Extensible Structural Analysis of Petri Net Product Lines

Abstract

Petri nets are a popular formalism to represent concurrent systems. However, their standard form does not offer variability support to model and effectively analyse large sets of variants of a given system. For this purpose, we propose a notion of product line of Petri nets to represent a set of similar concurrent systems. The formalization enriches Petri nets with a feature model characterizing the variability of the systems. Moreover, places, transitions and arcs can define presence conditions that determine the subset of system variants they belong to.

To enable an efficient analysis of the set of all net variants, we have lifted several structural analysis methods for Petri nets, to the product line level. Currently, we support the lifted checking of the marked graph, state-machine, and (extended) free-choice properties, which avoids their analysis on each particular net of the product line in isolation.

We demonstrate the feasibility of our proposal using examples in the domain of flexible assembly lines, and introduce an extensible tool infrastructure. The tool is based on Eclipse and FeatureIDE, and permits adding new analysis methods externally. Moreover, we present an evaluation that shows the efficiency gains of our method with respect to an enumerative approach that analyses the properties on every net within the product line separately.

Appendix

This appendix lifts the analysis of the state-machine property. A state-machine (SM) is a subclass of Petri net where each transition t has exactly one input and one output place, while each place may have multiple input and output transitions. SMs allow representing decisions, but not the synchronization of concurrent activities.

Definition 13

(State-machine, from [21]). A Petri net $PN=(P, T, A)$ is a state-machine, written $PN \,\models \, SM$, if $\forall t \in T:~|\,\!^\bullet {t}| = |{t}^\bullet | = 1$.

Next, we lift the definition of the SM property to the product line level. A PNPL is a strong (weak) SM if all (some) derivable nets are SMs.

Definition 14

(Strong and weak SM product line). A Petri net product line PNL is a strong state-machine iif $\forall PN_{\rho } \in Prod(PNL):~PN_{\rho } \,\models \, SM$. PNL is a weak state-machine iif $\exists PN_{\rho } \in Prod(PNL):~PN_{\rho } \,\models \, SM$.

Similar to the case of MGs, to ensure that all derivable nets are SMs, we check that the size of the lifted pre-set $\,\!^\circ {t} = \{ (p_0, \varPhi _{(p_0, t)}), ..., (p_n, \varPhi _{(p_n, t)}) \}$ and the lifted post-set ${t}^\circ = \{ (p_0, \varPhi _{(t, p_0)}), ..., (p_n, \varPhi _{(t, p_n)}) \}$ of a transition t is one in every configuration. The size of the lifted pre-set $\,\!^\circ {t}$ of a transition t is one if the following formula is true. The disjunction starts with false to consider the case when $\,\!^\circ {t}$ is empty. The formula $\varPhi _{{t}^\circ }$ to check that the size of the lifted post-set of a transition t is one is defined similarly.

$$\begin{aligned} \begin{aligned} \varPhi _{\,\!^\circ {t}} \triangleq&~false \\&~\vee ~(\varPhi _{(p_0, t)} \wedge \lnot \varPhi _{(p_1, t)} \wedge ... \wedge \lnot \varPhi _{(p_n, t)})\\&~\vee ~(\lnot \varPhi _{(p_0, t)} \wedge \varPhi _{(p_1, t)} \wedge ... \wedge \lnot \varPhi _{(p_n, t)})\\&~\vee ~... ~(\lnot \varPhi _{(p_0, t)} \wedge \lnot \varPhi _{(p_1, t)} \wedge ... \wedge \varPhi _{(p_n, t)}) \end{aligned} \end{aligned}$$

(7)

Hence, a PNPL includes some Petri net that is a SM if there is a feature configuration $\rho$ such that for every transition t in the PNPL:

• t is not in $PN_{\rho }$, therefore $\varPhi _t$ is false; or

• t is in $PN_{\rho }$, and therefore $\varPhi _{\,\!^\circ {t}}$ and $\varPhi _{{t}^\circ }$ need to be true.

Equation 8 shows the formula that captures the two previous conditions.

$$\begin{aligned} \varPhi _{SM} = \wedge _{t \in T}[\lnot \varPhi _t \vee (\varPhi _t \wedge \varPhi _{\,\!^\circ {t}} \wedge \varPhi _{{t}^\circ })] \end{aligned}$$

(8)

If there is a feature configuration such that $SAT(\varPsi$ $\wedge \varPhi _{SM})$ holds, then there is a derivable Petri net that is a SM, and the PNPL is a weak SM. On the contrary, the feature configurations that produce nets which are not SMs are those making the formula $\lnot \varPhi _{SM}$ true. Hence, the PNPL is a strong SM if $SAT (\varPsi \wedge \lnot \varPhi _{SM})$ does not hold.