Structure Theory of Petri Nets: the Free Choice Hiatus

Structure theory asks whether a relationship can be found between the behaviour of a marked net and the structure of the underlying unmarked net. From the rich body of structure theoretical results that exists in Petri net theory, this paper selects a few examples which are deemed to be typical. The class of free choice nets, whose structure theory is particularly agreeable, is studied in some detail. 1 I n t r o d u c t i o n By the 'structure' of a P/T-system we mean marking-independent properties depending on the way in which the places and the transitions of the underlying net are interconnected by the flow relation. By the 'behaviour' of a P/T-system we denote marking-dependent properties relating to the token flow effected by the transition rule, depending on the set of processes, the set of reachable markings, the reachability graph, and so on. The behaviour of a marked net is, in general, less easily analysable than its structure. But it is the behavioural properties that are of foremost interest in the analysis of systems. They include, for example, the property of deadlock-freeness, the existence of invariant assertions, safeness properties, the validity of intermediate assertions, and others. Structure theory asks whether a relationship can be found between the behaviour of a marked net and the structure of the underlying unmarked net. It asks questions such as: Can one deduce, from certain 'nice' structural properties of a net, that its behaviour will also be 'nice' ? Or, conversely: Does certain 'bad' behaviour preordain some 'bad' structure? In any case one may hope that the (behavioural) properties which are of interest may be reduced to easier-to-investigate (structural) properties. A rich body of structure theoretical results exists in net theory. From this body, we shall select some typical examples, neither too many in order not to let the paper grow out of size, nor too few let the reader get an idea of the kind of reasoning employed in structure theory (hopefully). There is a class of nets which has an interesting motivation and allows for a very satisfactory structure theory. This class is called free choice nets. While being a non-trivial class of nets, their theory is so nice that it has sometimes jokingly been said that every conjecture is true for free choice nets and false for other nets. Although we will exhibit some 'counterexamples' to this statement, a good part of these notes will be dedicated to the study of free choice nets. These notes are organised as follows. In section 2 we introduce and explain almost all notions we need, but we will rely on I39] for some definitions and explanations. We introduce some basic behavioural properties (liveness and safeness), and we show that they have an impact in terms of the connectedness of a system. Sometimes it is necessary to compare nets with each other and to state that one is 'similar' to another one. In section 3 we define a notion of simulation to capture this idea. In sections 4-6 we introduce various subclasses of nets (free choice nets, amongst others) and we investigate some basic properties of these classes. In sections 7 and 8 we deal almost exclusively with free choice nets, listing and explaining some more advanced results about their structure and behaviour.