Causal Behavioural Profiles  Efficient Computation, Applications, and Evaluation.
ABSTRACT Analysis of behavioural consistency is an important aspect of software engineering. In process and service management, consistency verification of behavioural models has manifold applications. For instance, a business process model used as system specification and a corresponding workflow model used as implementation have to be consistent. Another example would be the analysis to what degree a process log of executed business operations is consistent with the corresponding normative process model. Typically, existing notions of behaviour equivalence, such as bisimulation and trace equivalence, are applied as consistency notions. Still, these notions are exponential in computation and yield a Boolean result. In many cases, however, a quantification of behavioural deviation is needed along with concepts to isolate the source of deviation. In this article, we propose causal behavioural profiles as the basis for a consistency notion. These profiles capture essential behavioural information, such as order, exclusiveness, and causality between pairs of activities of a process model. Consistency based on these profiles is weaker than trace equivalence, but can be computed efficiently for a broad class of models. In this article, we introduce techniques for the computation of causal behavioural profiles using structural decomposition techniques for sound freechoice workflow systems if unstructured net fragments are acyclic or can be traced back to S or Tnets. We also elaborate on the findings of applying our technique to three industry model collections.

Article: Behavioral Equivalence and Compatibility of Business Process Models with Complex Correspondences
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ABSTRACT: Once multiple models of a business process are created for different purposes or to capture different variants, verification of behaviour equivalence or compatibility is needed. Equivalence verification ensures that two business process models specify the same behaviour. Since different process models are likely to differ with respect to their assumed level of abstraction and the actions that they take into account, equivalence notions have to cope with correspondences between sets of actions and actions that exist in one process but not in the other. In this paper, we present notions of equivalence and compatibility that can handle these problems. In essence, we present a notion of equivalence that works on correspondences between sets of actions rather than single actions. We then integrate our equivalence notion with work on behaviour inheritance that copes with actions that exist in one process but not in the other, leading to notions of behaviour compatibility. Compatibility notions verify that two models have the same behaviour with respect to the actions that they have in common. As such, our contribution is a collection of behaviour equivalence and compatibility notions that are applicable in more general settings than existing ones.Computer Journal. 01/2012; 
Conference Paper: Eyetracking the factors of process model comprehension tasks
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ABSTRACT: Understanding business process models has been previously related to various factors. Those factors were determined using statistical approaches either on model repositories or on experiments based on comprehension questions. We noticed that, when asking comprehension questions on a process model, usually the expert explores only a part of the entire model to provide the answer. This paper formalizes this observation under the notion of Relevant Region. We conduct an experiment using eyetracking to prove that the Relevant Region is indeed correlated to the answer given to the comprehension question. We also give evidence that it is possible to predict whether the correct answer will be given to a comprehension question, knowing the number and the time spent fixating Relevant Region elements. This paper sets the foundations for future improvements on model comprehension research and practice.Proceedings of the 25th international conference on Advanced Information Systems Engineering; 06/2013
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Fundamenta Informaticae XX (2011) 1–37
IOS Press
1
Causal Behavioural Profiles –
Efficient Computation, Applications, and Evaluation
Matthias WeidlichC
Hasso Plattner Institute, University of Potsdam
Prof.Dr.HelmertStr. 23, 14482 Potsdam, Germany
matthias.weidlich@hpi.unipotsdam.de
Artem Polyvyanyy
Hasso Plattner Institute, University of Potsdam
Prof.Dr.HelmertStr. 23, 14482 Potsdam, Germany
artem.polyvyanyy@hpi.unipotsdam.de
Jan Mendling
Humboldt University
Unter den Linden 6, D10099 Berlin, Germany
jan.mendling@wiwi.huberlin.de
Mathias Weske
Hasso Plattner Institute, University of Potsdam
Prof.Dr.HelmertStr. 23, 14482 Potsdam, Germany
mathias.weske@hpi.unipotsdam.de
Abstract. Analysis of behavioural consistency is an important aspect of software engineering. In
process and service management, consistency verification of behavioural models has manifold
applications. For instance, a business process model used as system specification and a corresponding
workflow model used as implementation have to be consistent. Another example would be the analysis
to what degree a process log of executed business operations is consistent with the corresponding
normative process model. Typically, existing notions of behaviour equivalence, such as bisimulation
and trace equivalence, are applied as consistency notions. Still, these notions are exponential in
computation and yield a Boolean result. In many cases, however, a quantification of behavioural
deviation is needed along with concepts to isolate the source of deviation.
In this article, we propose causal behavioural profiles as the basis for a consistency notion. These
profiles capture essential behavioural information, such as order, exclusiveness, and causality between
pairs of activities of a process model. Consistency based on these profiles is weaker than trace
equivalence, but can be computed efficiently for a broad class of models. In this article, we intro
duce techniques for the computation of causal behavioural profiles using structural decomposition
techniques for sound freechoice workflow systems if unstructured net fragments are acyclic or can
be traced back to S or Tnets. We also elaborate on the findings of applying our technique to three
industry model collections.
Keywords: Causal Behavioural Profiles, Formal Methods, Behavioural Abstraction, Structural
Decomposition, Exclusiveness, Concurrency, Order Relations, Causality, Optionality
CCorresponding author
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2M. Weidlich, A. Polyvyanyy, J. Mendling, M. Weske/Causal Behavioural Profiles
1.Introduction
Consistency verification is a central aspect of software engineering. Focussing on the domain of process
and service management, consistency between behavioural models needs to be assessed at different stages
of the implementation of business processes. First and foremost, process modelling has become one of
the most extensively used approaches for capturing business requirements [14]. These requirements are
typically refined and modified in an engineering process, resulting in a workflow model and software
artefacts. A workflow model often defines activities of the business process model in more detail, neglects
steps that are not implemented or do not need to be supported by the system, or adjusts behaviour to
the specifics of the workflow system. This raises the question to what degree a process model used as
specification and a workflow model used as implementation are behaviourally consistent.
