The Effect of Noise on Mined Declarative Constraints
Abstract
Declarative models are increasingly utilized as representational format in process mining. Models created from automatic process discovery are meant to summarize complex behaviors in a compact way. Therefore, declarative models do not define all permissible behavior directly, but instead define constraints that must be met by each trace of the business process. While declarative models provide compactness, it is up until now not clear how robust or sensitive different constraints are with respect to noise. In this paper, we investigate this question from two angles. First, we establish a constraint hierarchy based on formal relationships between the different types of Declare constraints. Second, we conduct a sensitivity analysis to investigate the effect of noise on different types of declarative rules. Our analysis reveals that an increasing degree of noise reduces support of many constraints. However, this effect is moderate on most of the constraint types, which supports the suitability of Declare for mining event logs with noise.
Keywords
Process mining Declarative workflows Noisy event logs1 Introduction
Automated process discovery is an important field of research in the area of process mining. The goal of process discovery is to generate a process model from the behavior captured in an event log. In this context, process models can be represented in different formats. There is ongoing research that aims at establishing which representations are best suited for describing the behaviour. While procedural languages like Petri nets have been found appropriate for structured processes, it is believed that declarative languages such as Declare yield a more compact representation for socalled Spaghetti processes, which are processes with a high degree of concurrency [1]. It has also been argued that Petri nets are better to communicate how a process can progress, while Declare models are good at describing the circumstances of execution of a particular activity [2, 3, 4].
Beyond these mutual strengths and weaknesses, one of the important matters of automated process discovery is robustness to noise. There has been extensive research into techniques to abstract from noise for procedural languages, which resulted among others in the heuristics miner [5], the fuzzy miner [6], and in an approach based on genetic mining [7]. In contrast to this, a detailed discussion of the effects of noise on declarative models is missing so far. Noisy logs/traces can be natural when discovering processes in unconventional scenarios, e.g., discovering “artful processes” carried out by knowledge workers through collaboration tools and email messages [8]. In such cases, logs are derived through object matching and text mining techniques applied to communication messages, and therefore the presence of noise is inevitable.
In this paper, we address this question from two angles. First, we investigate in how far different Declare constraints are robust to noise. To this end, we develop a constraint hierarchy that builds on formal relationships between the constraint types. Second, we conduct simulation experiments in order to study the degree of robustness of different Declare constraints. Based on these two perspectives, we gain insights into general properties of Declare with respect to noise.
Against this background, the remainder of this paper is structured as follows. Section 2 discusses the background of this research. In particular, we formally define Declare constraints. Section 3 discusses formal relationships between different constraint types and defines a hierarchy, which provides the basis for formulating experimental hypotheses listed in Sect. 4. Section 5 defines an experimental setup, which we use to investigate the hypotheses. Section 6 discusses our findings in the light of related work. Section 7 concludes the paper.
2 Declare Constraints
Declare constraints.


Existence constraints constrain single activities;

Cardinality constraints are existence constraints specifying the count of constrained activities;

Position constraints are existence constraints specifying the position of constrained activities;

Relation constraints constrain pairs of activities;

Coupling relation constraints are satisfied only when two relation constraints are satisfied;

Negative relation constraints negate coupling relation constraints.
As a consequence, existence constraints refer to single activities, whereas the other types constrain them in pairs. As explained in [13, 14], relation constraints are activated by the occurrence of an activity (named as “activation” in [13], or “implying” in [14]). When activated, they force the occurrence of the other activity in the pair (“target” [13] or “implied” [14]). If no activating task is performed during the process execution, the constraint imposes no condition on the rest of the enactment. For instance, if no \(\mathsf {a}\) is performed, \( RespondedExistence (\mathsf a ,\mathsf b )\) has no effect on the execution, and thus the occurrence of \(\mathsf {b}\) is not required. For coupling relation constraints and negative relation constraints, both involved activities are at the same time implying and implied.
2.1 Declare Constraint Templates as FOL Formulae
We provide the semantics of Declare templates as First Order Logic (FOL) formulae. The approach is inspired by the translation technique from Linear Temporal Logic (LTL) to FOL over finite linear ordered sequences, discussed in [16]. An exhaustive description of the rationale applied to Declare constraint templates can be found in [15]. Formulae 1a–1r are meant to be interpreted over finite traces. Therefore, they adopt variables \(i\), \(j\), \(k\) and \(l\) to indicate positions of events in traces. \( first \) and \( last \) are constants referring to the first and last position in a trace, respectively. \( Succ \) is a binary predicate specifying whether a position follows another. \( InTrace \) binary predicate states whether a given event occurs in the specified position.
Activations and targets for Declare relation constraints, coupling relation constraints, and negative relation constraints. \(\alpha (\mathcal {C})\) and \(\beta (\mathcal {C})\) are respectively the activation and target of constraint \(\mathcal {C}\)

In the following, we will refer to a constraint’s valid activation and target as \(\alpha (\mathcal {C})\) and \(\beta (\mathcal {C})\), respectively. Table 2 lists the activations and targets for each constraint. As the table shows, coupling relation constraints and negative relation constraints are such that both constrained activities play at the same time the roles of activation and target.
3 Constraints’ Properties
The effect of existence constraints and relation constraints on activities.

