Petri Net Classes Overview

• Petri Net classes are diverse modeling frameworks characterized by structural and behavioral restrictions that enable rigorous analysis of concurrency, causality, and resource management.
• They span classical (conflict-free, free-choice), graphical (marked graphs), and timed or stochastic variants, each tailored to ensure properties like liveness, reversibility, and decidability.
• Recent advances integrate categorical frameworks and dynamic extensions, expanding applicability in process synthesis, workflow mining, and collaboration modeling.

Petri nets are a foundational modeling formalism for concurrent, distributed, and resource-sensitive systems, supporting rigorous mathematical analysis of state, causality, and nondeterminism. Over decades of development, researchers have defined an extensive taxonomy of Petri net classes—each characterized by specific structural or behavioral restrictions, expressive power, and (un-)decidability properties. The study of these classes reveals a spectrum of model-theoretic, algorithmic, and categorical distinctions that are central to both theoretical investigation and practical application.

1. Classical Structural Classes: Choice, Conflict, and Causality

Several fundamental Petri net classes are defined by syntactic or semantic restrictions on their conflict and causal structure, often motivated by tractability or correspondence with specific event structure logics:

• Conflict-Free Nets (CF): Each place has at most one output transition; no two transitions share a preplace. This restriction yields maximal concurrency and no nondeterministic choice. Conflict-free nets correspond to prime event structures, and in the one-safe case, their partial order runs are unique up to interleaving (0912.4023).
• Free-Choice Nets (FC): Whenever two transitions share an input place, that place is their only input. Formally, for all $t \neq t'$, if ${}^\bullet t \cap {}^\bullet t' \neq \emptyset$, then ${}^\bullet t = {}^\bullet t'$. This structure centralizes choice to places, precluding interplay with synchronization. Free-choice nets have been shown to be persistent, and their reachability, liveness, and reversible behaviors admit specialized algorithms (Best et al., 25 Jan 2026, Hujsa et al., 2020).
• Extended Free-Choice Nets (EFC): Two places that share an output transition must have either disjoint or equal postsets. This class strictly generalizes FC, allowing 'blocks' of interconnected choices where choice dependency is global within blocks but isolated between them (Wimmel, 2019, 0912.4023).
• Asymmetric Choice (AC) and Dissymmetric Choice (DC) Nets: AC nets admit place postset inclusions (for all $p, p'$, if $p^\bullet \cap p'^\bullet \neq \emptyset$, then $p^\bullet \subseteq p'^\bullet$ or $p'^\bullet \subseteq p^\bullet$), while DC is the transition-dual of this condition. These classes model more general 'confusion' than FC/EFC but can still be tractably synthesized in certain subclasses (notably 'reduced AC') (Wimmel, 2019, Best et al., 25 Jan 2026).
• Equal-Conflict Nets (EC): Two transitions sharing a preplace must have identical input vectors ($F(\cdot, t) = F(\cdot, t')$). This class sits strictly between FC and DC and is significant in the analysis of persistent permutability (Best et al., 25 Jan 2026).

Structural conflict nets, as characterized in (Glabbeek et al., 2021), generalize the above conflict/choice-free conditions: a net is structural conflict-free if no two simultaneously enabled transitions ever share a preplace. This property yields uniqueness of 'abstract processes' (partial order run equivalence classes) even in nonsafe settings.

2. Marked Graphs and Graph-Structured Subclasses

Marked graphs (MG) and their weighted generalizations (WMG) occupy a foundational position as the principal subclass without choice:

• Marked Graphs (MG): Every place has at most one input and one output transition ($|\cdot p| \leq 1$, $|p^\bullet| \leq 1$), precluding both choice and synchronization. These nets model pipelined dataflow and are analytically tractable; liveness, reversibility, and boundedness are equivalent and efficiently testable via standard linear algebraic invariants (Hujsa et al., 2020, Hujsa et al., 2020).
• Weighted Marked Graphs with Relaxed Place Constraint (WMG<): Dropping the mutual exclusivity, allowing at most one input and one output per place, supports certain shared buffer modeling while retaining strong algebraic properties, notably PR–R equality (potentially reachable markings coincide with actually reachable ones) under liveness (Hujsa et al., 2020).

