Nets-within-Nets through the Lens of Data Nets

TL;DR

This paper analyzes the complexity of reachability in elementary object systems within the nets-within-nets paradigm, linking it to data nets and establishing bounds beyond primitive recursive classes.

Contribution

It characterizes conservative EOS coverability within data nets, bridging NWNs and data nets, and determines the complexity bounds of the problem.

Findings

cEOS coverability is equivalent to a fragment of data nets.

Complexity lies between F_{ω2} and F_{ω^ω}, beyond primitive recursive.

Bridges two approaches: data nets and NWNs.

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 $\nu$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 $\mathbf{F}_{\omega 2}$ and $\mathbf{F}_{\omega^\omega}$, two classes beyond Primitive Recursive.