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Analysing cycloids using linear algebra

Rüdiger Valk

TL;DR

The paper analyzes cycloids within Petri nets using a linear-algebraic framework called Cycloid Algebra. It establishes an equivalence criterion for points via lattice translations, proves three net isomorphisms, and derives a generalized closed-form for the minimal cycle length cyc, including several special cases and bounds. The results provide simpler, structurally grounded proofs for key properties of cycloids and extend prior work (Valk-2019) to broader classes of cycloids. The approach has potential practical impact for modeling distributed systems and sequential/circular processes using cycloids in Petri nets.

Abstract

Cycloids are particular Petri nets for modelling processes of actions or events. They belong to the fundaments of Petri's general systems theory and have very different interpretations, ranging from Einstein's relativity theory and elementary information processing gates to the modelling of interacting sequential processes. This article contains previously unpublished proofs of cycloid properties using linear algebra.

Analysing cycloids using linear algebra

TL;DR

The paper analyzes cycloids within Petri nets using a linear-algebraic framework called Cycloid Algebra. It establishes an equivalence criterion for points via lattice translations, proves three net isomorphisms, and derives a generalized closed-form for the minimal cycle length cyc, including several special cases and bounds. The results provide simpler, structurally grounded proofs for key properties of cycloids and extend prior work (Valk-2019) to broader classes of cycloids. The approach has potential practical impact for modeling distributed systems and sequential/circular processes using cycloids in Petri nets.

Abstract

Cycloids are particular Petri nets for modelling processes of actions or events. They belong to the fundaments of Petri's general systems theory and have very different interpretations, ranging from Einstein's relativity theory and elementary information processing gates to the modelling of interacting sequential processes. This article contains previously unpublished proofs of cycloid properties using linear algebra.
Paper Structure (5 sections, 4 theorems, 5 equations, 5 figures)

This paper contains 5 sections, 4 theorems, 5 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Petri Space and Cycloids
  3. Equivalence and Isomorphisms
  4. The Minimal Length of a Cycle
  5. Conclusion

Key Result

theorem 1

Two points $\vec{x}_1, \vec{x}_2\in X_1$ are equivalent $\vec{x}_1 \equiv \vec{x}_2$ if and only if for the difference $\vec{v} := \vec{x_2}-\vec{x_1}$ the parameter vector $\pi(\vec{v}) = \frac{1}{A} \cdot\mathbf{B} \cdot \vec{v}$ has integer values, where $A$ is the area and $\mathbf{B} = $. In an

Figures (5)

  • Figure 1: Three sequential processes synchronized by single-bit channels,
  • Figure 2: a) Petri space, b) circular traffic queue and c) time orthoid.
  • Figure 3: a) Fundamental parallelogram of $\mathcal{C}(4,2,2,3)$ and b) Petri space.
  • Figure 4: A shearing from $\mathcal{C}( 2,3,2,8)$ to $\mathcal{C}( 2,3,6,2)$.
  • Figure 5: Referenced in the proof of Theorem \ref{['minimal cycles']}.

Theorems & Definitions (10)

  • definition 1: Valk-2019
  • definition 2: Valk-2019
  • definition 3: Valk-2019
  • theorem 1: Valk-2020
  • proof
  • theorem 2: Valk-2019
  • proof
  • lemma 1: Valk-2019
  • theorem 3
  • proof