Object-Centric Petri Nets: A Comprehensive View

• Object-Centric Petri Nets (OCPNs) are advanced process models that explicitly integrate multiple co-evolving object types using typed tokens.
• They enable synchronized transitions and variable arcs to model complex, multi-object interactions in scenarios like order processing and event logging.
• OCPNs underpin sophisticated techniques in performance analysis, conformance checking, and data-aware process integration for modern process mining.

Object-Centric Petri Nets (OCPNs) generalize classical place/transition nets by enabling explicit, first-class modeling of multiple co-evolving object types within a single net, where tokens are carriers of object identities (and, in more advanced extensions, data attributes). This formalism is especially relevant in contexts where process executions and corresponding event logs refer simultaneously to multiple object instances—orders, items, customers, etc.—and where process steps synchronize or evolve multiple such objects. OCPNs are foundational to recent developments in object-centric process mining, performance analysis, soundness theory, and conformance verification, addressing limitations of single-case Petri nets in highly intertwined domains (Aalst et al., 2020, Lomazova et al., 2021, Park et al., 2022, Gianola et al., 2023).

1. Formal Structure and Semantics of OCPNs

An OCPN is defined as a tuple

$$\mathrm{OCPN} = (P, T, F, pt, F_{\mathrm{var}}, \ell)$$

where:

• $P$ is a finite set of places;
• $T$ is a finite set of transitions, $P \cap T = \emptyset$;
• $F \subseteq (P \times T) \cup (T \times P)$ is the set of directed arcs;
• $pt : P \to \Sigma$ assigns to each place an object type in a finite set $\Sigma$;
• $F_{\mathrm{var}} \subseteq F$ is the set of variable arcs, enabling transitions to consume/produce variable-cardinality multisets in one step;
• $\ell : T \to \mathcal{A} \cup \{\tau\}$ labels each transition with an activity (possibly silent).

A marking is a multiset $M \in \mathcal{B}(Q)$, where $P$0 with $P$1 the set of all objects and $P$2 giving the object’s type. A binding for a transition $P$3 is a partial map $P$4, where for each type $P$5 (the object types involved in $P$6), $P$7 selects the set of objects to be consumed/produced for that type. For variable arcs, $P$8 is unrestricted; for non-variable arcs, $P$9 (Aalst et al., 2020, Seidel et al., 18 Aug 2025, Benzin et al., 2023).

Transition firing is defined by multi-object synchronization: a transition may atomically consume tokens of different types and object identities, and variable arcs enable collective, non-deterministic synchronization of arbitrary subsets (Aalst et al., 2020, Lomazova et al., 2021).

2. Expressiveness and Modeling Capabilities

OCPNs transcend classical Petri nets by:

• Typed places and object-typed tokens: Each token identifies both location and object instance, natively supporting many-to-many interactions.
• Synchronized transitions: Transitions synchronize objects of multiple types, supporting multi-artifact business logic (e.g., "ship order" consumes one order and an arbitrary set of corresponding items) (Aalst et al., 2020, Lomazova et al., 2021).
• Variable arcs: Transitions may consume/produce variable subsets of objects, modeling batch or group operations. Well-formedness constraints (all arcs for a type on a transition must be consistently variable or not) avoid semantic ambiguity (Benzin et al., 2023, Lomazova et al., 2021).
• Event-correlation: Each step can be linked directly to events referencing many objects, providing alignment with multi-entity event logs (Aalst et al., 2020, Seidel et al., 18 Aug 2025).

OCPNs are strictly more expressive than WF-nets and colored Petri nets for object-centric synchronization tasks, but cannot, in their pure form, enforce persistent object-object constraints (such as stable many-to-one relations, e.g., a wheel always belongs to the same frame throughout execution) (Seidel et al., 18 Aug 2025).

3. Discovery and Analysis Methodology

OCPN discovery from object-centric event logs (OCEL) is a canonical procedure:

1. Per-object-type projection: For each object type, the log is "flattened" into case-centric slices; standard process discovery (e.g., Inductive Miner) is applied on each, yielding type-specific WF-nets.
2. Merging by activity label: Transition labels shared across types are merged into single transitions in the global net, enabling cross-type synchronization (e.g., "submit" fires for both "project" and "student" object types) (Aalst et al., 2020).
3. Variable arc inference: Arc variable-ness is inferred via replay-based heuristics or directly from log event multiplicities (Benzin et al., 2023).
4. Typed marking construction: Initial and final markings are assembled from object-instantiated places.

