A Petri net (also known as a place/transition net or P/T net) is one of several mathematical modeling languages for the description of distributed systems. A Petri net is a directed bipartite graph, in which the nodes represent transitions (i.e. events that may occur, signified by bars) and places (i.e. conditions, signified by circles). The directed arcs describe which places are pre- and/or postconditions for which transitions (signified by arrows). Some sources[1] state that Petri nets were invented in August 1939 by Carl Adam Petri – at the age of 13 – for the purpose of describing chemical processes.
Like industry standards such as UML activity diagrams, BPMN and EPCs, Petri nets offer a graphical notation for stepwise processes that include choice, iteration, and concurrent execution. Unlike these standards, Petri nets have an exact mathematical definition of their execution semantics, with a well-developed mathematical theory for process analysis.
A Petri net consists of places, transitions, and arcs. Arcs run from a place to a transition or vice versa, never between places or between transitions. The places from which an arc runs to a transition are called the input places of the transition; the places to which arcs run from a transition are called the output places of the transition.
Graphically, places in a Petri net may contain a discrete number of marks called tokens. Any distribution of tokens over the places will represent a configuration of the net called a marking. In an abstract sense relating to a Petri net diagram, a transition of a Petri net may fire whenever there are sufficient tokens at the start of all input arcs; when it fires, it consumes these tokens, and places tokens at the end of all output arcs. A firing is atomic, i.e., a single non-interruptible step.
Execution of Petri nets is nondeterministic: when multiple transitions are enabled at the same time, any one of them may fire. If a transition is enabled, it may fire, but it doesn't have to.
Since firing is nondeterministic, and multiple tokens may be present anywhere in the net (even in the same place), Petri nets are well suited for modeling the concurrent behavior of distributed systems.
Petri nets are state-transition systems that extend a class of nets called elementary nets.[2]
Definition 1. A net is a triple N = (P, T, F ) where:
The set P ∪ T are the net elements. The set of places define the local states of a net, however, the global state of a net can be defined by place subsets.
Definition 2. Given a net N = (P, T, F ), a configuration is a set C so that C ⊆ P.
Definition 3. An elementary net is a net of the form EN = (N, C ) where:
Definition 4. A Petri net is a net of the form PN = (N, M, W ), which extends the elementary net so that:
If a Petri net is equivalent to an elementary net, then Z can be the countable set {0,1} and those elements in P that map to 1 under M form a configuration. Similarly, if a Petri net is not an elementary net, then the multiset M can be interpreted as representing a non-singleton set of configurations. In this respect, M extends the concept of configuration for elementary nets to Petri nets.
In the diagram of a Petri net (see top figure left), places are conventionally depicted with circles, transitions with long narrow rectangles and arcs as one-way arrows that show connections of places to transitions or transitions to places. If the diagram were of an elementary net, then those places in a configuration would be conventionally depicted as circles, where each circle encompasses a single dot called a token. In the given diagram of a Petri net (see left), the place circles may encompass more than one token to show the number of times a place appears in a configuration. The configuration of tokens distributed over an entire Petri net diagram is called a marking.
In the top figure (see left), the place p1 is an input place of transition t; whereas, the place p2 is an output place to the same transition. Let PN0 (Fig. top) be a Petri net with a marking configured M0 and PN1 (Fig. bottom) be a Petri net with a marking configured M1. The configuration of PN0 enable transition t through the property that all input places have sufficient number of tokens (shown in the figures as dots) "equal to or greater" than the multiplicities on their respective arcs to t. Once and only once a transition is enabled will the transition fire. In this example, the firing of transition t generates a map that has the marking configured M1 in the image of M0 and results in Petri net PN1, seen in the bottom figure. In the diagram, the firing rule for a transition can be characterised by subtracting a number of tokens from its input places equal to the multiplicity of the respective input arcs and accumulating a new number of tokens at the output places equal to the multiplicity of the respective output arcs.
Remark 1. The precise meaning of "equal to or greater" will depend on the precise algebraic properties of addition being applied on Z in the firing rule, where subtle variations on the algebraic properties can lead to other classes of Petri nets; for example, Algebraic Petri nets (see[3]).
The following formal definition is loosely based on (Peterson 1981). Many alternative definitions exist.
A Petri net graph (called Petri net by some, but see below) is a 3-tuple , where
The flow relation is the set of arcs: . In many textbooks, arcs can only have multiplicity 1. These texts often define Petri nets using F instead of W. When using this convention, a Petri net graph is a bipartite multigraph with node partitions S and T.
The preset of a transition t is the set of its input places: ; its postset is the set of its output places: . Definitions of pre- and postsets of places are analogous.
A marking of a Petri net (graph) is a multiset of its places, i.e., a mapping . We say the marking assigns to each place a number of tokens.
A Petri net (called marked Petri net by some, see above) is a 4-tuple , where
The behavior of a Petri net is defined as a relation on its markings, as follows.
Note that markings can be added like any multiset:
The execution of a Petri net graph can be defined as the transition relation on its markings, as follows:
In words:
We are generally interested in what may happen when transitions may continually fire in arbitrary order.
We say that a marking is reachable from a marking M in one step if ; we say that it is reachable from M if , where is the reflexive transitive closure of ; that is, if it is reachable in 0 or more steps.
For a (marked) Petri net , we are interested in the firings that can be performed starting with the initial marking . Its set of reachable markings is the set
The reachability graph of N is the transition relation restricted to its reachable markings . It is the state space of the net.
