Frank Heitmann

Safeness for Object Nets

In this paper we discuss the concept of safeness for Elementary Object Nets (EOS). Object nets are Petri nets which have Petri nets as tokens - an approach known as the nets-within-nets paradigm. Object nets are called elementary if the net system has a two levelled structure. The well known p/t nets can be considered as a special case of EOS. For p/t nets the concept of safeness means that there is at most one token on each place. Since object nets have nested markings there are different possibilities to generalise this idea for EOS. In this paper we define different variants of EOS safeness, discuss their relationships, show that they all coincide for p/t-like EOS, and address the complexity of well known Petri net problems like reachability and liveness for this new class of object nets.

On the Expressiveness of Communication Channels for Object Nets

In this work we present object net systems, i.e. Petri nets with nets as token objects, which are equipped with channels that allow to transfer net-tokens in the vertical dimension of the nested marking. These channels are a modelling element powerful enough to describe a direct simulation of counter programs which shows that typical net problems like boundedness, coverability, and reachability are undecidable.

Liveness of Safe Object Nets

In this paper we study the complexity of the liveness problem for safe Elementary Object Nets (EOS). Object nets are Petri nets which have Petri nets as tokens. They are called elementary if the net system has a two levelled structure. The concept of safeness bounds the number of tokens which may reside on a place. PSPACE-hardness of the liveness problem for safe EOS directly follows from the related result for safe p/t nets. We then devise a polynomial space algorithm for this problem and indeed for every property that can be expressed in the temporal logic CTL.

Complexity results for elementary hornets

In this paper we study the complexity of Hornets, an algebraic extension of object nets. We define a restricted class: safe, elementary Hornets, to guarantee finite state spaces. We have shown previously that the reachability problem for this class requires at least exponential space, which is a major increase when compared to safe, elementary object nets, which require polynomial space. Here, we show how a safe, elementary Hornets can be simulated by a safe Eos, which establishes an upper bound for the complexity, since we know that that the reachability problem for safe Eos is PSpace-complete.

P- and t-systems in the nets-within-nets-formalism

We are concerned with (strongly deterministic) generalised state machines (GSMs), a restricted formalism of the nets-within-nets-family, and further restrict the involved nets to P- and T-nets. While GSMs with these restrictions are likely to be of little use in modelling applications, understanding them better might help in future attempts to analyse more sophisticated formalisms. We show that, given a strongly deterministic GSM where the system net and all object nets are P-nets, it is PSpace-complete to decide the reachability of a given marking. In past work we have already shown that the same restriction to T-nets remains solvable in polynomial time. We discuss this work in the context given here. At last we give some initial results concerning other combinations of restricting the system and/or the object nets to P- and/or T-nets. Throughout we also discuss the effect of dropping the restriction to strongly deterministic GSMs.

Conservative Elementary Object Systems

This contribution presents decidability results for the formalism of 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 paper we study the relationship of the reachability and the liveness problem. We prove that both problems are undecidable for EOS even for the subclass of conservative EOS while it is well known that both are decidable for classical p/t nets. Despite these undecidability results, boundedness can be decided for conservative EOS using a monotonicity argument similar to that for p/t nets.

An Upper Bound for the Reachability Problem of Safe, Elementary Hornets

In this paper we study the complexity of the reachability problem HORNETS, an algebraic extension of object nets. Here we consider the restricted class of safe, elementary HORNETS. In previous work we established the lower bound, i.e. reachability requires at least exponential space. In another work we have shown we can simulate elementary HORNETS with elementary object nets EOS, where reachability is known to be PSpace-complete. Since this simulation leads to a double exponential increase in the size of the simulating EOS, we obtain that for HORNETS the reachability problem is solvable in double exponential space. In this contribution we show that this kind of simulation is rather bad, since we show that exponential space is sufficient. Together with the known lower bound this shows that the upper is tight.

Structural and Dynamic Restrictions of Elementary Object Systems

Elementary object systems (EOS for short) are Petri nets in which tokens may be Petri nets again. Originally proposed by Valk for a two levelled structure, the formalism was later generalised for arbitrary nesting structures. However, even if restricted to a nesting depth of two, EOS are Turing-complete and thus many problems like reachability and liveness are undecidable for them. Nonetheless, since they are useful to model many practical applications a natural question is how to restrict the formalism in such a way, that the resulting restricted formalism is still helpful in a modelling context, but so that important verification problems like reachability become quickly decidable. In the last years several structural and dynamic restrictions for EOS have therefore been investigated. These investigations have been central to the first authors recent PhD thesis and have been published in past editions of this journal and on conferences. In this paper we add several new results and present them together with the old in a unified fashion highlighting the central message of these investigations.