3.1 How Constraints Affect the Activities
In the light of what stated by natural language (cf. Table 1) and FOL (cf. Formulae 1a–1r), Table 3 specifies how existence constraints and relation constraints affect the execution of activities. In particular, we distinguish between presence and absence for those tasks that are involved by constraints. For instance, \( AtMostOne (\mathsf {a})\) imposes that if the activating event, \(\mathsf {a}\), is found, not any other “\(\mathsf {a}\)” can occur in the trace (absence). With a slight abuse of terminology, we indicate \(\mathsf {a}\) as the activation, even though it is defined for relation constraints only, in the sense that if one \(\mathsf {a}\) occurs, the constraint has effect on the trace. In Table 3, any other occurrence of \(\mathsf {a}\) (resp. \(\mathsf {b}\)) in the trace is pointed at by \(\mathsf {a}'\) (\(\mathsf {b}'\)). \( Response (\mathsf {a},\mathsf {b})\) establishes that, if \(\mathsf {a}\) is found, then \(\mathsf {b}\) must occur afterwards (presence). \( Participation (\mathsf {a})\) has no activating event. However, it imposes the presence of \(\mathsf {a}\) in the trace. \( AlternateResponse (\mathsf {a},\mathsf {b})\) and \( ChainResponse (\mathsf {a},\mathsf {b})\) (resp. \( AlternatePrecedence (\mathsf {a},\mathsf {b})\) and \( ChainPrecedence (\mathsf {a},\mathsf {b})\)) not only constrain the presence of \(\mathsf {b}\) (resp. \(\mathsf {a}\)), as \( Response (\mathsf {a},\mathsf {b})\) (\({ Precedence }(\mathsf {a},\mathsf {b})\)), but also the absence of other \(\mathsf {a}\)’s (\(\mathsf {b}\)’s), under specific conditions. For the sake of comprehensiveness, we recall here that what stated for \( AtMostOne (\mathsf {a})\) and \( Participation (\mathsf {a})\) in Table 3 also applies to \( Absence (m, \mathsf {a})\) and \( Existence (n, \mathsf {a})\), respectively.
3.2 Constraints’ Interdependencies
Constraints under the relation of restriction.

\( forward \) and \( backward \) associations for the conjunction of coupling relation constraints against relation constraints.

Conjunction. Conjunction is a ternary relation among constraints \(\mathcal {C}, \mathcal {C}', \mathcal {C}''\) which holds when \( \mathcal {C}\models \mathcal {C}'\wedge \mathcal {C}''. \) \(\mathcal {C}(\rho ,\sigma )\) is the conjunction of \(\mathcal {C}'(\rho ,\sigma )\) and \(\mathcal {C}''(\rho ,\sigma )\) when only those traces that comply with both \(\mathcal {C}'(\rho ,\sigma )\) and \(\mathcal {C}''(\rho ,\sigma )\) are permitted by \(\mathcal {C}(\rho ,\sigma )\). As an example, \( Succession (\rho ,\sigma )\) is the conjunction of \( Response (\rho ,\sigma )\) and \({ Precedence }(\rho ,\sigma )\). Table 5 report the list of conjunction relations for the Declare constraints under analysis. The conjunction relation is represented by the \( forward \) and \( backward \) associations in Fig. 1. For the sake of readability, the associations are drawn only for the top elements in the hierarchy. They are meant to be inherited by the “descendant” constraints. The terms \( forward \) and \( backward \) refer to the verse in which the pairs of constrained activities become resp. activation and target for the constraints in the conjunction relation (cf. Table 2). For instance, \( CoExistence (\rho ,\sigma )\) is in conjunction relation with \( RespondedExistence (\rho ,\sigma )\) (\( forward \), being \(\rho \) the activation and \(\sigma \) the target) and \( RespondedExistence (\sigma ,\rho )\) (\( backward \), being \(\sigma \) the activation and \(\rho \) the target).
Negated relations for \( NegativeRelation \) constraints

We do not report formal proofs confirming the observations made so far, for the sake of space. However, they can be trivially verified by considering Formulae 1a–1r and the textual descriptions provided in Table 1.
4 Hypotheses on the Reaction of Constraints to Noise
Building upon the properties of Declare constraints, shown in Sects. 3.1 and 3.2, we have formulated ten hypotheses, relating the characteristics of constraints discussed so far to their sensitivity or resilience to noise in logs. For the formulation of hypotheses, we have considered two specific abstractions for the effects that noise can cause on logs: (i) presence of spurious events in traces (insertion errors), and (ii) events missing in traces (absence errors). The hypotheses have driven the experiments detailed in Sect. 5, conducted in order to have an experimental evidence of conclusions drawn from the theoretical analysis of Declare constraints.

