On Reduction and Synthesis of Petri's Cycloids
Rüdiger Valk, Daniel Moldt
TL;DR
This work addresses recovering the four cycloid parameters $(\alpha,\beta,\gamma,\delta)$ from a Petri net representation and deciding cycloid isomorphism via structured reductions. It introduces cycloid algebra and rewriting-style reduction rules that interpret reductions as shear mappings on the Petri space and yields closed-form synthesis results to obtain $(\alpha,\beta,\gamma,\delta)$ from path-based invariants and the net's area $A$. It proves that cycloid isomorphism coincides with $\beta\delta$-reduction equivalence (and symmetrically with $\alpha\gamma$-reductions) and shows how entire reduction chains can be reconstructed, enabling a linear-time decision procedure with $O(\log \beta \cdot \log \delta)$ bit-operations. The framework further extends to lbc-cycloids and canonical regular cycloids, providing practical methods to characterize and compare synchronous sequential processes via a compact parametric description.
Abstract
Cycloids are particular Petri nets for modelling processes of actions and events, belonging to the fundaments of Petri's general systems theory. Defined by four parameters they provide an algebraic formalism to describe strongly synchronized sequential processes. To further investigate their structure, reduction systems of cycloids are defined in the style of rewriting systems and properties of irreducible cycloids are proved. In particular the synthesis of cycloid parameters from their Petri net structure is derived, leading to an efficient method for a decision procedure for cycloid isomorphism.