Skip to content
Cohesion Dynamics (CD)

K-LENS-PN — Petri Net / Occurrence Net Lens

import { LinkCard } from ‘@astrojs/starlight/components’;

Paper K-LENS-PN establishes a representation theorem mapping the Cohesion Dynamics kernel grammar (K-KERN) to Petri net semantics — specifically safe Petri nets, occurrence nets, and asynchronous transition systems.

Summary

Petri nets provide a well-established mathematical framework for modeling concurrent systems with explicit resource and state management. K-LENS-PN demonstrates that K-KERN grammar can be represented within this framework under explicitly stated additional assumptions, particularly resource interpretation, conservation, and boundedness.

This mapping positions CD as compatible with executable carrier models and simulation frameworks, showing how K-KERN subsumes Petri net semantics.

Key Results

  1. K-KERN configurations map to Petri net places (resource locations)
  2. K-KERN resolutions map to Petri net transitions (state change events)
  3. K-KERN admissibility maps to Petri net enabling (sufficient tokens for firing)
  4. K-KERN histories map to Petri net reachable markings (state evolution)
  5. K-KERN irreconcilability maps to Petri net conflict (resource contention)

Status

  • Paper ID: K-LENS-PN
  • Series: K-series (Kernel Grammar & Invariants)
  • Subseries: K-LENS (Semantic Lenses)
  • Status: Staging
  • Dependencies: K-KERN, K-GOV, K-LENS-ORD (informative), K-LENS-ES (companion)

Epistemic Status

Conditional necessity per K-GOV: Petri net representation has been required in explored CD carrier implementations and M-series simulations. No claim of logical necessity or ontological primacy.

Petri Net Variants

Safe Petri Nets

K-KERN naturally maps to safe Petri nets (Boolean markings) due to global configuration indexing. Every K-KERN kernel can be represented as a safe Petri net.

Occurrence Nets

Require additional strong assumptions:

  • Acyclicity (no cyclic dependency)
  • Conflict-freeness (no irreconcilability)

Most CD models involve irreconcilability, so occurrence nets apply only in restricted regimes.

Asynchronous Transition Systems

Provide global state view of K-KERN dynamics. Natural for simulation implementations.

Executable Carriers and Simulations

K-LENS-PN provides the foundation for executable CD carriers:

  • Petri nets are directly executable (algorithmic firing rules)
  • Standard implementation strategy for carrier architectures
  • Enables M-series mechanism simulations
  • Underpins E-series empirical testing

Additional Assumptions

Required for Petri Net Representation

  • Configuration-place bijection (natural)
  • Resolution-transition bijection (natural)
  • Dependency-flow correspondence (natural)
  • Safe marking (natural from global indexing)

Additional Interpretive Commitments

  • Places as resources (semantic commitment)
  • Tokens as resource counts (interpretation)
  • Token conservation (not required by K-KERN)
  • Bounded marking (not required by K-KERN)

Read the Full Paper

See Also

Relationship to Other Formalisms

Petri Nets vs Event Structures

  • Event structures (K-LENS-ES): Emphasize causality and conflict
  • Petri nets (K-LENS-PN): Emphasize resource and state

Both are K-LENS lenses — neither is privileged.

Formal Connection: Occurrence nets are equivalent to prime event structures (Nielsen-Plotkin-Winskel theorem).

Petri Nets and A-series Carriers

A-series carriers (graph-based, order-theoretic) can be implemented as Petri nets. K-LENS-PN provides the semantic bridge between K-KERN grammar and executable carrier implementations.