Reachability in Higher-Order-Counters

  • Alexander Heußner
  • Alexander Kartzow
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8087)


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.


Model Check Reachability Analysis Tree Automaton Storage Type Reachability Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Alexander Heußner
    • 1
  • Alexander Kartzow
    • 2
  1. 1.Otto-Friedrich-Universität BambergGermany
  2. 2.Universität LeipzigGermany

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