  In this paper we propose a general framework to integrate supervised and
unsupervised examples with background knowledge expressed by a collection of
first-order logic clauses into kernel machines. In particular, we consider a
multi-task learning scheme where multiple predicates defined on a set of
objects are to be jointly learned from examples, enforcing a set of FOL
constraints on the admissible configurations of their values. The predicates
are defined on the feature spaces, in which the input objects are represented,
and can be either known a priori or approximated by an appropriate kernel-based
learner. A general approach is presented to convert the FOL clauses into a
continuous implementation that can deal with the outputs computed by the
kernel-based predicates. The learning problem is formulated as a
semi-supervised task that requires the optimization in the primal of a loss
function that combines a fitting loss measure on the supervised examples, a
regularization term, and a penalty term that enforces the constraints on both
the supervised and unsupervised examples. Unfortunately, the penalty term is
not convex and it can hinder the optimization process. However, it is possible
to avoid poor solutions by using a two stage learning schema, in which the
supervised examples are learned first and then the constraints are enforced.