3. Timed, Stochastic, and Extended Net Classes

The integration of timing, probabilities, or structural gadgets yields new expressive and computational properties:

• Time Petri Nets (TPN): Transitions are annotated with firing intervals $[\alpha(t), \beta(t)]$; time and enabling are tightly coupled—timers run only when all preplaces are marked. Coverability and bounded reachability remain decidable for bounded TPNs (Hélouët et al., 2022).
• Waiting Nets (WTPN): An extension of TPNs introducing two types of places—'standard' (start clocks) and 'control' (authorize firing only)—to decouple when timers start from when transitions become fully enabled. Waiting nets have strictly greater expressive power than TPNs (in terms of recognized timed languages), yet, unlike stopwatch TPNs, bounded WTPNs retain decidability for reachability, with construction of finite state-class graphs (Hélouët et al., 2022).
• Timed-Arc Petri Nets (TaPN): Arc ages and urgent deadlines induce expressiveness incomparable to WTPNs; mutual simulation does not hold due to unsimulable respective urgency features (Hélouët et al., 2022).
• Stopwatch Petri Nets (SwTPN): Adding stopwatch clocks makes even bounded nets Turing complete; reachability is undecidable (Hélouët et al., 2022).
• Product-Form Nets (Π³-nets): Layered weakly-reversible nets that permit efficient computation of steady-state distributions in both closed and certain infinite-state open cases. Liveness and reachability admit polynomial-time decision procedures; steady-state probabilities possess explicit product forms under simple ergodicity conditions (Bouyer et al., 2017).

4. Synthesis, Reachability, and Model-Theoretic Properties

The study of Petri net classes often focuses on algorithmic tractability for synthesis and analysis tasks:

• Synthesis: For CF and weighted EFC, synthesis from finite transition systems is decidable in polynomial time or pseudo-polynomial time. RAC synthesis admits practical algorithms with only bounded NP-completeness overhead. General AC synthesis remains open; the presence of chains of nested (nontrivial) choice structures or self-loops precludes general algorithms (Wimmel, 2019).
• Reachability and State Equation (PR–R): For classes with strong confluence properties (MG, WMG<, live and reversible H₁S-WMG), reachability is completely characterized by integer solutions of the state equation $M = M_0 + I Y$; no further restrictions are needed (PR–R equality) (Hujsa et al., 2020). In classical Petri nets, this does not hold due to counterexamples arising from siphon/trap emptiability and deadlocks.
• Liveness and Reversibility: In CF and WMG<, liveness, reversibility, and boundedness are all equivalent; in H₁S-WMG, liveness is characterized by an ILP of polynomial size, yielding co-NP completeness, with reversibility reducible to detection of a specific T-sequence (Hujsa et al., 2020).
• Persistence and Permutability: CF, FC, EC, and pure DC nets have the property that every persistently-permutable firing sequence implies global persistence, validating Ochmański’s conjecture for these classes. In AC nets, persistent permutability does not guarantee persistence, and counterexamples exist (Best et al., 25 Jan 2026).

5. Event Structure Correspondence and Configuration Theories

Safe Petri nets can be fully classified by their event structure semantics:

• Prime Event Structures: Correspond to conflict-free nets; pure causality, maximal concurrency, unique processes up to interleaving.
• Flow Event Structures: Correspond to free-choice nets; local nondeterministic choice resolves at places, but synchronization is globally controlled.
• Bundle Event Structures: Correspond to extended free-choice nets; choices may be grouped, and enabling is via disjunctive bundles. Translation to nets follows a systematic transformation of propositional (Horn/DNF) theories into inhibitor and flow arcs, providing clear logical–structural correspondence (0912.4023).

Abstract process semantics, and their collapse under swapping equivalence (collective token interpretation), isolate the precise range between free-choice nets, structural conflict nets, and arbitrary P/T systems, with critical implications for partial order semantics, run uniqueness, and compositional reasoning (Glabbeek et al., 2021).

