A Survey of Decidability Results for Elementary Object Systems

Abstract

This contribution presents recent results on Elementary Object Systems (EOS). Object nets are Petri nets which have Petri nets as tokens -an approach known as the nets-within-nets paradigm.In this work we study the relationship of EOS to existing Petri net formalisms. It turns out that EOS are equivalent to counter programs.But even for the restricted subclass of conservative EOS reachability and liveness are undecidable problems. On the other hand for other properties like boundedness are still decidable for conservative EOS.We also study the sub-class of generalised state machines, which is worth mentioning since it combines decidability of many theoretically interesting properties with a quite rich practical modelling expressiveness.states. The algebraic extension of objects nets -discussed in [20] -even allows operations on the nettokens, like sequential or parallel composition. This is a concise way to express the self-modification of net-tokens at run-time in an algebraic setting.Object Nets can be seen as the Petri net perspective on mobility, in contrast to the Ambient Calculus [4] or the π-calculus [36], which form the process algebra perspective.It is quite natural to use object nets to model mobility and mobile agents (cf. [16]). Each place of the system net describes a location that may hosts agents, which are net-tokens. Mobility can be modelled by moving net-tokens from one place to another. This hierarchy forms a useful abstraction of the system: on a high level the agent system and on a lower level of the hierarchy the agent itself.Example 1.1. Figure 1 shows an object net system of a simple mobile household robot inside a building with several rooms. The mobile robot is modelled as a net-token that moves from room to room in the system net, which models the building. The agent has a very simple plan: Firstly, it synchronises with the environment via the channel ch 1 and then via the channel ch 2 . The synchronisation for ch 1 is provided in the kitchen; the synchronisation for ch 2 is provided in the living room.One possible run is the following: Initially, the agent is in the state q 1 and is located at place p 1 in the bedroom (cf. Figure 1). The agent moves into the bath by t 1 and moves into the kitchen by t 2 . Here, it synchronises its transition u with the environment transition t 3 over the channel ch 1 . Then the agent is in the state q 2 and is located at p 6 . The agent moves into the living room by t 7 and synchronises v with t 8 over the channel ch 2 . Finally, the agent is located at p 5 and has reached its final state q 3 . w − → m ′ holds. The set of reachable markings is RS (m 0 ) := {m | ∃w ∈ T * : m 0 w − → m}. Elementary Object SystemsAn elementary object system (EOS) is composed of a system net, which is a p/t net N = ( P , T , pre, post) and a set of object nets N = {N 1 , . . . , N n }, which are p/t nets given as N = (P N , T N , pre N ,

A Categorical Semantics for Guarded Petri Nets

“…This is akin to calling subroutines in a piece of code. Moreover, we do not want to destroy the decidability of the reachability relation for our nets, as it happens for other hierarchical models such as the net-within-nets framework [11]. We consider this to be a very important requirement for practical reasons.…”