Petri net to WBS to schedule current working concept

Treat the project’s “physics” as a Petri net whose firing rule enforces what can actually happen given resources/conditions. Planning and scheduling then explore the reachable markings from an initial marking—the feasible state-space of configurations. Categorically, the same net freely generates a (very strict) symmetric/commutative monoidal process category where sequential composition is “do this then that” and the tensor is “do these independently”. Alternative WBS/schedule candidates show up as different factorizations/paths that still denote net-compliant executions. This cleanly separates the invariant constraint skeleton (the net) from a particular execution witness (the schedule). If richer state than token-counts is needed, you can lift to DPO-style rewriting on structured world-states.

Here’s a tightened, more technically accurate version

Keeping the intent, but correcting categorical and rewrite details:

A Petri net (places = resource/condition types; tokens = available units; transitions = activities) encodes feasibility via the firing rule: an activity can occur exactly when its required input multiset of tokens is present in the current marking, producing a new marking.
Under the collective-token reading (tokens indistinguishable), a Petri net (P) presents (i.e., freely generates) a commutative monoidal category (F_P) (a very strict form of symmetric monoidal category):
- objects are markings (token multisets),
- each transition (τ) gives a generating morphism (s(τ) → t(τ)),
- general morphisms are built from sequential composition and the monoidal product (often written (+) / (⊗)) for “doing things in parallel”.
With a chosen initial marking (m₀), the “possible project configurations” are the markings reachable from (m₀). Equivalently, (n) is reachable from (m₀) iff there exists a morphism (m₀ → n) in (F_P).
Each marking can be treated as a world state. A (resource-feasible, untimed) schedule/plan can be represented as either:
- a firing sequence (a path in the reachability graph of markings), or
- a morphism/string diagram in (F_P). Different ways of factoring a morphism into sequential vs parallel parts give different (still net-compliant) decompositions, which you can interpret as candidate WBS-like groupings—but note that this factorization/WBS is generally not unique.
If you want an explicit rewriting perspective: DPO rewriting is a general mechanism for applying “actions” (spans/rules) to “world states” modeled as C-sets; Petri-net firing is a simpler multiset-style rewriting discipline and can be embedded into richer DPO-style frameworks when you need more structure than plain markings.
Summary (generative claim, corrected): the Petri net generates the space of feasible executions; causality corresponds to composition, concurrency to the monoidal product, and dependencies/resource constraints to the enabling condition on markings. (If you mean schedules with durations/calendars, you’ll want a timed/weighted extension or a separate time semantics on top of the net.)