(a)
(b)
Get Project
Details (A)
Establish
Contact (F)
Request Offer from
Subcontractor (B)
Get Detailed
Requirements (H)
Create Project (G)
Clarify Requirement
Issues (J)
Negotiate
Contract (K)
Sign Precontract with
Subcontractor (D)
Create
Record
Get Project Details
from Marketing Module
Get Project Details
from PreSales Module
Enter Contact
Details
Load Request for Quote
Enter Project
Requirements
Attach Contract
& Close Record
Enter Project
Planning
Provide Technical Presentation (I)
Update Request for Offer (C)
Schedule Internal Resources (E)
Figure 1.
workflow implementation
Example of two Petri net process models, (a) focussing on the business perspective, (b) depicting the
Figure 1 illustrates the problem addressed in this article with two process models that relate to a
project handling process. Model (a) assumes a business perspective, whereas (b) shows the workflow
implementation of the process. Activities (or sets thereof) that correspond to each other are connected
by dashdotted lines. If process models assume different perspectives on a common business process,
such correspondences rarely express semantic equivalence between the matched activities. Consider
the activities ‘Establish Contact’ and ‘Enter Contact Details’ of the example in Figure 1. Apparently,
establishing a contact involves more than just entering the contact details. Still, we say that both activities
correspond to each other against the background of aligning process models that assume different
perspectives. As for the model in general, both activities assume either a business perspective or a
workflow perspective on a particular unit of work.
For this article, we assume that correspondences between activities are given. They may stem from
a system analyst inspecting the models or from automatic matching. Recently, techniques including
structural analysis and natural language processing to automatically identify such correspondences have
been introduced for the domain of business process modelling [18, 50, 64]. Moreover, techniques from
the area of schema and ontology matching can be exploited [27, 56]. Then, activities are regarded as
elements of a process model schema.
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M. Weidlich, A. Polyvyanyy, J. Mendling, M. Weske/Causal Behavioural Profiles3
Consistency verification between behavioural models is not only relevant during designtime. The
analysis to what degree a process log of executed business operations is consistent with the corresponding
normative process model is an example for consistency verification at runtime. Here, it may be the case
that the actual processing observed in information systems is not in line with the specified processing.
Deviations may stem from information systems that do not explicitly enforce the predefined execution
order of activities or from people working around the system [5].
For both use cases, existing notions of behaviour equivalence may be used as a consistency measure.
For instance, bisimulation and trace equivalence assume the set of all traces or the branching structure as
essential behavioural characteristics that have to be preserved. However, these notions are computationally
complex [29], which is particularly a problem for process models including many activities. Furthermore,
these notions only provide information whether behaviour is equivalent or not, but do not describe how
strong a deviation is in case of a mismatch. In many cases, however, a quantification of behavioural
deviation is needed and the source of deviation needs to be identified.
In this article, we argue that for certain scenarios of consistency verification, a criterion of behaviour
equivalence might be weakened in order to compensate for computational efficiency. To this end, we
define the notion of a causal behavioural profile. Such a profile represents a behavioural abstraction
that includes dependencies in terms of order, exclusiveness, or causality between pairs of activities. It is
computed efficiently using structural decomposition techniques for sound freechoice workflow systems if
unstructured net fragments are acyclic or can be traced back to S or Tnets.
This article is an extended and revised version of our previous work [69]. As an extension, we
included a broader discussion of the application of causal behavioural profiles. In particular, we do
not only illustrate how they form the basis of a consistency notion for related process models, but also
summarise recent work on their application for consistency analysis of process logs (aka conformance
checking). Further, we revised the section on the computation of causal behavioural profiles and integrated
the results obtained for structured fragments with those obtained for unstructured fragments. Hence, we
are able to present a complete computation algorithm for the proposed approach. Finally, we present an
extended validation of the techniques for the computation of causal behavioural profiles. In addition to the
models used in [69], we applied our approach to two more model collections from industry. Hence, we
provide further insights on the applicability of our approach in an industry setting.
The remainder of this article is structured as follows: Section 2 introduces our formal framework.
Causal behavioural profiles are defined in Section 3. We discuss applications of causal behavioural profiles
for consistency verification in Section 4. Section 5 elaborates on graph decomposition techniques that
are applied to workflow nets. How these techniques are used to compute causal behavioural profiles is
presented in Section 6. In Section 7, we report on the findings of applying our approach to three industry
model collections. Finally, Section 8 reviews related work, before Section 9 concludes the article.
2. Preliminaries
We use workflow (WF) systems [2] as our formal grounding, a class of Petri nets used for process
modelling and analysis. Note that Petri net based formalisations have been presented for (parts of)
common process modelling languages, such as the Business Process Model and Notation (BPMN), the
Web Service Business Process Execution Language (BPEL), Event Driven Process Chains (EPCs), and
UML Activity Diagrams, e.g., [19, 42, 38, 24, 47]. A survey of these formalisations can be found in [43].
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4 M. Weidlich, A. Polyvyanyy, J. Mendling, M. Weske/Causal Behavioural Profiles
Based on [2, 17], we recall basic definitions.
Definition 2.1. (WFnet Syntax)
◦ A net is a tuple N = (P,T,F) with P and T as finite disjoint sets of places and transitions, and
F ⊆ (P ×T)∪(T ×P) as the flow relation. We write X = (P ∪T) for all nodes. The irreflexive
transitive closure of F is denoted by F+.
◦ For a node x ∈ X, •x := {y ∈ X  (y,x) ∈ F} is the preset, x• := {y ∈ X  (x,y) ∈ F} is the
postset, and •(x•) := {z ∈ X  y ∈ X ∧ (x,y) ∈ F ∧ (z,y) ∈ F}.
◦ A tuple N?= (P?,T?,F?) is a subnet of a net N = (P,T,F), if P?⊆ P, T?⊆ T, and F?=
F ∩ ((P?× T?) ∪ (T?× P?)); N??= (P??,T??,F??) is a partial subnet of N, if P??⊆ P, T??⊆ T,
and F??⊆ F ∩ ((P??× T??) ∪ (T??× P??)).
◦ A net N is a Tnet, if ∀ p ∈ P [  • p ≤ 1 ≥ p •  ], and an Snet, if ∀ t ∈ T [  • t ≤ 1 ≥ t •  ].
◦ A net N is freechoice, iff ∀ p ∈ P with p •  > 1 holds •(p•) = {p}.
◦ A path is a nonempty sequence x1,...,xkof nodes, k ∈ N, k > 1, denoted by πN(x1,xk), which
satisfies (x1,x2),...,(xk−1,xk) ∈ F. By πN{x1,xk} = {x1,...,xk}, we denote the set of all
nodes on the path. We write xi ∈ πN, if xi ∈ πN{x1,xk}. A subpath π?
subsequence of πNthat is itself a path. A path πN(x1,xk) is a circuit, if (xk,x1) ∈ F and no node
is more than once part of the path.
