4 Excursion: On the Voting Example
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Fig. 4. A corrected negotiation for unanimous vote.
transition is enabled if the acceptance place carries all k tokens, whereas for each
possible distribution of k tokens on the two places we deﬁne a separate rejecting
transition. As above, we might exclude the rejecting transition consuming k
tokens from the acceptance place. Formally, we deﬁne a place/transition Petri
net with arc weights. Apparently this net has linear size with respect to k, the
number of agents. The net is k-bounded, i.e., no reachable marking assigns more
than k tokens to a place.
Another possibility of modelling the same behaviour is to provide k places for
acceptance, one for each agent, and k places for rejection, as in our ﬁrst Petri net.
Now we add, for each agent, a transition moving a token from the corresponding
acceptance place to the rejection place. This transition is labelled by the empty
word τ . No matter how the agents voted, we can reach the marking with all tokens
on the reject places by ﬁring these τ -labelled transitions. Therefore, it suﬃces to
have one acceptance transition that removes tokens from all acceptance places
and one rejection transition that removes tokens from all rejection places. Firing
a τ -labelled transition does not contribute to the observed behaviour of the Petri
net. So this net is at least language equivalent to the negotiation of Fig. 3.
Summarizing, we have provided a systematic way to construct a 1-safe Petri
net corresponding to a negotiation, which can be exponentially larger than the
negotiation. For the voting example, this Petri net has exponentially many transitions. For this example we also provided a linear-sized Petri net with the same
behaviour, which is, however, not 1-safe but k-bounded. Another Petri net with
this behaviour is 1-safe, but has τ -labelled transitions. We actually do not know
if, for negotiations in general, there always exist polynomial-sized Petri nets
with the same behaviour which are 1-safe, which are bounded, which have no
Negotiations and Petri Nets
217
τ -labelled transitions etc., i.e. all these problems are open. For the voting example, we did not ﬁnd a polynomial-sized equivalent 1-safe Petri net without τ labelled transitions.
4
4.1
Properties of the Net Associated with a Negotiation
S-components
An S-component of a Petri net is a subnet such that, for each place of the
subnet, all input- and output-transitions belong to the subnet as well, and such
that each transition of the subnet has exactly one input- and exactly one outputplace of the subnet [6]. It is immediate to see that the number of tokens in an
S-component never changes. A net is covered by S-components if each place
and each transition belongs to an S-component. Nets covered by S-components
carrying exactly one token are necessarily 1-safe. For example, every live and
1-safe free-choice net enjoys this nice property [6].
Petri nets associated with negotiations are not covered by S-components,
only because the end-transitions have no output places. However, if we add an
arc from each end-transition to each initially marked place, then the resulting
net is covered by S-components:
Proposition 4. The Petri net associated with a negotiation, with additional
arcs from each end-transition to each initially marked place, is covered by Scomponents.
Proof. We consider the Petri net with the additional arcs. For each agent a, the
subnet generated by all places [a, X] and all transitions labelled by (n, r) where
a ∈ Pn , is an S-component (being generated implies that the arcs of the subnet
are all arcs of the original net connecting nodes of the subnet). An arbitrary place
of the net belongs to one such subnet, because it corresponds to an agent. Each
transition has a label (n, r), and each atom n has a nonempty set of participants,
whence the transition belongs to the subnet of some agent.
4.2
Soundness
The following notion of sound negotiations was inspired by van der Aalst’s soundness of workﬂow nets [1]. It was ﬁrst deﬁned in [8].
Definition 9. A negotiation is sound if each outcome occurs in some initial
occurrence sequence and if, moreover, each finite occurrence sequence is a large
step or can be extended to a large step.
All the negotiations shown in the ﬁgures of this paper are sound. For an
example of an unsound negotiation, consider again the ping-pong negotiation
shown in Fig. 1 on the right hand side. Imagine that Daughter could choose to
start negotiating with Father or with Mother. This would formally be expressed
by replacing the arc from port D of n0 to port D of nFD by a hyperarc from port D
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of n0 to ports D of both nFD and nDM . If, in this modiﬁed distributed negotiation,
Daughter ﬁrst negotiates successfully with Mother, a marking is reached where
both Daughter and Mother can only engage in the ﬁnal atom nf , whereas Father
is still only able to participate in nFD . So the distributed negotiation has reached
a marking which is neither ﬁnal nor enables any outcome. We call such a marking
a deadlock. Clearly, sound negotiations have no reachable deadlocks.
