Temporal verification of reactive systems: safety
Temporal verification of reactive systems: safety
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Computing differential invariants of hybrid systems as fixedpoints
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ATVA'05 Proceedings of the Third international conference on Automated Technology for Verification and Analysis
Generating polynomial invariants for hybrid systems
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Generating invariants for non-linear hybrid systems by linear algebraic methods
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Proceedings of the 14th international conference on Hybrid systems: computation and control
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Automatica (Journal of IFAC)
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POPL '13 Proceedings of the 40th annual ACM SIGPLAN-SIGACT symposium on Principles of programming languages
Exponential-Condition-Based barrier certificate generation for safety verification of hybrid systems
CAV'13 Proceedings of the 25th international conference on Computer Aided Verification
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We present computational techniques for automatically generating algebraic (polynomial equality) invariants for algebraic hybrid systems. Such systems involve ordinary differential equations with multivariate polynomial right-hand sides. Our approach casts the problem of generating invariants for differential equations as the greatest fixed point of a monotone operator over the lattice of ideals in a polynomial ring. We provide an algorithm to compute this monotone operator using basic ideas from commutative algebraic geometry. However, the resulting iteration sequence does not always converge to a fixed point, since the lattice of ideals over a polynomial ring does not satisfy the descending chain condition. We then present a bounded-degree relaxation based on the concept of "pseudo ideals", due to Colon, that restricts ideal membership using multipliers with bounded degrees. We show that the monotone operator on bounded degree pseudo ideals is convergent and generates fixed points that can be used to generate useful algebraic invariants for non-linear systems. The technique for continuous systems is then extended to consider hybrid systems with multiple modes and discrete transitions between modes. We have implemented the exact, non-convergent iteration over ideals in combination with the bounded degree iteration over pseudo ideals to guarantee convergence. This has been applied to automatically infer useful and interesting polynomial invariants for some benchmark non-linear systems.