◦ For a net N = (P,T,F) and a partial subnet N?= (P?,T?,F?) a path πN(x1,xk), k > 1, where
all xiare distinct, of N is a handle of N?, iff πN{x1,xk} ∩ (P?∪ T?) = {x1,xk}.
◦ For a net N = (P,T,F) and two partial subnets N?= (P?,T?,F?), N??= (P??,T??,F??), a path
πN(x1,xk), k > 1 and all xiare distinct, of N is a bridge from N?to N??, iff πN{x1,xk} ∩ (P?∪
T?) = {x1} and πN{x1,xk} ∩ (P??∪ T??) = {xk}.
◦ A Petri net N = (P,T,F) is a workflow (WF) net, iff N has an initial place i ∈ P with •i = ∅, N
hasafinalplaceo ∈ P witho• = ∅, andtheshortciruit netN?= (P,T∪{tc},F∪{(o,tc),(tc,i)}),
tc/ ∈ T, of N is strongly connected.
Freechoiceness of a net implies that various behavioural analysis questions can be answered efficiently.
Note that the same effect can be achieved with a more relaxed notion, referred to as extended free
choiceness [10]. Any extended freechoice net can be transformed into a behaviour equivalent freechoice
net [10] – here, behaviour equivalence assumes that transitions inserted by the transformation are ignored.
For this article, we stick to the definition presented earlier.
Further, we speak of PP,TT,PT,TP handles and bridges, depending on the type (place or transition)
of the initial and the final node of the path. We define semantics for WFnets according to [2].
Nof a path πNis a
Definition 2.2. (WFnet Semantics)
Let N = (P,T,F) be a WFnet with initial place i and final place o.
◦ M : P ?→ N0is a marking of N, M denotes all markings of N. M(p) returns the number of
tokens in place p. [p] denotes the marking where place p contains just one token and all other places
contain no tokens.
◦ For any transition t ∈ T and any marking M ∈ M, t is enabled in M, denoted by (N,M)[t?, iff
∀ p ∈ •t [ M(p) ≥ 1 ].
◦ Marking M?is reachable from M by firing of t, denoted by (N,M)[t?(N,M?), such that M?=
M − •t + t•, i.e., one token is taken from each input place of t and one token is added to each
output place of t.
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M. Weidlich, A. Polyvyanyy, J. Mendling, M. Weske/Causal Behavioural Profiles5
◦ A sequence of transitions σ = t1,...,tn, n ∈ N0, is a firing sequence, iff there exist markings
M0,...,Mn∈ M, such that for all i ∈ N, 1 ≤ i ≤ n it holds (N,Mi−1)[ti?(N,Mi). We say that
σ is enabled in M0, denoted by (N,M0)[σ?. For n = 0, we refer to σ = ?? as the empty firing
sequence.
◦ For any two markings M,M?∈ M, M?is reachable from M in N, denoted by M?∈ [N,M?,
if there exists a firing sequence σ leading from M to M?. Firing of σ in M is denoted by
(N,M)[σ?(N,M?).
◦ A net system, or a system, is a pair (N,Mi), where N is a net and Miis the initial marking of N. A
WFsystem is a pair (N,Mi), where N is a WFnet with initial place i and Mi= [i].
Given a (freechoice, S, T) WFnet N with Mias its initial marking, the tuple S = (N,Mi) is a
(freechoice, S, T) WFsystem. The final marking is denoted by Mo= [o] with o being the final place
of a WFnet. Without stating it explicitly, we assume a net of a system to be defined as N = (P,T,F).
If the context is clear, we refer to WFsystems and shortcircuit nets as WFnets. Finally, we recall
the soundness property, which requires WFsystems (1) to always terminate, and (2) to have no dead
transitions [1]. Both requirements imply proper termination of the WFsystem, i.e., for all reachable
markings holds that a token in the final place implies the absence of tokens for all other places. Soundness
of a WFsystem is traced back to liveness and boundedness of its shortcircuit system, see [1].
Definition 2.3. (Liveness, Boundedness, Soundness)
◦ A system (N,Mi) is live, iff for every reachable marking M ∈ [N,Mi? and t ∈ T, there exists a
marking M?∈ [N,M? such that (N,M?)[t?.
◦ A system (N,Mi) is bounded, iff the set [N,Mi? is finite.
◦ A WFsystem (N,Mi) is sound, iff the shortcircuit system (N?,Mi) is live and bounded.
3. The Notion of a Causal Behavioural Profile
This section introduces causal behavioural profiles. They are based on the notion of behavioural profiles,
which we recall in Section 3.1. We introduced these profiles in an earlier work [66] to reason on order
constraints only. Optionality of transition execution or causality between transitions is not captured. These
aspects are addressed by the novel concept of a causal behavioural profile introduced in Section 3.2.
Section 3.3 discusses our concepts in the light of existing behavioural relations defined for Petri nets.
3.1.Execution Order Constraints: The Behavioural Profile
Behavioural profiles capture behavioural aspects in terms of order constraints of a process in a finegrained
manner [66]. They are grounded on the set of possible firing sequences of a WFsystem and the notion of
weak order.
Definition 3.1. (Weak Order)
Let (N,Mi), N = (P,T,F), be a WFsystem. A pair (x,y) ∈ (T × T) is in the weak order relation ?,
iff there exists a firing sequence σ = t1,...,tnwith (N,Mi)[σ? and indices j,k ∈ N, 1 ≤ j < k ≤ n,
for which holds tj= x and tk= y.
Two transitions t1,t2are in weak order, if there exists a firing sequence reachable from the initial marking
in which t1occurs before t2. Using weak order, we define three relations forming the behavioural profile.
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6M. Weidlich, A. Polyvyanyy, J. Mendling, M. Weske/Causal Behavioural Profiles
A
B
C
D
(a)
A
B
C
D
(b)
Figure 2.Optionality
Definition 3.2. (Behavioural Profile)
Let (N,Mi), N = (P,T,F), be a WFsystem and T?⊆ T a set of transitions. A pair of transitions
(x,y) ∈ (T?× T?) can be in the following profile relations:
◦ The strict order relation ?, if x ? y and y ?? x.
◦ The exclusiveness relation +, if x ?? y and y ?? x.
◦ The interleaving order relation , if x ? y and y ? x.
BT? = {?,+,} is the behavioural profile of (N,Mi) over T?.
