Petri Nets: Models & Applications

• Petri nets are bipartite graph models defined by places that hold tokens and transitions that fire to transform these tokens, establishing the core for concurrency modeling.
• They encompass a spectrum of variants—such as colored, stochastic, continuous, and reversible nets—enabling tailored representations for diverse systems like epidemiology and robotics.
• Their compositional frameworks and verification methods support analysis of properties like boundedness, liveness, and reachability in complex, dynamic environments.

In contemporary formal-methods, concurrency theory, systems biology, and executable modeling, “Petri” most commonly denotes the Petri-net family of models: bipartite graph formalisms in which places hold tokens, transitions transform token distributions, and system state is identified with a marking. Across the literature, Petri nets appear both as ordinary place/transition systems and as a large family of extensions—open, colored, stochastic, continuous, recursive, reversible, quantum, and application-specific variants—used to represent causality, synchronization, conflict, resource constraints, and compositional structure. A recurrent theme is that Petri nets are not a single fixed syntax but a spectrum of models whose semantic strength depends on how tokens, interfaces, timing, and symmetry are treated (Baez et al., 2021).

1. Ordinary place/transition semantics

In the ordinary place/transition setting, a net is defined as a triple $N=(S,T,F)$, where $S$ and $T$ are disjoint finite sets of places and transitions, and $F$ is an arc relation with no place-place or transition-transition arcs. A marking is a function $M:S\to \mathbb{N}$. A transition is enabled at marking $M$ if all its input places contain a token; when it fires, it removes one token from each input place and adds one token to each output place. The one-step firing relation is written $M \xrightarrow{t} M'$, and a sequence $\sigma$ yields $M_0 \xrightarrow{\sigma} M_n$. The reachability graph has reachable markings as nodes and enabled firings as labeled edges (Esparza et al., 2024).

A weighted formulation, common in applied modeling, treats a Petri net as a weighted bipartite graph $(P,T,A,w,x)$, where $S$0 is the set of places, $S$1 the set of transitions, $S$2 the arcs, $S$3 the arc-weight function, and $S$4 the marking. A transition $S$5 is enabled when each input place contains at least the required arc weight, formally $S$6 for all $S$7. Firing updates the marking by

$S$8

This weighted view is especially useful when encoding compartmental flows, resource multiplicities, or chemically inspired stoichiometries (Reckell et al., 2024).

Several papers emphasize a distinction that is often blurred in informal usage: the graph structure of a Petri net and the dynamics induced by enabling and firing are inseparable. A place does not merely denote a “state”; it stores tokens whose availability determines future behavior. Likewise, arcs do not merely denote connectivity; they encode consumption, production, and sometimes inhibition or read-only conditions. This is why properties such as boundedness, liveness, persistence, deadlock-freedom, and reachability are semantic rather than purely structural notions.

A common misconception is that any graph resembling a workflow diagram is already functioning as a Petri net. The literature instead treats the marking semantics as essential. In ordinary nets, the distinction between an enabled and a disabled transition, and between a reachable and an unreachable marking, is the core of the model rather than an auxiliary annotation.

2. Variants, token philosophies, and execution models

The Petri-net family includes multiple syntactic and semantic variants. One categorical synthesis distinguishes pre-nets, $S$9-nets, and Petri nets by how strongly they identify permutations of inputs and outputs. Pre-nets use ordered words $T$0, ordinary Petri nets use multisets $T$1, and $T$2-nets are defined by a functor

$T$3

where $T$4 is the free symmetric strict monoidal category on a set $T$5. This framework interpolates between the individual token philosophy, in which token identity and ordering matter, and the collective token philosophy, in which only multiplicities matter. In the corresponding execution semantics, pre-nets freely generate strict monoidal categories, $T$6-nets generate symmetric strict monoidal categories, and Petri nets generate commutative monoidal categories (Baez et al., 2021).

