4 Petri Nets
4.1 Basic Concepts
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Petri nets can be found in many different application fields like computer communications, business or production processes, operating systems, traffic light crossings, or systems biology. In this connection, a place models a state, for example, of an object or a condition while a transition models the change of several states, for example, activities or events.
Figure 4.1: Graphical representation of a place (left) and a transition (right)
This section serves as a basis for the implementation of the ordinary Petri net concept and its abbreviations and extensions which are necessary to model biological processes.
Abbreviations simplify the Petri net representation but can always be transformed to an ordinary Petri net. On the other hand, extensions cannot be represented by ordinary Petri nets and allow the usage of the Petri net formalism in a wider range of applications.
Definition 4.1 (net)
A net is the tuple , , , of a finite set of places , , … , , a finite set of transitions , , … , , where ∩ ∅, a set ⊆ of arcs from places to transitions, and a set ⊆ of arcs from transitions to places.
Definition 4.2 (Petri net)
The tuple , , , , is a Petri net if , , , is a net and : ∪ → is an arc weight function which assigns each arc a non-negative integer, whereby → denotes the arc from a place ∈ to a transition ∈ and → is the corresponding weight and
→ denotes the arc from to with the weight → .
Every place in a Petri net can contain an integer number of tokens. These tokens are graphically represented by little black dots or numbers in the places (see Figure 4.2). A concrete determination of the token number of a place is called marking of the place and a concrete determination of the token numbers of all places in a Petri net is called marking of the Petri net. If a Petri net contains only marked places, then it is called initially marked Petri net or following just Petri net.
Definition 4.3 (marking)
The tuple , , , , is a Petri net. A map : → is called marking of the Petri net and assigns each ∈ a concrete token number , called marking of the place . The map : → is the initial marking of the Petri net and the tuple , , , , , is called (initially marked) Petri net.
Figure 4.2: Tokens can be graphically represented by dots (left) or by numbers (right) in the places
Definition 4.4 (input, output)
The set of inputs of a Petri net element ∈ ∪ are defined as ≔ ∈
∪ | → ∈ ∪ and the set of outputs are ≔ ∈ ∪ | → ∈
∪ . The set of all input places of a transition ∈ is noted with ⊆ and the set of all output places is noted with ⊆ . Similarly, the set ⊆ contains all input transitions of a place ∈ and the set ⊆ contains all output transitions (see Figure 4.3). The number of inputs is noted by and the number of outputs by .
Figure 4.3: Input and output places, and input and output transitions
A transition is said to be active with regard to a concrete marking if all input places have at least as many tokens as their arc weights.
p1
t1
t2
. .. t1
t2
. ..
input
transitions output
transitions t1
p1
p2
. ..
p1
p2
. ..
input
places output
places
pnin pnout tnin tnout
Definition 4.5 (activation Petri net)
The tuple , , , , , is a Petri net. A transition ∈ is active with regard to a concrete marking if and only if
∀ ∈ ∶ → .
The set of all active input transitions of a place ∈ is denoted by and the set of all active output transitions is denoted by .
Active transitions can be enabled by connected places; if a transition is enabled by all its input places, it is firable and can fire tokens. Possibly, a place has not enough tokens to enable all output transitions. This is the case when its token number is less than the sum of output arc weights and a general conflict arises that has to be resolved.
Different possible approaches are available; two of them, priority and probabilistic choice are used within this work which accordingly have been defined and adapted based on the suggestions in (David and Alla 2005).
Definition 4.6 (general conflict)
The tuple , , , , , is a Petri net. A place ∈ has a general conflict with regard to a concrete marking if
→
∈
,
whereby the set contains two or more active output transitions.
Example 4.1
The places and of the Petri net in Figure 4.4 have a general conflict. can either enable or and can either enable 2 or 3. Transition 4 is not an option because it is not active due to missing tokens in .
Figure 4.4: General conflicts of discrete places (Example 4.1)
P1 P2 P3
T1 T2 T3 T4
1 1
Definition 4.7 (Petri net with priorities)
The tuple , , , , , , is a Petri net with priorities if , , , , , is a Petri net and : → 1,2, … , max is a priority function which assigns every arc from a place
∈ to a transition ∈ an enabling priority → under the condition that each priority from 1 to is only used once for each place
→ → ∀ , ∈
and that 1 is the best priority and is the worst priority.
