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Interorganizational Workflow nets (IWF-nets) can well model many concurrent systems such as web service composition, in which multiple processes interact via sending/receiving messages. Compatibility of IWF-nets is a crucial criterion for the correctness of these systems. It guarantees that a system has no deadlock, livelock, or dead tasks. In our previous work we proved that the compatibility problem is PSPACE-complete for safe IWF-nets. This paper defines a subclass of IWF-nets that can model many cases about interactions. Necessary and sufficient condition is presented to decide their compatibility, and it depends on the net structures only. Finally, an algorithm is developed based on the condition.

Petri nets are widely used to model concurrent/distributed systems due to both the intuitive descriptions for these systems and the diversified analysis methods. Researchers usually define different Petri net classes for different systems with different features.

For example, in flexible manufacturing systems [^{3}PRs [

Another important application of Petri nets is to model and analyze such concurrent systems as web services, in which multiple parallel processes interact/collaborate via sending/receiving messages. Interorganizational workflow nets (IWF-nets for short) [

Compatibility [

van der Aalst et al. have proven that the soundness problem is decidable for general WF-nets [

Additionally, many design patterns were proposed in order to standardize the system design and reduce the occurrence of some bad things (e.g., deadlocks) [

Therefore, how to decide compatibility for IWF-nets is interesting and important.

This paper defines a subclass of IWF-nets called

The remainder of this paper is organized as follows. Section

For readability, in this section Petri nets and WF-nets are recalled that are from [

First, let

A net is a 3-tuple

A net may be seen as a directed bipartite graph. Generally, a transition is represented by a rectangle and a place by a circle in a net graph. A

A transition

A

Notice that a marking may be viewed as a

If for all

A marking

A net

A Petri net

A transition

A transition

A net

A net

A net

In Definition

Let

for all

for all

for all

This definition was given in the early work of van der Aalst [

A subclass of nets called

A net

for all

Figures

(a) A compatible SIWF-net; (b) an incompatible SIWF-net; and ((c)-(d)) two sound acyclic FCWF-nets.

From the fourth item of Definition

Notice that Definition

Let

for all

for all

for all

For instance, Figure

For convenience, the

If a WF-net is also a free-choice net, then it is called

This section defines a subclass of IWF-nets named

for all

for all

In fact, Figures

Obviously, three constraints simplify an IWF-net.

First, it is acyclic. As we all know, if an IWF-net has a siphon that does not contain any source place, then the IWF-net is unsound (because all transitions associated with the siphon are dead at the initial marking) and this siphon must contain a circuit. Therefore, an SIWF-net is required to be acyclic, which guarantees that each siphon includes at least one source place.

Second, each basic WF-net is sound and free-choice. For an IWF-net combined by multiple basic WF-nets via a group of channel places, what this paper pays more attention to is the interaction among these basic WF-nets. Therefore, we suppose that these basic WF-nets are sound. In addition, FCWF-nets can not only model many basic structures of workflow, such as AND-split, AND-join, OR-split, and OR-join, but also own a nice property (i.e., their soundness is decidable in polynomial time [

Finally, Definition

It is worthy to note that this paper does not require an SIWF-net to observe the patterns proposed in [

Although SIWF-nets seem simple, they can model many interaction cases except for iterative structure. Indeed, iterative structure is sometimes frustrating. Some scholars consider limited iterations so that they can be unfolded into iteration-free structures [

This section gives a net-structure-based condition to decide compatibility for SIWF-nets. First, some concepts related to the net structures are defined.

Let

for all

Figure

((a)-(b)) All

Notice that because of for all

Let

for all

for all

Figure

All caps related to the

Obviously, each

A sound acyclic FCWF-net is covered by

Let

For each cap an enabled transition sequence

In fact, an acyclic FCWF-net is also sound if it is covered by

An acyclic FCWF-net is sound if and only if

it is covered by

for each cap there exists a

Let

for all

for all

The SIWF-net in Figure

((a)-(b))

A

If a transition is added to a

Let

for all

for all

Figure

(a) A cap of the SIWF-net in Figure

Notice that each

Let

For each enabled transition sequence

For each cap

(1) Because

(2) Because the cap

An SIWF-net is covered by

An SIWF-net

it is covered by

for each cap there exists a

(

By the second conclusion in Lemma

(

Case 1 does not hold: because

Case 2 does not hold either: let enabled transition sequence

Case 3 holds neither: let

Figure

Figure

(a) An incompatible SIWF-net; ((b)-(c)) two

A cap of an SIWF-net is maximal if there are no other caps properly containing it.

Figure

Notice that each

An SIWF-net is compatible if and only if

it is covered by

each maximal cap is a

(

(

In fact, the decision conditions in Theorem

If each maximal cap of an SIWF-net is a

(by contradiction) Assume that the SIWF-net is not covered by

The above cases indicate that if

An SIWF-net is compatible if and only if each maximal cap is a T-component.

The necessity is derived directly by Corollary

An SIWF-net is compatible if and only if for each cap there is a T-component containing it.

The SIWF-nets in Figures

Here a recursive algorithm is developed to solve the compability problem for an SIWF-net based on the decision conditions in Theorem

IsT-component(

};

};

};

Please notice that

Also please notice that if two transitions

Therefore, when this procedure is called in a main procedure,

This paper gives a necessary and sufficient condition to decide compatibility for a subclass of IWF-nets, which advances the state-of-the-art in the area of deciding the compatibility problem based on net structures. Future work may focus on some bigger subclasses of IWF-nets in which each basic WF-net may permit circuits.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This paper is supported in part by the Alexander von Humboldt Foundation and in part by the National Nature Science Foundation of China (Grant no. 61202016).

^{3}PR: complexity and decision