In applied modeling, the most prominent extensions are stochastic Petri nets (SPNs), continuous Petri nets (CPNs), colored Petri nets, and hybrid Petri nets. In epidemic modeling, a single Petri-net structure can be read stochastically, where transition rate functions define exponentially distributed waiting times and yield a CTMC semantics, or continuously, where places carry nonnegative real values and the semantics is a system of ODEs. For a continuous Petri net, one place equation is generated per place: $T$7 Colored formalisms compress repeated subnet structure by attaching color sets and variables to places, transitions, and arc inscriptions, while hybrid nets combine discrete and continuous state evolution (Connolly et al., 2021).

Specialized variants push the model in other directions. Recursive Petri nets enrich ordinary nets with dynamically created nested threads and cut transitions; their states are trees rather than flat markings. Reversing Petri nets attach named token instances, bonds, and a history function to support backtracking, causal reversing, and out-of-causal-order reversing. Quantum Petri nets equip nets with local quantum annotations compatible with quantum event structures. Application-oriented extensions also exist, such as nondeterministic-transition Petri nets for concurrency middleware and deterministic inhibitor nets used in small-universal-net constructions.

These variants differ not only in convenience but in ontology. Colored nets preserve higher-level repeated structure; recursive nets add tree-shaped control; reversible nets remember past participation of token instances; quantum nets attach Hilbert spaces and $T$8 maps; inhibitor nets allow explicit zero tests. Consequently, “Petri net” in current literature denotes a family resemblance rather than a single closed formalism.

3. Compositionality and open-system semantics

A major strand of modern Petri-net theory treats a net not as a closed state machine but as a component with interfaces. In this setting, an open Petri net is a cospan

$T$9

in $F$0, where $F$1 and $F$2 are interface sets and $F$3 maps a set to a transitionless net with those places. Composition is defined by gluing outputs of one open net to inputs of another via pushout, and parallel composition is given by disjoint union. Because pushouts are unique only up to isomorphism, the natural home for actual open nets is not merely a category but a symmetric monoidal double category $F$4 (Baez et al., 2018).

This open-system viewpoint supports two distinct semantics. The operational semantics sends a Petri net to the free commutative monoidal category $F$5, whose objects are markings and whose morphisms are processes built from transitions by sequential and parallel composition. This semantics is fully compositional. The reachability semantics instead associates to an open net the relation

$F$6

It is compositional only laxly: $F$7 The strict inequality arises because in the composite net tokens may cross the glued interface multiple times; ordinary relational composition cannot record such back-and-forth interaction.

A complementary line of work classifies additive invariants of open Petri nets, meaning $F$8-valued invariants additive under both sequential and parallel composition. In the unrestricted category of open Petri nets, every additive invariant is a weighted count of transitions by their input/output arities. In the monically open subcategory, additional interface-size invariants appear, and invariants are determined by values on transitionless boundary nets and one-transition body nets (Bumpus et al., 2023).

These results show that compositionality in Petri-net theory is not monolithic. Operational semantics, reachability semantics, and additive invariant theory preserve different amounts of structure. A frequent misconception is that “compositional reachability” should behave as cleanly as compositional process semantics. The open-net literature instead identifies an exact fault line: process categories compose strongly, whereas reachability relations generally under-approximate serial composition.

4. Verification, decidability, and algorithmic frontiers

The algorithmic theory of Petri nets is dominated by the distinction between properties that remain decidable, albeit at extreme complexity, and properties that become undecidable once one asks for equivalence or richer logics. A compact survey of roughly the first twenty-five years of decidability work presents the ordinary place/transition model and organizes results around three themes: individual properties, behavioral equivalences, and temporal logics. In that account, boundedness and reachability are the two foundational base problems. Boundedness is decidable by the Karp–Miller procedure; the survey cites the classical characterization that a net is unbounded iff there exists a reachable marking $F$9 and a sequence $M:S\to \mathbb{N}$0 such that

$M:S\to \mathbb{N}$1

for some nonzero marking $M:S\to \mathbb{N}$2. Reachability is decidable by Mayr’s theorem and later simplifications, but the proof is notoriously difficult (Esparza et al., 2024).