Definition 4.8 (Petri net with probabilities)
The tuple , , , , , 0, is a Petri net with probabilities if , , , , , 0 is a Petri net and : → 0,1 is a probability function which assigns every arc from a place ∈ to a transition ∈ an enabling probability → under the condition that the sum of probabilities is equal to one
→ 1 ∀ ∈
∈
.
The deterministic conflict resolution with priorities requires that all output transitions of a place are provided with a priority from 1 to . The transition with the priority 1 is proven at first. If it is active and the arc weight does not exceed the marking of the place, it is enabled by the place. Then the second prioritized transition is checked; if it is active and the arc weight sum is not greater than the marking, it is also enabled. The next transition proven is the one provided with the third priority and so on.
Definition 4.9 (enabling process Petri net with priorities)
The tuple , , , , , , is a Petri net with priorities. A place ∈ enables with regard to a concrete marking a subset, denoted by , of the transitions in . If has no general conflict according to Definition 4.6, it enables all transitions
in , i.e.
≡ .
Otherwise, enables as many transitions of the set as possible according to the priorities → so that the condition
→
∈
is still fulfilled.
The conflict resolution by probabilities proceeds non-deterministically. A probability is assigned to every output transition of a place. The enabled transitions are selected by a random process with due regard to the probabilities; thereby, it always has to be ensured that the current arc weight sum does not exceed the token number of the place.
Definition 4.10 (enabling process Petri net with probabilities)
The tuple , , , , , , is a Petri net with probabilities. A place ∈ enables with regard to a concrete marking a subset, denoted by , of the transitions in
. If has no general conflict according to Definition 4.6, it enables all transitions
in , i.e.
≡ .
Otherwise, enables randomly as many transitions of the set as possible according to the probabilities → so that the condition
→
∈
is still fulfilled.
Remark 4.1
All output transitions of a place are provided with a probability and the sum is equal to one. If not all output transitions are active, the probabilities have to be scaled by the following expression
→ →
∑ ∈ → .
It is also possible to mix the two concepts; then one part of the places enables their output transitions deterministically according to the assigned priorities and the other part randomly according to the assigned probabilities. Petri nets with priorities and probabilities are called simplified resolved Petri nets hereafter.
Definition 4.11 (resolved Petri net)
The tuple , , , , , , , is a Petri net with priorities and probabilities, simplified called resolved Petri net hereafter, if , , , , , is a Petri net, : → , is the resolution function that assigns every place either the resolution type priority or
probability, and : → 1,2, … , max : , 0,1 : is an
enabling function which assigns every arc from a place ∈ to a transition ∈
either an enabling priority or a probability → according to the resolution type of the place and under the condition that each priority from 1 to is only used once for each place with
→ → ∀ , ∈
and that 1 is the best priority and is the worst priority and that the sum of probabilities is equal to one for each place with
→ 1
∈
.
An active transition is firable if it is enabled by all input places. It fires by removing as much tokens as the arc weights from all input places and by adding as many tokens as the arc weights to all output places.
Definition 4.12 (firability Petri net)
The tuple , , , , , , , 0 is a resolved Petri net. An active transition ∈ is firable if and only if
∀ ∈ : ∈ .
When one or more transitions of a Petri net fire, the markings of the involved places have to be recalculated; thereby, the output firing sum is subtracted from the token number and the input firing sum is added to it.
Definition 4.13 (firing process Petri net)
The tuple , , , , , , , 0 is a resolved Petri net. A firable transition ∈ fires with regard to a concrete marking by removing as many tokens as the arc weights from all input places
→ ∀ ∈ and by adding as many tokens as the arc weights to all output places
→ ∀ ∈ .
The new marking of the place ∈ is recalculated by the following algebraic equation
→
∈
→
∈
,
whereby ⊆ is the set of all firing output transitions and ⊆ is the set of all firing input transitions. The sums ∑ ∈ → and ∑ ∈ → are called output firing sum and input firing sum, respectively.
