Parameterized Reachability for Register Machines with Data

2026-10-06

AI Summary

This study investigates the parameterized reachability problem for concurrent register machines over infinite data domains, determining whether a target state is reachable when n identical processes interact via local and shared registers. Leveraging formal verification, automata theory, and computational complexity analysis, the work delineates the decidability boundaries of this problem. Specifically, it establishes PSPACE-completeness in the general case and undecidability under the freshness assumption. For restricted scenarios involving only a single local or a single shared register, decidability is recovered with optimal complexity bounds. By systematically characterizing how resource-sharing mechanisms influence the complexity of reachability determination in concurrent register models, this research provides a rigorous theoretical foundation for the formal verification of concurrent systems.

Abstract

We investigate the parameterized reachability problem for concurrent register machines over infinite data domains. In this framework, each machine is a program equipped with a set of local registers, the communication across machines is mediated through a set of shared registers. Both local and shared registers can take values from an infinite data domain. The program's primitive operations include copying values between registers, assigning constants, comparing registers for (dis-)equality, and nondeterministic assignments that store an arbitrary domain value into a local register. The parameterized reachability problem considers a program and a target location, asking whether there exists some n in Naturals such that an execution of n identical machines (referred to as instances) results in at least one instance reaching the specified location. We show that this problem is Pspace-complete in the general case and it becomes undecidable if a freshness assumption (i.e., each assignment must produce a unique value distinct from all constants) is applied to nondeterministic assignments. This undecidability persists even for systems restricted to two shared and two local registers. Finally, we establish optimal decidability results for two restricted settings: when each thread is limited to a single local register, or when the system utilizes only one shared register.

Problem

• parameterized reachability
• register machines
• infinite data domains
• concurrent systems

Innovation

• Parameterized Reachability
• Register Machines
• Infinite Data Domains
• Freshness Assumption
• Decidability