  The appearance of strong CDCL-based propositional (SAT) solvers has greatly
advanced several areas of automated reasoning (AR). One of the directions in AR
is thus to apply SAT solvers to expressive formalisms such as first-order
logic, for which large corpora of general mathematical problems exist today.
This is possible due to Herbrand's theorem, which allows reduction of
first-order problems to propositional problems by instantiation. The core
challenge is choosing the right instances from the typically infinite Herbrand
universe. In this work, we develop the first machine learning system targeting
this task, addressing its combinatorial and invariance properties. In
particular, we develop a GNN2RNN architecture based on an invariant graph
neural network (GNN) that learns from problems and their solutions
independently of symbol names (addressing the abundance of skolems), combined
with a recurrent neural network (RNN) that proposes for each clause its
instantiations. The architecture is then trained on a corpus of mathematical
problems and their instantiation-based proofs, and its performance is evaluated
in several ways. We show that the trained system achieves high accuracy in
predicting the right instances, and that it is capable of solving many problems
by educated guessing when combined with a ground solver. To our knowledge, this
is the first convincing use of machine learning in synthesizing relevant
elements from arbitrary Herbrand universes.
