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Dualizing marked Petri nets results in tokens for transitions (t-tokens). A marked transition can strictly not be enabled, even if there are sufficient "enabling" tokens (p-tokens) on its input places. On the other hand, t-tokens can be moved by the firing of places. This permits flows of t-tokens which describe sequences of non-events. Their benefiit to simulation is the possibility to model (and observe) causes and effects of non-events, e.g. if something is broken down.

In this paper, we demonstrate by means of two examples how to work with probability propagation nets (PPNs). The fiirst, which comes from the book by Peng and Reggia [1], is a small example of medical diagnosis. The second one comes from [2]. It is an example of operational risk and is to show how the evidence flow in PPNs gives hints to reduce high losses. In terms of Bayesian networks, both examples contain cycles which are resolved by the conditioning technique [3].

The paper deals with a specific introduction into probability propagation nets. Starting from dependency nets (which in a way can be considered the maximum information which follows from the directed graph structure of Bayesian networks), the probability propagation nets are constructed by joining a dependency net and (a slightly adapted version of) its dual net. Probability propagation nets are the Petri net version of Bayesian networks. In contrast to Bayesian networks, Petri nets are transparent and easy to operate. The high degree of transparency is due to the fact that every state in a process is visible as a marking of the Petri net. The convenient operability consists in the fact that there is no algorithm apart from the firing rule of Petri net transitions. Besides the structural importance of the Petri net duality there is a semantic matter; common sense in the form of probabilities and evidencebased likelihoods are dual to each other.

Probability propagation nets
(2007)

A class of high level Petri nets, called "probability propagation nets", is introduced which is particularly useful for modeling probability and evidence propagation. These nets themselves are well suited to represent the probabilistic Horn abduction, whereas specific foldings of them will be used for representing the flows of probabilities and likelihoods in Bayesian networks.