If we do not restrict the set of transitions over which the behavioural profile is defined, we assume that it
is defined over all transitions. The inverse relation of strict order, ?−1= {(y,x) ∈ (T?× T?)  x ? y},
is referred to as reverse strict order. Computing the behavioural profile for all transitions of the system
(a) in Figure 1 reveals that, for instance, it holds C ? E as there exists no firing sequence, such that E
occurs before C. However, strict order does not imply the actual occurrence. There are firing sequences
containing only one of the two transitions, or even none of them. It holds D + E as both transitions will
never occur in a single firing sequence and BG as both transitions can occur in any order. Note that
the three relations are mutually exclusive and (together with reverse strict order) partition the Cartesian
product of transitions over which they are defined [66]. With respect to itself, a transition is either
exclusive (if it can occur at most once, e.g., D + D) or in interleaving order (if it can occur more than
once, e.g., BB).
3.2.Occurrence Constraints: The Causal Behavioural Profile
Behavioural profiles, as introduced above, relate pairs of transitions according to their order of potential
occurrence. Even though this information may be sufficient for certain consistency scenarios, these
profiles provide a rather coarsegrained behavioural abstraction. Information on ordering constraints is
not sufficient to draw conclusions on optionality and causality of transition occurrences.
Optionality of a transition is given, if there is a firing sequence leading from the initial to the final
marking of the system that does not contain the transition. Optionality can be lifted from single transitions
to sets of transitions. A set of transitions is considered to be jointly optional, if any firing sequence from
the initial to the final marking contains all or none of the transitions. As illustrated by Figure 2(a) and
Figure 2(b) this property cannot be derived from the knowledge about optionality of single transitions. In
both systems, B and C are optional, but only in Figure 2(b) the set {B,C} is jointly optional.
Closely related to optionality is causality, which requires that one transition can only occur after the
occurrence of another transition. Thus, causality comprises two aspects, a certain order of occurrences
and a causal coupling of occurrences. The former is addressed by the behavioural profile in terms of the
strict order relation, whereas the latter is not captured. For instance, B is a cause of C in Figure 2(b),
but not in Figure 2(a). Note that two transitions in interleaving order do not show causality according
to our definition. For both systems in Figure 3, it holds BC. We do not observe an ordering between
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M. Weidlich, A. Polyvyanyy, J. Mendling, M. Weske/Causal Behavioural Profiles7
A
DC
B
(a)
A
D
B
C
(b)
Figure 3.No causality for transitions (B,C) in a cycle
all occurrences of both transitions. Interleaving order is interpreted as the absence of any dependency
regarding the order of occurrence. Thus, it is reasonable to define causality as a dependency between
all occurrences of two transitions, instead of considering causal dependencies between their single
occurrences (cf., response or leadsto dependencies [23]). There is no causality between B and C in
either system in Figure 3.
To cope with the aforementioned aspects, we extend the behavioural profile yielding the causal
behavioural profile. The latter is a more finegrained behavioural abstraction that closer approximates
trace semantics of net systems. Hence, we can still achieve efficient computation for a broad class of
system, but have a closer approximation of behaviour equivalence for consistency verification. Technically,
the causal behavioural profile introduces a cooccurrence relation. Two transitions are cooccurring, if
any firing sequence from the initial to the final marking that contains the first transition contains also the
second transition.
Definition 3.3. (Causal Behavioural Profile)
Let (N,Mi), N = (P,T,F), be a WFsystem and T?⊆ T a set of transitions.
◦ A pair (x,y) ∈ (T?× T?) is in the cooccurrence relation ?, if for all firing sequences σ with
(N,Mi)[σ?(N,Mo), it holds x ∈ σ ⇒ y ∈ σ.
◦ CBT? = {?,+,,?} is the causal behavioural profile of (N,Mi) over T?.
Again, if the set of transitions is not restricted explicitly, we assume the causal behavioural profile to be
defined over all transitions. Trivially, it holds t ? t for all t ∈ T. We derive optionality and causality
as follows. A single transition t ∈ T is optional, if ti?? t for some ti∈ i• with i as the initial place.
A set T1⊆ T of transitions is optional, if all transitions themselves are optional and they are pairwise
cooccurring to each other, (T1×T1) ⊆ ?. Further, there is a causal dependency between two transitions
t1,t2∈ T, if they are in strict order (t1? t2) and occurrence of the first implies occurrence of the second
(t1? t2). Note that, in contrast to the behavioural profile, the causal behavioural profile differs for both
systems in Figure 2.
3.3. Relation to Existing Behavioural Relations
There is a large body of research on behavioural relations for formal models specifying dynamic systems
in general, and for Petri nets in particular. Focussing on the order of occurrence, the relations proposed
in [6] for workflow mining are close to our relations, yet different. We base our definitions on an indirect
weak order dependency, whereas the relations in [6] are grounded on a direct sequential order. As a result,
the notion of exclusiveness is restricted to ‘pairs of transitions that never follow each other directly’ [6],
whereas we capture exclusiveness for transitions that could potentially occur at different stages of a
firing sequence. The notion of direct sequential order is appropriate for workflow mining, but leads to
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8M. Weidlich, A. Polyvyanyy, J. Mendling, M. Weske/Causal Behavioural Profiles
undesired effects in our setting. For most scenarios of consistency verification, the relation associating
corresponding transitions of both models to each other is partial. Certain transitions of one model are
without counterpart in the other model. Figure 1 illustrates this phenomenon for the case of consistency
verification of business process models, both captured as WFnets, used as specification and workflow
models used as implementation. Consider, for instance, transitions G and K of model (a) in Figure 1.
They are exclusive according to the relations proposed in [6], whereas their counterparts in model (b)
are in a sequential order. The behavioural profile, in turn, yields equal relations in both models. The
respective transitions are in strict order in both models, (a) and (b). Therefore, indirect dependencies, as
defined by the behavioural profile, are more appropriate for consistency verification in the presence of
partial correspondence relations.
The wellknown notions of conflicting and concurrent transitions are related to our relations as well.
In a sound freechoice WFsystem, two transitions, which are in conflict and are not part of a circuit,
will be exclusive in the behavioural profile. This follows from Lemma 3 in [66] and the fact that sound
freechoice WFsystems are safe (a place carries at most one token in all markings, cf., Lemma 1 in [3]).
Similarly, all transitions that are enabled concurrently in some reachable marking (cf., the concurrency
relation [39]) are in interleaving order in the behavioural profile.