Since the Petri nets associated with negotiations are not workﬂow nets, we
cannot immediately compare the soundness notions of workﬂow nets and of negotiations. Instead we deﬁne additionally in/out-nets associated with negotiations,
which are obtained by a minor transformation from the originally constructed
Petri nets. These in/out-nets are a generalisation of workﬂow nets, as deﬁned in
[1]. Soundness, as deﬁned for workﬂow nets in [1], is generalized to in/out-nets
in the following deﬁnitions.
Definition 10. An in/out-net is a Petri net with two distinguished places pin
and pout such that pin has no input transition and pout has no output transition.
The initial marking of an in/out-net assigns one token to the place pin and
no token to all other places. In/out-nets also have a ﬁnal marking, assigning one
token to pout and no token to all other places.
An in/out-net net is sound if it has no dead transitions (i.e., each transition
belongs to an initially enabled occurrence sequence) and, moreover, each initially
enabled occurrence sequence is a prefix of an occurrence sequence leading to the
final marking.
A workﬂow net is an in/out-net such that, for each place or transition x,
there are directed paths from pin to x and from x to pout .
Now we associate in/out-nets with negotiations.
Definition 11. The in/out-net associated with a negotiation is obtained from
the Petri net associated with the negotiation by the following transformations:
1. The Petri net associated with the negotiation has, for each participating agent,
an initially marked place. We delete all except one of these places and adjacent
arcs and rename the remaining initially marked place to pin .
2. We add an initially unmarked place pout and arcs from all transitions labelled
by outcomes of the final atom nf (which we called end before) to this place.
In/out-nets associated with negotiations are not necessarily workﬂow nets
because not every element is necessarily on a path from the initial place to the
ﬁnal place. However, this condition holds if the negotiation is sound, as the
following proposition shows.
Proposition 5. The in/out-net associated with a sound negotiation is a workflow net.
Proof. By construction, the in/out-net has distinguished places pin and pout .
By deﬁnition of a distributed negotiation, the initial atom is not a possible
next atom for any atom and any agent, i.e., it does not belong to any X(n, a, r).
Negotiations and Petri Nets
219
Hence, by construction, the initially marked places of the Petri net associated
with the negotiation have no ingoing arcs. Since pin is one of these places, it has
no ingoing arc.
The new place pout has no outgoing arc.
Since, by soundness of the negotiation, every atom (and therefore every outcome) can be enabled, a token can be moved from the initial atom to any other
atom. Therefore, there is a directed path from the initial atom to any other
atom (more precisely, there is a path in the graph of the negotiation). By the
construction of the Petri net (and of the in/out-net), there are according paths
from the place pin to arbitrary places and transitions of the net.
Again by soundness of the negotiation, every occurrence sequence can be
extended to a large step, i.e., the ﬁnal atom can eventually be enabled and the
ﬁnal marking reached. So every token can be led to the ﬁnal atom, and therefore
there are paths in the graph of the negotiation from every atom to the ﬁnal
atom. By construction of the Petri net (and of the in/out-net), there are thus
paths from any element to an end-transition, where end is an outcome of nf ,
and – in the in/out-net – to the place pout .
Next we show that, for sound negotiations, the associated Petri net and
the associated in/out-net are behaviourally equivalent. To this end, we formally
introduce an equivalence relation on the set of Petri nets:
Definition 12. Two Petri nets N and N are in the relation R if
– N is obtained from N by the deletion of a place p and adjacent arcs and
– the reachability graphs of N and N are isomorphic.
The symmetrical, reflexive and transitive closure of R is called place equivalence.
Places p satisfying the condition of this deﬁnition are often called implicit.
Clearly, by construction place equivalence is an equivalence relation.
Lemma 3.
(a) Let N be a Petri net with two places p and p with identical sets of input
transitions, identical sets of output transitions and identical initial marking.
Then deletion of p together with adjacent arcs leads to a place-equivalent
net.
(b) Let N be a Petri net with a place p with no output transition. Assume that
there are no two distinct reachable markings m and m that disagree only
with respect to p, i.e., that satisfy
m(p) = m (p) and m(p ) = m (p ) for p = p .
