End · generate lawful options
Represent the refuge’s physical and operational logic, then generate and compare compliant executions, work structures and schedules. Keep necessary constraints without turning one chosen order into the only order.
ESSAY 06 · PETRI NETS → MONOIDAL STRUCTURE → WORK PLANS
Connect the work.
Keep the freedoms.
How can a Petri-net description of work become a family of lawful plans? This essay tackles one part: connect work packages without losing resource needs, early outputs or possible joint actions.
See the Petri netTHE ROUTE BACK TO ENDS / WAYS / MEANS
Why this step belongs in the foray.
Ways · follow the mechanism
Start with Petri nets and explicit starting conditions. Interpret their executions through symmetric monoidal categories (SMCs). Carry the resulting freedoms and dependencies into plans, with replayable reasons and clear limits.
Means · make it inspectable
Use small executable examples, independent checks and an interactive essay. Change the supply, inspect the tokens and follow the legal steps. The diagrams below show the same checked model as the work orders.
This essay’s step toward the End: determine what a reusable work package must retain so that joining it to another package preserves its lawful behaviour. The proposed boundary contract keeps resource demands, partial states, outputs and joint steps. The papers provide the foundations; this restricted contract and its checks are our construction.
Petri nets, open boundaries and monoidal structure — the three layers
Petri nets: what can happen?
A circle holds tokens representing resources or conditions. A task box consumes its input tokens and produces its outputs. Borrowing and returning a lift is a pair of ordinary arcs, not an unlimited permission to use it.
Lift : L → L ⊗ R
L is an available lift; R is an installed roof. The token’s return permits later reuse. This lab separately marks selected events as once-only.
Open Petri nets: how do pieces meet?
Expose places through named input and output ports. Connecting compatible ports identifies their places: the two packages now refer to the same resource. Two names must not become two lifts.
Baez & Master · Open Petri Nets
The lab uses restricted wiring and an explicit supply convention; the paper treats a more general construction.
Monoidal structure: how do actions combine?
Markings are objects; executions are arrows. Composition (∘) joins actions through matching intermediate markings. Tensor (⊗) combines resources or actions alongside one another. An identity carries untouched resources through a step.
Baez, Genovese, Master & Shulman · Categories of Nets
We use its commutative count-token interpretation, a particular SMC. Equal arrows alone do not establish that tasks can start together.
Connecting open nets assembles a mechanism; composing arrows describes an execution within it. These are related operations at different levels. The practical question comes from Wynn & Clarkson’s ASM2.0 work: can reusable process descriptions earn their maintenance cost when context changes?
Still to be achieved: this is a finite, once-only, selected-work construction—not a complete engineering model or general WBS generator. The earlier forward generator explores alternative work branches; the feedback essay handles review and rework. Here, timing is an added illustration, and a lawful execution need not fit one elementary WBS tree. Follow the full source → construction → limitation map.
01 / Retain
What each piece needs, what it releases, and which internal states are possible.
02 / Connect
Identify only the chosen, type-compatible ports. Account for each owned supply once.
03 / Challenge
Compare the composed contract with a separately glued process model, state by state and step by step.
THE EXECUTABLE ESSAY
Build a connection. Inspect what survives.
The pieces and their boundary
Change the starting supplies
Local tokens and unconnected input supplies belong to this scenario. A connected input’s old external supply is discharged; changing it cannot create a second token.
CURRENT FINDING
Preparing the example…
One state. Every admitted next step.
These are atomic decisions with no work already in progress. A joint step consumes all its inputs before producing any output.
The connected Petri net
What do these symbols mean?Circles hold the current token counts; boxes are work events. Solid green arrows consume tokens; dashed clay arrows produce them. A shared boundary is drawn as one place. This is the actual checked net, with the lab’s once-only event restriction.
● Green box: an admitted next event✓ Completed: cannot fire again here○ Pale box: waiting / not returned by the search
THE SAME STEP, READ AS AN SMC ARROW
From tokens to compositional execution
Inspect a possible step before taking it. Its task arrows are tensored with the identity on any resources left untouched. The step buttons above actually advance the marking.
Then, or alongside? Two consecutive step arrows compose as g ∘ f when the first output marking matches the second input marking. A joint step tensors the task arrows, but is admitted only if all input tokens are available at its start. With one shared lift, Frame then Roof and Roof then Frame can both be legal while Frame ⊗ Roof cannot start together.
These are count-token arrows in a commutative monoidal category. Occurrence identities and the joint-start check are retained separately: the category alone can forget token-reuse distinctions. See the interpretation and its limits.
Generate the lawful orders
Every selected task must finish, including tasks with no output. This essay composes a fixed work scope; the forward generator separately explores alternative work branches.
A representative schedule
Joint steps become batches; outputs are released at each task’s own finish. This is one replay-checked schedule under toy durations, not an optimum or proof of timed equivalence.
Retain the requirements, including blocked work
A standalone history records what happened with one supply. An open contract must also retain work waiting for its boundary inputs. We retain those demands, outputs and internally feasible states, then close the connection under its declared supplies.
What the equality does and does not establish
The two routes compare completed-event states, full count markings and jointly enabled steps for the declared finite fragments. The product and source-gluing constructions share a traversal routine; independent oracle tests separately check gluing, markings, labelled-token steps and complete orders. A cutoff means unknown. Equality is neither a theorem about arbitrary open nets nor evidence of engineering or human value.
Each admitted step interprets its event signatures through tensor, composition and untouched token context. Ownership and count semantics are explicit; no canonical token ancestry or exact elementary work tree is inferred.
Read the construction, counterexample and source limits →Inspect or edit the full construction
Change the fragments, event signatures, durations or named port connections. Checking validates the whole construction. An invalid edit keeps the last valid example available; it cannot silently become a result.
Keep and reopen a construction
Save one copy in this browser or download a portable JSON file. Opening a copy recomputes the result; imported findings are never trusted. No file, repository or remote model is overwritten.
No browser copy saved in this session.
WHY THIS MAY HELP
A reusable piece needs an explicit boundary.
The benefit is modest and testable: change a refuge method or resource arrangement, and see whether the combined work keeps its lawful alternatives. The source literature’s warning remains: context can make reusable process modules more trouble than they are worth. This example tests one piece of that problem.
Read the essay and source map · Build and independent challenge · Source code