In order to cope with concurrency and the interleaving problem, the unfolding of a Petri net (or its
complete prefix, respectively) may be exploited for behaviour analysis [25, 44]. A true concurrent model
is created in which a transition (i.e., an event) corresponds to a certain occurrence of a transition in the
original net. Events can be related as being in a weak causal predecessor, conflict, or concurrency relation.
Even though these relations resemble the relations of our casual behavioural profile, they are defined for
transition occurrences instead of transitions. Thus, we can derive our relations by lifting the relations of
the complete prefix unfolding to the level of transitions again. Recently, we introduced a computation
algorithm for behavioural profiles based on complete prefix unfoldings of bounded systems [65]. However,
usage of unfoldings is inappropriate w.r.t. the class of systems addressed in this article, as the construction
of unfoldings is computationally much harder than the approach introduced in the remainder of this article.
With respect to common notions of behaviour equivalence, we see that two WFsystems with equal
causal profiles are not necessarily trace equivalent. For instance, both systems in Figure 3 have the same
causal profile, whereas they are not trace equivalent. Evidently, the same holds true for bisimulation
equivalences, as the profile neglects the branching structure of a system. However, it is easy to see that
trace equivalence of two WFsystems implies equivalence of their causal behavioural profiles for all
transitions, as all behavioural relations formulate statements about the existence of firing sequences.
4. Applications of Causal Behavioural Profiles
We motivated the definition of causal behavioural profiles with the need to assess behavioural consistency
in an efficient and finegranular manner. This section reviews applications of causal behavioural profiles.
First, Section 4.1 discusses how a degree of consistency is determined for two WFsystems under the
assumption of a correspondence relation. Second, Section 4.2 summarises the application of causal
behavioural profiles for consistency analysis of process logs. Causal behavioural profiles have already
been applied in a much broader context, e.g., for query optimisation in complex event processing [71] and
process monitoring [72]. As such, causal behavioural profiles proved to be a valuable behavioural model
for many use cases. In this section, however, we focus on their application for consistency measurement.
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M. Weidlich, A. Polyvyanyy, J. Mendling, M. Weske/Causal Behavioural Profiles9
4.1. Consistency between Related Systems
Once a correspondence relation has been established between the transitions of two systems using
structural analysis and natural language processing (see [18, 50, 64]), the degree of consistent behaviour
can be quantified based on causal behavioural profiles. The general idea behind the consistency measure
is summarised as follows. Given a correspondence relation between the sets of transitions of two WF
systems, we consider all aligned transitions of both system, i.e., transitions in either model that have a
corresponding transitions in the other model. For each pair of aligned transitions, we check whether the
corresponding transitions show the same constraints as defined by the causal behavioural profile. Since
there can be complex n:m correspondences, see Figure 1, we have to count the correspondences from the
perspective of each model.
A consistency notion based on the behavioural profile has been introduced in [66]. The novel concept
of causal behavioural profiles is applied in the same manner and yields a degree of consistency that is more
finegrained (optionality and causality constraints are also considered) and closer to notions of behaviour
equivalence. The degree of consistency based on causal behavioural profiles is defined as follows.
Definition 4.1. (Degree of Consistency)
Let (N1,Mi1), N1= (P1,T1,F1), and (N2,Mi2), N2= (P2,T2,F2), be two WFsystems and CBT1=
{?1,+1,1,?1} and CBT2= {?2,+2,2,?2} their causal behavioural profiles. Let ∼ ⊆ T1× T2,
∼ ?= ∅, be a correspondence relation.
◦ The set T∼
is defined analogously.
◦ Two relations R1∈ {?1,+1,1,?−1
noted by R1? R2, iff either
− R1= ?1 ∧ R2= ?2,
− R1= ?−1
− R1= +1 ∧ R2= +2, or
− R1= 1 ∧ R2= 2.
◦ The set of behavioural profile consistent transition pairs CT∼
all pairs (tx,ty), such that
− if tx= ty, then ∀ ts∈ T∼
− if tx?= ty, then ∀ ts,tt∈ T∼
R1? R2or (2) tx∼ ttand ty∼ ts.
The set CT∼
◦ The set of causal behavioural profile consistent transition pairs CCT∼
contains all pairs (tx,ty), such that if tx?= tythen for all transitions ts,tt∈ T∼
ty∼ ttit holds either (1) tx?1ty⇔ ts?2ttor (2) tx∼ ttand ty∼ ts. The set CCT∼
is defined analogously.
◦ The degree of consistency of ∼ is defined as D∼=
Applying this degree to the scenario in Figure 1, we see that the order of potential occurrence is preserved
for all aligned transitions. Hence, the degree of consistency based on the behavioural profile proposed
in [66] yields a value of one. However, transition (A) is mandatory in model (a), whereas its counterparts
are optional in model (b). Consequently, causality between transition (A) and, for instance, transition (K)
is not preserved in model (b) either, which is taken into account in the causal behavioural profile. For
1= {t1∈ T1 ∃ t2∈ T2[ t1∼ t2]} contains all aligned transitions of (N1,Mi1). T∼
2
1} and R2∈ {?2,+2,2,?−1
2} are type equivalent, de
1
∧ R2= ?−1
2,
1⊆ (T∼
1×T∼
1) for (N1,Mi1) contains
2with tx∼ tsit holds (txR1tx ∧ tsR2ts) ⇒ R1? R2,
2with tx∼ tsand ty∼ ttit holds either (1) (txR1ty∧ tsR2tt) ⇒
2for S2is defined analogously.
1⊆ CT∼
1for (N1,Mi1)
2with tx∼ tsand
2for S2
CCT∼
(T∼
1+CCT∼
1)+(T∼
2
1×T∼
2×T∼
2).
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10M. Weidlich, A. Polyvyanyy, J. Mendling, M. Weske/Causal Behavioural Profiles
our example, the degree of consistency based on causal behavioural profiles is D∼=27+25
This value is derived as follows. Model (a) contains six transitions that are part of correspondences,
i.e., transitions {(A),(F),(G),(H),(J),(K)}. Also, model (b) contains six aligned transitions. Hence,
we consider 36 + 36 = 72 behavioural constraints between pairs of transitions overall. Checking the
constraints between transition pairs of model (a), we see that 27 out of 36 constraints are equal for all
pairs of corresponding transitions in model (b). For the 36 constraints between aligned transitions of
model (b), we observe that only 25 constraints are mirrored by the corresponding transitions in model (a).
36+36≈ 0.722.