H1 Cardinality constraints requiring the presence of an activity are resilient to insertion errors and sensitive to deletion errors on such an activity.

H2 Cardinality constraints requiring the absence of an activity are resilient to deletion errors and sensitive to insertion errors on the referred activity.

H3 Position constraints are resilient to insertion errors and sensitive to deletion errors on the constrained activity.^{2}

H4 All constraints having an activation are resilient to the absence of activation events.

H5 All constraints having an activation are sensitive to the presence of spurious activation events.

H6 All constraints requiring the presence of the target are resilient to the presence of spurious target events.

H7 All constraints requiring the presence of the target are sensitive to the absence of target events.

H8 Coupling relation constraints inherit the sensitivity of those constraints of which they are the conjunction.

H9 Negative relation constraints are sensitive to the presence of constrained activities and resilient to their absence.

H10 Along the restriction hierarchy, descendant constraints are more sensitive than ancestors to the presence of noise.
5 Evaluation
In order to observe the change in the mined models due to errors in logs, we have created errorinjected logs. Section 5.1 describes how we apply different categories of noise to logs complying to one constraint at a time, each representing a constraint template. Section 5.2 illustrates the experimental setup, in order to observe the reactions of constraints to different kinds of noise. Sections 5.3–5.10 present in detail the results for each of the hypotheses defined above. Section 5.11 summarizes the gathered insights and closes this section.
5.1 Noise Categories
 1.
Noise type: it can be either one of the following: (a) insertion of spurious events in the log; (b) deletion of events from the log; (c) random insertion/deletion of events.
 2.
Noise injection rate, ranging from \(0\) to \(100\,\%\).
 3.
Noise spreading policy; it can be either one of the following: (a) distribution of noise in every trace (tracebased); (b) distribution of noise over the entire log (logbased).
 4.
Faulty activity.
Setup of the experiments

