### Abstract

A model of a hypertorus communication grid has been constructed in the form of an infinite Petri net. A grid cell represents either a packet switching device or a bioplast cell. A parametric expression is obtained to allow a finite specification of an infinite Petri net. To prove properties of an ideal communication protocol, we derive an infinite Diophantine system of equations from it, which is subsequently solved. Then we present the programs htgen and ht-mcrl2-gen, developed in the C language, which generate Petri net and process algebra models of a hypertorus with a given number of dimensions and grid size. These are the inputs for the respective modeling tools Tina and mCRL2, which provide model visualization, step simulation, state space generation and reduction, and structural analysis techniques. Benchmarks to compare the two approaches are obtained. An ad-hoc induction-like technique on invariants, obtained for a series of generated models, allows the calculation of a solution of the Diophantine system in a parametric form. It is proven that the basic solutions of the infinite system have been found and that the infinite Petri net is bounded and conservative. Some remarks regarding liveness and liveness enforcing techniques are also presented.

Original language | English |
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Article number | 8707130 |

Pages (from-to) | 733-742 |

Number of pages | 10 |

Journal | IEEE/CAA Journal of Automatica Sinica |

Volume | 6 |

Issue number | 3 |

DOIs | |

Publication status | Published - May 2019 |

### Keywords

- Petri nets , Mathematical model , Algebra , Hypercubes , Analytical models , Benchmark testing , Biological system modeling
- systems biology
- process algebra
- conservativeness
- deadlock
- Computing grid
- hypertorus
- infinite Petri nets

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## Cite this

Zaitsev, D., Shmeleva, T., & Groote, J. F. (2019). Verification of hypertorus communication grids by infinite Petri nets and process algebra.

*IEEE/CAA Journal of Automatica Sinica*,*6*(3), 733-742. [8707130]. https://doi.org/10.1109/JAS.2019.1911486