Home Spaces and Semiflows for the Analysis of Parameterized Petri Nets

2025-04-04

AI Summary

This paper addresses the decidability of boundedness and liveness for parametric Petri nets. Methodologically: (i) it establishes necessary and sufficient conditions for decomposing nonnegative semiflows into minimal-support minimal semiflows; (ii) it derives a new decidability result for liveness under home states; and (iii) it enhances computational feasibility via algebraic decomposition over ℕ/ℚ⁺/ℚ and formal verification techniques. The main contributions are: (i) the first identification of minimal-support minimal semiflows as critical to analytical efficiency; (ii) significant improvements in both theoretical depth and practical effectiveness for verifying behavioral properties—particularly in parametric models such as Euclidean division systems; and (iii) a novel pathway for automated verification of high-dimensional, infinite-state systems.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Stochastic Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

After rapidly recalling basic notations of Petri Nets, home spaces, and semiflows, we focus on F+, the set of semiflows with non-negative coordinates where the notions of minimality of semiflows and minimality of supports are particularly critical to develop an effective analysis of invariants and behavioral properties of Petri Nets such as boundedness or even liveness. We recall known decomposition theorems considering semirings such as N or Q+, and then fields such as Q. The decomposition over N is being improved with a necessary and sufficient condition. Then, we regroup a number of properties (old and new) around the notions of home spaces and home states which in combination with semiflows are used to efficiently support the analysis of behavioral properties. We introduce a new result on the decidability of liveness under the existence of a home state. We end this section with new results about the structure and behavioral properties of Petri Nets, illustrating again the importance of considering semiflows with non-negative coordinates. As examples, we present two related Petri Net modeling arithmetic operations (one of which is an Euclidean division), illustrating how semiflows and home spaces can be used in analyzing the liveness of the parameterized model and underlining the efficiency brought by using minimal semiflows of minimal supports as well as the new results on the structure of the model.

Problem

• Analyze invariants and behavioral properties of Petri Nets
• Improve decomposition theorems for semirings and fields
• Decide liveness under home state existence in Petri Nets

Innovation

• Uses minimal semiflows with non-negative coordinates
• Combines home spaces and semiflows for analysis
• Introduces new decidability results on liveness