Then deletion of p and adjacent arcs leads to a place-equivalent net.2
2
Without the second condition, i.e., assuming only that p has no output transitions,
the derived net is a bisimular net. It has in particular identical occurrence sequences
as the original one, but it can have a smaller reachability graph because distinct
reachable markings might diﬀer only with respect to the place p.
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Proof.
(a) The nets N and N are obviously in the relation R as deﬁned in Deﬁnition 12.
(b) Clearly, removing p does not change the behaviour in terms of occurrence
sequences because a place can only restrict the enabledness of its output
transitions, but p has no output transitions. The second assumption implies
moreover that, for each reachable marking m, the number m(p) follows
uniquely from all m(p ), p = p . So we have a bijective mapping from reachable markings of the Petri net N to reachable markings of the reduced net,
which is formally given by the projection of markings to the set of places
without p. It is easy to see that this bijection actually induces an isomorphism between the two reachability graphs.
Using this lemma we now show that, at least for sound negotiations, the
associated Petri net and the associated in/out-net have the same behaviour.
Proposition 6. Let N be a sound negotiation. The reachability graph of its associated Petri net is isomorphic to the reachability graph of its associated in/outnet.
Proof. As argued in the proof of Proposition 5, the initially marked places of
the Petri net associated with the negotiation have no ingoing arcs. Since the
initial atom of the negotiation has all agents as participants, the transitions corresponding to its outcomes consume the tokens from all initially marked places.
Therefore, all these places have the same (empty) set of input transitions and
the same set of output transitions. So Lemma 3(a) applies and proves that the
transformation leads to a net with identical reachability graph.
Next we show that adding the place pout also does not change behaviour. We
argue considering the in/out-net with the place pout and show that removing
this place leads to a net with isomorphic reachability graph. We aim at applying
Lemma 3(b), and thus have to show that no two distinct reachable markings of
the in/out-net diﬀer only with respect to the marking of pout .
By construction of the Petri net and of the corresponding in/out-net associated to the negotiation, ﬁring a transition labelled with an outcome of the ﬁnal
atom removes all tokens from the net. This is because all agents participate in
the ﬁnal atom. Conversely, these transitions are the only transitions which do
not produce tokens on some places. Therefore, there are tokens in the Petri net
before one of these transitions occurs and there are no tokens in the Petri net
afterwords. In particular, there can only be one occurrence of such a transition.
In the in/out-net, occurrences of transitions representing ﬁnal outcomes add a
token to the place pout and no other transition changes the marking of this
place. Therefore, before the occurrence of a transition labelled by a ﬁnal outcome there are marked places (one for each participant) and pout is unmarked.
After the occurrence of a transition labelled by a ﬁnal outcome, pout is the only
marked place. So no two reachable markings diﬀer only with respect to pout , and
Lemma 3(b) applies.
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Unfortunately, soundness of a negotiation does not necessarily imply soundness of the associated in/out-net (which is, by Proposition 5, a workﬂow net).
The reason is that soundness requires that every atom can occur but not that
every branch of a hyperarc is actually used. If, for example, there was an additional hyperarc in Fig. 1 from the port F in n0 to the ports F in nFD and nf instead
of the arc from n0 to nFD , then the resulting negotiation would still be sound;
actually, the behaviour does not change at all. The associated in/out-net, however, would have an additional transition end with new input place [F, {nFD , nf }]
(and other input places) which is never enabled. This net is therefore not sound.
4.3
Deterministic Negotiations
In [9], we concentrate on deterministic negotiations, which are negotiations without proper hyperarcs.
Definition 13. A negotiation is deterministic if, for each atom n, agent a ∈ Pn
and result r ∈ Rn , X(n, a, r) contains at most one atom (and no atom only if
n = nf ).
The term deterministic is justiﬁed because there is no choice for an agent
with respect to the next possible atoms.
Since both, the exponential blow-up and the problem of useless arcs (branches
of hyperarcs) stem from proper hyperarcs, we can expect that deterministic
negotiations allow for better results. Actually, the Petri net associated with a
deterministic negotiation is in fact much smaller, because all its places have the
form [a, X], where a is an agent and X is a singleton set of atoms. So the set of
places is linear in agents and in atoms.
