# Abstraction of Elementary Hybrid Systems by Variable Transformation

arXiv: https://arxiv.org/abs/1403.7022 · HTML: https://mlsanity.com/paper/1403.7022/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Mar 2014.

## Built on by (most cited first, 4 of 4)

- [Joint Differentiable Optimization and Verification for Certified Reinforcement Learning](https://mlsanity.com/paper/2201.12243/) (2022)
- [Scalable Verification of Neural Control Barrier Functions Using Linear Bound Propagation](https://mlsanity.com/paper/2511.06341/) (2025)
- [Data-driven system identification using quadratic embeddings of nonlinear dynamics](https://mlsanity.com/paper/2501.08202/) (2025)
- [Set-Based Training of Neural Barrier Certificates for Safety Verification of Dynamical Systems](https://mlsanity.com/paper/2605.02526/) (2026)

## Abstract (arXiv)

Elementary hybrid systems (EHSs) are those hybrid systems (HSs) containing elementary functions such as exp, ln, sin, cos, etc. EHSs are very common in practice, especially in safety-critical domains. Due to the non-polynomial expressions which lead to undecidable arithmetic, verification of EHSs is very hard. Existing approaches based on partition of state space or over-approximation of reachable sets suffer from state explosion or inflation of numerical errors. In this paper, we propose a symbolic abstraction approach that reduces EHSs to polynomial hybrid systems (PHSs), by replacing all non-polynomial terms with newly introduced variables. Thus the verification of EHSs is reduced to the one of PHSs, enabling us to apply all the well-established verification techniques and tools for PHSs to EHSs. In this way, it is possible to avoid the limitations of many existing methods. We illustrate the abstraction approach and its application in safety verification of EHSs by several real world examples.