The complexity of downward closure comparisons

TL;DR

This paper investigates the computational complexity of inclusion and equivalence problems for downward closures of various language classes, revealing high complexity and hardness results across multiple automata models.

Contribution

It provides a comprehensive complexity analysis of downward closure inclusion and equivalence problems for several automata classes, including new hardness results.

Findings

Decidability of inclusion and equivalence for downward closures in various models

Complexity classifications ranging from completeness to Ackermann-hardness

Hardness results for Petri net and higher-order pushdown automata

Abstract

The downward closure of a language is the set of all (not necessarily contiguous) subwords of its members. It is well-known that the downward closure of every language is regular. Moreover, recent results show that downward closures are computable for quite powerful system models. One advantage of abstracting a language by its downward closure is that then equivalence and inclusion become decidable. In this work, we study the complexity of these two problems. More precisely, we consider the following decision problems: Given languages $K$ and $L$ from classes $\mathcal{C}$ and $\mathcal{D}$, respectively, does the downward closure of $K$ include (equal) that of $L$? These problems are investigated for finite automata, one-counter automata, context-free grammars, and reversal-bounded counter automata. For each combination, we prove a completeness result either for fixed or for arbitrary alphabets. Moreover, for Petri net languages, we show that both problems are Ackermann-hard and for higher-order pushdown automata of order~$k$, we prove hardness for complements of nondeterministic $k$-fold exponential time.