The presented degree of consistency shows the characteristics of a semimetric for the comparison of
two causal behavioural profiles. It is a nonnegative and symmetric measure that equals one (or zero if it is
subtracted from one, respectively), if and only if both profiles are equal. For the assessment of two profiles,
however, the degree of consistency is not a metric as it does not satisfy the triangle inequality. That is due
to the fact that the degree is a criterion for the quality of an alignment, i.e., a set of a correspondences.
Hence, it is normalised by the size (the number of transitions) of the alignment but independent of the size
of the respective WFsystems and, therefore, causal behavioural profiles. Still, we see that the relations of
the causal behavioural profile are transitive in the sense that equal relations between a first and a second
model, and the second and a third model imply the equivalence for the relations between the first and the
third model. Thus, triangle inequality holds for the comparison of the degree of consistency of different
alignments when considering solely those pairs of transitions that are part of all alignments.
Consistency measurement based on behavioural profiles has been applied successfully in a recent
case study on process variants [66]. Clearly, there is a need for a multitude of consistency criteria in
order to be able to graduate consistency requirements for a concrete setting. Still, an interval scale and
efficient computation methods have to be seen as core requirements on such notions. Being stronger
than behavioural profile consistency but weaker than trace equivalence, our proposal based on causal
behavioural profiles is one step further towards a spectrum of consistency notions.
A
X
B
AB
Figure 4.
ing transition X
Propagation of insert
It is worth to mention that consistency based on (causal) behavioural
profiles enables support for change propagation between related systems.
We formalised a change propagation approach using behavioural profiles
in [70]. The approach can be lifted to the causal behavioural profiles in
a straightforward manner. The general idea behind this approach can be
summarised as follows. Once a change in one system is localised by a
node or a flow, the location is manifested in the causal behavioural profile.
Consequently, the change has to be propagated to the other model so that
equal relations of the causal behavioural profile can be observed for the
corresponding nodes, respectively. We illustrate the approach to change propagation with the two systems
in Figure 4. Assume that both systems were equal before a change operation, i.e., insertion of transition X,
has been applied to the upper system. The causal behavioural profile of the upper system locates this
change as being in strict order with transition A and in reverse strict order with transition B. Leveraging
this information for the corresponding transitions in the lower model isolates the region in which the
change has to be applied. Taking the cooccurrence relation of the causal behavioural profile into account,
it even becomes clear how a corresponding transition X has to be inserted in the lower model. That is, its
execution has to be optional as there is no cooccurrence with neither transition A, nor transition B. Hence,
compared to existing work [70], causal behavioural profiles allow for more precise change propagation
under the assumption of a stricter notion of consistency.
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M. Weidlich, A. Polyvyanyy, J. Mendling, M. Weske/Causal Behavioural Profiles 11
4.2.Conformance of Process Logs
Besides consistency verification between related systems, causal behavioural profiles are applied to assess
consistency of recorded execution sequences with a normative model. The former are referred to as logs
and represent observed executions of transitions as recorded by information systems, e.g., Enterprise
Resource Planning (ERP) systems, Customer Relationship Management (CRM) systems, or Workflow
Management Systems (WFMS). Consistency of logs is evaluated by conformance measures that quantify
to what degree the behaviour of a log is captured in the respective model. These measures provide
feedback on cases that do not conform to the model and quantify any behaviour deviations. This may be
the case when the execution order of transitions is not explicitly enforced by information systems or when
people deliberately working around the supporting ITinfrastructure [5]. Conformance measures have to
be rather finegrained to distinguish marginally different logs from completely different logs. Further,
they have to be efficiently computable and shall provide diagnostics if nonconformance is observed.
In [67, 68], we showed how causal behavioural profiles are used
for conformance analysis of logs that meets the aforementioned re
quirements. We assess whether the behavioural constraints as imposed
by the model for pairs of transitions are satisfied in the log. Causal
behavioural profiles are used to capture behavioural constraints. The
share of constraints that is satisfied by a log is then used as a confor
mance measure. We illustrate conformance measurement based on
causal behavioural profiles with the example in Figure 5. Here, the
upper system is the normative model. Two logs have been recorded
as actual executions of the system. Evidently, the log L1represents
a valid execution sequence of the system. This is manifested by the conformance measure presented
in [67, 68], which yields a value of one. All constraints, e.g., the strict order between transitions A and
C or the cooccurrence between transitions B and C, are satisfied in the log L1. In contrast, the log
L2is not conformant. Constraints as imposed by the system and formalised in the causal behavioural
profile are violated. The system allows for at most one execution of transition A (it is exclusive to itself),
whereas the log records two executions. Further, the cooccurrence dependency between transitions B
and C is violated in the log, as the log contains transition B but no transition C. However, a transition
that is in strict order from transition C is already in the log, i.e., transition D. This shows that transition
C should have been observed already in the log. Overall, our conformance measure yields a value of
8+11
9+12≈ 0.90 for log L2. Eight out of nine constraints of the behavioural profile are satisfied (A + A
is violated), and 11 out of 12 cooccurrence constraints are in line with the process model (B ? C is
violated). For cooccurrence, we check 12 constraints since all pairs of transitions that are expected to
be in the log are considered, in our example this includes the constraints C ? A, C ? B, and C ? D,
whereas selfrelations are ignored. Our approach enables rootcause analysis for nonconformant cases,
see [68]. For the log L2, both observed violations would be considered to be independent root causes.
A
BC
D
Log L1: <A, B, C, D>
Log L2: <A, A, B, D>
Figure 5.
mance analysis
Example for confor
5. Graph Decomposition Techniques for WFSystems
This section discusses the application of graph decomposition techniques for WFsystems. First, Sec
tion 5.1 introduces the Refined Process Structure Tree (RPST), a structural decomposition technique for
workflow graphs. Second, Section 5.2 enriches the RPST for WFsystems with behavioural annotations.
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12 M. Weidlich, A. Polyvyanyy, J. Mendling, M. Weske/Causal Behavioural Profiles
B
A
C
D
E
I
F
H
K
P1
B1
P2
P3
P5
P4 B2
P6
P7
B3
P8
P9
P10
P11
P12
R1
G
J
(a)
P1
B1
P4P5
P6 P7
B2
B3
P2
P8
P9
P10
P11
R1
P3
P12
(b)
Figure 6.(a) A WFsystem and its canonical fragments, (b) the RPST of (a)
5.1.The Refined Process Structure Tree
The RPST [61, 54] is a technique for detecting the structure of a workflow graph. A workflow graph
can be parsed into a hierarchy of fragments with a single entry and a single exit, such that the RPST is a
containment hierarchy of canonical fragments of the graph. The RPST is unique for a given workflow
graph and can be computed in linear time [54, 61]. Although the RPST has been introduced for workflow
graphs, the technique can be applied to other graph based behavioural models such as WFnets in a
straightforward manner. Basic terms of the RPST are defined for WFnets as follows.
