Petri net


A Petri net, also known as a place/transition net, is one of several mathematical modeling languages for the description of distributed systems. It is a class of discrete event dynamic system. A Petri net is a directed bipartite graph, in which the nodes represent transitions and places. The directed arcs describe which places are pre- and/or postconditions for which transitions. Some sources 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, Business Process Model and Notation and event-driven process chains, 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.

Petri net basics

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 if it is enabled, i.e. there are sufficient tokens in all of its input places; when the transition fires, it consumes the required input tokens, and creates tokens in its output places. A firing is atomic, i.e. a single non-interruptible step.
Unless an execution policy is defined, the execution of Petri nets is nondeterministic: when multiple transitions are enabled at the same time, they will fire in any order.
Since firing is nondeterministic, and multiple tokens may be present anywhere in the net, Petri nets are well suited for modeling the concurrent behavior of distributed systems.

Formal definition and basic terminology

Petri nets are state-transition systems that extend a class of nets called elementary nets.
Definition 1. A net is a triple where:
  1. and are disjoint finite sets of places and transitions, respectively.
  2. is a set of arcs.
Definition 2. Given a net N =, a configuration is a set C so that C ⊆ P.
Definition 3. An elementary net is a net of the form EN = where:
  1. N = is a net.
  2. C is such that C ⊆ P is a configuration.
Definition 4. A Petri net is a net of the form PN =, which extends the elementary net so that:
  1. N = is a net.
  2. M : P → Z is a place multiset, where Z is a countable set. M extends the concept of configuration and is commonly described with reference to Petri net diagrams as a marking.
  3. W : F → Z is an arc multiset, so that the count for each arc is a measure of the arc multiplicity.
If a Petri net is equivalent to an elementary net, then Z can be the countable set 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, 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, 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, the place p1 is an input place of transition t; whereas, the place p2 is an output place to the same transition. Let PN0 be a Petri net with a marking configured M0, and PN1 be a Petri net with a marking configured M1. The configuration of PN0 enables transition t through the property that all input places have sufficient number of tokens "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.
The following formal definition is loosely based on. Many alternative definitions exist.

Syntax

A Petri net graph 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 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 is a 4-tuple, where
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 in one step if ; we say that it is reachable from if, where is the reflexive transitive closure of ; that is, if it is reachable in 0 or more steps.
For a 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 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 and initial marking is a sequence of transitions such that. The set of firing sequences is denoted as.

Variations on the definition

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, Desel and Juhás, is to allow capacities to be defined on places. This is discussed under extensions below.

Formulation in terms of vectors and matrices

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, write for the vector that maps every transition to its number of occurrences in. Then, we have
Note that it must be required that is a firing sequence; allowing arbitrary sequences of transitions will generally produce a larger set.

Mathematical properties of Petri nets

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.

Reachability

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 years before it was shown to be decidable at all. Papers continue to be published on how to do it efficiently. In 2018, Czerwinski et al. improved the lower bound and showed that the problem is not ELEMENTARY.
While reachability seems to 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. Linear temporal logic 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.

Liveness

Petri nets can be described as having different degrees of liveness. A Petri net is called -live if and only if 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, which additionally uses -live as a term for dead.

Boundedness

A place in a 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 said to be safe if it is 1-bounded; it is bounded if it is k-bounded for some k.
A Petri net is called k-bounded, safe, or bounded when all of its places are.
A Petri net is called 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.
Formally, Petri nets with place capacities can be defined as tuples, where is a Petri net, an assignment of capacities to 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.

Discrete, continuous, and hybrid Petri nets

As well as for discrete events, there are Petri nets for continuous and hybrid discrete-continuous processes that are useful in discrete, continuous and hybrid control theory, and related to discrete, continuous and hybrid automata.

Extensions

There are many extensions to Petri nets. Some of them are completely backwards-compatible with the original Petri net, some add properties that cannot be modelled in the original Petri net formalism. Although backwards-compatible models do not extend the computational power of Petri nets, they may have more succinct representations and may be more convenient for modeling. Extensions that cannot be transformed into Petri nets are sometimes very powerful, but usually lack the full range of mathematical tools available to analyse ordinary 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 such as Nets within 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.

Restrictions

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. Ordinary Petri nets are the nets where all arc weights are 1. Restricting further, the following types of ordinary Petri nets are commonly used and studied:
  1. In a state machine, every transition has one incoming arc, and one outgoing arc, and all markings have exactly one token. As a consequence, there can not be concurrency, but there can be conflict. Mathematically:
  2. In a marked graph, every place has one incoming arc, and one outgoing arc. This means, that there can not be conflict, but there can be concurrency. Mathematically:
  3. In a free choice net, every arc from a place to a transition is either the only arc from that place or the only arc to that transition, i.e. there can be both concurrency and conflict, but not at the same time. Mathematically:
  4. Extended free choice – a Petri net that can be transformed into an FC.
  5. In an asymmetric choice net, concurrency and conflict may occur, but not symmetrically. Mathematically:

    Work flow nets

s are a subclass of Petri nets intending to model the workflow of process activities.
The WF-net transitions are assigned to tasks or activities, and places are assigned to the pre/post conditions.
The WF-nets have additional structural and operational requirements, mainly the addition of a single input place with no previous transitions, and output place with no following transitions. Accordingly, start and termination markings can be defined that represent the process status.
WF-nets have the soundness property, indicating that a process with a start marking of k tokens in its source place, can reach the termination state marking with k tokens in its sink place. Additionally, all the transitions in the process could fire.
A general sound WF-net is defined as being k-sound for every k > 0.
A directed path in the Petri net is defined as the sequence of nodes linked by the directed arcs. An elementary path includes every node in the sequence only once.
A well-handled Petri net is a net in which there are no fully distinct elementary paths between a place and a transition, i.e., if there are two paths between the pair of nodes then these paths share a node.
An acyclic well-handled WF-net is sound.
Extended WF-net is a Petri net that is composed of a WF-net with additional transition t. The sink place is connected as the input place of transition t and the source place as its output place. Firing of the transition causes iteration of the process.
WRI WF-net, is an extended acyclic well-handled WF-net.
WRI-WF-net can be built as composition of nets, i.e., replacing a transition within a WRI-WF-net with a subnet which is a WRI-WF-net. The result is also WRI-WF-net. WRI-WF-nets are G-sound, therefore by using only WRI-WF-net building blocks, one can get WF-nets that are G-sound by construction.
The Design structure matrix can model process relations, and be utilized for process planning. The DSM-nets are realization of DSM-based plans into workflow processes by Petri nets, and are equivalent to WRI-WF-nets. The DSM-net construction process ensures the soundness property of the resulting net.

Other models of concurrency

Other ways of modelling concurrent computation have been proposed, including process algebra, the actor model, and trace theory. 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.

Application areas