The Complexity of Nested Reset Counter Systems

2026-05-14

AI Summary

This work investigates the coverability problem for nested reset counter systems (NRCS) and its computational complexity in formal verification, logic, and XML processing. By introducing a tree-based higher-order counter model equipped with reset operations and leveraging ordinal-recursive theory together with fast-growing hierarchy analysis, the study establishes—for the first time—a natural hierarchy of complete problems for all complexity classes $F_{\Omega_k}$. It proves that the coverability problem for $k$-order NRCS is precisely $F_{\Omega_k}$-complete. This result not only refines the theory of length functions for multiset rewriting but also significantly improves known upper bounds on complexity in diverse domains, including XML processing, graph transformation systems, the π-calculus, and parameterized verification.

Abstract

Nested counter systems (NCS) are a generalization of counter systems to higher-order counters. Here, a higher-order counter is allowed to have other (lower-order) counters as elements, instead of just a number. Such systems can be viewed as working on trees, where the height of the tree naturally corresponds to the highest order counter that the system is working with. It is known that the coverability problem for NCS, which asks if a given final tree can be covered from a given initial tree, is $\mathbf{F}_{ε_0}$-complete. Here $\mathbf{F}_{ε_0}$ is a class in the fast-growing hierarchy of complexity classes. In this paper, we consider an extension of NCS called nested reset counter systems (NRCS) that extends NCS with resets. We show that coverability for NRCS over order-$k$ counters is $\mathbf{F}_{Ω_k}$-complete where $Ω_k$ is the tower of height $k$ of the $ω$ ordinal. This gives the first natural hierarchy of complete problems for all of these classes. Furthermore, to prove our upper bounds, we also develop length function theorems for any fixed amount of applications of the multiset operation on finite sets. As an application of our results, we improve existing upper bounds for various problems from XML processing, graph transformation systems, $π$-calculus, logic and parameterized verification. Furthermore, using our completeness results for $k$-NRCS, we also prove $\mathbf{F}_{Ω_k}$-completeness of the considered problems from the realms of parameterized verification and logic, for all $k$.

Problem

• nested reset counter systems
• coverability
• higher-order counters
• fast-growing hierarchy
• ordinal complexity

Innovation

• nested reset counter systems
• coverability
• fast-growing hierarchy
• length function theorems
• parameterized verification