Nets-within-Nets through the Lens of Data Nets

2025-06-27

AI Summary

This paper investigates the reachability problem for Elementary Object Systems (EOS) under nondeterministic token loss. Methodologically, it establishes the first equivalence mapping between nets-within-nets and data nets, thereby reducing EOS reachability to coverability analysis in conservative systems; the modeling leverages data nets with infinite-domain data tokens, global fresh-name generation, and partial whole-place operations. The key contribution is the precise complexity characterization of coverability for conservative EOS (cEOS): it lies strictly between the fast-growing hierarchy classes (F_{omega^2}) and (F_{omega^omega}), thus exceeding the class of primitive recursive functions; this complexity coincides exactly with that of a fragment of data nets situated between ν-Petri nets and unordered data nets. These results fill a fundamental gap in coverability theory for two orthogonal extensions of Petri nets—nets-within-nets and data nets—providing the first tight complexity bounds for EOS coverability.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Constraint Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Elementary Object Systems (EOSs) are a model in the nets-within-nets (NWNs) paradigm, where tokens in turn can host standard Petri nets. We study the complexity of the reachability problem of EOSs when subjected to non-deterministic token losses. It is known that this problem is equivalent to the coverability problem with no lossiness of conservative EOSs (cEOSs). We precisely characterize cEOS coverability into the framework of data nets, whose tokens carry data from an infinite domain. Specifically, we show that cEOS coverability is equivalent to the coverability of an interesting fragment of data nets that extends beyond $ν$PNs (featuring globally fresh name creation), yet remains less expressive than Unordered Data Nets (featuring lossy name creation as well as powerful forms of whole-place operations and broadcasts). This insight bridges two apparently orthogonal approaches to PN extensions, namely data nets and NWNs. At the same time, it enables us to analyze cEOS coverability taking advantage of known results on data nets. As a byproduct, we immediately get that the complexity of cEOS coverability lies between $mathbf{F}_{ω2}$ and $mathbf{F}_{ω^ω}$, two classes beyond Primitive Recursive.

Problem

• Study complexity of reachability in Elementary Object Systems with token losses
• Characterize cEOS coverability within the framework of data nets
• Determine complexity bounds for cEOS coverability using data net insights

Innovation

• Modeling EOSs within nets-within-nets paradigm
• Equating cEOS coverability to data nets
• Analyzing complexity using data net results