# Petri net

Post-publication activity

Curator: Wolfgang Reisig

Figure 1: Example of a production net.

A Petri net is a graphical tool for the description and analysis of concurrent processes which arise in systems with many components(distributed systems). The graphics, together with the rules for their coarsening and refinement, were invented in August 1939 by the German Carl Adam Petri -- at the age of 13 -- for the purpose of describing chemical processes, such as Figure 1.

## Anatomy of a Petri net

The <review>elementary</review> components of these nets are called states (for substances) and transitions (for reactions). The round symbols denote states $$S$$ (also called places or, more abstractly, conditions); the rectangular shapes denote transitions $$T\ .$$ They are interconnected by arrows which show the direction of flow. <review>Arrows only connect elementary components of different type - that is, an arrow never joins two circles, or two rectangles.</review> The activities of transitions are accordingly subdivided in give and take. Black marks (or tokens) are <review>used to denote (or mark) conditions that hold, and are</review> moved by the occurrence of transitions; they describe the present state-of affairs. In the picture above, the tokens show the initial[/itex] state. Each mark may stand for one molecule but also for 1 Mol consisting of over $$6 x 10^{23}$$ molecules. Note that $$CO_2$$ is present twice, for the two separate roles of $$CO_2\ .$$

By convention, the state symbols of regularly built nets are often omitted, in order to show their abstract structure more clearly, which may result in surprising shapes ( Figure 2).

Figure 2:
Figure 3:

Interchanging $$S$$ and $$T$$ in the production net gives a totally different structure, called an organization net ( Figure 3). Here, the transitions denote offices and the states denote channels, capable of holding messages (such as commands) until they are taken away <review>(used)</review> by a subordinate office. The net shows an organizational structure with two offices $$A\ ,$$ $$B$$ of equal rank who can issue orders independently. Offices like $$D$$ and $$F$$ are rare in human organizations but they do exist, e.g. in the military.

The most important class of nets are cyclic nets in which each arrow lies on a directed circuit. If well structured, they express the idea of an infinitely repeatable process ( Figure 1). A finite piece of their history can be described by an occurrence graph (a production net).

Figure 4:

This net has the same structure as a mechanical oscillator (where momentum is caused by the displacement of a mass owing to a retracting force).

## General net theory

The algebraic aspect of distributed systems was described by Petri in 1962 in his doctoral thesis "Communication with Automata" (Petri, 1966), in which it was argued that the then prevailing theory of automata had to be replaced by a new theory which <review>took into account</review> the results of modern physics, in particular those of Special Relativity and the Uncertainty Principle, to yield a theory of distributed systems consistent with the laws of Physics (basic net theory).

That doctoral thesis was followed by a series of papers by Petri himself and his research group on the applicability of nets to various areas of science (economics, mechanics, computer science, logic, organization, biology, telecommunication protocols...). The aim was to create a tool for the interdisciplinary transfer of structural knowledge. Higher-level nets were introduced later, on the axiomatic foundation of general net theory (GNT), introducing a general definition of net, and the basic transition rule.

In particular, the GNT introduced the following axioms:

• 1.1 The empty set is not a net (relevant to theorem proving)
• 1.2 $$S$$ and $$T$$ are <review>disjoint</review> sets
• 1.3 There are no isolated elements in a net
• 1.4 There are no short circuits (both forward and backward arrows between two elements)

An event is a set of concurrently executable transitions. Event axioms:

• 2.1 (Every net can be marked so that) the class $$E$$ of events is not empty
• 2.2 Transitions in the same event have non-overlapping sets of connected states
• 2.3 Each transition in an event e has all incoming states marked, but unmarked at the state reached by the occurrence of e
• 2.4 Each transition in an event e has all outgoing states unmarked, but marked at the state reached by the occurrence of e

Mappings from one net to another are governed by the axioms

• 3.1 A map does not invert arrows
• 3.2 A map is continuous in the sense of topology: it does not destroy neighbourhood.

The <review>incidence</review> matrix $$C$$ of a basic net is an $$SxT$$ matrix with entries $$(0,+1,-1)$$

Meaningful sequences of nets, denoting a multi-dimensional object, are constructed thus:

• 4.1 $$T$$ of net $$N(n+1)$$ is identical with $$S$$ of $$Net(n)$$
• 4.2 $$C$$ of net $$N(n+1)$$ multiplied by $$C$$ of $$Net(n)$$ yields a zero matrix

Axioms *3 describe the conceptual step from basic nets to higher level nets in which arrows can carry labels, states $$S$$ can carry any number of tokens, transitions $$T$$ can mean any composite process, and short circuits are allowed if their meaning is defined. The Event Axioms have to be re-defined accordingly.

