Abstract
Higher-order counter automata (HOCS) can be either seen as a restriction of higher-order pushdown automata (HOPS) to a unary stack alphabet, or as an extension of counter automata to higher levels. We distinguish two principal kinds of HOCS: those that can test whether the topmost counter value is zero and those which cannot.
We show that control-state reachability for level k HOCS with 0-test is complete for (k − 2)-fold exponential space; leaving out the 0-test leads to completeness for (k − 2)-fold exponential time. Restricting HOCS (without 0-test) to level 2, we prove that global (forward or backward) reachability analysis is P-complete. This enhances the known result for pushdown systems which are subsumed by level 2 HOCS without 0-test.
We transfer our results to the formal language setting. Assuming that P \(\subsetneq\) PSPACE \(\subsetneq\) EXPTIME, we apply proof ideas of Engelfriet and conclude that the hierarchies of languages of HOPS and of HOCS form strictly interleaving hierarchies. Interestingly, Engelfriet’s constructions also allow to conclude immediately that the hierarchy of collapsible pushdown languages is strict level-by-level due to the existing complexity results for reachability on collapsible pushdown graphs. This answers an open question independently asked by Parys and by Kobayashi.
The second author is supported by the DFG research project GELO. We both thank M. Bojańczyk, Ch. Broadbent, and M. Lohrey for helpful discussions and comments.
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
Bouajjani, A., Esparza, J., Maler, O.: Reachability analysis of pushdown automata: Application to model-checking. In: Mazurkiewicz, A., Winkowski, J. (eds.) CONCUR 1997. LNCS, vol. 1243, pp. 135–150. Springer, Heidelberg (1997)
Bouajjani, A., Meyer, A.: Symbolic reachability analysis of higher-order context-free processes. In: Lodaya, K., Mahajan, M. (eds.) FSTTCS 2004. LNCS, vol. 3328, pp. 135–147. Springer, Heidelberg (2004)
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.) ICALP 2012, Part II. LNCS, vol. 7392, pp. 165–176. Springer, Heidelberg (2012)
Carayol, A.: Regular sets of higher-order pushdown stacks. In: Jedrzejowicz, J., Szepietowski, A. (eds.) MFCS 2005. LNCS, vol. 3618, pp. 168–179. Springer, Heidelberg (2005)
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)
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)
Engelfriet, J.: Iterated stack automata and complexity classes. Inf. Comput. 95(1), 21–75 (1991)
Göller, S.: Reachability on prefix-recognizable graphs. Inf. Process. Lett. 108(2), 71–74 (2008)
Hague, M., Ong, C.-H.L.: Symbolic backwards-reachability analysis for higher-order pushdown systems. LMCS 4(4) (2008)
Heußner, A., Kartzow, A.: Reachability in higher-order-counters. CoRR, arxiv:1306.1069 (2013), http://arxiv.org/abs/1306.1069
Jancar, P., Sawa, Z.: A note on emptiness for alternating finite automata with a one-letter alphabet. Inf. Process. Lett. 104(5), 164–167 (2007)
Kartzow, A.: Collapsible pushdown graphs of level 2 are tree-automatic. Logical Methods in Computer Science 9(1) (2013)
Kartzow, A., Parys, P.: Strictness of the collapsible pushdown hierarchy. In: Rovan, B., Sassone, V., Widmayer, P. (eds.) MFCS 2012. LNCS, vol. 7464, pp. 566–577. Springer, Heidelberg (2012)
Kobayashi, N.: Pumping by typing. To appear in Proc. LICS (2013)
Maslov, A.N.: The hierarchy of indexed languages of an arbitrary level. Sov. Math., Dokl. 15, 1170–1174 (1974)
Maslov, A.N.: Multilevel stack automata. Problems of Information Transmission 12, 38–43 (1976)
Parys, P.: Variants of collapsible pushdown systems. In: Proc. of CSL 2012. LIPIcs, vol. 16, pp. 500–515 (2012)
Slaats, M.: Infinite regular games in the higher-order pushdown and the parametrized setting. PhD thesis, RWTH Aachen (2012)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Heußner, A., Kartzow, A. (2013). Reachability in Higher-Order-Counters. In: Chatterjee, K., Sgall, J. (eds) Mathematical Foundations of Computer Science 2013. MFCS 2013. Lecture Notes in Computer Science, vol 8087. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40313-2_47
Download citation
DOI: https://doi.org/10.1007/978-3-642-40313-2_47
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-40312-5
Online ISBN: 978-3-642-40313-2
eBook Packages: Computer ScienceComputer Science (R0)