5.2 Experiment Setup
We have created 18 groups of 9,300 synthetic logs each (see Table 7). Every group was generated in order to comply with one constraint at a time, among the 18 templates involving a, as the implying activity, and (optionally) b, as the implied (i.e., \( Participation (\mathsf {a})\), \( AtMostOne (\mathsf {a})\), ..., \( RespondedExistence (\mathsf {a},\mathsf {b})\), \( Response (\mathsf {a},\mathsf {b})\), ...). The alphabet comprised \(6\) more nonconstrained activities (c, d, ..., h), totaling 8. Logs have been generated by a specifically adhoc developed software module that utilizes the dk.bricks.automaton library.^{3} This Java tool is capable of generating random strings that comply with userdefined Regular Expressions (REs). In particular, we adopted the DeclaretoRE translation map, discussed in our previous works [8, 14, 17]. We chose a as the faulty activity. The faulty activity plays thus both the role of activation in, e.g., \( Response (\mathsf a ,\mathsf b )\), and the role of target in, e.g., \({ Precedence }(\mathsf a ,\mathsf b )\). Then, we have injected errors in the synthetic logs, with all possible combinations of the aforementioned parameters: (i) insertion, deletion or random noise type, (ii) tracebased or logbased noise spreading policy, (iii) noise injection rate, ranging between \(0\,\%\) and \(30\,\%\). Thereupon, we have run the technique for process discovery presented in [15], on the resulting altered logs. We have collected the results and, for each of the \(18\) groups of logs, analyzed the trend of the support for the generating constraint. In other words, given the only constraint which had to be verified, we have looked at how its support is lowered, w.r.t. the increasing percentage of introduced noise.
5.3 \( Participation \) (H1)
\( Participation \) imposes the presence of the referred activity in every case. Therefore, Fig. 2a and b show that missing occurrences of \(\mathsf {a}\) undermine the detectability of \( Participation (\mathsf {a})\) in the log. Spurious \(\mathsf {a}\)’s do not have any effect on the support of that constraint, as \( Participation (\mathsf {a})\) requires that at least one occurrence of \(\mathsf {a}\) is read in every trace. Therefore, Fig. 2 gives an experimental evidence of H1, as \( Participation \) is a cardinality constraints requiring the presence of the constrained activity.
5.4 \( AtMostOne \) (H2)
\( AtMostOne \) entails a behavior which is dual w.r.t. \( Participation \), as it requires that at least one occurrence of \(\mathsf {a}\) is read in every trace. This is reflected in the opposite receptiveness to the different noise types: for \( AtMostOne (\mathsf {a})\), spurious \(\mathsf {a}\)’s lower the computed support, whereas missing \(\mathsf {a}\)’s have no effect on it. This supports H2, as \( AtMostOne \) is a cardinality constraints requiring the absence of the constrained activity (Fig. 3).
5.5 \( Init \) and \( End \) (H3)
5.6 \( Response \) (H4, H5)
5.7 \( Precedence \) (H6, H7)
5.8 \( Succession \) (H8)
5.9 \( NotCoExistence \) (H9)
Negative relation constraints require that when one of the two referred activities occurs in the trace, the other misses, by definition. Therefore, when any of the two is missing, negative relation constraints will be more likely to be satisfied, either because none of the two is probably in the log, or because at least one of the two misses. This is the reason why Fig. 8b shows that support for \( NotCoExistence (\mathsf {a},\mathsf {b})\) remains fixed to its maximum value, when expunging \(\mathsf {a}\)’s. Vice versa, the insertion of spurious \(\mathsf {a}\)’s makes the support decrease, almost linearly w.r.t. the noise injection rate (see Fig. 8a). This is due to the fact that the newly inserted \(\mathsf {a}\)’s can fall into traces where a \(\mathsf {b}\) lay. The shown behavior supports hypothesis H9.
5.10 The Restriction Hierarchy Under \( CoExistence \) (H10)
5.11 Summary of Experiments
With the tests conducted, we have obtained experimental evidence for all formulated hypotheses. In particular, we have observed that constraints become less resilient to errors, in terms of trend of decreasing support compared to the increasing percentage of introduced noise, along the restriction hierarchy. In general terms, the expunction of activation tasks from traces does not diminish the support of constraints, whereas the insertion of spurious ones can cause traces to become not compliant. Constraints thus tend to be resistant to insertion errors as well as receptive to deletion errors, or viceversa. Nevertheless, we have also seen that those constraints that are the conjunction of other two, inherit the sensitivity of both to noise. All such reactions to noise reflect the characteristics discussed in precedence, referred to the constraints’ definition of activation and target, effects on activities in traces, and interdependencies. This has been extensively explained within the comments to gathered results along this section. Experimental data, though, also show that the effect of noise on support is moderate on most of the constraint types. This supports the suitability of Declare for mining event logs with noise.
6 Related Work
Process Mining, a.k.a. Workflow Mining [1], is the set of techniques that allow the extraction of process descriptions, stemming from a set of recorded real executions (event logs). ProM [18] is one of the most used plugin based software environments for implementing workflow mining techniques. Process Mining mainly covers three different aspects: process discovery, conformance checking and operational support. The first aims at discovering the process model from logs. Controlflow mining in particular focuses on the causal and sequential relations among activities. The second focuses on the assessment of the compliance of a given process model with event logs, and the possible enhancement of the process model in this regard. The third is finally meant to assist the enactment of processes at runtime, based on given process models.
From [19] onwards, many techniques have been proposed for the controlflow mining: pure algorithmic (e.g., \(\alpha \) algorithm, drawn in [20] and its evolution \(\alpha ^{++}\) [21]), heuristic (e.g., [5]), genetic (e.g., [7]), etc. A very smart extension to the previous research work was achieved by the twosteps algorithm proposed in [22]. Differently from the former approaches, which typically provide a single process mining step, it splits the computation in two phases: (i) the configurable mining of a Transition System (TS) representing the process behavior and (ii) the automated construction of a Petri Net bisimilar to the TS [23, 24]. In the field of conformance checking, Fahland et al. [25, 26] have proposed techniques capable of realigning imperative process models to logs.
