OmegaPRM: Outcome-Oriented Process Supervision

• OmegaPRM is an outcome-oriented prescriptive process supervision framework that employs temporal logic patterns to guide flexible business process execution.
• It utilizes a decision tree-based machine learning model to estimate outcome probabilities and generate interpretable temporal constraints in real time.
• Evaluations on diverse event logs demonstrate that OmegaPRM achieves high reliability (F1 > 90% in most benchmarks) while maintaining operational flexibility.

Process Supervision OmegaPRM is an outcome-oriented prescriptive process monitoring framework that leverages temporal logic patterns to provide flexible, real-time recommendations during the execution of business processes. Designed to maximize the likelihood of achieving desired outcomes, OmegaPRM employs a transparent, interpretable mechanism that moves beyond rigid prescriptive controls by recommending temporal relations among process activities, thus providing both reliability and operational flexibility (Donadello et al., 2022).

1. Mathematical Formulation and Problem Definition

Let $\Sigma$ denote the finite set of activity names in a business process. A trace $\sigma = \langle a_1, \ldots, a_n \rangle$, with $a_i \in \Sigma$, represents a sequence of activities. The event log $L \subseteq \Sigma^*$ comprises observed completed traces, each labeled with $y(\sigma)\in \{0,1\}$ indicating whether the trace yielded a desirable outcome (1) or not (0).

Given a historical labeled log $$L_{\text{train}} = \{(\sigma_1, y(\sigma_1)), \ldots, (\sigma_N, y(\sigma_N))\}$$, the prescriptive process monitoring task is as follows: for each ongoing process case, at prefix $\sigma_k$ (i.e., the first $k$ executed activities), OmegaPRM outputs at each decision point a set $\mathcal{R}_k$ of temporal logic constraints---each constraint associated with a requirement to be satisfied or violated---such that compliance with $\mathcal{R}_k$ maximizes the probability $\sigma = \langle a_1, \ldots, a_n \rangle$0.

OmegaPRM is formulated to satisfy two desiderata:

• Reliability: If the recommendations $\sigma = \langle a_1, \ldots, a_n \rangle$1 are obeyed, the probability of achieving the positive outcome is high (empirically, $\sigma = \langle a_1, \ldots, a_n \rangle$2 in 18 of 22 benchmarks).
• Flexibility: Recommendations are temporal relations (not rigid activity sequences), maintaining operational freedom.

2. Temporal Logic Pattern Encoding

OmegaPRM builds on a catalog of Linear Temporal Logic over finite traces (LTL$\sigma = \langle a_1, \ldots, a_n \rangle$3) patterns, known as Declare templates, to express process-level constraints. Patterns include:

• Existence: $\sigma = \langle a_1, \ldots, a_n \rangle$4 (“Activity $\sigma = \langle a_1, \ldots, a_n \rangle$5 must eventually occur”)
• Absence: $\sigma = \langle a_1, \ldots, a_n \rangle$6 (“Activity $\sigma = \langle a_1, \ldots, a_n \rangle$7 must not occur”)
• Response: $\sigma = \langle a_1, \ldots, a_n \rangle$8 (“If $\sigma = \langle a_1, \ldots, a_n \rangle$9 occurs, $a_i \in \Sigma$0 must eventually occur afterwards”)
• Precedence: $a_i \in \Sigma$1 (“$a_i \in \Sigma$2 can only occur if $a_i \in \Sigma$3 has already occurred”)

A feature mapping $a_i \in \Sigma$4 projects each prefix to a binary vector indicating which patterns are satisfied.

3. Machine Learning Model for Outcome Estimation

OmegaPRM utilizes a machine learning classifier $a_i \in \Sigma$5---specifically, a decision tree (DT)---trained to estimate $a_i \in \Sigma$6, where $a_i \in \Sigma$7. Training uses the Gini impurity on splits, with hyperparameters controlling tree depth and minimum node size. The model learns mappings between combinations of satisfied/violated temporal logic patterns and outcome probabilities.

The classifier $a_i \in \Sigma$8 is trained as: $a_i \in \Sigma$9 where $L \subseteq \Sigma^*$0 denotes the 0--1 loss approximated by Gini impurity.

