The complexity of downward closures of indexed languages

2026-01-27

AI Summary

This work resolves the long-standing open problem of whether the downward closure of indexed languages admits a primitive recursive construction. By transforming indexed grammars into context-free grammars that preserve the same downward closure, and by integrating finite semigroup abstractions with stack structure replacement techniques, the authors present the first effective construction of automata recognizing these closures. The method establishes an upper bound of nondeterministic triple-exponential and deterministic quadruple-exponential time complexity for this construction. Furthermore, the paper proves that these bounds are asymptotically tight, thereby precisely characterizing the computational complexity of the downward closure for indexed languages.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Natural Language Processing: Discourse, Pragmatics & Argument Mining

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Models for Web evolution

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 was shown by Zetzsche (ICALP 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.

Problem

• indexed languages
• downward closure
• computational complexity
• primitive-recursive
• automata construction

Innovation

• downward closure
• indexed languages
• complexity bounds
• semigroup theory
• pushdown automata