Many semantic properties reduce to reachability. Liveness is recursively equivalent to reachability and hence decidable. Deadlock-freedom is reducible in polynomial time to reachability. Home-state problems, promptness, strong promptness, persistence, and semilinearity are also decidable in the ordinary model, though their complexity is often enormous. By contrast, most behavioral equivalences—marking equivalence, trace equivalence, language equivalence, and many forms below bisimulation in the linear/branching spectrum—are undecidable for general labeled nets. The survey’s overall picture is that property checking is often decidable, while equivalence checking is mostly not.

Temporal logic introduces another sharp boundary. Even weak branching-time logics over reachability graphs become undecidable in general. In linear time, certain fragments remain decidable, especially when place predicates are absent or severely constrained, but branching-time model checking is characterized as largely hopeless for general Petri nets. This corrects another frequent overstatement: Petri nets are not a universal cure for model checking; rather, they occupy a delicate position where verification is feasible only for carefully delimited questions.

Extended models can preserve or destroy this balance in surprising ways. Recursive Petri nets strictly extend ordinary Petri nets and context-free grammars, yet coverability, termination, boundedness, and finiteness remain EXPSPACE-complete, matching ordinary Petri nets. At the same time, boundedness and finiteness are no longer equivalent, because the number of threads may grow unboundedly even when each local marking stays bounded (Finkel et al., 2021). This suggests that additional expressiveness does not uniformly worsen monotone verification problems, but it does change their structural interpretation.

5. Modeling practice across scientific and engineering domains

Petri nets are widely used as executable modeling languages precisely because the same structural ingredients—places, transitions, tokens, markings, and enabling rules—can be reinterpreted across domains without changing the underlying logic of concurrency and resource flow.

In epidemiology, Petri nets are used to encode compartmental models such as SIR and SIRS. Places represent compartments $M:S\to \mathbb{N}$3, $M:S\to \mathbb{N}$4, and $M:S\to \mathbb{N}$5; tokens represent population counts; transitions represent infection, recovery, and loss of immunity. When a Petri-net SIRS implementation is compared numerically with the standard ODE system,

$M:S\to \mathbb{N}$6

the comparison is not automatic. Discrete Petri nets require integer token movements, so real-valued rates must be rounded, and time must be refined by a parameter $M:S\to \mathbb{N}$7, the number of Petri-net time steps per ODE unit time. The study reports that rescaling rates by $M:S\to \mathbb{N}$8 and using a residual-carrying rounding method are critical for convergence; otherwise naive rounding can cause unrealistic oscillatory “2-cycle” behavior or suppress infection entirely. With appropriate procedures, the maximum observed RRMSE at $M:S\to \mathbb{N}$9 becomes approximately $M$0 over the full tested range, and below $M$1 for biologically realistic parameters (Reckell et al., 2024).

In pandemic modeling, colored stochastic and continuous Petri nets support orthogonal extension along epidemic structure, population stratification, and geography. A basic SIR net can be lifted to age-stratified models, metapopulation travel graphs, or combined systems such as $M$2, whose unfolded form has 1,440 places and 10,380 transitions. Structural Petri-net notions such as conservativeness, P-invariants, siphons, and traps become validation tools for epidemiological models rather than only abstract algebraic artifacts (Connolly et al., 2021).

In systems neuroscience, a timed, ordinary, infinite-capacity Petri net with stochastic inputs has been used to model the aggression circuit involving OFC, medial amygdala, LNET, HA, SGPA, and EXPRE. Places represent nuclei or control modules, tokens represent packets of neural information or inhibition-related signals, and synchronized firing into SGPA represents the concurrence condition for aggressive expression. The model includes non-biological control places such as TTL and Recap to emulate inhibition, timing comparison, and reverberation (Moreno et al., 2015).

