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The Complexity of Downward Closures of Indexed Languages

Authors Richard Mandel , Corto Mascle , Georg Zetzsche

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ACM Subject Classification
  • Theory of computation → Regular languages
Keywords
  • Higher-order pushdown automata
  • well quasi-orders
  • semigroup algebra

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Abstract

Indexed languages are a classical notion in formal language theory, which has attracted attention in recent decades due to its role in higher-order model checking: They are precisely the languages accepted by order-2 pushdown automata.
The downward closure of an indexed language - the set of all (scattered) subwords of its members - is well-known to be a regular over-approximation. It is known since 2015 that the downward closure of a given indexed language is effectively computable. However, the algorithm comes with no complexity bounds, and it has remained open whether a primitive-recursive construction exists.
We settle this question and provide a triply (resp. quadruply) exponential construction of a non-deterministic (resp. deterministic) automaton. We also prove (asymptotically) matching lower bounds. For the upper bounds, we rely on recent advances in semigroup theory, which let us compute bounded-size summaries of words with respect to a finite semigroup. By replacing stacks with their summaries, we are able to transform an indexed grammar into a context-free one with the same downward closure, and then apply existing bounds for context-free grammars.

Cite As Get BibTex

Richard Mandel, Corto Mascle, and Georg Zetzsche. The Complexity of Downward Closures of Indexed Languages. In 41st Annual Symposium on Logic in Computer Science (LICS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 380, pp. 69:1-69:28, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.LICS.2026.69

Author Details

Richard Mandel
  • Max Planck Institute for Software Systems (MPI-SWS), Kaiserslautern, Germany
Corto Mascle
  • Max Planck Institute for Software Systems (MPI-SWS), Kaiserslautern, Germany
Georg Zetzsche
  • Max Planck Institute for Software Systems (MPI-SWS), Kaiserslautern, Germany

Funding

Funded by the European Union (ERC, FINABIS, 101077902). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them.

Acknowledgements

We are grateful to Alexander Kartzow for answering questions about his work [Alexander Kartzow, 2011], and to Hermann Gruber for making us aware of literature on pumping constants.

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