6. Categorical Frameworks and Modular Extensions

Recent categorical treatments generalize Petri net classes through their functorial semantics and monoidal category correspondences:

• Pre-nets, Σ-nets, and Petri nets: Pre-nets admit sequential arc ordering (strict monoidal categories), Σ-nets generalize to partial symmetries (symmetric monoidal categories), ordinary Petri nets quotient all arc orderings (commutative monoidal categories). All are embedded in the Σ-net framework, unifying individual and collective token philosophies into an ambient diagram of adjunctions and semantic functors (Baez et al., 2021).
• Whole-Grain Petri Nets (WGPet): Kock's whole-grain nets fit as those Σ-nets in the image of pre-nets, supporting both individualized and commuted process semantics.
• Petri Net Receptors and Effectors: Recent work defines modular extensions for closed-loop control by adding/removing 'receptor' and 'effector' gadgets via operations of fusion/defusion; these do not extend expressiveness but add dynamic feedback modeling capabilities (Chunikhin et al., 2019).

7. Application-Driven and Specialized Classes

Petri net classes continue to proliferate in response to modeling and mining challenges in workflow, service, and collaboration domains:

• Workflow and Group-Choice Nets: Free-choice workflow nets serve as the preferred model for process mining tools; encodings into process calculi (e.g., CCS) preserve behavioral equivalence and support formal verification. 'Group-choice' nets generalize synchronizing choice scenarios and are also covered by such encodings (Bogø et al., 2024).
• Petri Net Classes for Collaboration Mining: Marketplace and multi-agent process modeling motivate classes like colored nets, object systems (nets-within-nets), interaction Petri nets, and resource-augmented workflow nets. Systematic assessment matrices and design guidelines balance expressiveness, decidability, modularity, and mining algorithm support (Benzin et al., 2023).

Class	Structural Restriction	Key Decision Properties
Marked Graph	$|\cdot p| \leq 1$, $|p^\bullet| \leq 1$	PR–R equality, Liveness = Bdd = Rev
Choice-Free (CF)	$|p^\bullet| \leq 1$	Persistent, Synthesis in P
Free-Choice (FC)	If $p^\bullet \cap p'^\bullet \neq \emptyset$, then $p^\bullet = p'^\bullet$	Persistent, Synthesis in P/NP (plain)
EFC/Eq.Conflict	Weighted $0/1$, equal postsets	RAC, Synthesis in NP
AC/DC	Postset or preset inclusion	Synthesis open/tractable in RAC
Waiting Net (WTPN)	Decoupled enabling/timing places	Decidable bounded reachability
Open Π³-net	Layered, weakly-reversible	Poly-time liveness/reachability, prod. form steady-state
Structural Conflict	Never enable competing transitions with shared preplaces	Unique abstract process semantics

• Waiting nets and expressive timed extensions (Hélouët et al., 2022).
• Configuration structures and event-structure–net correspondences (0912.4023).
• Weighted marked graphs, choice-free, and single shared-place nets (Hujsa et al., 2020, Hujsa et al., 2020).
• Persistent permutability and choice net generalizations (Best et al., 25 Jan 2026).
• Synthesis and hierarchy of (reduced) asymmetric-choice nets (Wimmel, 2019).
• Product-form nets and stochastic analysis (Bouyer et al., 2017).
• Abstract processes and structural conflict (Glabbeek et al., 2021).
• Categorical foundations and Σ-nets (Baez et al., 2021).
• Receptors/effectors and dynamic net manipulation (Chunikhin et al., 2019).
• Process mining classes and collaboration workflow assessment (Benzin et al., 2023).
• Encodability into process calculi (Bogø et al., 2024).
• Finite place/transition net expressibility via process calculi (Gorrieri et al., 2010).

These sources constitute a foundational body of knowledge on Petri net classes, highlighting the central taxonomic, semantic, and computational bifurcations which underlie contemporary applications and open research directions.