Paper: The Complexity of Downward Closures of Indexed Languages (at LICS 2026)

Authors: Richard Mandel Corto Mascle Georg Zetzsche

Open access: https://doi.org/10.4230/LIPIcs.LICS.2026.69

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.

BibTeX

  @InProceedings{MandelMascleZetzsch-TheComplexityofDown,
    author = 	 {Richard Mandel and Corto Mascle and Georg Zetzsche},
    title = 	 {The Complexity of Downward Closures of Indexed Languages},
    booktitle =  {Proceedings of the Forty-First Annual Symposium on Logic in Computer Science (LICS 2026)},
    year =	 {2026},
    month =	 {July}, 
    pages =      {69:1--69:28},
    location =   {Lisbon, Portugal}, 
    publisher =	 {Schloss Dagstuhl -- Leibniz-Zentrum für Informatik},
    doi =        {10.4230/LIPIcs.LICS.2026.69}
  }