Abstract

AbstractHybrid systems are models for complex physical systems and are defined as dynamical systems with interacting discrete transitions and continuous evolutions along differential equations. With the goal of developing a theoretical and practical foundation for deductive verification of hybrid systems, we introduce a dynamic logic for hybrid programs, which is a program notation for hybrid systems. As a verification technique that is suitable for automation, we introduce a free variable proof calculus with a novel combination of real-valued free variables and Skolemisation for lifting quantifier elimination for real arithmetic to dynamic logic. The calculus is compositional, i.e., it reduces properties of hybrid programs to properties of their parts. Our main result proves that this calculus axiomatises the transition behaviour of hybrid systems completely relative to differential equations. In a case study with cooperating traffic agents of the European Train Control System, we further show that our calculus is well-suited for verifying realistic hybrid systems with parametric system dynamics.

Highlights

  • Which constraints for SB?∀MA ∃SB [Train]safe continuous evolution along differential equations + discrete change differential dynamic logic dL = DL + HP dL Motives: Regions in First-order Logic differential dynamic logic dL =.

  • DL Motives: State Transitions in Dynamic Logic differential dynamic logic dL = FOL + DL.

  • ∀t after(train-runs(t))(v 2 ≤ 2b(MA − z)) [train-runs]v 2 ≤ 2b(MA − z) dL Motives: Hybrid Programs as Uniform Model differential dynamic logic dL = FOL + DL + HP [train-runs]v 2 ≤ 2b(MA − z).

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Summary

Which constraints for SB?

∀MA ∃SB [Train]safe continuous evolution along differential equations + discrete change differential dynamic logic dL = DL + HP dL Motives: Regions in First-order Logic differential dynamic logic dL =. DL Motives: State Transitions in Dynamic Logic differential dynamic logic dL = FOL + DL. ∀t after(train-runs(t))(v 2 ≤ 2b(MA − z)) [train-runs]v 2 ≤ 2b(MA − z) dL Motives: Hybrid Programs as Uniform Model differential dynamic logic dL = FOL + DL + HP [train-runs]v 2 ≤ 2b(MA − z). ]v 2 ≤ 2b(MA − z) far neg far ST negot SB corr MA cor fsa rec cor not compositional fsa Differential Logic dL: Syntax. (continuous evolution within invariant region) (discrete jump) (conditional execution)

Dynamic Rules
Distance Profile d vdes vdes m
Future Work

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