  Many software analysis techniques attempt to determine whether bugs are
reachable, but for security purpose this is only part of the story as it does
not indicate whether the bugs found could be easily triggered by an attacker.
The recently introduced notion of robust reachability aims at filling this gap
by distinguishing the input controlled by the attacker from those that are not.
Yet, this qualitative notion may be too strong in practice, leaving apart bugs
which are mostly but not fully replicable. We aim here at proposing a
quantitative version of robust reachability, more flexible and still amenable
to automation. We propose quantitative robustness, a metric expressing how
easily an attacker can trigger a bug while taking into account that he can only
influence part of the program input, together with a dedicated quantitative
symbolic execution technique (QRSE). Interestingly, QRSE relies on a variant of
model counting (namely, functional E-MAJSAT) unseen so far in formal
verification, but which has been studied in AI domains such as Bayesian
network, knowledge representation and probabilistic planning. Yet, the existing
solving methods from these fields turn out to be unsatisfactory for formal
verification purpose, leading us to propose a novel parametric method. These
results have been implemented and evaluated over two security-relevant case
studies, allowing to demonstrate the feasibility and relevance of our ideas.