Definition 5.1. (Edges, Entry, Exit, Canonical Fragment)
Let N = (P,T,F) be a WFnet.
◦ For a node x ∈ X of a net N = (P,T,F), inN(x) = {(n,x) ∈ F  n ∈ •x} are its incoming
edges and outN(x) = {(x,n) ∈ F  n ∈ x•} are its outgoing edges.
◦ A node x ∈ X?of a connected subnet N?= (P?,T?,F?) of a net N is a boundary node, if
∃ e ∈ inN(x)∪outN(x) [ e / ∈ F?]. If x is a boundary node, it is an entry of N?, if inN(x)∩F?= ∅
or outN(x) ⊆ F?, or an exit of N?, if outN(x) ∩ F?= ∅ or inN(x) ⊆ F?.
◦ Any connected subnet ω of N, is a fragment, if it has two boundary nodes: one entry, denoted by
ω?, and one exit, denoted by ω?.
◦ A fragment is place bordered if its boundary nodes are places.
◦ A fragment is transition bordered if its boundary nodes are transitions.
◦ Afragmentω = (Pω,Tω,Fω)iscanonicalinasetofallfragmentsΣofN, iff∀γ = (Pγ,Tγ,Fγ) ∈
Σ [ ω ?= γ ⇒ (Fω∩ Fγ= ∅) ∨ (Fω⊂ Fγ) ∨ (Fγ⊂ Fω) ].
Figure 6 exemplifies the RPST for the WFsystem from Figure 1(a). Figure 6(a) illustrates its canonical
fragments, each of them formed by a set of edges enclosed in or intersecting the region with a dotted
border. Each edge itself is a fragment, which is neglected in Figure 6 by showing only fragments that
comprise more than one edge. As an example, consider fragments P4 and B2. The former is a subnet that
consists of transition B, the place in its preset, the place in its postset, and two edges, which connect
these places with transition B. The place in the preset of transition B is the entry of this fragment, the
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M. Weidlich, A. Polyvyanyy, J. Mendling, M. Weske/Causal Behavioural Profiles13
place in the postset of transition B is the exit. Fragment B2, in turn, comprises two transitions, B and C,
the two places in their presets and postsets, and four edges that connect these places with both transitions.
Fragment B2 has the same entry and the same exit as fragment P4. All fragments visualised in Figure 6
are canonical. Assume that the subnet B2 is extended by transition A and the edge between A and the
place in the preset of transition B. Then, this subnet would be a fragment, but would not be canonical in
the set of fragments illustrated in Figure 6.
For a set of canonical fragments, the containment relation provides us with a hierarchy of fragments.
It is illustrated in Figure 6(b), in which each node represents a canonical fragment and edges hint at
containment relation of fragments. Observe that one obtains a tree structure—the RPST. Fragment B2
mentioned earlier comprises two fragments, P4 and P5.
If the RPST is computed for a normalized workflow graph, i.e., a
workflow graph that does not contain nodes with multiple incoming
and multiple outgoing edges, each canonical fragment can be classified
to one out of four structural classes [53, 54]: A trivial (T) fragment
consists of a single edge. A polygon (P) represents a sequence of
fragments. This type of fragment represents the most simple control
flow structure, a sequence of transitions or fragments, respectively.
A bond (B) stands for a collection of fragments that share common
boundary nodes. This type of fragment represents a wellstructured part of the graph, see [37, 52] for
a discussion of wellstructuredness of workflow graphs. For such a fragment, the behavioural relation
between the contained transitions or subnets depends on the entry of the fragment, since there are no
edges between the children of a bond fragment. Any fragment that does not match the requirements
for the aforementioned classes is a rigid (R). In this article, we use fragment names that hint at their
structural class, e.g., R1 is a rigid fragment. Every workflow graph can be normalized by performing a
nodesplitting preprocessing step, illustrated for WFnets in Figure 7. The WFsystem in Figure 6(a) is
normalized.
Figure 7.Nodesplitting
5.2.An Annotated RPST: The WFTree
The structural patterns derived by the RPST can be related to behavioural properties of the underlying
WFsystem. In the previous section, we already mentioned that the structural characteristics, especially of
polygon and bond fragments, are close to control flow routing concepts. In this section, we concretise
RPST fragments by annotating them with behavioural characteristics. As such, we explicitly establish
the relation between structural and behavioural characteristics, which is the basis for our approach to the
computation of causal behavioural profiles.
We refer to the containment hierarchy of annotated canonical fragments of a WFsystem as the RPST
with behavioural annotations, or WFtree for short. The WFtree is defined for sound freechoice WF
systems. It is wellknown that the freechoice and soundness properties are required to derive behavioural
statements from the structure of a system, as both together imply a tight coupling of syntax and semantics
(cf., [3, 36]).
Definition 5.2. (WFTree)
Let (N,Mi) be a sound freechoice WFsystem. The RPST with behavioural annotations, the WFTree of
N, is a tuple TN= (Ω,χ,t,b), where:
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14M. Weidlich, A. Polyvyanyy, J. Mendling, M. Weske/Causal Behavioural Profiles
◦ Ω is a set of all canonical fragments of N,
◦ χ : Ω → P(Ω) is a function that assigns to fragment its child fragments,
◦ t : Ω → {T,P,B,R} is a function that assigns a type to a fragment,
◦ b : ΩB→ {B◦,B?,L}, ΩB= {ω ∈ Ω  t(ω) = B}, is a function that assigns a refined type to a
bond fragment, where B◦, B?, and L types stand for place bordered, transition bordered, and loop
bonds, respectively.
The intuition behind the annotation of bond fragments is summarised as follows. If the bond is acyclic, a
placebordered bond represents a wellstructured part that incorporates an exclusive choice. That is, there
is a conflict between the transitions that are part of different children of the fragment for a token in the
place that represents the entry of the bond. A transitionbordered bond represents a wellstructured part
that enables concurrent processing. Firing of the transition that is the entry of such a fragment marks a
place for each of the children fragments. A loop bond is a cyclic bond. There is at least one path from the
entry to the exit, and vice versa. Since there are no paths between nodes of different children, therefore, a
loop bond represents a wellstructured control flow cycle.
