classical example of the hybrid system community (https://github.com/dreal/dreal2/tree/master/benchmarks/hybrid_systems/bouncing_ball)

implemented here with reactions and events

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Statistical evaluation by sampling the parameter space

This is time consuming since it involves making one simulation for each parameter set

The FO-LTL formula defines the amplitude range of x in the domain of the free variable h

The FO-LTL formula defines the height of the first bounce of x in the domain of the free variable h

The amplitude 3.96 exceeds here the objective of 3.5, the satisfaction degree is thus greater than one showing some formula robustness

The height 3.96 exceeds here the objective of 3.5, the satisfaction degree is thus greater than one showing some formula robustness

However the model robustness w.r.t. (default) parameter perturbations is below one (0.88) showing that some parameter perturbations destroy the amplitude objective.

However the model robustness w.r.t. (default) parameter perturbations is below one (0.88) showing that some parameter perturbations destroy the height objective.

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# Formula robustness

With a satisfaction degree greater than 1 the formula is already satisfied with some margin

Formula robustness can be further optimized by parameter optimization with formula robustness as objective function with no extra cost and much faster computation time than for estimating model robustness

It gives an amplitude of 4.84 which obviously improves the satisfaction degree of the formula

It gives a height of 4.84 which obviously improves the satisfaction degree of the formula

This is shown to also improve the model robustness (0.93) to parameter perturbations as expected in many examples