# Examples#

In this section, we will go over several examples of properties, and how to check them on a network. We will also discuss the basics of the property DSL.

We have made several DNN verification benchmarks available in DNNP and ONNX formats in dlshriver/dnnv-benchmarks. This benchmark repository includes both ERAN-MNIST and the ACAS Xu benchmark, ready to run with DNNV!

## Local Robustness#

Local robustness specifies that, given an input, $$x$$, to a DNN, $$\mathcal{N}$$, any other input within some distance, $$\epsilon$$, of that input will be classified to the same class. Formally:

$\forall \delta \in [0, \epsilon]^n. \mathcal{N}(x) = \mathcal{N}(x \pm \delta)$

This property can be specified in our DSL as follows:

from dnnv.properties import *
import numpy as np

N = Network("N")
x = Image(Parameter("input", type=str))
epsilon = Parameter("epsilon", float, default=1.0)

output_class = np.argmax(N(x))

Forall(
x_,
Implies(
((x - epsilon) < x_ < (x + epsilon)),
np.argmax(N(x_)) == output_class,
),
)


## ACAS Xu#

Properties other than local robustness can also be specified in DNNP. For example, the properties for the ACAS Xu aircraft collision avoidance network (as introduced in the evaluation of Reluplex) can easily be encoded in DNNP.

Here we write the specification for ACAS Xu Property $$\phi_3$$. The specification states that if an intruding aircraft is directly ahead and moving towards our aircraft, then the score for a Clear-of-Conflict classification (class 0) will not be minimal (this network recommends the class with the minimal score).

In this property, we also see how inputs can be pre-processed. The ACAS Xu networks expects inputs to be normalized by subtracting a pre-computed mean value, and dividing by the given range. We apply that normalization to the input bounds before bounding the network input, x.

from dnnv.properties import *
import numpy as np

N = Network("N")
# x: $\rho$, $\theta$, $\psi$, $v_{own}$, $v_{int}$
x_min = np.array([[1500.0, -0.06, 3.10, 980.0, 960.0]])
x_max = np.array([[1800.0, 0.06, 3.141593, 1200.0, 1200.0]])

x_mean = np.array([[1.9791091e04, 0.0, 0.0, 650.0, 600.0]])
x_range = np.array([[60261.0, 6.28318530718, 6.28318530718, 1100.0, 1200.0]])

x_min_normalized = (x_min - x_mean) / x_range
x_max_normalized = (x_max - x_mean) / x_range

Forall(
x, Implies(x_min_normalized <= x <= x_max_normalized, argmin(N(x)) != 0),
)