  Temporal logic inference is the process of extracting formal descriptions of
system behaviors from data in the form of temporal logic formulas. The existing
temporal logic inference methods mostly neglect uncertainties in the data,
which results in limited applicability of such methods in real-world
deployments. In this paper, we first investigate the uncertainties associated
with trajectories of a system and represent such uncertainties in the form of
interval trajectories. We then propose two uncertainty-aware signal temporal
logic (STL) inference approaches to classify the undesired behaviors and
desired behaviors of a system. Instead of classifying finitely many
trajectories, we classify infinitely many trajectories within the interval
trajectories. In the first approach, we incorporate robust semantics of STL
formulas with respect to an interval trajectory to quantify the margin at which
an STL formula is satisfied or violated by the interval trajectory. The second
approach relies on the first learning algorithm and exploits the decision tree
to infer STL formulas to classify behaviors of a given system. The proposed
approaches also work for non-separable data by optimizing the worst-case
robustness in inferring an STL formula. Finally, we evaluate the performance of
the proposed algorithms in two case studies, where the proposed algorithms show
reductions in the computation time by up to four orders of magnitude in
comparison with the sampling-based baseline algorithms (for a dataset with 800
sampled trajectories in total).
