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Reachability in Higher-Order-Counters

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Mathematical Foundations of Computer Science 2013 (MFCS 2013)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8087))

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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.

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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

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  • 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

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