Net Condition/Event systems (NCES)

The formalism of Net Condition/Event systems exists in several dialects and have been applied for modeling, verification and synthesis of controllers and control systems of various types.

Figure 1. Graphical notation of a module.

From today's point of view, the general idea which is common to each particular "dialect" is quite simple, namely the way of thinking of and modeling a system as a set of modules with a particular dynamic behavior and their interconnection via signals. This way of modeling is a very intuitive one, and the modules can be pre-tailored and used over and over again. Each module is equipped with inputs and outputs which are of two types:

1. Condition inputs/outputs carrying state information, and

2. Event inputs/outputs carrying state transition information.

This way of extension of the system with inputs and outputs clearly reflects the duality of Petri nets, namely the clear distinction between states and states transitions with their own graphical representation, semantics, and formal properties. An illustrative example of the graphical notation of a module is provided in Figure 1 .

Condition input signals as well as event input signals are connected with transitions inside the module. Whether a transition of a module fires does not only depend on the current marking (as it is the case in classical Petri nets) but also on the incoming condition and event signals. Incoming condition signals enable/disable a transition by their values in addition to the current marking. Incoming event signals force transitions to fire if they are enabled by marking and by condition signals. Hence, we get a modeling concept that can represent enabling/disabling of transitions by signals as well as enforcing transitions by signals. More than this, the concept provides a basis for a compositional approach to build larger models from smaller components. "Composition" is performed by "glueing" inputs of one module with outputs of another module as depicted in Figure 2 .

Figure 2. Modular composition.

Result of the composition of two NCES N₁ and N₂ is an NCES N_c obtained as a union of the components and which can be represented as a new module. Inputs and outputs of the "composition" are unions of the components' inputs and outputs, except for those which are interconnected to each other, and hereby "glued", i.e. substituted by the corresponding condition and event arcs, as shown in Figure 3 .

Figure 3. Result of the modular composition.