Fig. 2

Event net examples (a and b) and intensity net examples (c, d, e and f). a Let us assume that the stoichiometry of a given reaction R is not completely known and can be expressed as R: nA + 2nB → C where n is known to be between 10 and 12. That is, the production of one molecule of type C requires a quantity of molecules of type A that is in [10,12] and twice as many molecules of type B. Such a reaction can be modeled by the event net in A together with the set of inequalities 10r ≤ a ≤ 12r; b = 2a; c = r associated with event handler v₁. R can be expressed as R:nA + 2nB → C where n is known to be in [10,12]. The inequalities in v₁ are interpreted as follows: If one reaction occurs, then r = 1 and one action in R is used to change the marking in places. In order to satisfy the inequalities, a, which is the number of tokens consumed from A, is between 10 and 12 (the actual value of a is chosen in a non-deterministic way for each occurrence of R), b, which is the number of tokens consumed from B, is b = 2a, and c, which is the number of tokens produced in C, is c = r = 1. Assume that the initial marking of places is m₀[A] = 20, m₀[B] = 22, m₀[C] = 0. The occurrence of R drives the system non-deterministically either to m[A] = 10, m[B] = 2, m[C] = 1, or to m[A] = 9, m[B] = 0, m[C] = 1. Notice that, for this initial marking, a cannot take a value higher than 11 as it would lead to a negative m[B]. The consumption and production of tokens is instantaneous and coincident. b Net modeling a reaction R with two alternative set of reactants and products, the set of equalities associated with the event handlers are v₁:a = p = b = r, v₂:c = p = d = r; either A and P are reactants and B is product, or C and P are reactants and D is product. In a chemical context, this net might model a reaction R that either takes A and P as reactants and B as product, or C and P as reactants and D as product, i.e., R is either A + P → B or C + P → D. In the first case, the occurrence of the reaction, i.e., the action in R, is implemented by the edge connected to v₁, and in the second case by the edge connected to v₂. The selection of the event handler that uses the action is done non-deterministically for every occurrence of the reaction. This net can be interpreted in different ways: (a) The transition could model an event that can be observed, and whose occurrence can produce different marking changes which are modeled by the event handlers. For instance, the transition can model a biochemical event such as phosphorylation, but it is not possible for the observer to determine whether molecule A or C has been phosphorylated. (b) The transition could model an input action whose effect on the system is not fully controllable as any of the connected event handlers can make use of the reaction. c Net establishing a synchronization between the tokens in A and the tokens in B by means of the equation s₁:2a = b = 2r associated with s₁. More precisely, when a token in A synchronizes with two tokens of B, the intensity in transition R is increased by one unit. d Let s₁:a = r₁ = r₂, then a token in A produces simultaneously an intensity unit both in R1 and R2, i.e., it can be informally said that the intensities of R1 and R2 are synchronized by the tokens in A. e Let s₁:a = b = r and s₂:c = d = r, then a token in A together with a token in B produce an increase of one unit in the intensity of R. Similarly, a token in C together with a token in D produce a decrease of one unit in the intensity of R. Thus, the tokens in A and B can be seen as positive modulators and the tokens in C and D as negative modulators. f Let s₁:10a ≤ r ≤ 12a and s₂:a = b = r. The net models a choice in place A, i.e., a token in A can be used either to produce an intensity within the interval [10,12] in R1 or, together with a token in B, an intensity in R2 of one unit. The choice of the intensity handler that uses the tokens in A is non-deterministic; in contrast to the actions used by event handlers, it can change over time. In other words, a given token in A can modulate the intensity in R1 during a given time period and then synchronize with a token in B to modulate intensity in R2