A Taxonomy of Persistent and Nonviolent Steps

A concurrent system is persistent if throughout its operation no activity which become enabled can subsequently be prevented from being executed by any other activity. This is often a highly desirable (or even absolutely necessary) property, in particular, if the system is to be implemented in hardware. Over the past 40 years, persistence has been investigated and applied in practical implementations assuming that each activity is a single atomic action which can be represented, for example, by a single transition of a Petri net used as a formal representation of a concurrent system. Recently, it turned out that such a notion of persistence is restricted and in dealing with the synthesis of GALS systems one also needs to consider activities represented by steps which are sets of simultaneously executed transitions. Moving into the realm of step based execution semantics creates a wealth of new fundamental problems and intriguing questions. In particular, there are different ways in which the standard notion of persistence could be lifted from the level of sequential semantics to the level of step semantics. Moreover, at a local level, one may consider steps which are persistent and cannot be disabled by other steps, as well as steps which are nonviolent and cannot disable other steps. In this paper, we aim at providing a classification of different types of persistence and nonviolence, both for steps and markings of pt-nets. We also investigate behavioural and structural properties of such notions both for the general pt-nets and their subclasses.