Abstract
We propose a generic method for deciding the language inclusion problem between context-free languages and deterministic contextfree languages. Our method extends a given decision procedure for a subclass to another decision procedure for a more general subclass called a refinement of the former. To decide \(\mathcal{L}_0 \subseteq \mathcal{L}_1\), we take two additional arguments: a language \(\mathcal{L}_2\) of which \(\mathcal{L}_1\) is a refinement, and a proof of \(\mathcal{L}_0 \subseteq \mathcal{L}_2\). Our technique then refines the proof of \(\mathcal{L}_0 \subseteq \mathcal{L}_2\) to a proof or a refutation of \(\mathcal{L}_0 \subseteq \mathcal{L}_1\). Although the refinement procedure may not terminate in general, we give a sufficient condition for the termination. We employ a type-based approach to formalize the idea, inspired from Kobayashi’s intersection type system for model-checking recursion schemes. To demonstrate the usefulness, we apply this method to obtain simpler proofs of the previous results of Minamide and Tozawa on the inclusion between context-free languages and regular hedge languages, and of Greibach and Friedman on the inclusion between context-free languages and superdeterministic languages.
Chapter PDF
References
Alur, R., Madhusudan, P.: Visibly pushdown languages. In: Babai, L. (ed.) STOC, pp. 202–211. ACM (2004)
Alur, R., Madhusudan, P.: Adding nesting structure to words. J. ACM 56(3), 1–43 (2009)
Asveld, P.R.J., Nijholt, A.: The inclusion problem for some subclasses of context-free languages. Theor. Comput. Sci. 230(1-2), 247–256 (2000)
Caucal, D.: Synchronization of Pushdown Automata. In: Ibarra, O.H., Dang, Z. (eds.) DLT 2006. LNCS, vol. 4036, pp. 120–132. Springer, Heidelberg (2006)
Greibach, S.A., Friedman, E.P.: Superdeterministic PDAs: A subcase with a decidable inclusion problem. J. ACM 27(4), 675–700 (1980)
Kobayashi, N.: Model-checking higher-order functions. In: Porto, A., López-Fraguas, F.J. (eds.) PPDP, pp. 25–36. ACM (2009)
Kobayashi, N.: Types and higher-order recursion schemes for verification of higher-order programs. In: Shao, Z., Pierce, B.C. (eds.) POPL, pp. 416–428. ACM (2009)
Kobayashi, N.: A Practical Linear Time Algorithm for Trivial Automata Model Checking of Higher-Order Recursion Schemes. In: Hofmann, M. (ed.) FOSSACS 2011. LNCS, vol. 6604, pp. 260–274. Springer, Heidelberg (2011)
Kobayashi, N., Ong, C.H.L.: A type system equivalent to the modal mu-calculus model checking of higher-order recursion schemes. In: LICS, pp. 179–188. IEEE Computer Society (2009)
Kobayashi, N., Sato, R., Unno, H.: Predicate abstraction and CEGAR for higher-order model checking. In: Hall, M.W., Padua, D.A. (eds.) PLDI, pp. 222–233. ACM (2011)
Kobayashi, N., Tabuchi, N., Unno, H.: Higher-order multi-parameter tree transducers and recursion schemes for program verification. In: Hermenegildo, M.V., Palsberg, J. (eds.) POPL, pp. 495–508. ACM (2010)
Minamide, Y., Tozawa, A.: XML Validation for Context-Free Grammars. In: Kobayashi, N. (ed.) APLAS 2006. LNCS, vol. 4279, pp. 357–373. Springer, Heidelberg (2006)
Møller, A., Schwarz, M.: HTML Validation of Context-Free Languages. In: Hofmann, M. (ed.) FOSSACS 2011. LNCS, vol. 6604, pp. 426–440. Springer, Heidelberg (2011)
Murata, M.: Hedge automata: a formal model for XML schemata (1999), http://www.xml.gr.jp/relax/hedge_nice.html
Nguyen, V.T., Ogawa, M.: Alternate stacking technique revisited: Inclusion problem of superdeterministic pushdown automata. IPSJ Transactions on Programming 1(1), 36–46 (2008)
Nowotka, D., Srba, J.: Height-Deterministic Pushdown Automata. In: Kučera, L., Kučera, A. (eds.) MFCS 2007. LNCS, vol. 4708, pp. 125–134. Springer, Heidelberg (2007)
Ong, C.H.L.: On model-checking trees generated by higher-order recursion schemes. In: LICS, pp. 81–90. IEEE Computer Society (2006)
Pair, C., Quéré, A.: Définition et etude des bilangages réguliers. Information and Control 13(6), 565–593 (1968)
Tsukada, T., Kobayashi, N.: Untyped Recursion Schemes and Infinite Intersection Types. In: Ong, L. (ed.) FOSSACS 2010. LNCS, vol. 6014, pp. 343–357. Springer, Heidelberg (2010)
Tsukada, T., Kobayashi, N.: A type-theoretic proof of the decidability of the language containment between context-free languages and superdeterministic languages. IPSJ Transactions on Programming 4(2), 31–47 (2011) (in Japanese)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 IFIP International Federation for Information Processing
About this paper
Cite this paper
Tsukada, T., Kobayashi, N. (2012). An Intersection Type System for Deterministic Pushdown Automata. In: Baeten, J.C.M., Ball, T., de Boer, F.S. (eds) Theoretical Computer Science. TCS 2012. Lecture Notes in Computer Science, vol 7604. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33475-7_25
Download citation
DOI: https://doi.org/10.1007/978-3-642-33475-7_25
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-33474-0
Online ISBN: 978-3-642-33475-7
eBook Packages: Computer ScienceComputer Science (R0)