A firing sequence for a Petri net with graph G and initial marking is a sequence of transitions such that . The set of firing sequences is denoted as .
As already remarked, a common variation is to disallow arc multiplicities and replace the bag of arcs W with a simple set, called the flow relation, . This doesn't limit expressive power as both can represent each other.
Another common variation, e.g. in, e.g. Desel and Juhás (2001),[4] is to allow capacities to be defined on places. This is discussed under extensions below.
The markings of a Petri net can be regarded as vectors of nonnegative integers of length .
Its transition relation can be described as a pair of by matrices:
Then their difference
can be used to describe the reachable markings in terms of matrix multiplication, as follows. For any sequence of transitions w, write for the vector that maps every transition to its number of occurrences in w. Then, we have
Note that it must be required that w is a firing sequence; allowing arbitrary sequences of transitions will generally produce a larger set.
One thing that makes Petri nets interesting is that they provide a balance between modeling power and analyzability: many things one would like to know about concurrent systems can be automatically determined for Petri nets, although some of those things are very expensive to determine in the general case. Several subclasses of Petri nets have been studied that can still model interesting classes of concurrent systems, while these problems become easier.
An overview of such decision problems, with decidability and complexity results for Petri nets and some subclasses, can be found in Esparza and Nielsen (1995).[5]
The reachability problem for Petri nets is to decide, given a Petri net N and a marking M, whether .
Clearly, this is a matter of walking the reachability graph defined above, until either we reach the requested marking or we know it can no longer be found. This is harder than it may seem at first: the reachability graph is generally infinite, and it is not easy to determine when it is safe to stop.
In fact, this problem was shown to be EXPSPACE-hard[6] years before it was shown to be decidable at all (Mayr, 1981). Papers continue to be published on how to do it efficiently.[7]
While reachability seems to a be a good tool to find erroneous states, for practical problems the constructed graph usually has far too many states to calculate. To alleviate this problem, linear temporal logic is usually used in conjunction with the tableau method to prove that such states cannot be reached. LTL uses the semi-decision technique to find if indeed a state can be reached, by finding a set of necessary conditions for the state to be reached then proving that those conditions cannot be satisfied.
Petri nets can be described as having different degrees of liveness . A Petri net is called -live iff all of its transitions are -live, where a transition is
Note that these are increasingly stringent requirements: -liveness implies -liveness, for .
These definitions are in accordance with Murata's overview,[8] which additionally uses -live as a term for dead.
A place in Petri net is called k-bounded if it does not contain more than k tokens in all reachable markings, including the initial marking; it is safe if it is 1-bounded; it is bounded if it is k-bounded for some k.
A (marked) Petri net is called k-bounded, safe, or bounded when all of its places are. A Petri net (graph) is called (structurally) bounded if it is bounded for every possible initial marking.
Note that a Petri net is bounded if and only if its reachability graph is finite.
Boundedness is decidable by looking at covering, by constructing the Karp–Miller Tree.
It can be useful to explicitly impose a bound on places in a given net. This can be used to model limited system resources.
Some definitions of Petri nets explicitly allow this as a syntactic feature.[9] Formally, Petri nets with place capacities can be defined as tuples , where is a Petri net, an assignment of capacities to (some or all) places, and the transition relation is the usual one restricted to the markings in which each place with a capacity has at most that many tokens.
For example, if in the net N, both places are assigned capacity 2, we obtain a Petri net with place capacities, say N2; its reachability graph is displayed on the right.
Alternatively, places can be made bounded by extending the net. To be exact, a place can be made k-bounded by adding a "counter-place" with flow opposite to that of the place, and adding tokens to make the total in both places k.
As well as discrete events, there are Petri nets for continuous and hybrid discrete-continuous processes and useful in discrete, continuous and hybrid control theory.[10] and related to discrete, continuous and hybrid automata.
There are many extensions to Petri nets. Some of them are completely backwards-compatible (e.g. coloured Petri nets) with the original Petri net, some add properties that cannot be modelled in the original Petri net (e.g. timed Petri nets). If they can be modelled in the original Petri net, they are not real extensions, instead, they are convenient ways of showing the same thing, and can be transformed with mathematical formulas back to the original Petri net, without losing any meaning. Extensions that cannot be transformed are sometimes very powerful, but usually lack the full range of mathematical tools available to analyse normal Petri nets.
The term high-level Petri net is used for many Petri net formalisms that extend the basic P/T net formalism; this includes coloured Petri nets, hierarchical Petri nets, and all other extensions sketched in this section. The term is also used specifically for the type of coloured nets supported by CPN Tools.
A short list of possible extensions:
There are many more extensions to Petri nets, however, it is important to keep in mind, that as the complexity of the net increases in terms of extended properties, the harder it is to use standard tools to evaluate certain properties of the net. For this reason, it is a good idea to use the most simple net type possible for a given modelling task.
Instead of extending the Petri net formalism, we can also look at restricting it, and look at particular types of Petri nets, obtained by restricting the syntax in a particular way. The following types are commonly used and studied:
Other ways of modelling concurrent computation have been proposed, including process algebra, the actor model, and trace theory.[17] Different models provide tradeoffs of concepts such as compositionality, modularity, and locality.
An approach to relating some of these models of concurrency is proposed in the chapter by Winskel and Nielsen.[18]
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