Formal Methods in System Design

, Volume 45, Issue 2, pp 165–188 | Cite as

Quantifier-free encoding of invariants for hybrid systems

  • Alessandro Cimatti
  • Sergio MoverEmail author
  • Stefano Tonetta


Hybrid systems are a clean modeling framework for embedded systems, which feature integrated discrete and continuous dynamics. A well-known source of complexity comes from the time invariants, which represent an implicit quantification of a constraint over all time points of a continuous transition.

Emerging techniques based on Satisfiability Modulo Theory (SMT) have been found promising for the verification and validation of hybrid systems because they combine discrete reasoning with solvers for first-order theories. However, these techniques are efficient for quantifier-free theories and the current approaches have so far either ignored time invariants or have been limited to hybrid systems with linear constraints.

In this paper, we propose a new method that encodes a class of hybrid systems into transition systems with quantifier-free formulas. The method does not rely on expensive quantifier elimination procedures. Rather, it exploits the sequential nature of the transition system to split the continuous evolution enforcing the invariants on the discrete time points. This way, we can encode all hybrid systems whose invariants can be expressed in terms of polynomial constraints. This pushes the application of SMT-based techniques beyond the standard linear case.


Satisfiability modulo theory Hybrid systems Bounded model checking 


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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Alessandro Cimatti
    • 1
  • Sergio Mover
    • 1
    Email author
  • Stefano Tonetta
    • 1
  1. 1.Fondazione Bruno KesslerTrentoItaly

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