  To verify safety and robustness of neural networks, researchers have
successfully applied abstract interpretation, primarily using the interval
abstract domain. In this paper, we study the theoretical power and limits of
the interval domain for neural-network verification.
  First, we introduce the interval universal approximation (IUA) theorem. IUA
shows that neural networks not only can approximate any continuous function $f$
(universal approximation) as we have known for decades, but we can find a
neural network, using any well-behaved activation function, whose interval
bounds are an arbitrarily close approximation of the set semantics of $f$ (the
result of applying $f$ to a set of inputs). We call this notion of
approximation interval approximation. Our theorem generalizes the recent result
of Baader et al. (2020) from ReLUs to a rich class of activation functions that
we call squashable functions. Additionally, the IUA theorem implies that we can
always construct provably robust neural networks under $\ell_\infty$-norm using
almost any practical activation function.
  Second, we study the computational complexity of constructing neural networks
that are amenable to precise interval analysis. This is a crucial question, as
our constructive proof of IUA is exponential in the size of the approximation
domain. We boil this question down to the problem of approximating the range of
a neural network with squashable activation functions. We show that the range
approximation problem (RA) is a $\Delta_2$-intermediate problem, which is
strictly harder than $\mathsf{NP}$-complete problems, assuming
$\mathsf{coNP}\not\subset \mathsf{NP}$. As a result, IUA is an inherently hard
problem: No matter what abstract domain or computational tools we consider to
achieve interval approximation, there is no efficient construction of such a
universal approximator.
