4.3 Experimental Evaluation
4.3.3 A Comparison to a Normalization-Based Approach
A naïve solution to the scale problem could be to rescale the signals used in specification at the same scale. This is a normalization approach. For example, instead of falsifying (¬(gear = 4) ∨speed > 35), we can falsify(¬(gear =4) ∨ (γ ·speed >γ ·35)), whereγ is arescaling factorthat normalizes the robustness value ofspeed > 35 to the same scale as that ofgear =4.
The Sbenchalready gives an implementation of the approach with manual selections of rescaling factors. We thus can compare the performance of our approach to this possible baseline. In some cases, this baseline approach performs quite well. For example, the performance of AT1−2in Table 4.2 are the cases wherespeed is rescaled by 0.01. In these cases, the falsification performance in terms of SR is quite good
4.3 Experimental Evaluation 75
0 5 10 15 20 25 30
0 50 100
throttle
0 5 10 15 20 25 30
0 200 400
brake
0 5 10 15 20 25 30
0 50 100
speed
0 5 10 15 20 25 30
time 0
5000
RPM
(a) Input signalsthrottle,brakeand output sig- nalsspeed,RPM ofa sampleduring falsifying AT50
6
0 5 10 15 20 25 30
50 100
150 speed[t] < 135
false true Quant. sat Bool. sat
0 5 10 15 20 25 30
0 1000 2000 3000
4000 RPM[t] < 4780
false true
0 5 10 15 20 25 30
0 50 100 150
200 (speed[t] < 135) and (RPM[t] < 4780)
false true
0 5 10 15 20 25 30
time 0
50
100 alw_[0,30] ((speed[t] < 135) and (RPM[t] < 4780))
false true
(b) Boolean satisfaction and quantitative ro- bustness of the sample in Fig. 4.5a to sub- formulas of AT50
6
Figure 4.5: A sample from hill-climbing optimization during falsification to AT50
6 ≡ (speed < 135∧rpm< 4780)
(SR being 30/30), compared to the cases with other rescaling factors. However, the baseline approach does not always work well. For example, the specifications of AT5 give restriction tospeedandrpmtogether, and these properties suffer from the scale problem asspeedis one order of magnitude less thanrpm. However, from Table 4.2, we observe that the baseline approach (i.e., running Breach over AT51) is not effective, as SR is 0.4/30, that is much lower than the original SR 14.1/30 of the unscaled approach using Breach. Our approach, instead, raises SR to 28.4/30 and to 27.6/30 using the two proposed versions. By monitoring Breach execution, we notice that the baseline approach fails because it tries to falsifyrpm < 4780, which, however, is not falsifiable;
our approach, instead, understands that it should try to falsifyspeed<ρthanks to the application of MAB algorithms.
Here, it gives rise to the problem: how to select the rescaling factor. The quick answer is by comparing the scales of different signals. However, the example of AT5 proves that this method does not work. Instead, the experimental results in Table 4.2 show that 10−2(0.01)is the best choice. Let us take a further look into the example.
Fig. 4.5 presents a sample during the process falsifying AT50
6. We can see that the final robustness comes from the robustness to the sub-formularpm <4780. This is opposite to our intuition, since robust semantics for conjunctive (see Def. 8) selects the minimum
76 Chapter 4. Multi-Armed Bandits for Boolean Connectives in STL
one between sub-formulas, butrpmis the one of larger magnitude. The explanation is as follows: althoughrpmhas a larger scale, it is less variant thanspeed; in other words, JM(u),rpm< 4780Kis more likely to be smaller thanJM(u),speed <135K, because it is not hard to driverpmto a high value. Therefore, the larger rescaling factor to speedwe select, the more probably thatrpmtakes the final robustness. That is why it performs the best when the rescaling factor is 10−2(0.01).
77
Constraining Counterexamples via 5
Search Space Transformation
In this chapter, we consider the falsification problem in the presence of logical constraints on input signals. Typical hill-climbing optimization algorithms rely on random samplings, therefore, introduction of input constraints increases the occurrence of infeasible samplings and thus makes the search more difficult.
We firstly show two naive penalty-based approaches that are, though able to solve the problem, not very effective. We then present the main contribution of this chapter, that is, a framework based on search space transformation. It consists of a space mappingthat maps points in an unconstrained search space to the constrained input space, and a fitness definition that assigns fitness values to points in the search space according to the robustness values of points in the input space. In this way, it allows the search performing in an unconstrained space, and when a negative fitness is detected, it returns the mapped point in the input space as a falsifying input. An
The material in this chapter is based on [82] and [90]