This pipeline is implemented in process mining frameworks (e.g., PM4PY, ocpa) and supports both holistic and type-specific process perspectives, providing drill-down on single object types and cross-type interactions (Aalst et al., 2020, Benzin et al., 2023, Liss et al., 2023).

4. Soundness and Behavioral Properties

Object-centric soundness adapts and strictly generalizes classical WF-net soundness:

• Option to complete: For any marking, the net must allow all tracked (i.e., focal) object types to reach designated completion states, preventing dead objects and guaranteeing completion in presence of arbitrarily many helpers ("resource" objects of other types).
• No dead transitions: Every transition should be firable on some run involving the tracked object.
• No object-specific garbage: Completion must clean up all tokens of the tracked object type except in the final place; tokens of other types are treated as resources (Lomazova et al., 2021, Benzin et al., 2023).

Decidability of OCPN soundness was established for the acceptor nets generated by discovery pipelines, leveraging reduction to well-structured transition systems (WSTS) and classic Petri net coverability. The complexity is at least as high as general Petri-net coverability (EXPSPACE-hard) (Lomazova et al., 2021). Unsoundness can arise from improper merging or loop structures; recent process discovery frameworks include constructs to prevent such behaviors in collaborative, multi-object scenarios (Benzin et al., 2023).

5. Conformance Checking, Alignments, and Quality Metrics

OCPNs support advanced conformance analysis:

• Fitness and precision: Context-aware generalizations measure the degree to which the net permits and restricts observed event sequences, taking into account multi-object historical context, joint prefixes (for every event, the activity sequences of all involved object instances), and possible next actions in both log and model (Adams et al., 2021).
• Object-centric alignments: Optimal alignments are partial-orderings of "moves" (log moves, model moves, and synchronous moves) at the object-set level, constructed as directed acyclic graphs rather than sequential traces. The alignment cost reflects the minimal set of deviations (object-level insertions, deletions, or rebindings) required to explain observed events in the model (Liss et al., 2023).
• Complexity and implementation: Conformance checking is NP-hard in the number of objects, process steps, and alignment costs. Practical solvers leverage A* and Dijkstra search in the state space of product nets; scalable implementations exist in Python (ocpa) and integrate directly with mining libraries (Liss et al., 2023).

6. Extensions: Identifiers, Data, and Nested Object-Centric Nets

OCPNs form the conceptual core for several advanced formalisms:

• OPIDs/OCPNs with identifiers: These extend OCPNs with explicit object-identifier management, supporting exact object-object synchronization, cardinality constraints, and persistent relationships. Transition arcs are inscribed with identifier variables or list variables, with synchronization enforcing stepwise binding of compound object tuples (Gianola et al., 2023, Gianola et al., 21 May 2025, Werf et al., 2022).
• Data-aware DOPIDs: These further extend OPIDs by allowing tokens to carry both object identifiers and structured data values, and transitions to be guarded by SMT-expressible constraints. Guards can encode data-dependent process logic, e.g., value-based branching or synchronization based on object attributes (Gianola et al., 21 May 2025).
• Nested nets and EOS: Elementary Object Systems (EOS) generalize the "token-as-net" paradigm—an EOS is a net whose tokens are themselves Petri nets. This hierarchical model allows representation of object nets as tokens and supports symmetry reduction in state-space analysis via automorphism-induced canonical forms (Köhler-Bussmeier et al., 2024). Symmetries in net structure and markings are exploited to minimize the reachability graph, drastically reducing state explosion for process systems exhibiting regularities or symmetric roles.

7. Performance Analysis and Practical Applications

OCPNs underpin object-centric performance analytics in process mining (Park et al., 2022):

• Object-centric metrics: Beyond classical per-case metrics, OCPNs support synchronization time, pooling time, lagging time, waiting time, and sojourn time, accounting for multi-object interactions and temporal dependencies.
• Tooling and execution: Integrated replay engines support token visit extraction, per-activity timing annotation, and model-based time decomposition per object type.
• Practical case studies: In loan application or manufacturing scenarios, OCPNs provide interpretable and accurate performance projections that would be unattainable (or misleading) under case-flattened approaches.

---

In summary, OCPNs and their extensions constitute a robust framework for multi-object process modeling, mining, conformance analysis, and performance insights, and form the basis for ongoing research targeting data integration, advanced soundness, symbolic representation, and modular state-space reduction (Aalst et al., 2020, Park et al., 2022, Lomazova et al., 2021, Benzin et al., 2023, Köhler-Bussmeier et al., 2024, Gianola et al., 2023, Gianola et al., 21 May 2025, Seidel et al., 18 Aug 2025, Liss et al., 2023).