Before discussing soundness of deterministic negotiations, we make a structural observation. For the deﬁnition of free-choice nets used here, see [6].
Proposition 7. The Petri net associated with a deterministic negotiation is a
free-choice net, i.e., every two places either share no output transitions, or they
share all their output transitions. The same holds for the in/out-net associated
with a deterministic negotiation.
Proof. Since, in a net associated with a deterministic negotiation, each place
has the form [a, X], where X is a singleton set {n}, all its output transitions are
labelled by (n, r), r being a possible result of n. By construction, every other
place [b, {n}] has exactly the same output transitions as [a, {n}] whereas all other
places have no common output transition with [a, {n}].
The transformations of Deﬁnition 11 do not destroy the free-choice
property.
Proposition 8. A deterministic negotiation is sound if and only if its associated
in/out-net is sound.
Proof. The translation from a negotiation to its associated Petri net can be
rephrased in a much simpler way if the negotiation is deterministic, as follows:
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– For each atom n and each a in Pn , we add a place [a, n].
– For each atom n and result r ∈ Rn , we add a transition (n, r) (no two transitions correspond to the same outcome (n, r), so transitions can be identiﬁed
with their previously used labels).
– Arcs connect all places [a, n] with all transitions (n, r).
– For each transition (n, r) with n = nf and each a ∈ Pn there is an arc from
(n, r) to [a, n ], where n is the unique atom in X(n, a, r).
– All places [a, n0 ] carry one token initially; all other places are initially
unmarked.
It is immediate that to see that the behaviour of the negotiation is precisely
mimicked by this Petri net. So the negotiation is sound if and only if the net has
no dead transitions and moreover can reach the ﬁnal (empty) marking from any
reachable marking.
The result follows since the Petri net can, as above, be transformed into a
behaviourally equivalent in/out-net.
Combining Propositions 5 and 8 yields:
Corollary 4. If a negotiation is deterministic and sound then its associated
in/out-net is a sound workflow net.
5
From Petri Nets to Negotiations
In this section we study the converse direction: Given a labelled Petri net, is there
a negotiation such that the net is associated with the negotiation? Obviously,
for a positive answer the net has to enjoy all the properties derived before. In
particular, it must have disjoint S-components and initially marked input places.
However, in the general case it appears to be diﬃcult to characterise nets that
have corresponding negotiations.
We will provide an answer for the case of sound deterministic negotiations
and sound free-choice workﬂow nets.
Proposition 9. Every sound free-choice workflow net is place equivalent to a
net which is associated with a sound deterministic negotiation.
Proof. A workﬂow net is sound if and only if the net with an additional feedback transition moving the token from pout back to pin is live and 1-safe [1]. Live
and 1-safe free-choice nets are covered by S-components [6]. Therefore a sound
free-choice workﬂow net is covered by S-components as well. However, these
S-components have not necessarily disjoint sets of places. Consequently, we
cannot easily ﬁnd candidates for agents involved in the negotiation to be constructed.
Instead we proceed as follows: We choose a minimal set of S-components
that cover the net. Since each S-component of a live net has to carry a token, all
these S-components contain the place pin . Each S-component corresponds to an
agent of the net to be constructed. Each conﬂict cluster, i.e., each set of places
Negotiations and Petri Nets
223
sharing the same output transitions, corresponds a negotiation atom (remember
that the net is free-choice and therefore any two places either share all output
transitions or do not share any).
Each place p of the net is contained in at least one S-component of the cover.
Let Cp be the set of all S-components of the derived minimal cover containing
p. If Cp contains more than one S-component, we duplicate the place p, getting
a new place p with input and output transitions like p.
The new net still has a cover by S-components, where one of the
S-components containing p now contains p instead. Repetition of this procedure
eventually leads to a net where each place p belongs to exactly one S-component
Cp of the cover. Finally we delete the place pout . Both operations, duplication
of places and deletion of pout , lead to place-equivalent nets by Lemma 3.
The resulting net is associated with the following negotiation: The set of
agents is the set of S-components of the minimal cover. The atoms are the
conﬂict clusters of the net. The results of an atom are the transitions of the
corresponding conﬂict cluster. The X-function can be derived from the arcs of
the Petri net leading from transitions to places.