In plant regulatory biology, a stochastic Petri net has been used to model phosphate-starvation-induced root hair elongation in Arabidopsis thaliana. The formal basis is

$M$3

augmented with read arcs, inhibitor arcs, modifier arcs, stochastic timing, and deterministic output transitions. Tokens represent either amount/concentration or regulatory information depending on context. The model recovers the reported $M$4-fold increase in root hair length under low Pi and uses model-data discrepancies to suggest an important role for RSL2 in regulating or shaping effective RSL4-dependent elongation (Fijn et al., 10 Mar 2025).

These case studies show that Petri nets are neither confined to software verification nor reducible to stylized workflow diagrams. They function as executable abstractions of systems in which concurrency, synchronization, or state-dependent resource flow is the primary mathematical difficulty.

6. Toolchains, implementation ecosystems, and current directions

The contemporary Petri-net ecosystem spans graphical editors, structural analyzers, simulation environments, synthesis tools, and domain-specific translators. In pandemic and systems-biology modeling, the PetriNuts ecosystem combines Snoopy, Patty, Charlie, Marcie, Spike, and MC2. Snoopy supports graphical construction of qualitative, stochastic, continuous, and hybrid nets; Charlie computes invariants, siphons, traps, and explicit CTL/LTL model checking for uncolored nets; Spike provides reproducible simulation of large-scale colored nets; MC2 performs trace-based Monte Carlo model checking (Connolly et al., 2021).

For analysis and synthesis of labeled place/transition nets and transition systems, APT provides a stand-alone Java toolbox. It supports structural checks such as plainness, pureness, marked-graph and T-net properties, invariant and siphon/trap computation, reachability or coverability graph construction, and region-theoretic synthesis of Petri nets from labeled transition systems under constraints such as pure, plain, output-nonbranching, conflict-free, $M$5-bounded, safe, and distributed via locations (Best et al., 2015).

A separate current direction translates external computational artifacts into Petri nets for verification. Petrify encodes Java bytecode into ordinary place/transition nets via an intermediate language, then uses a PN model checker such as LoLA to check deadlock, livelock, and termination. Its encoding is flow-sensitive, context-sensitive, and path-insensitive; locks become dedicated lock places, thread control is represented by control places, and the resulting PN is a sound over-approximation under explicit assumptions about alias analysis, unsupported features, and non-reentrant locks (Shenoy et al., 1 Jul 2026). In autonomous robotics, SkiNet compiles Skillset architectures into priority Petri nets that make resource-state machines, skill-state machines, guards, and effects available to LTL and CTL verification in the Tina toolbox (Pelletier et al., 2022). For reversible systems, a formal translation maps acyclic low-level Reversing Petri Nets into bounded Colored Petri Nets, allowing CPN Tools to analyze backtracking, causal, and out-of-causal-order reversibility (Barylska et al., 2023).

Recent foundational work also pushes Petri nets into new semantic regimes. Quantum Petri Nets seek to recover, in a quantum setting, the classical triangle between Petri nets, occurrence nets, and event structures by defining Local Quantum Occurrence Nets whose induced global semantics satisfies functoriality, obliviousness, and a localized drop condition. This line explicitly responds to the criticism that earlier “quantum Petri nets” lacked rigorous concurrent semantics, sound quantum interpretation, and unfolding theory (Joachim et al., 20 Aug 2025).

A plausible implication is that the future of “Petri” lies less in a single dominant formalism than in a stable methodological core—token-based concurrency with explicit causal structure—combined with multiple semantic envelopes. The literature already supports this reading: ordinary nets remain central for decidability and verification theory, open nets for compositional semantics, colored and hybrid nets for scalable scientific modeling, and specialized translations for software, robotics, reversibility, and quantum concurrency. Across these developments, the persistent technical attraction of Petri nets is that they keep local causal structure explicit while still supporting global analysis.