4. Real-Time Supervision and Recommendation Procedure

At runtime, for an ongoing case at prefix $L \subseteq \Sigma^*$1:

1. Encode: Compute $L \subseteq \Sigma^*$2 indicating satisfied patterns for the prefix.
2. Query: Feed $L \subseteq \Sigma^*$3 into the DT classifier to identify “positive” paths (paths to leaves with the majority of training labels $L \subseteq \Sigma^*$4).
3. Score: For each such path $L \subseteq \Sigma^*$5, compute:
   • Fitness: fraction of split conditions in $L \subseteq \Sigma^*$6 that $L \subseteq \Sigma^*$7 satisfies,
   • Purity: $L \subseteq \Sigma^*$8 Gini impurity of the leaf,
   • Support: fraction of training positives reaching $L \subseteq \Sigma^*$9.

Score each path as

$y(\sigma)\in \{0,1\}$0

1. Select the highest scoring path $y(\sigma)\in \{0,1\}$1. For any split $y(\sigma)\in \{0,1\}$2 along $y(\sigma)\in \{0,1\}$3 not satisfied by $y(\sigma)\in \{0,1\}$4, output the recommendation:
   • “Pattern $y(\sigma)\in \{0,1\}$5 must be satisfied” if $y(\sigma)\in \{0,1\}$6 and $y(\sigma)\in \{0,1\}$7,
   • “Pattern $y(\sigma)\in \{0,1\}$8 must be violated” otherwise.

Recommendations are prioritized by split order in $y(\sigma)\in \{0,1\}$9. This produces a ranked set $$L_{\text{train}} = \{(\sigma_1, y(\sigma_1)), \ldots, (\sigma_N, y(\sigma_N))\}$$0 of temporal constraints.

5. Experimental Evaluation and Performance

OmegaPRM was evaluated on 22 real-world event logs from the process mining literature, covering business, healthcare, and production domains. Metrics included:

• Classifier Quality: Precision, recall, F1-score on complete traces (achieving F1 > 90% on 18/22 datasets).
• Prescriptive Quality: Offline “what-if” experimentation, scoring F1 > 90% in prescriptive peaks, showing no loss of reliability versus more rigid next-activity recommenders.
• Efficiency: Each recommendation is generated in milliseconds, even for long traces.
• Pattern Selection: Response and precedence patterns, and collections covering all patterns, delivered the best prescriptive accuracy.

A typical running case: In a sepsis-management log, if the current prefix lacks occurrence of ReleaseA, OmegaPRM recommends “◇ ReleaseA must be satisfied”—in LTL$$L_{\text{train}} = \{(\sigma_1, y(\sigma_1)), \ldots, (\sigma_N, y(\sigma_N))\}$$1, expressing the clinical imperative that a specific intervention should eventually be performed.

6. Deployment Considerations and Best Practices

Deployment involves:

• Offline: Mining labeled logs, selecting a moderate set of frequent LTL$$L_{\text{train}} = \{(\sigma_1, y(\sigma_1)), \ldots, (\sigma_N, y(\sigma_N))\}$$2 patterns (e.g., via Apriori), and DT training/tuning.
• Online: Maintaining current prefixes, incrementally encoding satisfied patterns, executing the recommendation algorithm in real time, and serving recommendations in either formulaic or controlled natural language (e.g., “Please ensure that eventually ReleaseA occurs”).
• Scenarios: Optimal for knowledge-intensive, unpredictable domains (e.g., healthcare, crisis management) where prescriptive flexibility is essential.
• Scalability: Recommendations are generated in real time (milliseconds per case).
• Extensibility: For data-heavy traces where data-payload correlations are important, extend the pattern set $$L_{\text{train}} = \{(\sigma_1, y(\sigma_1)), \ldots, (\sigma_N, y(\sigma_N))\}$$3 to capture data-aware temporal logic.

7. Context within Process Supervision and Limitations

OmegaPRM represents a class of process supervision systems that operationalize flexible, interpretable prescriptive interventions via learned temporal-logic rules. Unlike static “next-activity” recommenders, OmegaPRM's constraints capture temporal dependencies and allow end-users to select implementation strategies—as long as high-level temporal goals are met. A limitation is the reliance on well-selected LTL$$L_{\text{train}} = \{(\sigma_1, y(\sigma_1)), \ldots, (\sigma_N, y(\sigma_N))\}$$4 patterns; inadequate pattern sets may underrepresent latent process relationships. Additionally, effectiveness hinges on the quality and representativeness of the labeled training log. Future directions include extending pattern sets to handle complex data correlations and integrating continuous learning in evolving process contexts (Donadello et al., 2022).

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OmegaPRM demonstrates the integration of symbolic temporal logic, interpretable machine learning, and real-time monitoring to deliver outcome-oriented, user-friendly process supervision in business-process environments.