Axioms *4 define the dimensional step from basic nets having only two kinds ($$S$$ and $$T$$) of elements to net piles with any required number of kinds. Assume we want to express the shape of a cube by nets: Take the net of corners and edges, and pile up the net of edges and faces - consistent orientation is required. We get a net pile with three kinds of elements: corners $$(dim 0)\ ,$$ edges $$(dim 1)$$ and faces $$(dim 2)\ .$$ Thus, we are prepared to tackle the ten or eleven dimensions of String Theory.

## Marking class, safety, security, and fairness

Given an initial marking, there is often the task to enquire not only the future but also the past of the history of the net. Therefore, the marking class is defined as the set of markings which are reachable from the initial marking by forward and backward execution of transitions.

A net structure is called safe if there are no contact situations in its marking class. Contact is given when a collision of tokens threatens because axioms 2*3 and 2*4 are not satisfied.

A net structure is called secure if there are no transjunctions in its marking class. A transjunction is a conjunction of conditions across a transition: an input condition is marked and also an output condition. Tokens denoting electromagnetic signals might interfere.

Figure 5:

Some nets allow two or more independent processes kept apart by a mechanism of mutual exclusion. This mechanism ensures that only one of the processes can enter a critical phase on each repetition. The working of the mechanism is described below: the mutual exclusion is embodied by a dead transition which can never occur. Dead transitions are called facts because they represent factual statements about the marking class.

The schedule on the left can be misused by a fast competitor through usurping the system all for himself. To ensure a fair competition, the cycles must be linked by a buffer. A buffer of capacity 1 would enforce alternating use; in practice, larger buffers allow more freedom.

Figure 6:

The example above ( Figure 6) indicates the general method to define all valid logical statements about the marking class by a set of facts.

General net theory features the following innovations:

1. The replacement of temporal order by causal order, in keeping with the laws of Physics. Simultaneity is replaced by concurrency (mutual causally independent occurrence).
2. Merging of discrete and continuous modeling, in keeping with the laws of topology. This is done by equipping all nets with a topological structure: The closure of a state is defined as that state plus all transitions connected to it, irrespective of direction of arrows. Moreover, all rules of mathematical topology are to be applied by definition. As a result, every finite connected net is a continuous space.
3. Derivation of the physical primitives for computation from the smallest structures of particle movement periodic in time (by re-orientation).

This highly abstract and laborious procedure, which Petri performed in cooperation with Konrad Zuse, is as follows:

The background of this table is an Integer Minkowski Grid, implemented between a pulsating light source and a mirror. The yellow patches are cut out and their opposite sides are glued together to form a torus on which certain tokens (particles) oscillate periodically in specific modes ( Figure 7). LT-compatible means that the Lorentz transformation can be applied literally to the patterns, while this is not immediately possible for the degenerate patterns. It turned out, to the surprise of the inventors, that those periodical patterns of movement corresponded one-to-one to a complete set of loss-free computing primitives (by changing the orientation of arrows). Moreover, loss-free means that they have exact reverses (each is even its own reverse!).

Figure 7:
Figure 8:

Figure 9:

The detailed procedure for constructing the Quine Transfer which performs the conditional interchange of two bits are shown in Figure 9. Quine's function $$Q(x, y, z)$$ is defined as

$$Q(a,b,c) = \begin{cases} c, & \mbox{if }b\mbox{ =1} \\ a, & \mbox{otherwise} \end{cases}$$

Two copies of the variable $$b$$ are needed in order to perform the transfer in one step.

There exist systematic mappings to generate information flow graphs from basic nets ( Figure 9). They are the tool for the actual construction of nets fit for computing arbitrary functions from a given set of bits under physically feasible conditions.

Returning to the derivation of the smaller transfers necessary for computation in general, the synchronization of bit streams, and the transfer computing exclusive OR (XOR) are given explicitly. The Information Graph shorthand for both is shown on bottom of the constructs:

Figure 10:

Here is a universal construct which performs bit addition modulo 2 and bit multiplication in three steps. On the right, the general form of information flow graphs is demonstrated.

Figure 11:

Finally, it is pointed out that all basic computing primitives can be composed by joining noisy channels only (example: composition of XOR). As all real-world channels are noisy, there is ample supply of them. We hold that noise is not different from other physical influences, that is, caused by signals. Once a method is found for joining two or four noisy channels physically, there is no obstacle for a large-scale production of computing-power.

Figure 12:

Since 1968, many authors have helped to develop net theory into a well-established body of knowledge, with many computer tools available for detecting the properties of nets hidden in the net graphics, such as behavioural invariants, or possibilities of decomposition.

## References

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