani, A., Touili, T.: On the Reachability Analysis of Acyclic Networks of Pushdown Systems. In: van Breugel, F., Chechik, M. (eds.) CONCUR 2008. LNCS, vol. 5201, pp. 356–371. Springer, Heidelberg (2008)
Axelsson, R., Heljanko, K., Lange, M.: Analyzing Context-Free Grammars Using an Incremental SAT Solver. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part II. LNCS, vol. 5126, pp. 410–422. Springer, Heidelberg (2008)
Bachmeier, G., Luttenberger, M., Schlund, M.: Finite Automata for the Sub- and Superword Closure of CFLs: Descriptional and Computational Complexity. CoRR abs/1410.2737 (2014). http://arxiv.org/abs/1410.2737
Bar-Hillel, Y., Perles, M., Shamir, E.: On Formal Properties of Simple Phrase Structure Grammars. Zeitschrift für Phonetik, Sprachwissenschaft und Kommunikationsforschung 14, 143–172 (1961)
Brabrand, C., Giegerich, R., Møller, A.: Analyzing Ambiguity of Context-Free Grammars. Sci. Comput. Program. 75(3), 176–191 (2010)
Courcelle, B.: On Constructing Obstruction Sets of Words. Bulletin of the EATCS 44, 178–186 (1991)
Esparza, J., Luttenberger, M., Schlund, M.: FPsolve: A Generic Solver for Fixpoint Equations over Semirings. In: Holzer, M., Kutrib, M. (eds.) CIAA 2014. LNCS, vol. 8587, pp. 1–15. Springer, Heidelberg (2014)
Ganty, P., Majumdar, R., Monmege, B.: Bounded underapproximations. Formal Methods in System Design 40(2), 206–231 (2012)
Gruber, H., Holzer, M., Kutrib, M.: More on the Size of Higman-Haines Sets: Effective Constructions. Fundam. Inf. 91(1), 105–121 (2009)
Habermehl, P., Meyer, R., Wimmel, H.: The Downward-Closure of Petri Net Languages. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds.) ICALP 2010. LNCS, vol. 6199, pp. 466–477. Springer, Heidelberg (2010)
Higman, G.: Ordering by Divisibility in Abstract Algebras. Proc. London Math. Soc. s3–2(1), 326–336 (Jan 1952)
Lange, M., Leiß, H.: To CNF or not to CNF? An Efficient Yet Presentable Version of the CYK Algorithm. Informatica Didactica 8 (2009)
van Leeuwen, J.: Effective constructions in well-partially-ordered free monoids. Discrete Mathematics 21(3), 237–252 (1978)
Long, Z., Calin, G., Majumdar, R., Meyer, R.: Language-Theoretic Abstraction Refinement. In: de Lara, J., Zisman, A. (eds.) Fundamental Approaches to Software Engineering. LNCS, vol. 7212, pp. 362–376. Springer, Heidelberg (2012)
Mohri, M., Nederhof, M.J.: Regular Approximation of Context-Free Grammars through Transformation. In: Junqua, J.C., van Noord, G. (eds.) Robustness in Language and Speech Technology. Text, Speech and Language Technology, vol. 17, pp. 153–163. Springer, Netherlands (2001)
Nederhof, M., Satta, G.: New Developments in Formal Languages and Applications. Studies in Computational Intelligence, pp. 229–258. Springer, Heidelberg (2008)
Okhotin, A.: On the State Complexity of Scattered Substrings and Superstrings. Fundam. Inform. 99(3), 325–338 (2010)
Rampersad, N., Shallit, J., Xu, Z.: The Computational Complexity of Universality Problems for Prefixes, Suffixes, Factors, and Subwords of Regular Languages. Fundam. Inform. 116(1–4), 223–236 (2012)
Schmitz, S.: Conservative Ambiguity Detection in Context-Free Grammars. In: Arge, L., Cachin, C., Jurdziński, T., Tarlecki, A. (eds.) ICALP 2007. LNCS, vol. 4596, pp. 692–703. Springer, Heidelberg (2007)
Vasudevan, N., Tratt, L.: Detecting Ambiguity in Programming Language Grammars. In: Erwig, M., Paige, R.F., Van Wyk, E. (eds.) SLE 2013. LNCS, vol. 8225, pp. 157–176. 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
Bachmeier, G., Luttenberger, M., Schlund, M. (2015). Finite Automata for the Sub- and Superword Closure of CFLs: Descriptional and Computational Complexity. In: Dediu, AH., Formenti, E., Martín-Vide, C., Truthe, B. (eds) Language and Automata Theory and Applications. LATA 2015. Lecture Notes in Computer Science(), vol 8977. Springer, Cham. https://doi.org/10.1007/978-3-319-15579-1_37
Download citation
DOI: https://doi.org/10.1007/978-3-319-15579-1_37
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-15578-4
Online ISBN: 978-3-319-15579-1
eBook Packages: Computer ScienceComputer Science (R0)