# History-dependent Petri nets

TL;DR: It is shown that some classes of history-dependent nets can be automatically converted to classical Petri nets for analysis purposes and some classes are characterized by the form of the guards and sometimes the additional requirement that the underlying Classical Petri net is either bounded or has finite synchronization distances.

Abstract: Most information systems that are driven by process models (e.g., workflow management systems) record events in event logs, also known as transaction logs or audit trails. We consider processes that not only keep track of their history in a log, but also make decisions based on this log. To model such processes we extend the basic Petri net framework with the notion of history and add guards to transitions evaluated on the process history. We show that some classes of history-dependent nets can be automatically converted to classical Petri nets for analysis purposes. These classes are characterized by the form of the guards (e.g., LTL guards) and sometimes the additional requirement that the underlying classical Petri net is either bounded or has finite synchronization distances.

## 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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### 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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###### 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.