Edge, Event and State Removal: The Complexity of Some Basic Techniques that Make Transition Systems Petri Net Implementable

Abstract

In Petri net synthesis we ask whether a given transition system A can be implemented by a Petri net N. Depending on the level of accuracy, there are three ways how N can implement A: an embedding, the least accurate implementation, preserves only the diversity of states of A; a language simulation already preserves exactly the language of A; a realization, the most accurate implementation, realizes the behavior of A exactly. However, independent of the implementation sought, a corresponding net does not always exist. In this case, it was suggested to modify the input behavior –of course as little as possible. Since transition systems consist of states, events and edges, these components appear as the natural choice for modifications. In this paper we show that the task of converting an unimplementable transition system into an implementable one by removing as few states or events or edges as possible is NP-complete –regardless of what type of implementation we are aiming for.