Fernando Rosa-Velardo

Decidability and complexity of Petri nets with unordered data

Abstract We prove several decidability and undecidability results for ν -PN, an extension of P/T nets with pure name creation and name management. We give a simple proof of undecidability of reachability, by reducing reachability in nets with inhibitor arcs to it. Thus, the expressive power of ν -PN strictly surpasses that of P/T nets. We encode ν -PN into Petri Data Nets, so that coverability, termination and boundedness are decidable. Moreover, we obtain Ackermann-hardness results for all our decidable decision problems. Then we consider two properties, width-boundedness and depth-boundedness, that factorize boundedness. Width-boundedness has already been proven to be decidable. Here we prove that its complexity is also non-primitive recursive. Then we prove undecidability of depth-boundedness. Finally, we prove that the corresponding “place version” of all the boundedness problems is undecidable for ν -PN. These results carry over to Petri Data Nets.

On the Expressiveness of Mobile Synchronizing Petri Nets

In recent papers we have introduced Mobile Synchronizing Petri Nets, a new model for mobility based on coloured Petri Nets. It allows the description of systems composed of a collection of (possibly mobile) hardware devices and mobile agents, both modelled in a homogenous way and abstracting from middleware details. Our basic model introduced a colour to describe localities, but still lacked appropriate primitives to deal with security, and in fact it was equivalent to P/T nets. Then, we introduced the primitives to cope with security: a new colour for identifiers, basically corresponding to the natural numbers, that are created by means of a special transition. This mechanism allows us to deal with authentication issues. In this paper we discuss the expressiveness of the extended model with the authentication primitives. More specifically, we study several instances of the classical reachability and coverability problems. Finally, we also study a more abstract version of the mechanism to create identifiers, using abstract names, close to those in the @p-calculus or the Ambient Calculus. We have proved that both models are strictly in between P/T nets and Turing machines.

Mobile Synchronizing Petri Nets: A Choreographic Approach for Coordination in Ubiquitous Systems

The term Ubiquitous Computing was coined by Mark Weiser almost two decades ago. Despite all the time that has passed since Weisers vision, ubiquitous computing still has a long way ahead to become a pervasive reality. One of the reasons for this may be the lack of widely accepted formal models capable of capturing and analyzing the complexity of the new paradigm. We propose a simple Petri Net based model to study some of its main characteristics. We model both devices and software components as a special kind of coloured Petri Nets, located in locations, that can move to other locations and synchronize with other co-located nets, offering and requesting services. We obtain an amenable model for ubiquitous computing, due to its graphical representation. We present our proposal in a progressive way, first presenting a basic model where coordination is formalized by the synchronized firing of pairs of compatible transitions that offer and request a specific service, and ad hoc networks are modeled by constraining mobility by the dynamic acquisition of locality names. Next, we introduce a mechanism for the treatment of robust security properties, namely the generation of fresh private names, to be used for authentication properties.

Coding Mobile Synchronizing Petri Nets into Rewriting Logic

Mobile Synchronizing Petri Nets (MSPNs) are a model for mobility and coordination based on coloured Petri Nets, in which systems are composed of a collection of (possibly mobile) hardware devices and mobile agents, both modelled homogenously. In this paper we approach their verification, for which we have chosen to code MSPNs into rewriting logic. In order to obtain a representation of MSPN systems by means of a rewrite theory, we develop a class of them, that we call @n-Abstract Petri nets (@n-APNs), which are easily representable in that framework. Moreover, the obtained representation provides a local mechanism for fresh name generation. Then we prove that, even if @n-APNs are a particular class of MSPN systems, they are strong enough to capture the behaviour of any MSPN system. We have chosen Maude to implement @n-APNs, as well as the translation from MSPNs to @n-APNs, for which we make intensive use of its reflective features.

