The complexity of coverability in ν-Petri nets

We show that the coverability problem in ν-Petri nets is complete for ‘double Ackermann’ time, thus closing an open complexity gap between an Ackermann lower bound and a hyper-Ackermann upper bound. The coverability problem captures the verification of safety properties in this nominal extension of Petri nets with name management and fresh name creation. Our completeness result establishes ν-Petri nets as a model of intermediate power among the formalisms of nets enriched with data, and relies on new algorithmic insights brought by the use of well-quasi-order ideals

Ordinal Theory for Expressiveness of Well Structured Transition Systems

Abstract. To the best of our knowledge, we characterize for the first time the importance of resources (counters, channels, alphabets) when measuring expressiveness of WSTS. We establish, for usual classes of wpos, the equivalence between the existence of order reflections (nonmonotonic 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 solve some open problems and to unify the existing proofs of the WSTS classification

Forward analysis for petri nets with name creation

Pure names are identifiers with no relation between them, except equality and inequality. In previous works we have extended P/T nets with the capability of creating and managing pure names, obtaining ν-APNs and proved that they are strictly well structured (WSTS), so that coverability and boundedness are decidable. Here we use the framework recently developed by Finkel and Goubault-Larrecq for forward analysis for WSTS, in the case of ν-APNs, to compute the cover, that gives a good over approximation of the set of reachable markings. We prove that the least complete domain containing the set of markings is effectively representable. Moreover, we prove that in the completion we can compute least upper bounds of simple loops. Therefore, a forward Karp-Miller procedure that computes the cover is applicable. However, we prove that in general the cover is not computable, so that the procedure is non-terminating in general. As a corollary, we obtain the analogous result for Transfer Data nets and Data Nets. Finally, we show that a slight modification of the forward analysis yields decidability of a weak form of boundedness called width-boundedness

Depth boundedness in multiset rewriting systems with name binding

In this paper we consider ν-MSR, a formalism that combines the two main existing approaches for multiset rewriting, namely MSR and CMRS. In ν-MSR we rewrite multisets of atomic formulae, in which some names may be restricted. ν-MSR are Turing complete. In particular, a very straightforward encoding of π-calculus process can be done. Moreover, pν-PN, an extension of Petri nets in which tokens are tuples of pure names, are equivalent to ν-MSR. We know that the monadic subclass of ν-MSR is a Well Structured Transition System. Here we prove that depth-bounded ν-MSR, that is, ν-MSR systems for which the interdependance of names is bounded, are also Well Structured, by following the analogous steps to those followed by R. Meyer in the case of the π-calculus. As a corollary, also depth-bounded pν-PN are WSTS, so that coverability is decidable for them

Language-based Comparison of Petri Nets with Black Tokens, Pure Names and Ordered Data

We apply language theory to compare the expressive power of models that extend Petri nets with features like colored tokens and/or whole place operations. Specifically, we consider extensions of Petri nets with transfer and reset operations defined for black indistinguishable tokens (Affine Well-Structured Nets), extensions in which tokens carry pure names dynamically generated with special ν-transitions (ν-APN), and extensions in which tokens carry data taken from a linearly ordered domain (Data nets and CMRS). These models are well-structured transitions systems. In order to compare these models we consider the families of languages they recognize, using coverability as accepting condition. With this criterion, we prove that ν-APNs are in between AWNs and Data Nets/CMRS. Moreover, we prove that the family of languages recognized by ν-APNs satisfies a good number of closure properties, being a semi-full AFL. These results extend the currently known classification of the expressive power of well-structured transition systems with new closure properties and new relations between extensions of Petri nets