6
Conclusions
This contribution presented a translation from a distributed negotiation to a
behaviourally equivalent Petri net. The chosen notion of behavioural equivalence
is very strong, namely isomorphism of the reachability graphs.
In the worst case, the translation yields a Petri net exponentially larger than
the negotiation. We conjecture that this exponential blow-up is unavoidable, but
currently we do not have a proof. The problem of the succinctness of negotiations
with respect to weaker equivalence notions like bisimulation or language equivalence is also open. On the other hand, we have shown that for deterministic
negotiations the translation only causes a linear growth. Further, for deterministic negotiations soundness and non-soundness is respected by the transformation
to workﬂow-like Petri nets, whence in this case the reverse translation is possible
as well.
The translation to Petri nets is implicitly used in [8,9], and in a recent paper
on the analysis of Coloured Workﬂow Nets [11]. On the one hand, the fact that
deterministic negotiations are so closed to workﬂow free-choice nets guided our
eﬀorts to obtain a reduction algorithm for the analysis of soundness and the
input/output relation of negotiations. On the other hand, in [11] we transferred
the reduction procedure back to Petri nets. The resulting reduction procedure
has been successfully applied to a collection of industrial workﬂows.
Since we do not currently have a large suite of negotiation models, while such
suites exist for workﬂow Petri nets, we have used negotiations mostly as a theoretical formalism to design new analysis techniques that can be later translated
to workﬂow nets. In future work we plan to analyze the connection between
negotiations and languages for the description of business processes. Negotiations could become an intermediate language between business processes and
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Petri nets, oﬀering more compact descriptions and cleaner analysis procedures
and the possibility to apply the highly developed tool support for Petri net
analysis. This includes in particular model checking tools that can verify properties formulated in an appropriate Temporal Logic. Application of such tools to
negotiations requires not only prior transformation of the model but also of the
formula. So we are interested in appropriate languages for formalizing relevant
behavioural properties of negotiations.
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A Formal Framework for Diagnostic Analysis
for Errors of Business Processes
Suman Roy1(B) and A.S.M. Sajeev2
1
Infosys Ltd., #44 Electronics City, Hosur Road, Bangalore 560 100, India
Suman Roy@infosys.com
2
Melbourne Institute of Technology, Sydney, NSW 2000, Australia
asmsajeev2@gmail.com
Abstract. Business process models expressed in languages such as
BPMN (Business Process Model and Notation), play a critical role in
implementing the workﬂows in modern enterprises. However, control
ﬂow errors such as deadlocks and lack of synchronization, and syntactic
errors arising out of poor modeling practices often occur in industrial
process models. A major challenge is to provide the means and methods
to detect such errors and more importantly, to identify the location of
each error. In this work, we develop a formal framework of diagnosing
errors by locating their occurrence nodes in business process models at
the level of sub-processes and swim-lanes. We use graph-theoretic techniques and Petri net-based analyses to detect syntactic and control ﬂowrelated errors respectively. While syntactic errors can be easily located on
the processes themselves, we project control-related errors on processes
using a mapping from Petri nets to processes. We use this framework to
analyze a sample of 174 industrial BPMN process models having 1262
sub-processes in which we identify more than 2000 errors. We are further able to discover how error frequencies change with error depth, how
they correlate with the size of the sub-processes and swim-lane interactions in the models, and how they can be predicted in terms of process
metrics like sub-process size, coeﬃcient of connectivity, sequentiality and
structuredness.
Keywords: Veriﬁcation · Formal methods · Processes · BPM Notation ·
Errors · Soundness · Petri nets · Workﬂow nets · Woﬂan · Diagnosis ·
Metrics
1
Introduction
Modern-day enterprises rely on streamlined business processes for implementing
the workﬂows in the operation. This is particularly important for internet-based
businesses where on-line processes such as accepting orders need to be seamlessly
integrated with physical processes like delivery of products. Correct implementation of process models can result in signiﬁcant cost savings in industry. For
example, Hammer [Ham10] reports a computer manufacturer reducing time to
c Springer-Verlag Berlin Heidelberg 2016
M. Koutny et al. (Eds.): ToPNoC XI, LNCS 9930, pp. 226–261, 2016.
DOI: 10.1007/978-3-662-53401-4 11