Strictness of the Collapsible Pushdown Hierarchy
Conference paper
Abstract
We present a pumping lemma for each level of the collapsible pushdown graph hierarchy in analogy to the second author’s pumping lemma for higher-order pushdown graphs (without collapse). Using this lemma, we give the first known examples that separate the levels of the collapsible pushdown graph hierarchy and of the collapsible pushdown tree hierarchy, i.e., the hierarchy of trees generated by higher-order recursion schemes. This confirms the open conjecture that higher orders allow one to generate more graphs and more trees.
Full proofs can be found in the arXiv version[10] of this paper.
Keywords
Context Free Grammar Recursion Scheme Pushdown Automaton Collapsible System Monadic Theory
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References
- 1.Aehlig, K., de Miranda, J.G., Ong, C.-H.L.: Safety Is not a Restriction at Level 2 for String Languages. In: Sassone, V. (ed.) FOSSACS 2005. LNCS, vol. 3441, pp. 490–504. Springer, Heidelberg (2005)CrossRefGoogle Scholar
- 2.Blumensath, A.: On the structure of graphs in the caucal hierarchy. Theor. Comput. Sci. 400(1-3), 19–45 (2008)MathSciNetMATHCrossRefGoogle Scholar
- 3.Broadbent, C., Carayol, A., Hague, M., Serre, O.: A Saturation Method for Collapsible Pushdown Systems. In: Czumaj, A., Mehlhorn, K., Pitts, A., Wattenhofer, R. (eds.) Automata, Languages, and Programming, Part II. LNCS, vol. 7392, pp. 165–176. Springer, Heidelberg (2012)CrossRefGoogle Scholar
- 4.Carayol, A., Wöhrle, S.: The Caucal Hierarchy of Infinite Graphs in Terms of Logic and Higher-Order Pushdown Automata. In: Pandya, P.K., Radhakrishnan, J. (eds.) FSTTCS 2003. LNCS, vol. 2914, pp. 112–123. Springer, Heidelberg (2003)CrossRefGoogle Scholar
- 5.Caucal, D.: On Infinite Terms Having a Decidable Monadic Theory. In: Diks, K., Rytter, W. (eds.) MFCS 2002. LNCS, vol. 2420, pp. 165–176. Springer, Heidelberg (2002)CrossRefGoogle Scholar
- 6.Gilman, R.H.: A shrinking lemma for indexed languages. Theor. Comput. Sci. 163(1&2), 277–281 (1996)MathSciNetMATHCrossRefGoogle Scholar
- 7.Hague, M., Murawski, A.S., Ong, C.-H.L., Serre, O.: Collapsible pushdown automata and recursion schemes. In: LICS, pp. 452–461. IEEE Computer Society (2008)Google Scholar
- 8.Hayashi, T.: On derivation trees of indexed grammars. Publ. RIMS, Kyoto Univ. 9, 61–92 (1973)MATHCrossRefGoogle Scholar
- 9.Kartzow, A.: A pumping lemma for collapsible pushdown graphs of level 2. In: Bezem, M. (ed.) CSL. LIPIcs, vol. 12, pp. 322–336. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2011)Google Scholar
- 10.Kartzow, A., Parys, P.: Strictness of the collapsible pushdown hierarchy. CoRR, abs/1201.3250 (2012)Google Scholar
- 11.Knapik, T., Niwiński, D., Urzyczyn, P.: Higher-Order Pushdown Trees Are Easy. In: Nielsen, M., Engberg, U. (eds.) FOSSACS 2002. LNCS, vol. 2303, pp. 205–222. Springer, Heidelberg (2002)CrossRefGoogle Scholar
- 12.Maslov, A.N.: The hierarchy of indexed languages of an arbitrary level. Soviet Math. Dokl. 15, 1170–1174 (1974)MATHGoogle Scholar
- 13.Maslov, A.N.: Multilevel stack automata. Problems of Information Transmission 12, 38–43 (1976)Google Scholar
- 14.Parys, P.: Collapse operation increases expressive power of deterministic higher order pushdown automata. In: Schwentick, T., Dürr, C. (eds.) STACS. LIPIcs, vol. 9, pp. 603–614. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2011)Google Scholar
- 15.Parys, P.: On the significance of the collapse operation. In: To appear in LICS (2012)Google Scholar
- 16.Parys, P.: A pumping lemma for pushdown graphs of any level. In: Dürr, C., Wilke, T. (eds.) STACS. LIPIcs, vol. 14, pp. 54–65. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2012)Google Scholar
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