catcolab-petri-nets

# CatColab Petri Nets: Concurrent Systems **Trit**: +1 (PLUS - generator) **Color**: Magenta (#FF00FF) ## Overview Petri nets in CatColab model concurrent and distributed systems: - **Places**: States holding tokens (resources, conditions) - **Transitions**: Events that fire when enabled - **Arcs**: Flow of tokens between places and transitions - **Tokens**: Resources consumed/produced by transitions Petri nets are the categorical foundation for process algebra, workflow modeling, and chemical reaction networks. ## Mathematical Foundation A Petri net is a bipartite graph: ``` ┌─────────────────────────────────────────────────────┐ │ PETRI NET │ ├─────────────────────────────────────────────────────┤ │ Places (States): │ │ (P1) Ready (P2) Running (P3) Done │ │ │ │ Transitions (Events): │ │ [T1] Start [T2] Finish │ │ │ │ Arcs (Token Flow): │ │ P1 → T1 → P2 → T2 → P3 │ │ │ │ Diagram: │ │ (P1)●──►[T1]──►(P2)──►[T2]──►(P3) │ │ ●=token │ └─────────────────────────────────────────────────────┘ ``` ## Free Symmetric Monoidal Category Petri nets are the **free symmetric monoidal category** on a signature: ``` Objects: P₁, P₂, ..., Pₙ (places) Morphisms: Generated by transitions Tensor: P₁ ⊗ P₂ (concurrent resources) Composition: Sequential firing Example: T: P₁ ⊗ P₂ → P₃ ⊗ P₄ "Transition T consumes tokens from P₁, P₂ and produces tokens in P₃, P₄" ``` ## Double Theory ```rust // Petri net double theory pub fn th_petri_net() -> DiscreteDblTheory { let mut cat = FpCategory::new(); // Object type cat.add_ob_generator(name("Place")); // Morphism type (transitions as structured morphisms) cat.add_mor_generator(name("Transition"), name("Place"), name("Place")); // Monoidal structure (parallel composition) // P ⊗ Q represents "P and Q concurrently" cat.into() } ``` ## CatColab Implementation ### Place Declaration ```typescript { "type": "ObDecl", "name": "Idle", "theory_type": "Place", "description": "process waiting for input" } ``` ### Transition Declaration ```typescript { "type": "MorDecl", "name": "activate", "inputs": ["Idle", "Resource"], "outputs": ["Active"], "theory_type": "Transition", "description": "process starts when resource available" } ``` ## Practical Examples ### Example 1: Producer-Consumer ``` Places: Buffer (capacity N), Produced, Consumed Transitions: produce: Empty → Buffer + Produced consume: Buffer → Consumed Tokens: Buffer starts empty, count items produced/consumed ``` ### Example 2: Dining Philosophers ``` Places: Thinking(i), Eating(i), Fork(i) for i=1..5 Transitions: pickup(i): Thinking(i) ⊗ Fork(i) ⊗ Fork(i+1) → Eating(i) putdown(i): Eating(i) → Thinking(i) ⊗ Fork(i) ⊗ Fork(i+1) Deadlock possible when all pick up left fork simultaneously ``` ### Example 3: Chemical Reaction (A + B → C) ``` Places: A, B, C (species concentrations as token counts) Transitions: react: A ⊗ B → C Stochastic: Rate = k·[A]·[B] (mass action) ``` ## Connection to Stock-Flow Petri nets and stock-flow diagrams are **equivalent**: ``` Petri Net ≅ Stock-Flow Place ↔ Stock Transition ↔ Flow Token count ↔ Population Firing rate ↔ Mass-action rate ``` ## AlgebraicPetri.jl Integration ```julia using AlgebraicPetri using Catlab # Define SIR as Petri net sir_petri = @acset LabelledPetriNet begin S = 3 # places: S, I, R T = 2 # transitions: infect, recover tname = [:infect, :recover] sname = [:S, :I, :R] is = [1, 2] # infect inputs: S, I os = [2, 2] # infect outputs: I, I it = [1, 1] # transition indices ot = [1, 1] end # Compose Petri nets combined = sir_petri ⊕ vaccination_petri ``` ## GF(3) Triads ``` catcolab-regulatory-networks (-1) ⊗ topos-catcolab (0) ⊗ catcolab-petri-nets (+1) = 0 ✓ crn-topology (-1) ⊗ catcolab-stock-flow (0) ⊗ catcolab-petri-nets (+1) = 0 ✓ ``` ## Commands ```bash # Create Petri net just catcolab-new petri-net "producer-consumer" # Analyze reachability just catcolab-analyze producer-consumer --reachability # Check for deadlock just catcolab-analyze producer-consumer --deadlock # Export to PNML just catcolab-export producer-consumer --format=pnml # Simulate token dynamics just catcolab-simulate producer-consumer --stochastic ``` ## References - Baez & Master (2020) "Open Petri nets" - Patterson et al. (2022) "Categorical data structures for technical computing" - [AlgebraicPetri.jl](https://algebraicjulia.github.io/AlgebraicPetri.jl/) - [CatColab Petri Net Help](https://catcolab.org/help/logics/petri-net) (planned) --- **Skill Name**: catcolab-petri-nets **Type**: Concurrent Systems / Process Algebra **Trit**: +1 (PLUS) **GF(3)**: Balances the CatColab skill ecosystem