Example 4.2
Figure 4.5 displays an example of a Petri net with seven places 1, 2, 3, 4, 5, 6, 7 and four transitions 1, 2, 3, 4 . The numbers at the arcs are the weights and the black dots in the places are the tokens. Place 1 has two tokens 1 2 , 2 has five tokens 2 5 , 4 has one token 4 1 , and 7 has two tokens 7 2 . All others have no tokens 3 5 6 0 and, hence, the initial marking of this Petri net is 2,5,0,1,0,0,2 .
Figure 4.5: Petri net (Example 4.2)
Activation:
- 1: The input places are 1 and 2. 1 must have at least two tokens (weight of the arc from 1 to 1 ( 1 → 1 2)) which it actually has and 2 must have at least one token
2 → 1 1 and it has five; hence, transition is active.
- 2: The only input place is 2. 2 must have at least one token ( 2 → 2 1) which is fulfilled; hence, transition is active.
- 3: The input places are 4 and 5. 4 must have at least one token ( 4 → 3 1) which it actually has but 5 must have at least two tokens and it has none; hence, transition is not active.
- 4: The input places are 3 and 6. Both places have no tokens; hence, transition is not active.
Enabling:
- 1 enables its only output transition 1.
P1
T1
P2
P3
P4
P5
P6
P7
T2
T4
T3
2 3
2
2 2
1
1 2
3 2
1 1
- 2 has enough tokens to enable both output transitions 1 and 2 2 5 2 → 1 2 → 2 2 . There is no general conflict.
Firing: T1 and T2 are firable and fire by
- Removing 2 1 → 1 tokens from 1
- Removing 2 2 → 1 2 → 2 tokens from 2 - Adding 3 1 → 3 tokens to 3
- Adding 2 1 → 4 tokens to 4 - Adding 2 2 → 5 tokens to 5.
The new marking of the Petri net is 0,3,3,3,2,0,2 and it is displayed in Figure 4.6. Now the transitions 2 and 3 are active and can be enabled by all their input places. Hence, 2 and 3 are firable and the firing of 2 and 3 leads to the new marking 0,2,3,2,2,1,2 .
Figure 4.6: Petri net of Figure 4.5 after firing and (Example 4.2)
Example 4.3
Both transitions of the Petri net example in Figure 4.7 are active because of
1 2 1 → 1 1
and
1 2 1 → 2 2.
But 1 can only enable one of them due to its token number 1 2 ≱ 1 → 1 1 → 2 3 . Hence, 1 has a general conflict. If the enabling is performed by priorities and 2 has priority 1 1 → 2 1 and 1 has priority 2 1 → 1 2 , 1 enables 2. 2 is firable and fires by removing two tokens from 1 and adding one to 3. If the enabling is performed at random, one of the transitions is enabled according to their assigned probabilities, e.g. 1 has the probability 1 → 1 0.7 and 2 has the probability
1 → 2 0.3.
P1
T1
P2
P3
P4
P5
P6
P7
T2
T4
T3
2 3
2
2 2
1
1 2
3 2
1 1
Figure 4.7: Petri net with a general conflict (Example 4.3)
Example 4.4
All transitions of the Petri net in Figure 4.8 are active but 1 and 2 can only enable one of their output transitions due to their token numbers. Four cases can occur if probabilistic enabling is applied:
- Case 1: 1 enables 1 and 2 enables 3
1 and 3 are firable and fire by removing one token from 1 and 2 and adding one to 3 and 5. The new marking of the Petri net is 0,0,1,0,1 .
- Case 2: 1 enables 1 and 2 enables 2
Only 1 is firable and fires by removing one token from 1 and adding one to 3. The new marking of the Petri net is 0,1,1,0,0 .
- Case 3: 1 enables 2 and 2 enables 3
Only 3 is firable and fires by removing one token from 2 and adding one to 5. The new marking of the Petri net is 1,0,0,0,1 .
- Case 4: 1 enables 2 and 2 enables 2
2 is firable and fires by removing one token from 1 and 2 and adding one to 4. The new marking of the Petri net is 0,0,0,1,0 .
Figure 4.8: Petri net with general conflicts (Example 4.4)