  Integrating symbolic knowledge and data-driven learning algorithms is a
longstanding challenge in Artificial Intelligence. Despite the recognized
importance of this task, a notable gap exists due to the discreteness of
symbolic representations and the continuous nature of machine-learning
computations. One of the desired bridges between these two worlds would be to
define semantically grounded vector representation (feature embedding) of logic
formulae, thus enabling to perform continuous learning and optimization in the
semantic space of formulae. We tackle this goal for knowledge expressed in
Signal Temporal Logic (STL) and devise a method to compute continuous
embeddings of formulae with several desirable properties: the embedding (i) is
finite-dimensional, (ii) faithfully reflects the semantics of the formulae,
(iii) does not require any learning but instead is defined from basic
principles, (iv) is interpretable. Another significant contribution lies in
demonstrating the efficacy of the approach in two tasks: learning model
checking, where we predict the probability of requirements being satisfied in
stochastic processes; and integrating the embeddings into a neuro-symbolic
framework, to constrain the output of a deep-learning generative model to
comply to a given logical specification.
