  Property Specification Language (PSL) is a form of temporal logic that has
been mainly used in discrete domains (e.g. formal hardware verification). In
this paper, we show that by merging machine learning techniques with PSL
monitors, we can extend PSL to work on continuous domains. We apply this
technique in machine learning-based anomaly detection to analyze scenarios of
real-time streaming events from continuous variables in order to detect
abnormal behaviors of a system. By using machine learning with formal models,
we leverage the strengths of both machine learning methods and formal semantics
of time. On one hand, machine learning techniques can produce distributions on
continuous variables, where abnormalities can be captured as deviations from
the distributions. On the other hand, formal methods can characterize discrete
temporal behaviors and relations that cannot be easily learned by machine
learning techniques. Interestingly, the anomalies detected by machine learning
and the underlying time representation used are discrete events. We implemented
a temporal monitoring package (TEF) that operates in conjunction with normal
data science packages for anomaly detection machine learning systems, and we
show that TEF can be used to perform accurate interpretation of temporal
correlation between events.
