Reasoning on Labelled Petri Nets and Their Dynamics in a Stochastic Setting

Abstract

Interest in stochastic models for business processes has been revived in a recent series of studies on uncertainty in process models and event logs, with corresponding process mining techniques. In this context, variants of stochastic labelled Petri nets, that is with duplicate labels and silent transitions, have been employed as a reference model. Reasoning on the stochastic, finite-length behaviours induced by such nets is consequently central to solve a variety of model-driven and data-driven analysis tasks, but this is challenging due to the interplay of uncertainty and the potentially infinitely traces (including silent transitions) induced by the net. This explains why reasoning has been conducted in an approximated way, or by imposing restrictions on the model. The goal of this paper is to provide a deeper understanding of such nets, showing how reasoning can be properly conducted by leveraging solid techniques from qualitative model checking of Markov chains, paired with automata-based techniques to suitably handle silent transitions. We exploit this connection to solve three central problems: computing the probability of reaching a particular final marking; computing the probability of a trace or that a temporal property, specified as a finite-state automaton, is satisfied by the net; checking whether the net stochastically conforms to a probabilistic Declare model. The different techniques have all been implemented in a proof-of-concept prototype.