Propositional dynamic logic for Petri nets

Abstract

Propositional Dynamic Logic (PDL) is a multi-modal logic used for specifying and reasoning on sequential programs. Petri Net is a widely used formalism to specify and to analyse concurrent programs with a very nice graphical representation. In this work, we propose a PDL to reasoning about Petri Nets. First we define a compositional encoding of Petri Nets from basic nets as terms. Second, we use these terms as PDL programs and provide a compositional semantics to PDL Formulas. Finally, we present an axiomatization and prove completeness w.r.t. our semantics. The advantage of our approach is that we can do reasoning about Petri Nets using our dynamic logic and we do not need to to translate it to other formalisms. Moreover our approach is compositional allowing for construction of complex nets using basic ones.

A Calculus for Automatic Verification of Petri Nets Based on Resolution and Dynamic Logics

“…The language of Petri-PDL [11] is constructed from a denumerable set of propositional symbols Φ = {p, q, . .…”

“…Petri-PDL [11] is a dynamic logic derived from PDL where each modality is a Marked Petri Net. If π is a Petri Net with markup s, then a Petri-PDL formula s, π ϕ means that after the running of π with the initial markup s, ϕ will possibly be true (the 2-like modality, given by [s, π], is also possible, where [s, π]ϕ is an abbreviation for ¬ s, π ¬ϕ).…”

“…However, this restriction in the language does not reflect on the class of problems that can be expressed in Petri-PDL: as shown in [1], the combination of those three types of transitions is enough to represent any possible Marked Petri Net. The axiomatisation of Petri-PDL is sound and complete, and the satisfiability problem is decidable [11]. Other systems have been proposed in the literature to reason about the properties and behaviour of Petri Nets, but they lack some of these properties, as, for instance, the Trace Theory [12,13], which is incomplete and undecidable.…”

“…By embedding Petri Nets as part of the language, Petri-PDL provides a natural way of specifying complex systems and the properties to be verified. Also, the provision of such abstractions at the (logical) specification level comes at no extra cost: the complexity of the satisfiability problem for Petri-PDL is the same as the complexity of the satisfiability problem for PDL, that is, EXPTIME-complete [11].…”

A Dynamic Logic for Every Season

“…One could mention its role in model validation (as in e.g. [10]), or the whole family of variants tailored to specific programming languages (as in e.g. [11,1]), or its important extensions to new computing domains, namely probabilistic [8] or continuous [13,14].…”