78 Chapter 5. Constraining Counterexamples via Search Space Trans.
throttle
brake
throttle
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brake2[0,1]
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throttle<latexit sha1_base64="VRBD2DQF6mP8/L7R2s3IdDIiU80=">AAACAXicbVC7SgNBFJ2NrxhfqzaCzWAQrMKuCNoEgjaWEcwDkiXMTmaTIbMzy8xdISyx8VdsLBSx9S/s/BsnyRaaeKvDeXDvPWEiuAHP+3YKK6tr6xvFzdLW9s7unrt/0DQq1ZQ1qBJKt0NimOCSNYCDYO1EMxKHgrXC0c1Ubz0wbbiS9zBOWBCTgeQRpwQs1XOPYKgV2FjV61ovDjUZMVzFXs8texVvNngZ+Dkoo3zqPfer21c0jZkEKogxHd9LIMiIBk4Fm5S6qWEJoSMyYB0LJYmZCbLZBxN8apk+juwBkZKAZ+zvREZiY8ZxaJ0xgaFZ1Kbkf1onhegqyLhMUmCSzhdFqcCg8LQO3OeaURBjCwjV3N6K6ZBoQsGWVrIl+IsvL4PmecX3Kv7dRbl2nddRRMfoBJ0hH12iGrpFddRAFD2iZ/SK3pwn58V5dz7m1oKTZw7Rn3E+fwAWFZX5</latexit><latexit sha1_base64="VRBD2DQF6mP8/L7R2s3IdDIiU80=">AAACAXicbVC7SgNBFJ2NrxhfqzaCzWAQrMKuCNoEgjaWEcwDkiXMTmaTIbMzy8xdISyx8VdsLBSx9S/s/BsnyRaaeKvDeXDvPWEiuAHP+3YKK6tr6xvFzdLW9s7unrt/0DQq1ZQ1qBJKt0NimOCSNYCDYO1EMxKHgrXC0c1Ubz0wbbiS9zBOWBCTgeQRpwQs1XOPYKgV2FjV61ovDjUZMVzFXs8texVvNngZ+Dkoo3zqPfer21c0jZkEKogxHd9LIMiIBk4Fm5S6qWEJoSMyYB0LJYmZCbLZBxN8apk+juwBkZKAZ+zvREZiY8ZxaJ0xgaFZ1Kbkf1onhegqyLhMUmCSzhdFqcCg8LQO3OeaURBjCwjV3N6K6ZBoQsGWVrIl+IsvL4PmecX3Kv7dRbl2nddRRMfoBJ0hH12iGrpFddRAFD2iZ/SK3pwn58V5dz7m1oKTZw7Rn3E+fwAWFZX5</latexit><latexit sha1_base64="VRBD2DQF6mP8/L7R2s3IdDIiU80=">AAACAXicbVC7SgNBFJ2NrxhfqzaCzWAQrMKuCNoEgjaWEcwDkiXMTmaTIbMzy8xdISyx8VdsLBSx9S/s/BsnyRaaeKvDeXDvPWEiuAHP+3YKK6tr6xvFzdLW9s7unrt/0DQq1ZQ1qBJKt0NimOCSNYCDYO1EMxKHgrXC0c1Ubz0wbbiS9zBOWBCTgeQRpwQs1XOPYKgV2FjV61ovDjUZMVzFXs8texVvNngZ+Dkoo3zqPfer21c0jZkEKogxHd9LIMiIBk4Fm5S6qWEJoSMyYB0LJYmZCbLZBxN8apk+juwBkZKAZ+zvREZiY8ZxaJ0xgaFZ1Kbkf1onhegqyLhMUmCSzhdFqcCg8LQO3OeaURBjCwjV3N6K6ZBoQsGWVrIl+IsvL4PmecX3Kv7dRbl2nddRRMfoBJ0hH12iGrpFddRAFD2iZ/SK3pwn58V5dz7m1oKTZw7Rn3E+fwAWFZX5</latexit><latexit sha1_base64="VRBD2DQF6mP8/L7R2s3IdDIiU80=">AAACAXicbVC7SgNBFJ2NrxhfqzaCzWAQrMKuCNoEgjaWEcwDkiXMTmaTIbMzy8xdISyx8VdsLBSx9S/s/BsnyRaaeKvDeXDvPWEiuAHP+3YKK6tr6xvFzdLW9s7unrt/0DQq1ZQ1qBJKt0NimOCSNYCDYO1EMxKHgrXC0c1Ubz0wbbiS9zBOWBCTgeQRpwQs1XOPYKgV2FjV61ovDjUZMVzFXs8texVvNngZ+Dkoo3zqPfer21c0jZkEKogxHd9LIMiIBk4Fm5S6qWEJoSMyYB0LJYmZCbLZBxN8apk+juwBkZKAZ+zvREZiY8ZxaJ0xgaFZ1Kbkf1onhegqyLhMUmCSzhdFqcCg8LQO3OeaURBjCwjV3N6K6ZBoQsGWVrIl+IsvL4PmecX3Kv7dRbl2nddRRMfoBJ0hH12iGrpFddRAFD2iZ/SK3pwn58V5dz7m1oKTZw7Rn3E+fwAWFZX5</latexit> = 0_brake= 0
Figure 5.1: Feasible areas without/with considering input constraints
instance of space mapping, namedproportional transformation, is then defined. We propose three approaches that make use of the proportional transformation, and we experimentally show that the one based on Multi-Armed Bandit (MAB) model performs better than others.