Typed Mobile Ambients in Maude

Maude has revealed itself as a powerful tool for implementing different kinds of semantics so that quick prototypes are available for trying examples and proving properties. In this paper we show how to define in Maude two semantics for Cardelli and Gordons Ambient Calculus. The first one is the operational (reduction) semantics which requires the definition of Maude strategies in order to avoid infinite loops. The second one is a type system defined by Cardelli and Gordon to avoid communication errors. The correctness of that system was not formally proved. We enrich the operational semantics with error rules and prove that well-typed processes do not produce such errors. The type system is highly non-deterministic. We show here one possible way of implementing such non-determinism in the rules.

Name creation vs. replication in Petri net systems

We study the relationship between name creation and replication in a setting of infinitestate communicating automata. By name creation we mean the capacity of dynamically producing pure names, with no relation between them other than equality or inequality. By replication we understand the ability of systems of creating new parallel identical threads, that can synchronize with each other. We have developed our study in the framework of Petri nets, by considering several extensions of P/T nets. In particular, we prove that in this setting name creation and replication are equivalent, but only when a garbage collection mechanism is added for idle threads. However, when simultaneously considering both extensions the obtained model is, a bit surprisingly, Turing complete and therefore, more expressive than when considered separately.

On the coverability and reachability languages of monotonic extensions of Petri nets

We apply language theory to compare the expressive power of infinite-state models that extend Petri nets with features like coloured tokens and/or whole place operations. Specifically, we consider extensions of Petri nets in which tokens carry pure names dynamically generated with special @n-transitions (@n-PN) and compare their expressiveness with transfer and reset nets with black indistinguishable tokens (Affine Well-Structured Nets), and nets in which tokens carry data taken from a linearly ordered domain (Data nets and CMRS). All these models are well-structured transition systems. In order to compare these models we consider the families of languages they recognize, using coverability as the accepting condition. With this criterion, we prove that @n-PNs are in between AWNs and Data Nets/CMRS, but equivalent to an extension of @n-PN with whole-place operations. These results extend the currently known classification of the expressive power of well-structured transition systems. Finally, we study several problems regarding (coverability) languages of AWN and @n-PN.

Dynamic soundness in resource-constrained workflow nets

Workflow Petri nets (wf-nets) are an important formalism for the modeling of business processes. For them we are typically interested in the soundness problem, that intuitively consists in deciding whether several concurrent executions can always terminate properly. Resource-Constrained Workflow Nets (rcfw-nets) are wf-nets enriched with static places, that model global resources. In this paper we prove the undecidability of soundness for rcwf-nets when there may be several static places and in which instances are allowed to terminate having created or consumed resources. In order to have a clearer presentation of the proof, we define an asynchronous version of a class of Petri nets with dynamic name creation. Then, we prove that reachability is undecidable for them, and reduce it to dynamic soundness in rcwf-nets. Finally, we prove that if we restrict our class of rcwf-nets, assuming in particular that a single instance is sound when it is given infinitely many global resources, then dynamic soundness is decidable by reducing it to the home space problem in P/T nets for a linear set of markings.

Ordinal recursive complexity of Unordered Data Nets

Abstract Data Nets are a version of colored Petri nets in which tokens carry data from an infinite and linearly ordered domain. This is a very expressive class, though coverability and termination remain decidable. Those problems have recently been proven complete for the F ω ω ω class in the fast growing complexity hierarchy. We characterize the exact complexity of Unordered Data Nets (UDN), a subclass of Data Nets with unordered data. We bound the length of bad sequences in well-quasi orderings of multisets over tuples of naturals by adapting the analogous result by Schmitz and Schnoebelen for words over a finite alphabet. These bounds imply that both problems are in F ω ω . We prove that this result is tight by constructing UDN that weakly compute fast-growing functions and their inverses. This is the first complete problem for F ω ω with an underlying wqo not based on finite words over a finite alphabet.

Ordinal theory for expressiveness of well-structured transition systems

We characterize the importance of resources (like counters, channels, or alphabets) when measuring the expressiveness of Well-Structured Transition Systems (WSTS). We establish, for usual classes of well partial orders, the equivalence between the existence of order reflections (non-monotonic order embeddings) and the simulations with respect to coverability languages. We show that the non-existence of order reflections can be proved by the computation of order types. This allows us to extend the current classification of WSTS, in particular solving some open problems, and to unify the existing proofs.