  The study of machine learning-based logical query-answering enables reasoning
with large-scale and incomplete knowledge graphs. This paper further advances
this line of research by considering the uncertainty in the knowledge. The
uncertain nature of knowledge is widely observed in the real world, but
\textit{does not} align seamlessly with the first-order logic underpinning
existing studies. To bridge this gap, we study the setting of soft queries on
uncertain knowledge, which is motivated by the establishment of soft constraint
programming. We further propose an ML-based approach with both forward
inference and backward calibration to answer soft queries on large-scale,
incomplete, and uncertain knowledge graphs. Theoretical discussions present
that our methods share the same complexity as state-of-the-art inference
algorithms for first-order queries. Empirical results justify the superior
performance of our approach against previous ML-based methods with number
embedding extensions.
