# History-dependent Petri nets

## Summary (3 min read)

### 1 Introduction

- Numerous state-of-the-art enterprise information systems contain a workflow engine, which keeps track of all events as a part of its basic functionality.
- To ensure safety, the authors require that b can fire only if the right traffic light is red, i.e., transitions d and e have fired the same number of times.
- Global history is in fact a special case of token history for transparent systems where all components are aware of the actions of other components.
- By introducing history-dependent guards, the authors increase the expressive power.
- Since, the authors are interested not only in modeling but also in verification, they identify a number of important classes of global history nets (e.g. nets with LTL guards) that can be transformed to bisimilar classical Petri nets and provide corresponding transformations.

### 2 Preliminaries

- The authors identify a bag with all elements occurring only once with the set containing the elements of the bag.
- The authors overload the set notation, writing ∅ for the empty bag and ∈ for the element inclusion.
- For sequences of elements over a set P the authors use the following notation:.
- The authors will drop N and write R(m) when no ambiguity can arise.
- A marked net (N, m0) is called bounded if its reachability set is finite.

### 3 Event History and History Logic

- In this section the authors present the general notion of event history.
- In the coming sections the authors investigate two kinds of nets that use event history: token history nets and global history nets.
- Information on the relative order of events registered by different components might be missing.
- Sets of formulae, terms and label expressions over Σ are denoted as FΣ, QΣ and LΣ, respectively.
- One can show that for closed terms and formulae, i.e., terms and formulae where all variables appear in the scope of #, the result of the evaluation does not depend on ν.

### 4 Token History Nets

- In this section the authors introduce token history nets as a special class of colored Petri nets [11] with history as color.
- Recall that the union of two histories is defined for consistent histories only.
- The following lemma states that consistency of markings is an invariant property (observe that a transition firing cannot destroy consistency).
- To conclude this section the authors illustrate the semantics of token history nets.
- The transition labeled d can fire consuming tokens [(p, H1)] and [(q, H3)] since the tokens share event e1 in their history.

### 5 Global History Nets

- Where history is a separate object accessible when the guards of transitions are evaluated.the authors.
- One can simulate inhibitor arcs by adding the condition m0(p) − #p + #p = 0. Since inhibitor nets are known to be Turing complete (cf. [17]), global history nets with unique labels are Turing complete as well.the authors.
- Observe that in general it is impossible to derive the corresponding token histories from the history of a global history net.
- It is easy to show that both nets are indeed bisimilar.

### 6 Global History Nets with Counting Formulae Guards

- I.e., formulae that do not explore the precedence of events ≺.the authors.
- Note that global history nets with counting formulae guards are Turing complete since they allow zero testing on the marking of a place.
- To facilitate simulation and validation of these nets, the authors show that every global history net with counting formulae guards can be transformed into a bisimilar inhibitor net.
- Furthermore, the authors identify conditions on the global history net implying that the net can be translated to a bisimilar classical Petri net.

### 6.1 Nets with Counting Formulae as Guards vs. Inhibitor Nets

- For the sake of brevity the authors call these expressions basic counting formulae (over A and B).
- The process terminates when all guards are true, i.e. the authors obtained a regular inhibitor net.
- Figure 5 shows the basic idea of the eliminating a transition with guard g(t).
- First the authors transform their net to a net where all guards are conjunctions of basic counting formulae by applying the following construction:.
- The latter observation motivates their interest in the existence of a classical Petri net bisimilar to a global history net.

### 6.2 Guards Depending on the Marking Only

- In this subsection the authors give conditions on the guards that allow a transformation into an equivalent bounded Petri net.
- So global history nets satisfying these conditions will accept regular languages.
- The authors consider here guards that depend only on the marking.

### 6.3 Counting Formulae with Bounded Synchronization Distance

- The authors use here an important concept in Petri nets introduced by Carl Adam Petri: synchronization distance [4,7,13].
- The authors use a generalization of this notion, the so-called y-distance [16].
- If the authors take the underlying classical Petri net, the transitions are completely independent of each other and the y-distance is ∞ for any weight vector with at least one positive component.
- (Idea) Disjunctions and conjunctions are taken care of as in Theorem 19.

### 7 Global History Nets with LTL Guards

- Now the authors consider the class of global history nets with LTL guards.
- While LTL formulae over infinite traces can be translated to Büchi automata, LTL formulae over finite traces can be translated to finite automata. [5] presents a translation algorithm that modifies standard LTL to Büchi automata conversion techniques to generate finite automata that accept finite traces satisfying LTL formulae.
- The main aspect of modification there is the selection of accepting conditions.
- Fig. 8 shows a simplistic example for a credit card company, where a credit card can be issued, reported lost, used for a payment or cancelled.
- Note that this net can perform an arbitrary sequence of steps, and the place “true” has a token when the guard on the history should be evaluated to true and “false” when the guard should be evaluated to false.

### 9 Conclusion

- In this paper the authors emphasize the importance of taking history into account while modelling processes.
- Historical information is present in most state-of-the-art enterprise information systems.
- The guard language will allow to evaluate conditions both on separate tokens and on their combinations.
- The authors are going to develop a method for transforming broader subclasses of global history nets to classical and inhibitor Petri nets.
- The authors are grateful to Jan Hidders and Jan Paredaens for a number of fruitful discussions at the early stages of this research.

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##### Citations

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### Cites background from "History-dependent Petri nets"

...[120] define history-dependent Petri nets in which each token records its history....

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### Cites background or methods from "History-dependent Petri nets"

...Global history nets [12] further extend Petri nets by assuming the availability of a history record, registering all firings and the time of firing together with the information which process performed it....

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...Our approach relies on the framework of adaptive workflow nets [10,12]....

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10 citations

6 citations

### Cites background from "History-dependent Petri nets"

...In this paper, which is a revised version of [17], we consider processes that not only record the events but also make choices based on the previous events, i.e. based on its history....

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##### References

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### "History-dependent Petri nets" refers background in this paper

...Therefore, they can be made deterministic and minimized with standard algorithms [10]....

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##### Frequently Asked Questions (2)

###### Q2. What have the authors contributed in "History-dependent petri nets" ?

The authors consider processes that not only keep track of their history in a log, but also make decisions based on this log. The authors show that some classes of historydependent nets can be automatically converted to classical Petri nets for analysis purposes.