The need for flexibility in the definition of some types of process, such as the knowledgeintensive processes [27], lead to an alternative to the classical “imperative” approach: the “declarative” approach. Rather than using a procedural language for expressing the allowed sequences of activities (“closed” models), it is based on the description of workflows through the usage of constraints: the idea is that every task can be performed, except what does not respect such constraints (“open” models). The work of van der Aalst et al. [11] showed how the declarative approach (such as the one adopted by Declare [28]) could help in obtaining a fair tradeoff between flexibility in managing collaborative processes and support in controlling and assisting the enactment of workflows. Maggi et al. [13] first outlined an algorithm for mining Declare processes implemented in ProM (Declare Miner), based on LTL verification over finite traces. [29] proposed an evolution of [13], to address at the same time the issues of efficiency of the computation and efficacy of the results. Logicbased approaches to declarative process mining have been proposed by [30, 31, 32, 33]. However, they rely on the presence of prelabeled traces, stating whether they were compliant or not to the correct process execution. For further insight and details, the reader can refer to the work of Montali [34]. Di Ciccio et al. [14, 15, 35] have proposed a further alternative approach, based on heuristicdriven statistical inference over temporal and causal characteristics of the log. De Leoni et al. [36] have first proposed a framework for assessing the conformance of a declarative process to a given log.
In Process Mining, logs are thus usually considered the ground truth from which the process can be discovered. To the best of our knowledge, this is the first study aiming at systematically defining the effect of noise on mined models. In fact, RoggeSolti et al. [37, 38, 39] have tackled the challenge of repairing logs on the basis of statistical information derived from correct logs and imperative process models. In their study, the process model is known a priori, and the objective is to derive a reliable log from one containing missing or incorrect information. Our analysis, instead, tries to shed light on what would happen when mining a previously unknown process from noisy logs, i.e., when no ground truth is provided.
In the area of controlflow mining, proposed approaches such as [5, 7] for imperative models and [14, 29, 33] for declarative ones, allowed for thresholdbased techniques that filter possible outliers out of noisy logs. However, the value for such threshold is left to the choice of the user, who is probably unaware of the best setup. Furthermore, our studies put in evidence how different constraints react with a different degree of sensitivity to noise. Therefore, a single threshold for all constraints could end up being inaccurate.
First studies on their mutual interdependencies have been reported in [14, 40, 41]. The first two were aimed at exploiting such connections in order to make the declarative process mining result more readable, i.e., avoiding redundancies in the returned model. The third elaborated on such analysis to refine compliance models and prune irrelevant constraints out. This paper instead builds upon the characteristics of constraints, in order to have theoretical bases on top of which the level of resilience of constraints to noise is estimated. Experimental results actually support our hypotheses.
7 Conclusion
Throughout this paper, we have analyzed how much the errors affecting event logs have an impact on the discovery of declarative processes. In particular, we have formulated ten hypotheses about the resilience and sensitivity of different Declare constraints, and verified our hypotheses on a set of over 160,000 synthetically generated traces. The specific technique used for discovering control flows out of the traces has no impact on the results, therefore the presented study about the effect of noise in event logs has general validity.
Noisy logs are quite natural when applying workflow discovery techniques to unconventional scenarios, such as inferring collaboration processes out of email messages and/or social network interactions, mining of habits in smart environments (in which sensors may provide faulty measures), etc. The more process discovery techniques will be applied in such scenarios, the more existing techniques, which mainly assume errorfree logs, should be improved in order to cope with noisy logs. Our study is a preliminary, yet foundational step towards the comprehension of how logs are affected by noise and how this impacts the mined constraints, thus providing a solid basement for the development of new more resilient techniques.
Starting from the present study, we aim at investigating in future work the applicability of the presented analysis to declarative languages other than Declare. We will also conduct a dedicated analysis on the effect on mined constraints of a specific category of noise that van der Spoel et al. name sequence noise in [42], i.e., the occurrence of events in a trace in a wrong order. The problem of defining an automated approach for the selfadjustment of userdefined thresholds in process discovery techniques, on the basis of the nature of each discovered constraint, is a future objective too. Intuitively, indeed, a more “robust” constraint should be considered valid in the log (and therefore for the process) if its support exceeds a higher threshold. However, the threshold should be diminished for more “sensitive” ones. We also aim at mixing such an approach with the analysis of different metrics, pertaining to the number of times an event occurred in the log. The intuition is that the more an event is frequent in the log, the less it can be considered subject to errors. Such metrics have been already considered in literature [29] for assessing the relevance of discovered constraints. We want to exploit them for estimating the reliability of constraints in mined processes as well.
Footnotes
 1.
For the sake of readability, we will use the following notation: \(\rho \) and \(\sigma \), to indicate general activities; \(\mathsf {a}, \mathsf {b}, \mathsf {c}, \ldots \), to identify actual activities as well as events in the trace.
 2.
Position constraints behave like cardinality constraints requiring the presence of an activity – cf. H1.
 3.
 4.
We recall that assigning a constraint the support of \(0.5\) would be equivalent to asserting that such constraint would held if, tossing a coin, a cross were shown in the end. Thereby, \(0.75\) is the least value of the topmost half of the “reliable” range.
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