Extended Petri Nets

At first, the places enable as many output transitions as their minimum capacities allow. 1 has no general conflict and enables 1 and 2 has a general output conflict and can either enable 2 or 3.

- Case 1: 2 enables 2

3 has a general input conflict due to its maximum capacity and can either enable 1 or 2. 3 is not an option because it is not enabled by its input place 2. If 3 enables 1, 1 is firable and fires by removing one token from 1 and adding one to 3 and on the other hand, if it enables 2, 2 is firable and fires by removing one token from 2 and adding one to 3.

- Case 2: 2 enables 3

3 has a general input conflict due to its maximum capacity and can either enable 1 or 3. 2 is not an option because it is not enabled by its input place 2. If 3 enables 1, 1 is firable and fires by removing one token from 1 and adding one to 3 and on the other hand, if it enables 3, 3 is firable and fires by removing one token from 2 and adding one to 3.

Figure 4.9: Capacitive Petri net with a general output conflict (Example 4.5)

The test arc is represented by a dashed line, the inhibitor arc has a small circle at its end, and the read arc has black square at its end (see Figure 4.10 and Figure 4.11).

If places are connected with test, inhibitor, or read arcs to transitions, their markings are not changed during the firing processes. In the case of test and inhibitor arcs the markings are only read to influence the activation process while read arcs do not influence the activation process nor the firing process. Hence, read arcs have no weights; they only indicate the usage of the token number in the connected transition, for example, for firing conditions or arc weight functions (see Section 4.3).

The same place can be connected with the same transition by a test and normal arc as well as by an inhibitor and normal arc. These arcs are called double arcs.

Definition 4.23 (extended Petri net, double arc)

The tuple , , , , , , , , , , is an extended Petri net if , , , , , , , is a resolved Petri net, ⊆ is a set of test arcs, ⊆ is a set of inhibitor arcs,

⊆ is a set of read arcs, and the arc weight function is modified such that : ∪ ∪ ∪ → , whereby → is the weight of the test arc → and → is the weight of the inhibitor arc → . If → ∈ and →

∈ or → ∈ and → ∈ then the arc is called double arc.

A transition in an extended Petri net is active if

 all input places connected by normal arcs have at least as many tokens as the arc weights,

 all input places connected by test arcs have more tokens than the arc weights, and

 all input places connected by inhibitor arcs have fewer tokens than the arc weights.

Places connected by read arcs do not influence the activation of a transition. The places connected by test, inhibitor, and read arcs enable all active output transitions because tokens are not changed during the firing process.

Definition 4.24 (activation extended Petri net)

The tuple , , , , , , , , , , ₀ is an extended Petri net. A transition ∈ is active with regard to a concrete marking if and only if

∀ ∈ :

→ → ∈

→ → ∈

→ → ∈ .

The enabling process by priorities or probabilities, the firability definition, and the firing process are not affected by these extensions and have been adopted from the basic concepts (Definition 4.9, Definition 4.10, Definition 4.12, and Definition 4.13).

Example 4.6

The Petri nets at the top in Figure 4.10 contain test arcs and the Petri nets at the bottom inhibitor arcs. Transition 1 is active with regard to a concrete marking because the token number of 2 is above the weight of the test arc 2 3 2 → 1 2 . However, 2 is not active because the marking of 5 is less than the arc weight 5 1 ≯

5 → 2 2 . 3 is also not active because the token number of 8 is greater than the weight of the inhibitor arc 8 3 ≮ 8 → 3 2 . However, 4 is active because the marking of 11 is less than the arc weight 11 1 11 → 4 2 .

Figure 4.10: Extended Petri nets with test arcs (top) and inhibitor arcs (bottom) (Example 4.6). and are active and and are not active.

Example 4.7

Figure 4.11 shows three different biological reactions modeled by extended Petri nets. The first reaction is inhibited by the inhibitor which is modeled by an inhibitor arc from place to transition . This reaction can only proceed when is less than a specific bound which is represented by the arc weight → .

The second reaction activated by the activator which is modeled by a test arc from place to transition . This reaction can only proceed when is greater than a specific bound represented by the arc weight → .

The third reaction is catalyzed by the enzyme . This enzyme is not consumed in the reaction but is needed because it influences the amount of substrate molecules which can be converted to product molecules. This is modeled by a read arc to indicate that the token

P1

P2

P3 1

2

1

P4

P5

P6 1

2

1

P7

P8

P9 1

2

T3 1

T1 T2

P10

P11

P12 1

2

T4 1

amount of place is needed for the arc weights of the transition (these are functional arc weights see also Section 4.3, Example 4.8, and Example 4.9). This arc is only for visualization and does not influence the firing process of the transition .

Figure 4.11: Modeling of different biological reactions with extended Petri nets; top: the reaction is inhibited by the inhibitor , it can only proceed when is less than a specific bound, middle: the reaction is activated by the activator , it can only proceed, when is greater than a specific bound, bottom: the reaction is catalyzed by the enzyme , the information about the amount of is needed to determine how many substrates molecules can be converted to product molecules (Example 4.7).