Further, we define auxiliary concepts for the WFtree.
Definition 5.3. (Parent, Child, Root, Ancestor, Descendant, LCA, Path)
Let TN= (Ω,χ,t,b) be the WFtree.
◦ For any fragment ω ∈ Ω, ω is a parent of γ and γ is a child of ω, if γ ∈ χ(ω). By χ+we denote
the irreflexive transitive closure of χ.
◦ The fragment ω ∈ Ω is a root of T , denoted by ωr, if it has no parent.
◦ The partial function ρ : Ω \ {ωr} → Ω assigns parents to fragments.
◦ For any fragment ω ∈ Ω, ω is an ancestor of ϑ and ϑ is a descendant of ω, if ϑ ∈ χ+(ω).
◦ For any two fragments ω,γ ∈ Ω their lowest common ancestor (LCA), denoted by lca(ω,γ), is
the shared ancestor of ω and γ that is located farthest from the root of the WFtree. By definition,
lca(ω,ω) = ω.
◦ For any fragment ω0∈ Ω and its descendant ωn∈ Ω, a downward path from ω0to ωn, denoted by
πT(ω0,ωn), is a sequence (ω0,ω1,...,ωn), such that ωiis a parent of ωi+1for all i ∈ 0...n−1. In
addition, πT(ω0,ωn,i) = ωiand πT{ω0,ωn} is a set which contains all fragments of πT(ω0,ωn).
Figure 8 shows the WFtree of the WFsystem from Figure 6(a).
The WFtree is isomorphic to the RPST of the WFsystem, cf.,
Figure 6(b). Given the RPST, adding the behavioural annotation is
a trivial task for most fragments, except of the following cases: A
bond fragment γ = (Pγ,Tγ,Fγ) ∈ dom(b) of TN = (Ω,χ,t,b)
is assigned the L type, if there exists a fragment ω ∈ Ω, such that
it is a child of γ and γ?= ω?. Otherwise, b(γ) = B◦if γ?∈ Pγ,
or b(γ) = B?if γ?∈ Tγ.
Children of a polygon fragment are arranged with respect
to their execution order. A partial function order : Ω?→ N0,
Ω?= {ω ∈ Ω \ {ωr}  t(ρ(ω)) = P} assigns to children of
polygon fragments their respective order positions; order(ω) = 0,
if ω?= γ?with γ = ρ(ω) being the parent, and order(ω) = i, i ∈ N, if ω?= ϑ?for some ϑ ∈ Ω, such
that order(ϑ) = i − 1. Observe that the orders of two nodes are only comparable if they share a common
P1
B 1
P4P5
P6P7
L1
B○1
P2
P8
P9
P10
P11
R1
P3
P12
Figure 8.The WFtree
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M. Weidlich, A. Polyvyanyy, J. Mendling, M. Weske/Causal Behavioural Profiles15
parent. For instance, in Figure 8, order(L1) = 1 and order(B◦1) = 2. This means that fragment L1
is always executed before fragment B◦1 inside of polygon P2. Note that a trivial fragment resides at
position 0 in fragment P2. The layout of child fragments of polygon fragments in Figure 8 hints at their
order relations.
Children of a loop fragment are classified as forward (⇒) or backward (⇐). A partial function
? : Ω??→ {⇐,⇒} with Ω??= {ω ∈ Ω \ {ωr}  b(ρ(ω)) = L} assigns an orientation to children of
loop fragments. ?(ω) =⇒ if ω?= γ?with γ = ρ(ω), otherwise ?(ω) =⇐. In Figure 8, P4 and P5 are
forward and backward fragments, respectively, which is visualised by the direction of edges.
We introduce two lemmas that prove the completeness of the codomain of function b by showing that
a bond fragment is either place or transition bordered, and that each loop fragment is place bordered. Note
that a rigid fragment bordered with a place and a transition can still be freechoice and sound (see [4]).
Lemma 5.1. Let TN = (Ω,χ,t,b) be the WFtree of a sound freechoice WFsystem (N,Mi), N =
(P,T,F). No bond fragment ω ∈ Ω, t(ω) = B, has {p,t} boundary nodes, where p ∈ P and t ∈ T.
Proof:
Assume ω is a bond fragment with {p,t} boundary nodes. There exists a circuit Γ in a shortcircuit net
of N that contains {p,t}. Let Γωbe a subpath of Γ inside ω. There exists a child fragment γ of ω that
contains Γω. A bond fragment has k ≥ 2 child fragments, cf., [54, 53]. Let ϑ be a child of ω, ϑ ?= γ. We
distinguish two cases:
◦ Let H be a path from p to t contained in ϑ. H is a PThandle of Γ. In a live and bounded freechoice
system, H is bridged to Γωthrough a TPbridge K, cf., Proposition 4.2 in [26]. This implies that
ϑ = γ; otherwise bond fragment ω contains path K that is not inside of a single child fragment,
cf., [53, 54]. Thus, ω has a single child fragment, a contradiction with the assumption of ω being a
bond fragment.
◦ Let H be a path from t to p contained in ϑ. H is a TPhandle of Γ. In a live and bounded freechoice
system, no circuit has TPhandles, cf., Proposition 4.1 in [26], which yields a contradiction with
our assumptions.
? ?
Lemma 5.2. Let TN= (Ω,χ,t,b) be the WFtree of a sound freechoice WFsystem, (N,Mi), N =
(P,T,F). A loop fragment ω = (Pω,Tω,Fω) ∈ Ω, b(ω) = L, is place bordered, i.e., {ω?,ω?} ∈ P.
Proof:
Because of Lemma 5.1, ω is either place or transition bordered. Assume ω is transition bordered. There
exists place p such that p ∈ •ω?∩ Pω, Mi(p) = 0. Transition ω?is enabled if there exists a marking
M ∈ [(N,Mi)? with M(p) > 0. Since ω is a connected subnet, for all t ∈ Tω\ {ω?,ω?} all edges are in
ω, i.e., (inN(t)∪outN(t)) ⊆ Fω. Thus, every path from i to p visits ω?. M(p) > 0 is only possible, if ω?
has fired before. We reached a contradiction. Transition ω?is never enabled and N is not live, and hence,
not sound. Since any loop fragment is not transition bordered, it is place bordered (Lemma 5.1).
? ?
For sound freechoice WFsystems, the WFtree can be derived efficiently.
Corollary 5.1. The following problem can be solved in linear time.
Given a sound freechoice WFsystem, to compute its WFtree.