Petri-Net-Notes-Expanded

# Petri-Net-Notes-Expanded - NotesonPetriNets Introduction ....

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Computational Thinking 1 Dennis Kafura Notes on Petri Nets Introduction Petri nets were invented by Carl Adam Petri in 1939 at the age of 13. This work was the foundation for his 1962 doctoral dissertation entitled Kommunikation mit Automaten . Petri nets have been used in a variety of fields including computer science, chemistry, and biology. He retired from the Theoretical Foundation of Computer Science group at the University of Hamburg in 1991. Figure 1: Carl Adam Petri Petri nets are a graphical for representing a system in which there are multiple independent activities in progress at the same time. The ability to model multiple activities differentiates Petri nets from finite state machines. In a finite state machine there is always a single “current” state that determines which action can next occur. In Petri nets there may be several states any one of which may evolve by changing the state of the Petri net. Alternatively, some, of even all, of these states may evolve in parallel causing several independent changes to the Petri net to occur at once. Basic Structure A Petri net consists of four elements: places, transitions, edges, and tokens. Graphically, places are represented by circles, transitions by rectangles, edges by directed arrows, and tokens by small solid (filled) circles. There are a wide variety of extensions to Petri nets. These extensions add features to model probabilistic behavior, allow weighted edges, or have tokens of various colors among others. Only the most basic Petri net concepts will be covered here. A basic Petri net is shown in Figure 2. This Petri net has four places, labeled P0 through P4, and three transitions, labeled T0 through T2. Notice that places P0 and P2 each have a single token represented by the black dot inside each place. Edges, represented as directed arcs, connect places to transitions and transitions to places. In a properly formed Petri net, places cannot be directly connected to other places and transitions cannot be directly connected to other transitions. Also notice that the Petri net may contain cycles. The Petri net in Figure 2 contains two cycles. One cycle contains P0, T0, P1, T1, P3, and

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Computational Thinking 2 Dennis Kafura T3. The other cycle contains T1, P4, T2, and P2. Cycles are common in Petri nets which represent activities that happen repeatedly. For example, a web server repeated services incoming requests to deliver web page content to different clients. Figure 2: A Basic Petri Net The state of a Petri net is represented by the occurrence of the tokens at various places. The state of the Petri net in Figure 2 has tokens at places P0 and P2. It will be shown that in another state of this Petri net there are tokens at states P1 and P2. Yet another state has tokens at states P3 and P4. Not all placements of tokens at places represent a possible state of the system. For example, the Petri net in Figure 2 will never have as a possible state one in which the only tokens are at places P1 and P4. Which
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